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FQXi FORUM
February 11, 2012
I realize this forum hasn't gotten much action lately, but I'm hoping enough people pop on here that I get some feedback on this (particularly some of the membership who may have worked on or thought about related problems).
What I am interested in from a non-speculative standpoint are those places that mathematics *diverges* from the physical world, i.e. those places in which mathematics describes something that is very clearly not physically possible. Now, this obviously does not mean any instance of infinity since, as an example, infinity is necessary to resolve such physical paradoxes as those conjectured by Zeno. But there *are* instances where mathematics in some infinite limit ends up describing something that is not physically possible. There are probably plenty of examples that don't involve infinity.
More succinctly, is this divergence necessarily always a discrete point or can it be continuous? For instance, take something simple like a set whose elements represent some physical object (perhaps we're simply numbering these objects and the numbers are the elements of the set). Call the number of elements N. As long as N is finite, it is presumably physically possible (regardless of how unlikely that may be) to realize N of these objects. But in the infinite limit it is *not* possible if both the universe and these objects possess a finite, non-zero volume.
Incidentally, could this be one way to talk about the many-worlds interpretation? In other words, maybe math diverges from a single physical universe and only perfectly matches reality if there are an infinite number of parallel universes (e.g. limits to infinity such as the one just mentioned could be physically realizable in this case). This would get at the heart of whether mathematics is discovered or invented. If it is always true that it diverges somewhere and that either there's only one universe or it can be shown that it still diverges even with an infinite number of them, then it seems as if it would be invented. Otherwise it could be discovered.
One motivation for this is the Quine-Putnam indispensability argument which basically says that if we believe in the concreteness of the physical theories described by mathematical objects then we also ought to believe in the concreteness of those mathematical objects themselves.
I am very curious to get some thoughts on this from people.
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What I am interested in from a non-speculative standpoint are those places that mathematics *diverges* from the physical world, i.e. those places in which mathematics describes something that is very clearly not physically possible. Now, this obviously does not mean any instance of infinity since, as an example, infinity is necessary to resolve such physical paradoxes as those conjectured by Zeno. But there *are* instances where mathematics in some infinite limit ends up describing something that is not physically possible. There are probably plenty of examples that don't involve infinity.
More succinctly, is this divergence necessarily always a discrete point or can it be continuous? For instance, take something simple like a set whose elements represent some physical object (perhaps we're simply numbering these objects and the numbers are the elements of the set). Call the number of elements N. As long as N is finite, it is presumably physically possible (regardless of how unlikely that may be) to realize N of these objects. But in the infinite limit it is *not* possible if both the universe and these objects possess a finite, non-zero volume.
Incidentally, could this be one way to talk about the many-worlds interpretation? In other words, maybe math diverges from a single physical universe and only perfectly matches reality if there are an infinite number of parallel universes (e.g. limits to infinity such as the one just mentioned could be physically realizable in this case). This would get at the heart of whether mathematics is discovered or invented. If it is always true that it diverges somewhere and that either there's only one universe or it can be shown that it still diverges even with an infinite number of them, then it seems as if it would be invented. Otherwise it could be discovered.
One motivation for this is the Quine-Putnam indispensability argument which basically says that if we believe in the concreteness of the physical theories described by mathematical objects then we also ought to believe in the concreteness of those mathematical objects themselves.
I am very curious to get some thoughts on this from people.
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Dear Member Ian Durham,
When did mathematics “diverge”? You might take issue concerning any opinion I uttered for instance in my essays 369 and 527. Being a retired teacher of fundamentals including complex theory of signal processing to students of engineering, I am trying to shed light into several related matters.
My basic argument was: Future data are not available. There is no...
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When did mathematics “diverge”? You might take issue concerning any opinion I uttered for instance in my essays 369 and 527. Being a retired teacher of fundamentals including complex theory of signal processing to students of engineering, I am trying to shed light into several related matters.
My basic argument was: Future data are not available. There is no...
view entire post
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Anonymous was me, Eckard Blumschein who offers alternative explanations for a few oddities that are bothering the mainstream. Since you Ian Durham meant "infinity is necessary to resolve such physical paradoxes as those conjectured by Zeno." I would like to add that Zeno of Elea (490-430) was a disciple of the monist Parmenides of Elea (515-445) who denied time, motion, and the split of the continuum into points. Zeno's paradoxes relied on missing ability to distinguish between abstracted ideal notions like point, line, number, infinitely divisible continuum and the obvious reality. You are quite right: Continuity implies infinite accuracy. In other words, both ideal ideas (continuity and infinity) are likewise unphysical. For instance, E(R=0) of a point charge is mathematically infinite. In so far, mathematical models diverge from reality from the very beginning. However I do not consider this the crucial divergence.
While I imagine you will need some time as to read a few times and scrutinize my essays and the added files, I would eventually highly appreciate your reply.
Eckard
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While I imagine you will need some time as to read a few times and scrutinize my essays and the added files, I would eventually highly appreciate your reply.
Eckard
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Eckard,
Thanks for your reply. You are correct in saying that I will need time to digest your ideas. I was a practicing mechanical engineer before I became a theoretical physicist so I understand where you're coming from.
Ian
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Thanks for your reply. You are correct in saying that I will need time to digest your ideas. I was a practicing mechanical engineer before I became a theoretical physicist so I understand where you're coming from.
Ian
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Dear Member Ian Durham,
"...What I am interested in from a non-speculative standpoint are those places that mathematics *diverges* from the physical world, i.e. those places in which mathematics describes something that is very clearly not physically possible. ..."
I understand that your question probably addresses advanced areas of theoretical physics where even theoretical physicsts might openly question the physical meanings of their mathematical expressions. So much of what I see discussed in the Blog section already strikes me as having long since left reality behind. It makes me wonder: How can we know when theoretical physics has gone too far astray? I think it happened a great many moons ago. Many theoretical physicsits appear to me to believe that they are still well within the bounds of reality. It is as if internally consistant mathematics is the test for reality.
My opinion does not fit within the boundaries of your inquiry. I think that as soon as theoretical physics began to guess about the nature of cause, they went astray. I don't know yet how to prove this. The only avenue I see available is to show that fundamental theoretical unity is possible right from the start of development of theoretical physics. Afterall, if the fundamental causes of theoretical physics are not necessary to interpret empirical evidence, then perhaps theoretical physics has been chasing its own tail. This approach hasn't been received well. Yet, when I see the many exotic theoretical paths that theoretical physics has veered off into, I wonder: How did things get so far? Anyway, you now have at least one response. My response could be dismissed as being speculative. My response to that is: Theoretical physics has been speculative right from its start.
James
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"...What I am interested in from a non-speculative standpoint are those places that mathematics *diverges* from the physical world, i.e. those places in which mathematics describes something that is very clearly not physically possible. ..."
I understand that your question probably addresses advanced areas of theoretical physics where even theoretical physicsts might openly question the physical meanings of their mathematical expressions. So much of what I see discussed in the Blog section already strikes me as having long since left reality behind. It makes me wonder: How can we know when theoretical physics has gone too far astray? I think it happened a great many moons ago. Many theoretical physicsits appear to me to believe that they are still well within the bounds of reality. It is as if internally consistant mathematics is the test for reality.
My opinion does not fit within the boundaries of your inquiry. I think that as soon as theoretical physics began to guess about the nature of cause, they went astray. I don't know yet how to prove this. The only avenue I see available is to show that fundamental theoretical unity is possible right from the start of development of theoretical physics. Afterall, if the fundamental causes of theoretical physics are not necessary to interpret empirical evidence, then perhaps theoretical physics has been chasing its own tail. This approach hasn't been received well. Yet, when I see the many exotic theoretical paths that theoretical physics has veered off into, I wonder: How did things get so far? Anyway, you now have at least one response. My response could be dismissed as being speculative. My response to that is: Theoretical physics has been speculative right from its start.
James
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You have a mathematical model that has been derived under some formal system.
Any model of anything is just that, a model; it will include some assumptions, some explicit, others implicit.
It may or may not represent reality or more likely it is an approximation to some degree.
The creator and/or user of the model must decide to what extent they can interpret or equate facets of the model with the reality they are trying to model.
Their own personal preexisting worldviews will influence this essentially creative process.
So I would say that there is no hard and fast rule; if there is a simple logical equivalence there shouldn't be a debate.
Unfortunately such models are few. Infinities of one sort or another are a common though not the only source of difficulty.
I doubt very much that problematic cases can be reduced to a divergence at any particular point, that sounds like a mathematicians attempt to define the problem.
A healthy dose of physical insight not to mention experiments if possible are better ways to test models.
If mathematics is discovered, this implies that the objects in question preexisted and existed before anyone was there to have thought about them. Believing such a thing seems like an act of faith (or perhaps confusing the model with reality).
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Any model of anything is just that, a model; it will include some assumptions, some explicit, others implicit.
It may or may not represent reality or more likely it is an approximation to some degree.
The creator and/or user of the model must decide to what extent they can interpret or equate facets of the model with the reality they are trying to model.
Their own personal preexisting worldviews will influence this essentially creative process.
So I would say that there is no hard and fast rule; if there is a simple logical equivalence there shouldn't be a debate.
Unfortunately such models are few. Infinities of one sort or another are a common though not the only source of difficulty.
I doubt very much that problematic cases can be reduced to a divergence at any particular point, that sounds like a mathematicians attempt to define the problem.
A healthy dose of physical insight not to mention experiments if possible are better ways to test models.
If mathematics is discovered, this implies that the objects in question preexisted and existed before anyone was there to have thought about them. Believing such a thing seems like an act of faith (or perhaps confusing the model with reality).
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MI,
You wrote: "If mathematics is discovered, this implies that the objects in question preexisted and existed before anyone was there to have thought about them. Believing such a thing seems like an act of faith (or perhaps confusing the model with reality)."
Non sequitur.
Georg Cantor made a celebrated statement: "The essence is just its freedom." He did not manage escaping the small city Halle where he died in a madhouse, not by means of having got the CH directly from god, and not even by means of this populism.
I vote for the opposite: Mathematics must avoid arbitrary fabrications. Meanwhile the seems to be a lot of arbitrarily founded guesswork in mathematics and even more in its applications on physics. Let me mention a bit of very obvious nonsense:
Rotate or blow up a point (Confusing the point with a phasor or a tiny sphere)
Singular treatment of a real number, e.g. |sign(0)| nonsensically = 0.
Denial of restriction for physical quantities to positive real values
Spacetime is thought to from minus infinity to plus infinity, amen.
Heaviside was cheeky enough as to call mathematics an empirical science. Indeed, it arose from applied reasoning. Many animals are already intelligent enough as to learn by trial and error repetitious patterns. Consequent further steps were the abilities to count and to trust in causal explanation. I see it neither justified to deny laws of nature nor to follow Quine/Putnam.
Evidently the relationship between e, pi, and i was discovered step by step. However, you must not infer that objects exist forward and backward "in time".
Eckard
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You wrote: "If mathematics is discovered, this implies that the objects in question preexisted and existed before anyone was there to have thought about them. Believing such a thing seems like an act of faith (or perhaps confusing the model with reality)."
Non sequitur.
Georg Cantor made a celebrated statement: "The essence is just its freedom." He did not manage escaping the small city Halle where he died in a madhouse, not by means of having got the CH directly from god, and not even by means of this populism.
I vote for the opposite: Mathematics must avoid arbitrary fabrications. Meanwhile the seems to be a lot of arbitrarily founded guesswork in mathematics and even more in its applications on physics. Let me mention a bit of very obvious nonsense:
Rotate or blow up a point (Confusing the point with a phasor or a tiny sphere)
Singular treatment of a real number, e.g. |sign(0)| nonsensically = 0.
Denial of restriction for physical quantities to positive real values
Spacetime is thought to from minus infinity to plus infinity, amen.
Heaviside was cheeky enough as to call mathematics an empirical science. Indeed, it arose from applied reasoning. Many animals are already intelligent enough as to learn by trial and error repetitious patterns. Consequent further steps were the abilities to count and to trust in causal explanation. I see it neither justified to deny laws of nature nor to follow Quine/Putnam.
Evidently the relationship between e, pi, and i was discovered step by step. However, you must not infer that objects exist forward and backward "in time".
Eckard
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Ian,
There are, in my mind, two kinds of maths; the abstract one and the real one.
The abstract maths goes as follow. I have 1 dollar in my pocket and 1 dollar in the bank. I have 2 dollars total.
The real maths, on the other hand, is pretty much how we first learn about it in school. We count objects by gathering them together. The actual addition consists in bringing closer together the items added. Similarly, planets gain weight by actual aggregation of matter, not by banking on the locally available matter. As we can see, the real “metaphysical” addition carries a geometric component while the abstract addition does not. The metaphysical addition is never “completed” because it comes down to close juxtaposition. For example, 1 + 1 becomes (1+1) but never two or one because they are real entities, they never completely melt into an undefined mush total (one). It would seem that the fact of existence is a discrete very small scale process. their effect add up to as more mass but their discreteness remains. (But their number will vary as in a star collapsed to a neutron star....)
The question is; are our maths truly metaphysical in that they represent the real universe or, are they just abstract representations of our perceptual reality? Physics being empirical, I would think the latter is the case.
Marcel,
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There are, in my mind, two kinds of maths; the abstract one and the real one.
The abstract maths goes as follow. I have 1 dollar in my pocket and 1 dollar in the bank. I have 2 dollars total.
The real maths, on the other hand, is pretty much how we first learn about it in school. We count objects by gathering them together. The actual addition consists in bringing closer together the items added. Similarly, planets gain weight by actual aggregation of matter, not by banking on the locally available matter. As we can see, the real “metaphysical” addition carries a geometric component while the abstract addition does not. The metaphysical addition is never “completed” because it comes down to close juxtaposition. For example, 1 + 1 becomes (1+1) but never two or one because they are real entities, they never completely melt into an undefined mush total (one). It would seem that the fact of existence is a discrete very small scale process. their effect add up to as more mass but their discreteness remains. (But their number will vary as in a star collapsed to a neutron star....)
The question is; are our maths truly metaphysical in that they represent the real universe or, are they just abstract representations of our perceptual reality? Physics being empirical, I would think the latter is the case.
Marcel,
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Marcel,
I'm not entirely sure I agree (actually, I don't have an opinion yet - I tend to be torn on the issue), but thanks for the well-posed response. The empirical nature of physics is something Bas van Fraassen has explored a bit. I wonder if he'd agree with you or not.
Thanks for the response!
Ian
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I'm not entirely sure I agree (actually, I don't have an opinion yet - I tend to be torn on the issue), but thanks for the well-posed response. The empirical nature of physics is something Bas van Fraassen has explored a bit. I wonder if he'd agree with you or not.
Thanks for the response!
Ian
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Ian,
Allow me to explain…
Science is empirical. What does it mean? It means that we recognize not knowing about the underlying reality. It means that we accept this ignorance because we have found about 300 years ago a pragmatic approach to this situation. We simply treat this universe as a black box. We ignore the content of the box and concentrate our study on our interaction or experience (empirical) with the box. By studying our experiences with the box we have come up with regularities and some possible image and idea of what the box contains. These are our laws of physics and the models that we can infer from them. But no matters how pointed our empirical method is, no matters how sharp and detailed our models are, they are still modeled and framed on the requirements of proof within the empirical system. In other words, the empirical method was meant to study our experience of the box, never to find its content, which must be addressed in a metaphysical approach. No matter how wonderful our science may appear, it is just child’s play. Without knowing the content of the box, we do not have any idea of what we are really doing. This is the limit of physics. We don’t do or understand as much as we could and should. The content of the box is about the two following metaphysical questions; what is the universe made of and what makes it evolve by itself? The two pillars of metaphysics: substance and cause.
Somehow on the way, we forgot half of the question. This ignorance is in fact an oversight…. Your maths pertain to our experience of the box, not to its content.
Marcel,
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Allow me to explain…
Science is empirical. What does it mean? It means that we recognize not knowing about the underlying reality. It means that we accept this ignorance because we have found about 300 years ago a pragmatic approach to this situation. We simply treat this universe as a black box. We ignore the content of the box and concentrate our study on our interaction or experience (empirical) with the box. By studying our experiences with the box we have come up with regularities and some possible image and idea of what the box contains. These are our laws of physics and the models that we can infer from them. But no matters how pointed our empirical method is, no matters how sharp and detailed our models are, they are still modeled and framed on the requirements of proof within the empirical system. In other words, the empirical method was meant to study our experience of the box, never to find its content, which must be addressed in a metaphysical approach. No matter how wonderful our science may appear, it is just child’s play. Without knowing the content of the box, we do not have any idea of what we are really doing. This is the limit of physics. We don’t do or understand as much as we could and should. The content of the box is about the two following metaphysical questions; what is the universe made of and what makes it evolve by itself? The two pillars of metaphysics: substance and cause.
Somehow on the way, we forgot half of the question. This ignorance is in fact an oversight…. Your maths pertain to our experience of the box, not to its content.
Marcel,
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Hi Ian,
I'm not exactly sure what "divergence" you're asking about here... Starting from the reasonable premise that pure mathematics is not part of the physical world, then the question isn't when they diverge, so much as when they can be said to be vaguely analogous. The branch of physics that deals with the rough overlap between math and reality is measurement theory (which tells one how to map physical events onto mathematical structures and vice-versa). And of course there are an infinite number of (presumably incorrect) measurement theories where the math diverges from the physical reality.
But still, I think I have a rough idea of what you're asking... so consider these two examples which might help clarify the issues you raise:
1) When you solve the (time-independent) Schrodinger equation for the Hydrogen atom, you get a continuum of solutions that just happen to mostly be unnormalizable. It's the measurement theory -- which demands normalization to make physical sense of the solution -- that effectively "eliminates" most of these solutions on grounds that they're not physical. So without this "divergence" between math and reality, it's not even obvious that quantization would emerge in the first place. (One can run across many examples of how physical reasoning impacts the "allowable" mathematical solution space in just about any field of physics.)
2) In quantum measurement theory, one does not map the mathematical wavefunction directly onto physical reality, but rather the "probability" of various real outcomes. Still, once the measurement is made, the probability of the actual outcome jumps to 100%, meaning that it is perfectly acceptable for a (mathematical) superposition to get mapped to one particular physical outcome. But quantum measurement theory also governs state preparation -- via a non-probabilistic rule. Here the particular physical outcome is always mapped onto a mathematical pure state, never a superposition between that state and others. So the map between physics and math is now asymmetric; one uses one set of rules when going from reality->math, and another set of rules when one goes from math->reality. If this asymmetry is real, then the premise of a 11 equivalence map between math and physics is demonstrably false. (For the record, I'd rather keep the math/physics map, and symmetrize quantum measurement theory...)
Ken
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I'm not exactly sure what "divergence" you're asking about here... Starting from the reasonable premise that pure mathematics is not part of the physical world, then the question isn't when they diverge, so much as when they can be said to be vaguely analogous. The branch of physics that deals with the rough overlap between math and reality is measurement theory (which tells one how to map physical events onto mathematical structures and vice-versa). And of course there are an infinite number of (presumably incorrect) measurement theories where the math diverges from the physical reality.
But still, I think I have a rough idea of what you're asking... so consider these two examples which might help clarify the issues you raise:
1) When you solve the (time-independent) Schrodinger equation for the Hydrogen atom, you get a continuum of solutions that just happen to mostly be unnormalizable. It's the measurement theory -- which demands normalization to make physical sense of the solution -- that effectively "eliminates" most of these solutions on grounds that they're not physical. So without this "divergence" between math and reality, it's not even obvious that quantization would emerge in the first place. (One can run across many examples of how physical reasoning impacts the "allowable" mathematical solution space in just about any field of physics.)
2) In quantum measurement theory, one does not map the mathematical wavefunction directly onto physical reality, but rather the "probability" of various real outcomes. Still, once the measurement is made, the probability of the actual outcome jumps to 100%, meaning that it is perfectly acceptable for a (mathematical) superposition to get mapped to one particular physical outcome. But quantum measurement theory also governs state preparation -- via a non-probabilistic rule. Here the particular physical outcome is always mapped onto a mathematical pure state, never a superposition between that state and others. So the map between physics and math is now asymmetric; one uses one set of rules when going from reality->math, and another set of rules when one goes from math->reality. If this asymmetry is real, then the premise of a 11 equivalence map between math and physics is demonstrably false. (For the record, I'd rather keep the math/physics map, and symmetrize quantum measurement theory...)
Ken
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Hi Ken,
Why do you deny some obvious restrictions in reality, which got lost in mathematics? In particular I refer to extensions of natural numbers.
Hopefully you will agree that there are no negative or imaginary tangible items. You might have debts but definitely no negative coins in your pocket.
For some implications you might read or reread at least my essays 369 and 527.
Eckard
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Why do you deny some obvious restrictions in reality, which got lost in mathematics? In particular I refer to extensions of natural numbers.
Hopefully you will agree that there are no negative or imaginary tangible items. You might have debts but definitely no negative coins in your pocket.
For some implications you might read or reread at least my essays 369 and 527.
Eckard
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Many thanks to everyone for their thoughts (and keep them coming!). Ken: (good to see you checking these forums, by the way) since you began from the premise that mathematics is not part of the physical world, I counter with the Quine-Putnam indispensability argument: if we believe in the concreteness of the physical theories described by mathematical objects then we also ought to believe in the concreteness of those mathematical objects themselves.
Also, regarding your second point, I don't see that as "divergent" in a sense. Even if the wavefunction doesn't directly represent something physically real, it still represents a probability that can be mapped to physical reality in some way. The same can be said of complex numbers. While they aren't directly physically observable, they are useful - even indispensable - in certain valid physical theories (in fact a colleague of mine even argues that certain physical effects with lasers are indirect evidence of complex numbers).
So, what I'm talking about is when math describes something that we know is very clearly physically impossible, then working backwards to where the same *type* of math starts to describe something physically possible and determining whether the cutoff is continuous or discrete. (And, for some math, perhaps you can't even do this.)
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Also, regarding your second point, I don't see that as "divergent" in a sense. Even if the wavefunction doesn't directly represent something physically real, it still represents a probability that can be mapped to physical reality in some way. The same can be said of complex numbers. While they aren't directly physically observable, they are useful - even indispensable - in certain valid physical theories (in fact a colleague of mine even argues that certain physical effects with lasers are indirect evidence of complex numbers).
So, what I'm talking about is when math describes something that we know is very clearly physically impossible, then working backwards to where the same *type* of math starts to describe something physically possible and determining whether the cutoff is continuous or discrete. (And, for some math, perhaps you can't even do this.)
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Physical evidence of numbers? I see such kind of articulation close to blown up points, rotated points, 3D points, and the like. Isn't the design of my home quite different from the really existing home? I do not refer to disagreements but to the fact that a plan is always limited to the realm of imagination while my home exists in reality.
I would like to distinguish between unavoidable abstraction and unjustified additional gaps between mathematics an reality. These gaps arose due to the fallacy that widening of mathematical notions implies enlarged degrees of freedom in physics too. In particular I blame the unrestricted unphysical use of negative and complex numbers. People in pubs describe mathematics as something ridiculous where three out of two people left a room and therefore someone has to come in as to make the room empty. Ask your laser colleague which of the two alternative complex representations he considers the physically true one. Both together is impossible, and so far the decision for one out of the two is arbitrary. Already the engineer Raffael Bombelli (1526-1573) understood: Any complex number always occurs together with its conjugate (col suo Residuo, cf. Gericke's Geschichte des Zahlbegriffs, p. 61). When Gauss in 1831 promoted complex numbers by representing them orthogonally in complex plane, he omitted the conjugacy.
No expert questions the usefulness of complex calculus. We should deal with the question whether or not final complex representations are really indispensable. Engineers do not have scruples when they e.g. use the complex relationship between two quantities. However, they learned to transform single quantities back into reality where the imaginary numbers vanish. Physicists tend to neglect such trivialities. The team Erwin Schroedinger and Peterle Weyl did not obey the due obligation of correct transformation into complex domain and return.
It might be a bit exaggerated when I am nonetheless calling Hermann Weyl the last at least consequently and honestly thinking physicist when he uttered "at the moment is no explanation in sight" and later "we are less certain than ever about the ultimate foundations of mathematics".
Eckard Blumschein
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I would like to distinguish between unavoidable abstraction and unjustified additional gaps between mathematics an reality. These gaps arose due to the fallacy that widening of mathematical notions implies enlarged degrees of freedom in physics too. In particular I blame the unrestricted unphysical use of negative and complex numbers. People in pubs describe mathematics as something ridiculous where three out of two people left a room and therefore someone has to come in as to make the room empty. Ask your laser colleague which of the two alternative complex representations he considers the physically true one. Both together is impossible, and so far the decision for one out of the two is arbitrary. Already the engineer Raffael Bombelli (1526-1573) understood: Any complex number always occurs together with its conjugate (col suo Residuo, cf. Gericke's Geschichte des Zahlbegriffs, p. 61). When Gauss in 1831 promoted complex numbers by representing them orthogonally in complex plane, he omitted the conjugacy.
No expert questions the usefulness of complex calculus. We should deal with the question whether or not final complex representations are really indispensable. Engineers do not have scruples when they e.g. use the complex relationship between two quantities. However, they learned to transform single quantities back into reality where the imaginary numbers vanish. Physicists tend to neglect such trivialities. The team Erwin Schroedinger and Peterle Weyl did not obey the due obligation of correct transformation into complex domain and return.
It might be a bit exaggerated when I am nonetheless calling Hermann Weyl the last at least consequently and honestly thinking physicist when he uttered "at the moment is no explanation in sight" and later "we are less certain than ever about the ultimate foundations of mathematics".
Eckard Blumschein
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I think I may have found a partial answer in Florin Moldoveanu's Fourth Prize essay for the recent essay contest, "Heuristic rule for constructing physics axiomatization."
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Dear Ian Durham,
I presume that the question you raised is intended for Professional level discussion. I am not a physicist. My first message was a general statement of my opinion that the use of mathematics by theoretical physicists has been only loosely connected to reality. It is true that the equations make excellent predictions. This is as it should be since they are modeled to fit the patterns observed in empirical evidence. It is the patterns that do the work of leading us to successful predictions.
Where I see theory and reality separate is in any attempt to invent indefinable properties and their units. It is through those invented units that the invented properties become solidified into the equations. Before this act is commited, the equations accurately represent empirical knowledge. After that act of inventing properties is begun, the equations veer away from empirical knowledge and become subservient, even captive, to theoretical ideas. If any added properties and their units of measurement cannot be defined in terms of the original empirical evidence, they are guesses.
I will end this message here. There were no comments made with regard to my first message. So, if this line of thought is not relevent to what you want to explore, then I will not pursue it. I have re-read your essay in order to better understand your viewpoint. Congratulations again on winning a prize and membership.
James
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I presume that the question you raised is intended for Professional level discussion. I am not a physicist. My first message was a general statement of my opinion that the use of mathematics by theoretical physicists has been only loosely connected to reality. It is true that the equations make excellent predictions. This is as it should be since they are modeled to fit the patterns observed in empirical evidence. It is the patterns that do the work of leading us to successful predictions.
Where I see theory and reality separate is in any attempt to invent indefinable properties and their units. It is through those invented units that the invented properties become solidified into the equations. Before this act is commited, the equations accurately represent empirical knowledge. After that act of inventing properties is begun, the equations veer away from empirical knowledge and become subservient, even captive, to theoretical ideas. If any added properties and their units of measurement cannot be defined in terms of the original empirical evidence, they are guesses.
I will end this message here. There were no comments made with regard to my first message. So, if this line of thought is not relevent to what you want to explore, then I will not pursue it. I have re-read your essay in order to better understand your viewpoint. Congratulations again on winning a prize and membership.
James
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James,
Actually, I think you've got the gist of what I'm going for and I actually agree with you that theoretical physicists have, perhaps, abused the connection between mathematics and reality just a bit. I think I need time to digest some of this (I'm a "processor" as opposed to a "reactor") but your ideas are indeed along the lines of what I was getting at (though not quite as specific as I had hoped).
> Congratulations again on winning a prize and membership.
Thanks! It shocked the heck out of me.
Ian
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Actually, I think you've got the gist of what I'm going for and I actually agree with you that theoretical physicists have, perhaps, abused the connection between mathematics and reality just a bit. I think I need time to digest some of this (I'm a "processor" as opposed to a "reactor") but your ideas are indeed along the lines of what I was getting at (though not quite as specific as I had hoped).
> Congratulations again on winning a prize and membership.
Thanks! It shocked the heck out of me.
Ian
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Dear Ian Durham,
My two messages were meant to test the water. It seemed most likely that you might be interested in comments that are based upon the acceptance of most of theoretical physics as representing reality. In that case, I would not pursue a discussion based upon the possibility that theoretical physics might have gone astray right from its beginning. Actually, everything I bring up in the blogs, forums, and contests are ideas for which I have already developed the specifics. My essay in the first contest put some of those specifics forward.
That essay was a limited, out of context, test to see if there was any interest by others in the possibility that theoretical physics went astray right from the start. I work alone and place my work on the Internet at my own website, because, that has been the only way available to proceed. I have written a great deal, with specifics, about everything I mention to others. I just don't bother bothering others. Others are either interested or they are not.
I will post a sample of my view of how the mathematical representation of theoretical physics diverged early from reality. Please say whatever you honestly feel about it. This time I will show how I think theory might have been started without introducing indefinable theoretical properties. It will be a simple, but also seriously different perspective. If you deem it to be unhelpful, then I will drop it. It will be my next message.
James
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My two messages were meant to test the water. It seemed most likely that you might be interested in comments that are based upon the acceptance of most of theoretical physics as representing reality. In that case, I would not pursue a discussion based upon the possibility that theoretical physics might have gone astray right from its beginning. Actually, everything I bring up in the blogs, forums, and contests are ideas for which I have already developed the specifics. My essay in the first contest put some of those specifics forward.
That essay was a limited, out of context, test to see if there was any interest by others in the possibility that theoretical physics went astray right from the start. I work alone and place my work on the Internet at my own website, because, that has been the only way available to proceed. I have written a great deal, with specifics, about everything I mention to others. I just don't bother bothering others. Others are either interested or they are not.
I will post a sample of my view of how the mathematical representation of theoretical physics diverged early from reality. Please say whatever you honestly feel about it. This time I will show how I think theory might have been started without introducing indefinable theoretical properties. It will be a simple, but also seriously different perspective. If you deem it to be unhelpful, then I will drop it. It will be my next message.
James
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Dear Ian Durham,
Here is an excerpt from an essay I wrote. The basic idea is that our only source of empirical knowledge is via photons that carry information about changes of distance with respect to time. My premise is that all properties should be expressible in terms of these two empirical properties. In other words, all theory should be clearly traceable, through their adopted units of...
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Here is an excerpt from an essay I wrote. The basic idea is that our only source of empirical knowledge is via photons that carry information about changes of distance with respect to time. My premise is that all properties should be expressible in terms of these two empirical properties. In other words, all theory should be clearly traceable, through their adopted units of...
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"Many thanks to everyone for their thoughts (and keep them coming!). Ken: (good to see you checking these forums, by the way) since you began from the premise that mathematics is not part of the physical world, I counter with the Quine-Putnam indispensability argument: if we believe in the concreteness of the physical theories described by mathematical objects then we also ought to believe in the concreteness of those mathematical objects themselves."
So, if I believe in quarks, I have to believe in sets, is that it?
What about applicable (useful) math that is inconsistent? Does this mean that you believe in inconsistent math?
It is possible that you have to have a sort of Platonist mindset to come up with some of the outlandish math that is around, however that doesn't make it right in an applied context and certainly doesn't make it real.
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So, if I believe in quarks, I have to believe in sets, is that it?
What about applicable (useful) math that is inconsistent? Does this mean that you believe in inconsistent math?
It is possible that you have to have a sort of Platonist mindset to come up with some of the outlandish math that is around, however that doesn't make it right in an applied context and certainly doesn't make it real.
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Well, maybe not, but certainly there is enough mathematics out there that *seems* inherent to make this debate worth having by a lot of people. But just to give an example of where I'm coming from, it is clear that certain animals can do very, very basic math (adding and subtracting not to mention recognition of certain geometric shapes as Pavlov demonstrated). So, at some level, it's not an entirely human construct which means it is at least partly inherent (discovered).
Perhaps mathematics isn't a single thing, then, i.e. maybe some mathematics is inherent (discovered) and some isn't. But then where is the cut-off between the two types? Or is it a gradual change? It's a much harder problem than it looks.
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Perhaps mathematics isn't a single thing, then, i.e. maybe some mathematics is inherent (discovered) and some isn't. But then where is the cut-off between the two types? Or is it a gradual change? It's a much harder problem than it looks.
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Ian, You wrote:
"Perhaps mathematics isn't a single thing, then, i.e. maybe some mathematics is inherent (discovered) and some isn't. But then where is the cut-off between the two types? Or is it a gradual change? It's a much harder problem than it looks."
I think so, and I would like to add: It might be worthwhile dealing with this challenge. Here you are:
- I expect "natural" mathematics to be globally consistent and free of arbitrary choices. Invented theorems, axioms etc. are at best candidates for contributing to the unique puzzle of appropriate instruments. Usual textbooks on algebra do not fulfill this criterion. An ugly mathematical language that is overly sophisticated and worrying indicates to me a lack of deep understanding of those who fabricated it.
- Having already looked into much original work, I am still reading the thick book Labyrinth of Thought by Ferreiros. I did not yet find compelling arguments for abandoning Euclid's notions of number and point, respectively, and introducing point-sets instead. While I am incompetent in so far I am not a mathematician, I feel entitled to judge that four mutually excluding pieces of arbitrary advice from four experts cannot be correct but are possibly wrong altogether. While I do not expect the "gods" learning from me I am nonetheless claiming to suggest a reasonable way out.
- There are more or less equivalent mathematical descriptions of the same matter. This is well known to physicists for matrices used by Heisenberg/Born and Dirac, which correspond to the picture by Schroedinger/Weyl.
- According to Ockham, mathematics without redundancy deserves preference. I discovered something, which is strictly speaking trivial: The additional degree of freedom in C has no bearing in applications where the variables are restricted to R+. While it is advantageous to arbitrarily refer for instance pressure to 20 micro Pascal, there is in principle no mathematical reason to use negative or complex values.
Regards,
Eckard
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"Perhaps mathematics isn't a single thing, then, i.e. maybe some mathematics is inherent (discovered) and some isn't. But then where is the cut-off between the two types? Or is it a gradual change? It's a much harder problem than it looks."
I think so, and I would like to add: It might be worthwhile dealing with this challenge. Here you are:
- I expect "natural" mathematics to be globally consistent and free of arbitrary choices. Invented theorems, axioms etc. are at best candidates for contributing to the unique puzzle of appropriate instruments. Usual textbooks on algebra do not fulfill this criterion. An ugly mathematical language that is overly sophisticated and worrying indicates to me a lack of deep understanding of those who fabricated it.
- Having already looked into much original work, I am still reading the thick book Labyrinth of Thought by Ferreiros. I did not yet find compelling arguments for abandoning Euclid's notions of number and point, respectively, and introducing point-sets instead. While I am incompetent in so far I am not a mathematician, I feel entitled to judge that four mutually excluding pieces of arbitrary advice from four experts cannot be correct but are possibly wrong altogether. While I do not expect the "gods" learning from me I am nonetheless claiming to suggest a reasonable way out.
- There are more or less equivalent mathematical descriptions of the same matter. This is well known to physicists for matrices used by Heisenberg/Born and Dirac, which correspond to the picture by Schroedinger/Weyl.
- According to Ockham, mathematics without redundancy deserves preference. I discovered something, which is strictly speaking trivial: The additional degree of freedom in C has no bearing in applications where the variables are restricted to R+. While it is advantageous to arbitrarily refer for instance pressure to 20 micro Pascal, there is in principle no mathematical reason to use negative or complex values.
Regards,
Eckard
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Of course I meant textbooks of analysis, not of algebra.
Considering some influences on Dedekind including those of Cauchy, Dirichlet, Gauss, Gudermann, Hankel, Heine, Herbart, Jacobi, Moebius, Martin Ohm, Pluecker, Riemann, Steiner, von Stern, and Weierstrass, I got aware that already Gauss "regarded the interpretation of (complex) numbers as points (in a plane)..." and Cantor himself indicated that the term Maechtigkeit (cardinality) was taken from one of Steiner's work.
Eckard
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Considering some influences on Dedekind including those of Cauchy, Dirichlet, Gauss, Gudermann, Hankel, Heine, Herbart, Jacobi, Moebius, Martin Ohm, Pluecker, Riemann, Steiner, von Stern, and Weierstrass, I got aware that already Gauss "regarded the interpretation of (complex) numbers as points (in a plane)..." and Cantor himself indicated that the term Maechtigkeit (cardinality) was taken from one of Steiner's work.
Eckard
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Ian,
you solicited thoughts on " ... the Quine-Putnam indispensability argument which basically says that if we believe in the concreteness of the physical theories described by mathematical objects then we also ought to believe in the concreteness of those mathematical objects themselves."
"Concrete" as opposed to what? Abstract? Then is there such a thing as "concrete" language? That is, do symbols stand for themselves only, or are they independent of the objects for which they stand?
My view: language is independent of meaning. It follows therefore that physical (mathematical) theories are no more concrete than the language (mathematics) that describes them. There is a distinction to be made between objects and meaning; there is no such distinction between a (physical) theory that maps symbols to symbols and a (mathematical) theory that maps symbols to objects. In other words, physical objects and mathematical objects are both symbolic representations independent of phenomenological observation. Thus, physical science is necessarily an open system, progressing toward what Popper called verisimilitude, an asymptotic approach to truth, and never offering a completely closed judgment of truth, a proof of its conclusions. Mathematical science necessarily offers closed judgments based on axiomatic deduction, and theorems (true mathematical statements) are proven in the domains to which they apply. The physical domain--being the whole observed, and even perhaps the unobserved, universe--may not be subject to such axiomatization. How would we know in any case?--Goedel taught us that no set of axioms is sufficient to prove its own self-consistency; there always exist true statements that cannot be deduced from the axioms.
That being said, there are serious attempts to recast mathematics as an experimental science. Chaitin, Wolfram, et al, may really lead us to a common point of closure between what we say about the world and what the world says back.
Tom
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you solicited thoughts on " ... the Quine-Putnam indispensability argument which basically says that if we believe in the concreteness of the physical theories described by mathematical objects then we also ought to believe in the concreteness of those mathematical objects themselves."
"Concrete" as opposed to what? Abstract? Then is there such a thing as "concrete" language? That is, do symbols stand for themselves only, or are they independent of the objects for which they stand?
My view: language is independent of meaning. It follows therefore that physical (mathematical) theories are no more concrete than the language (mathematics) that describes them. There is a distinction to be made between objects and meaning; there is no such distinction between a (physical) theory that maps symbols to symbols and a (mathematical) theory that maps symbols to objects. In other words, physical objects and mathematical objects are both symbolic representations independent of phenomenological observation. Thus, physical science is necessarily an open system, progressing toward what Popper called verisimilitude, an asymptotic approach to truth, and never offering a completely closed judgment of truth, a proof of its conclusions. Mathematical science necessarily offers closed judgments based on axiomatic deduction, and theorems (true mathematical statements) are proven in the domains to which they apply. The physical domain--being the whole observed, and even perhaps the unobserved, universe--may not be subject to such axiomatization. How would we know in any case?--Goedel taught us that no set of axioms is sufficient to prove its own self-consistency; there always exist true statements that cannot be deduced from the axioms.
That being said, there are serious attempts to recast mathematics as an experimental science. Chaitin, Wolfram, et al, may really lead us to a common point of closure between what we say about the world and what the world says back.
Tom
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Tom,
Awesome reply! I agree in that regard that mathematics - as given by the symbols we employ to carry out analyses - is, indeed, a language (certainly, like language, our choice of symbols is arbitrary). But are there "mathematical objects" that lie "beneath" that language? For instance, while the word "water" isn't itself a real object in the way we are talking (since in Spanish its agua, for example), but it is very clear that what this word describes is very real. So how much of what underlies mathematics is "real?" Since much of mathematics can be described as a process, are these underlying processes real?
Ian
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Awesome reply! I agree in that regard that mathematics - as given by the symbols we employ to carry out analyses - is, indeed, a language (certainly, like language, our choice of symbols is arbitrary). But are there "mathematical objects" that lie "beneath" that language? For instance, while the word "water" isn't itself a real object in the way we are talking (since in Spanish its agua, for example), but it is very clear that what this word describes is very real. So how much of what underlies mathematics is "real?" Since much of mathematics can be described as a process, are these underlying processes real?
Ian
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Ian,
You wrote, "So how much of what underlies mathematics is 'real?' Since much of mathematics can be described as a process, are these underlying processes real?"
In those terms, I think we _have_ to take mathematics as an experimental science. That is, as you implied elsewhere in your post, there is no reality in arbitrarily chosen symbols and their manipulation; as an art, mathematics is not "about" anything, any more than natural language--that the syntax of a statement may be entirely correct, does not imply meaning.
If we take meaning as real (that is what I mean by 'what the world says back to us'), then the meaning that comes from a computer programming language is as real as the substrate on which the program runs, because that is the process, the mechanical throughput, that mimics all physical processes. I.e., information flows continuously over the substrate, though the exchange of information between nodes is in discrete units. The bad news is that Chaitin has discovered an algorithmically defined but uncomputable number (Omega) whose digital expansion is normal but whose value depends on the computer language running the algorithm. So even our arithmetic, in this context, possesses a degree of built in uncertainty. This raises the question of whether a complete one-to-one relation between language and meaning is even possible.
Tom
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You wrote, "So how much of what underlies mathematics is 'real?' Since much of mathematics can be described as a process, are these underlying processes real?"
In those terms, I think we _have_ to take mathematics as an experimental science. That is, as you implied elsewhere in your post, there is no reality in arbitrarily chosen symbols and their manipulation; as an art, mathematics is not "about" anything, any more than natural language--that the syntax of a statement may be entirely correct, does not imply meaning.
If we take meaning as real (that is what I mean by 'what the world says back to us'), then the meaning that comes from a computer programming language is as real as the substrate on which the program runs, because that is the process, the mechanical throughput, that mimics all physical processes. I.e., information flows continuously over the substrate, though the exchange of information between nodes is in discrete units. The bad news is that Chaitin has discovered an algorithmically defined but uncomputable number (Omega) whose digital expansion is normal but whose value depends on the computer language running the algorithm. So even our arithmetic, in this context, possesses a degree of built in uncertainty. This raises the question of whether a complete one-to-one relation between language and meaning is even possible.
Tom
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Tom,
I think I need to read some of Chaitin's writing on this topic...
Ian
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I think I need to read some of Chaitin's writing on this topic...
Ian
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Ian,
There are plenty of technical primary and secondary sources available; however, I think the most comprehensive view and most fun to read, though philosophical and not technical, is Chaitin's collection of essays:
Thinking About Godel and Turing: essays on complexity, 1970-2007.
Tom
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There are plenty of technical primary and secondary sources available; however, I think the most comprehensive view and most fun to read, though philosophical and not technical, is Chaitin's collection of essays:
Thinking About Godel and Turing: essays on complexity, 1970-2007.
Tom
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Tom,
Thanks for the reference. I'll have to get my hands on a copy (seems like something I ought to order for our library...).
Ian
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Thanks for the reference. I'll have to get my hands on a copy (seems like something I ought to order for our library...).
Ian
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Can I get a scholar to do the arithmatic.
?
to get the equation for the big bang we look at a Godel universe where time is a contradiction.
And we look at the penrose equation for a black hole in a Godel universe.
And reverse it to get a non contradictory equation for a big bang in our universe.
That is without the meanningless infnities.
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?
to get the equation for the big bang we look at a Godel universe where time is a contradiction.
And we look at the penrose equation for a black hole in a Godel universe.
And reverse it to get a non contradictory equation for a big bang in our universe.
That is without the meanningless infnities.
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Ian,
Mathematics is not fundamental because they are born out of logic. Logic is more fundamental to the universe. The universe only requires logic in order to work. But the unnatural vantage point of the observer creates and requires numbers greater than one (1).
Mathematics is the extension of logic for the conscious mind. Sure, nice geometric and mathematical structures emerges in nature from the effect of logic on large numbers. But again, the appreciation of those same structures is but for the conscious mind.
Marcel,
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Mathematics is not fundamental because they are born out of logic. Logic is more fundamental to the universe. The universe only requires logic in order to work. But the unnatural vantage point of the observer creates and requires numbers greater than one (1).
Mathematics is the extension of logic for the conscious mind. Sure, nice geometric and mathematical structures emerges in nature from the effect of logic on large numbers. But again, the appreciation of those same structures is but for the conscious mind.
Marcel,
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Marcel,
I guess I see logic as being mathematics on some level. But, anyway, while I'm not necessarily disagreeing with you, you have to admit that if numbers greater than one (1) are a result of the conscious mind then consciousness as we understand it isn't unique to humans.
Ian
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I guess I see logic as being mathematics on some level. But, anyway, while I'm not necessarily disagreeing with you, you have to admit that if numbers greater than one (1) are a result of the conscious mind then consciousness as we understand it isn't unique to humans.
Ian
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Ian,
"I guess I see logic as being mathematics on some level" sounds not very logical to me. Didn't you need staying outside yourself for such statement? Incidentally you may refer to Hilbert who also tried to subordinate logic below mathematics.
In order to get a feeling how ordinary people like me react on frequent use of something like "on some level" or "to some extent" read Wolfram's essay.
Do not get me wrong. I do not intend offending you or him. I just suggest focusing on the essentials.
Best,
Eckard
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"I guess I see logic as being mathematics on some level" sounds not very logical to me. Didn't you need staying outside yourself for such statement? Incidentally you may refer to Hilbert who also tried to subordinate logic below mathematics.
In order to get a feeling how ordinary people like me react on frequent use of something like "on some level" or "to some extent" read Wolfram's essay.
Do not get me wrong. I do not intend offending you or him. I just suggest focusing on the essentials.
Best,
Eckard
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Eckard,
I will be the first to admit that statement was rather vague. I guess I always envisioned logic as being the foundation upon which mathematics is built and, to me, all extensions of logic are just mathematics. But maybe I have a broader view of what mathematics is (honestly, it's just a label we give to a type of language, I guess you could say).
Actually, that might be another intriguing question: is there a branch of logic or a part of logic that you *can't* build mathematics on? That would be quite interesting if it were true (actually it would be interesting if it were not true as well since it would give us a glimpse of the relationship between mathematics and logic).
Ian
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I will be the first to admit that statement was rather vague. I guess I always envisioned logic as being the foundation upon which mathematics is built and, to me, all extensions of logic are just mathematics. But maybe I have a broader view of what mathematics is (honestly, it's just a label we give to a type of language, I guess you could say).
Actually, that might be another intriguing question: is there a branch of logic or a part of logic that you *can't* build mathematics on? That would be quite interesting if it were true (actually it would be interesting if it were not true as well since it would give us a glimpse of the relationship between mathematics and logic).
Ian
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Ian,
Maybe “consciousness” is a bit too much to ask. Lets just say an integrated geometric point of view. A camera with proper optics may capture a point of view e.g. the honeycomb structure in a picture. Does it appreciate it ??? Don’t think so. Why “integrated” ? Because, if we could discriminate every single incoming photon, all we would see is a bunch of scintillations… Like the film emulsion or the CCD chip, we must accumulate the incoming data in order to be able to form an image. We have to remember also that the capturing of photons to make a picture is not related to the actual distance to the origin of the photon. Otherwise, all we’d see is a slice of the landscape at some distance. In other words, we can see at a glance in the same moment of perception both the Sun at 8 light minutes from us and the Moon at about half a light second away. These are the specs we use to view the universe.
So, either you are one (1) minding your own business, or you are a spectator with a specific point of view that allows you to count numbers higher than one. Yes, it is kind of a metaphysical reasoning: You are, or you watch what is. I can see four levels here. 1) you are one minding your business. 2) you are a spectator that has a 2D point of view 3) your mind allows you to figure out a 3D world out of your 2D point of view. 4) you forget about the D’s because you know the universe is not a point of view and you understand it as 1) does.
Marcel,
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Maybe “consciousness” is a bit too much to ask. Lets just say an integrated geometric point of view. A camera with proper optics may capture a point of view e.g. the honeycomb structure in a picture. Does it appreciate it ??? Don’t think so. Why “integrated” ? Because, if we could discriminate every single incoming photon, all we would see is a bunch of scintillations… Like the film emulsion or the CCD chip, we must accumulate the incoming data in order to be able to form an image. We have to remember also that the capturing of photons to make a picture is not related to the actual distance to the origin of the photon. Otherwise, all we’d see is a slice of the landscape at some distance. In other words, we can see at a glance in the same moment of perception both the Sun at 8 light minutes from us and the Moon at about half a light second away. These are the specs we use to view the universe.
So, either you are one (1) minding your own business, or you are a spectator with a specific point of view that allows you to count numbers higher than one. Yes, it is kind of a metaphysical reasoning: You are, or you watch what is. I can see four levels here. 1) you are one minding your business. 2) you are a spectator that has a 2D point of view 3) your mind allows you to figure out a 3D world out of your 2D point of view. 4) you forget about the D’s because you know the universe is not a point of view and you understand it as 1) does.
Marcel,
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Marcel,
Interesting. I'll have to think about that. In the meantime, I will say that, to some extent, distant to an object *does* have an effect - remember that light redshifts the longer it travels. So, true, the human mind isn't capable of differing between the light coming from the Sun and that from the Moon, but there are things in the cosmos we can't see with the naked eye because they have been redshifted right out of the visible spectrum.
Ian
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Interesting. I'll have to think about that. In the meantime, I will say that, to some extent, distant to an object *does* have an effect - remember that light redshifts the longer it travels. So, true, the human mind isn't capable of differing between the light coming from the Sun and that from the Moon, but there are things in the cosmos we can't see with the naked eye because they have been redshifted right out of the visible spectrum.
Ian
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I know that you stated that you wanted to approach the subject from a non-speculative standpoint but some of the things you are querying are dependant on the Ontologial status one places on Mathermatics.
Is Mathematics something that is discovered or created? Is mathematics used as an inferential model of reality or a description of it?
What is the mathematical model under scrutiny meant to describe? What is the aim and scope? As an example, the foundation of much of the mathematics employed in modern theoretical physics is based on the implcit and explcit use of the imaginary number system and it's subsequent analysis. There is no physical or real correlation with imaginary numbers and phenomenon or measureable quanitities in the real world, but the theoretical use of the imaginary number system is critical to the mathematical formalism present in much of Physics.
Also, historically, when singularities or divergences are encountered in mathematical physics, they are simply ignored or brushed aside. A classic case is renormalization in QED and field theories.
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Is Mathematics something that is discovered or created? Is mathematics used as an inferential model of reality or a description of it?
What is the mathematical model under scrutiny meant to describe? What is the aim and scope? As an example, the foundation of much of the mathematics employed in modern theoretical physics is based on the implcit and explcit use of the imaginary number system and it's subsequent analysis. There is no physical or real correlation with imaginary numbers and phenomenon or measureable quanitities in the real world, but the theoretical use of the imaginary number system is critical to the mathematical formalism present in much of Physics.
Also, historically, when singularities or divergences are encountered in mathematical physics, they are simply ignored or brushed aside. A classic case is renormalization in QED and field theories.
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Well, actually what I'm trying to do is see if it is possible to actually answer the question of whether mathematics is discovered or created. So its ontological status is part of what I'm trying to figure out.
I also wouldn't say that we always ignore singularities or divergences. Certainly in relativity we've spent a great deal of time studying the consequences of singularities (and have been led down such strange roads as wormholes and baby universes).
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I also wouldn't say that we always ignore singularities or divergences. Certainly in relativity we've spent a great deal of time studying the consequences of singularities (and have been led down such strange roads as wormholes and baby universes).
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Also, on another note, the methodology and use of mathematics in physics is self-correcting in that one of the implicit tenets of mathematical physics is that the results gleaned from the mathematics must represent a physically admisable solution. If the result is telling you something that is nonsensical from an emperical standpoint, you toss it out as an inadmissable solution. For example, if you are performing a calculation in classical mechancis and you end up with something like negative mass, you know that you either erred in your solution or the results are invalid and do not correlate with any real phenomenon.
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Yes, Bubba Gump. Some worst theoreticians lost this willingness to correct nonsensical solutions to the extent that they even allow for negative probability.
I consider electrical engineering in principle close enough to reality as to decide which solution is nonsensical while much of possibly questionable mathematical preconditions is required in measurement of properties that are attributed to single particles.
Measured functions of time are always realistic in that they are not imaginary and do not include future. Correspondingly, one has to use Heaviside's trick of analytic continuation as to expand them into the future before performing a spectral analysis by means of the complex Fourier transform on an nonsensical time scale between minus infinity and plus infinity.
Resulting quantities in complex plane must be unrealistic in that they are complex functions of positive as well as negative frequencies in order to correspond to realistic functions in real domain. Realistic frequencies of frequency correspond to so called analytic (complex) bilateral functions of time.
Eckard
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I consider electrical engineering in principle close enough to reality as to decide which solution is nonsensical while much of possibly questionable mathematical preconditions is required in measurement of properties that are attributed to single particles.
Measured functions of time are always realistic in that they are not imaginary and do not include future. Correspondingly, one has to use Heaviside's trick of analytic continuation as to expand them into the future before performing a spectral analysis by means of the complex Fourier transform on an nonsensical time scale between minus infinity and plus infinity.
Resulting quantities in complex plane must be unrealistic in that they are complex functions of positive as well as negative frequencies in order to correspond to realistic functions in real domain. Realistic frequencies of frequency correspond to so called analytic (complex) bilateral functions of time.
Eckard
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Absolutely. Unfortunately a lot of mathematical physicists these days - and even theoretical computer scientists (which blows my mind) - have lost sight of this. Now, I'm "tolerant" enough to take crazy results and pursue them for awhile to see where they might lead because sometimes they do lead to genuinely useful physical insights. But at some point one has to say that the abstraction is just too much and one tosses it out.
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Yes, what represents a physically admissable solution obviously depends on context. Sometimes, especially in the highly abstract world of mathematical physics, what represents 'physically admissable' can be hard to understand. I guess a classic case would be the physicist coming to terms with the singularity in the field equations of GR for a highly compact, dense object. Obviously, there was some angst over the notion of a singularity present in nature and there was temptation to write it off as implausible.
Anwyays, I wanted to expand on the topic of mathematical inference a bit but did not want to hijack the thread by going off target. For some reason I cannot create a new thread and I see no options for doing this. What gives?
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Anwyays, I wanted to expand on the topic of mathematical inference a bit but did not want to hijack the thread by going off target. For some reason I cannot create a new thread and I see no options for doing this. What gives?
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Eckard,
I have worked with “negative probabilities”. For example, the probability of finding a particle in a position that decreases as time of observation increases.
In normal probability of position, two factors are at play. 1) First, the amount of time one spends observing that position and 2) the amount of time the object actually spend in that position with respect (relative) to the amount of time spent elsewhere. Now, for an equal observation time ( factor 1 is constant) and an equal relative time spent in that position by the particle (factor 2 is constant), the probability of finding the particle is always the same. If we increase the time of observation, this probability of finding the particle in that position normally increases and that would be a positive probability.
But if the particle is released in position A in a gravitational field, the probability of finding it in position A decreases with time or, a negative probability. The maximum of probability of finding this particle is greater toward the ground.
We may see this particle in a gravitational field as being in a probability gradient where the probability for it to go upward is, no matter the amount of time one waits, an impossibility. And the possibility that it moves toward the ground is in fact a certainty. (unless one stops it!)
In a gravitational field, this negative probability in point A is coupled with higher probability in one direction; toward the ground. But there are places with negative probability with no preference of direction. Things in such a place simply “want” to get out of there. This is an explosion, and the ontological passage of time is such an explosion.
(from here you can go to my essay and find out about the rest of the story)
Marcel,
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I have worked with “negative probabilities”. For example, the probability of finding a particle in a position that decreases as time of observation increases.
In normal probability of position, two factors are at play. 1) First, the amount of time one spends observing that position and 2) the amount of time the object actually spend in that position with respect (relative) to the amount of time spent elsewhere. Now, for an equal observation time ( factor 1 is constant) and an equal relative time spent in that position by the particle (factor 2 is constant), the probability of finding the particle is always the same. If we increase the time of observation, this probability of finding the particle in that position normally increases and that would be a positive probability.
But if the particle is released in position A in a gravitational field, the probability of finding it in position A decreases with time or, a negative probability. The maximum of probability of finding this particle is greater toward the ground.
We may see this particle in a gravitational field as being in a probability gradient where the probability for it to go upward is, no matter the amount of time one waits, an impossibility. And the possibility that it moves toward the ground is in fact a certainty. (unless one stops it!)
In a gravitational field, this negative probability in point A is coupled with higher probability in one direction; toward the ground. But there are places with negative probability with no preference of direction. Things in such a place simply “want” to get out of there. This is an explosion, and the ontological passage of time is such an explosion.
(from here you can go to my essay and find out about the rest of the story)
Marcel,
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Marcel,
I did not find an essay of you on the topic time, and I looked in vain into your essay on the topic"what is ultimately possible" for a more understandable to me explanation of your negative probability.
As an engineer I am familiar with negative values of resistance (du/di, not u/i). I also know that the same distance can be attributed to a negative x coordinate as well as to a positive radius. In all, there are several cases where negative values are reasonable. Velocity can change direction. Nonetheless I maintain that basic physical quantities are scalar ones that have a natural reference zero and primarily no negative values.
Could you please give a reference that might help me to understand your negative probability? If I understood you correctly, you consider a small box (part of space) that might or might not contain at least one a particle. Correct? Well you wrote probability of finding instead of probability of being in the box. However, you did not yet reveal to me your method and criterion of finding.
Given you did consider as usual the particle(s) contained or found in any case if only it was in the box at least once within the given timespan. Then I did not doubt that you understood: Extending the timespan of observation can only increase the probability. Are you familiar with conditional probability?
Given you did count the total time a single particle might be expected to be in summa located in the box on average, and the likelihood of being contained decreasing with time, then I could also imagine the average probability reduced or even equal to zero but never negative.
Incidentally, do you share Cantor's opinion that there are more real numbers than rational numbers? Are there more positive and negative numbers altogether as compared to just the positive ones? I am challenging your forensic experience.
Eckard
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I did not find an essay of you on the topic time, and I looked in vain into your essay on the topic"what is ultimately possible" for a more understandable to me explanation of your negative probability.
As an engineer I am familiar with negative values of resistance (du/di, not u/i). I also know that the same distance can be attributed to a negative x coordinate as well as to a positive radius. In all, there are several cases where negative values are reasonable. Velocity can change direction. Nonetheless I maintain that basic physical quantities are scalar ones that have a natural reference zero and primarily no negative values.
Could you please give a reference that might help me to understand your negative probability? If I understood you correctly, you consider a small box (part of space) that might or might not contain at least one a particle. Correct? Well you wrote probability of finding instead of probability of being in the box. However, you did not yet reveal to me your method and criterion of finding.
Given you did consider as usual the particle(s) contained or found in any case if only it was in the box at least once within the given timespan. Then I did not doubt that you understood: Extending the timespan of observation can only increase the probability. Are you familiar with conditional probability?
Given you did count the total time a single particle might be expected to be in summa located in the box on average, and the likelihood of being contained decreasing with time, then I could also imagine the average probability reduced or even equal to zero but never negative.
Incidentally, do you share Cantor's opinion that there are more real numbers than rational numbers? Are there more positive and negative numbers altogether as compared to just the positive ones? I am challenging your forensic experience.
Eckard
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I believe that a glaring problem in Physics education today is that instructors are too concerned with presenting mathematical formalisms divorced from any context of physical intuition. Much of modern Physics has become a branch of Applied Mathematics and, as such, should now fall under the tutelage of University Mathematics departments. Many Physics programs around the world might as well close down the Freshman laboratories and send everyone packing to the Mathematics building. There simply is not enough concentration on having students think intuitively or pragmatically in an attempt to understand physical phenomenon.
Even less priority is placed on giving students any rudimentary understanding of the origins of the theories they are presented with. Obviously, education in Physics must include a reliance on the theoretical tools which physicists employ. As such, a great majority of time must be devoted to mathematical formalism. However, too often, universities are churning out applied mathematicians, not scientists. They are not thinking like scientists. They are thinking like mathematicians. The fact that there are String Theorists stating that perhaps we should reevaluate what it means for a theory to be verified kind of gets to the heart of the problem. In many cases, it appears that some communities in Theoretical Physics are losing sight of the larger picture and have become too laden down in mathematical formalism. The actual Physics is nowhere to be seen.
This exclusive reliance on abstract mathematical formalism causes many to lose sight of the fact that Physics is an empirical science and always will be one. Most students of Physics will have one rudimentary freshman lab requirement and then perhaps a laboratory course in electronics. After that, they will never step foot inside of a laboratory for the rest of their professional lives.
IMO, Maxwell's, 'A Dynamical Theory of the Electromagnetic Field' should be required reading in E&M and portions of Newton's 'Principia' should be required in the obligatory Mechanics course.
What was the physical intuition Maxwell used? How about Newton? What was he really thinking and from where did the impetus for his ideas come? How did he arrive at what he did?
You would be surprised at how much your own understanding of the classic subjects is enhanced by thinking along with the original source(s) instead of relying exclusively on the highly generalized, modern version of the subject that is presented. Having an instructor throw out the equations and tell students to sit down and do the problem sets totally sterilizes the subject matter and turns our future scientists into automatons.
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Even less priority is placed on giving students any rudimentary understanding of the origins of the theories they are presented with. Obviously, education in Physics must include a reliance on the theoretical tools which physicists employ. As such, a great majority of time must be devoted to mathematical formalism. However, too often, universities are churning out applied mathematicians, not scientists. They are not thinking like scientists. They are thinking like mathematicians. The fact that there are String Theorists stating that perhaps we should reevaluate what it means for a theory to be verified kind of gets to the heart of the problem. In many cases, it appears that some communities in Theoretical Physics are losing sight of the larger picture and have become too laden down in mathematical formalism. The actual Physics is nowhere to be seen.
This exclusive reliance on abstract mathematical formalism causes many to lose sight of the fact that Physics is an empirical science and always will be one. Most students of Physics will have one rudimentary freshman lab requirement and then perhaps a laboratory course in electronics. After that, they will never step foot inside of a laboratory for the rest of their professional lives.
IMO, Maxwell's, 'A Dynamical Theory of the Electromagnetic Field' should be required reading in E&M and portions of Newton's 'Principia' should be required in the obligatory Mechanics course.
What was the physical intuition Maxwell used? How about Newton? What was he really thinking and from where did the impetus for his ideas come? How did he arrive at what he did?
You would be surprised at how much your own understanding of the classic subjects is enhanced by thinking along with the original source(s) instead of relying exclusively on the highly generalized, modern version of the subject that is presented. Having an instructor throw out the equations and tell students to sit down and do the problem sets totally sterilizes the subject matter and turns our future scientists into automatons.
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Amen to that. We try to give our students a broad background and, though I'm a theorist, I always make an attempt to make it "physical." In fact, Carl Caves once said essentially the same thing (we can't lose site of the physicality).
You should really read Moore's Six Ideas That Shaped Physics. It really tries to teach what you're talking about - that conceptual under-current that is so crucial.
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You should really read Moore's Six Ideas That Shaped Physics. It really tries to teach what you're talking about - that conceptual under-current that is so crucial.
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Eckard,
I missed the contest on the nature of time and the essay on the limits of physics does not mention negative probability.
…. “that have a natural reference zero”… If you think in absolute terms yes, but in relative terms, there may be a negative values..
I re-read my post and it is not all too clear.. The idea of a negative probability came to me when looking for a wave-like distribution that would represent motion as for an associated wave. I made the graph of the function (gauss minus SIN). It gave a kind of a skewed distribution with a negative first half wave and an above normal for the second half, for a motion in the plus x – axis. This was what I was looking for as a distribution representing a wave made of a variation of the rate of passage of time. Just as the time rate gradient causes gravitational fall, the structure of this distribution represented inertia as a probability distribution based on a wave of variation in the rate of passage of time. ( to understand this passage of time you must go back to the essay that explains the logical origin of the substantial passage of time)
Calling the first half of the (gauss – sin) graph a “negative probability” appeared natural to me because it was below the x axis. The second half of the graph is higher than the normal probability, which suggests a relative higher probability. One may understand the directionality and spontaneity of this pair by the conjunction of this negative probability and higher than normal probability…
The original idea was that pushing a particle would change its gaussian normal probability to a skewed (gauss – sin) type of probability representing now its inertial motion. (this is an absolute metaphysical explanation, so, forget in this context the empirical Relativity point of view)
I don’t know about conditional probability and do not know about Cantor’s opinion. But I am interesting in a natural origin to the structure of numbers i.e. even – odd – prime numbers. Can you tell me in short if and how this conditional probability could apply to the above exercise?
Thanks,
Marcel,
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I missed the contest on the nature of time and the essay on the limits of physics does not mention negative probability.
…. “that have a natural reference zero”… If you think in absolute terms yes, but in relative terms, there may be a negative values..
I re-read my post and it is not all too clear.. The idea of a negative probability came to me when looking for a wave-like distribution that would represent motion as for an associated wave. I made the graph of the function (gauss minus SIN). It gave a kind of a skewed distribution with a negative first half wave and an above normal for the second half, for a motion in the plus x – axis. This was what I was looking for as a distribution representing a wave made of a variation of the rate of passage of time. Just as the time rate gradient causes gravitational fall, the structure of this distribution represented inertia as a probability distribution based on a wave of variation in the rate of passage of time. ( to understand this passage of time you must go back to the essay that explains the logical origin of the substantial passage of time)
Calling the first half of the (gauss – sin) graph a “negative probability” appeared natural to me because it was below the x axis. The second half of the graph is higher than the normal probability, which suggests a relative higher probability. One may understand the directionality and spontaneity of this pair by the conjunction of this negative probability and higher than normal probability…
The original idea was that pushing a particle would change its gaussian normal probability to a skewed (gauss – sin) type of probability representing now its inertial motion. (this is an absolute metaphysical explanation, so, forget in this context the empirical Relativity point of view)
I don’t know about conditional probability and do not know about Cantor’s opinion. But I am interesting in a natural origin to the structure of numbers i.e. even – odd – prime numbers. Can you tell me in short if and how this conditional probability could apply to the above exercise?
Thanks,
Marcel,
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Marcel,
Given a salesman offers 50% off and an additional 40 % reduction. Does this mean you have to pay just 10%? Why do you feel entitled to simply add probabilities?
You may add velocities but you have to multiply reductions of price as well as probabilities. Ergo, probability can neither get smaller than zero nor larger than one. You were muddling probability with velocity.
What about the question whether or not the positive plus negative numbers together are more than only the positive ones, Cantor found it correctly out but he admitted to Dedekind: I cannot believe it. Take his doubt as an other example for a typical fallacy due to superficial thinking.
Forget unnecessary musing about even, odd, and prime numbers. You should rather ask yourself why on cannot put zero and infinity in these drawers. The original structure of numbers, as already be found in Euclid's definitions, was carelessly abandoned in the 19th century mathematics.
Eckard
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Given a salesman offers 50% off and an additional 40 % reduction. Does this mean you have to pay just 10%? Why do you feel entitled to simply add probabilities?
You may add velocities but you have to multiply reductions of price as well as probabilities. Ergo, probability can neither get smaller than zero nor larger than one. You were muddling probability with velocity.
What about the question whether or not the positive plus negative numbers together are more than only the positive ones, Cantor found it correctly out but he admitted to Dedekind: I cannot believe it. Take his doubt as an other example for a typical fallacy due to superficial thinking.
Forget unnecessary musing about even, odd, and prime numbers. You should rather ask yourself why on cannot put zero and infinity in these drawers. The original structure of numbers, as already be found in Euclid's definitions, was carelessly abandoned in the 19th century mathematics.
Eckard
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Eckard,
"Given a salesman offers 50% off and an additional 40 % reduction. Does this mean you have to pay just 10%? Why do you feel entitled to simply add probabilities?"
--- In dealing with the probability of position of a particle… the sum must be one; the particle exists somewhere. But, it is more of an image. Gravity is often portrayed as a dip in a sheet, an attractor for balls rolling in the vicinity. Consider now a bump in the same sheet. Place a ball on the bump and it will leave in whatever direction… this bump is my negative probability.
"You may add velocities but you have to multiply reductions of price as well as probabilities. Ergo, probability can neither get smaller than zero nor larger than one. You were muddling probability with velocity."
--- I know I am muddling something in my explanation, but it is not what you say.
"What about the question whether or not the positive plus negative numbers together are more than only the positive ones, Cantor found it correctly out but he admitted to Dedekind: I cannot believe it. Take his doubt as an other example for a typical fallacy due to superficial thinking."
--- Superficial reading: Since the early Greek philosophers, we have understood the distinction between two important concepts: the underlying reality and our perceptual experience. Over the centuries, we have always mixed the two concepts at the same time and amounted to nothing. Around the time of Newton, Descartes and others, the empirical method was born. We would forget for now/for now about the underlying reality and would consider the universe as a black box. We would concentrate our study on our experience of the black box, i.e. the empirical concept and approach and find the laws that best described our experiences. But no matter how successful the empirical concept is in this year 2010, the other concept (underlying reality) is still sitting on the back burner where we left it 300 years ago. Because we do not know what the universe is made of (what is the substance) and what makes it work by itself (the cause), all of our best science remains an educated guess on outcomes. And, that is the limit of physics. Your superficial reading of my essay did not reveal this to you. Unless one understands this state of affair, he cannot ask the question and the answer to this question certainly means nothing to him. The future of physics lies with metaphysics.
"Forget unnecessary musing about even, odd, and prime numbers. You should rather ask yourself why on cannot put zero and infinity in these drawers. The original structure of numbers, as already be found in Euclid's definitions, was carelessly abandoned in the 19th century mathematics. "
----- My monistic description of the universe suggests that there is a natural and logical structure under the even odd and prime numbers… and maybe not… not loosing too much sleep over that one.
Marcel,
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"Given a salesman offers 50% off and an additional 40 % reduction. Does this mean you have to pay just 10%? Why do you feel entitled to simply add probabilities?"
--- In dealing with the probability of position of a particle… the sum must be one; the particle exists somewhere. But, it is more of an image. Gravity is often portrayed as a dip in a sheet, an attractor for balls rolling in the vicinity. Consider now a bump in the same sheet. Place a ball on the bump and it will leave in whatever direction… this bump is my negative probability.
"You may add velocities but you have to multiply reductions of price as well as probabilities. Ergo, probability can neither get smaller than zero nor larger than one. You were muddling probability with velocity."
--- I know I am muddling something in my explanation, but it is not what you say.
"What about the question whether or not the positive plus negative numbers together are more than only the positive ones, Cantor found it correctly out but he admitted to Dedekind: I cannot believe it. Take his doubt as an other example for a typical fallacy due to superficial thinking."
--- Superficial reading: Since the early Greek philosophers, we have understood the distinction between two important concepts: the underlying reality and our perceptual experience. Over the centuries, we have always mixed the two concepts at the same time and amounted to nothing. Around the time of Newton, Descartes and others, the empirical method was born. We would forget for now/for now about the underlying reality and would consider the universe as a black box. We would concentrate our study on our experience of the black box, i.e. the empirical concept and approach and find the laws that best described our experiences. But no matter how successful the empirical concept is in this year 2010, the other concept (underlying reality) is still sitting on the back burner where we left it 300 years ago. Because we do not know what the universe is made of (what is the substance) and what makes it work by itself (the cause), all of our best science remains an educated guess on outcomes. And, that is the limit of physics. Your superficial reading of my essay did not reveal this to you. Unless one understands this state of affair, he cannot ask the question and the answer to this question certainly means nothing to him. The future of physics lies with metaphysics.
"Forget unnecessary musing about even, odd, and prime numbers. You should rather ask yourself why on cannot put zero and infinity in these drawers. The original structure of numbers, as already be found in Euclid's definitions, was carelessly abandoned in the 19th century mathematics. "
----- My monistic description of the universe suggests that there is a natural and logical structure under the even odd and prime numbers… and maybe not… not loosing too much sleep over that one.
Marcel,
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Marcel,
You wrote: "In dealing with the probability of position of a particle… the sum must be one; the particle exists somewhere. ... Gravity is often portrayed as a dip in a sheet, an attractor for balls rolling in the vicinity. ... Place a ball on the bump and it will leave in whatever direction… this bump is my negative probability."
-- I understand dip and bump as attractive or repulsive force, respectively, which are factors that influence an otherwise random motion. Addition is only allowed as a linear approximation for small values.
I am not familiar with monism. I merely know that it relates to Ostwald, Haeckel, Parmenides, and Zeno. My old dictionary calls it a naive kind of materialism. Admittedly, I did not yet read your essay at all. I just used the search option for negative probability as to clarify whether there is any reasonable use of negative probability.
Yes, around 1650, the time between Descartes (1596-1650) and Newton (1642-1727), Guericke contributed to the experimental method. His intention was to understand the forces that rule the motion of planets, and eventually this gave rise to steam engine and electricity. Guericke did not like mere speculation. You wrote: "The universe as a black box". Did you not understand that we are living within this box and have no possibility to look at it from outside? The very notion universe is in this sense a prison we cannot escape. This is however no relevant limit to physics if we accept that within the notion universe there is enough to investigate. It is perhaps possible to find almost any causes of something tangible but definitely impossible and therefore useless to deal with "the" primary cause. Causality and time are nearly synonymous in that they structure all events that already happened (probability one), to be seen from the chosen point of view (probability of correct observation smaller than one). Future causality and future time are not yet observable and more or less uncertain. Do you object?
Eckard
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You wrote: "In dealing with the probability of position of a particle… the sum must be one; the particle exists somewhere. ... Gravity is often portrayed as a dip in a sheet, an attractor for balls rolling in the vicinity. ... Place a ball on the bump and it will leave in whatever direction… this bump is my negative probability."
-- I understand dip and bump as attractive or repulsive force, respectively, which are factors that influence an otherwise random motion. Addition is only allowed as a linear approximation for small values.
I am not familiar with monism. I merely know that it relates to Ostwald, Haeckel, Parmenides, and Zeno. My old dictionary calls it a naive kind of materialism. Admittedly, I did not yet read your essay at all. I just used the search option for negative probability as to clarify whether there is any reasonable use of negative probability.
Yes, around 1650, the time between Descartes (1596-1650) and Newton (1642-1727), Guericke contributed to the experimental method. His intention was to understand the forces that rule the motion of planets, and eventually this gave rise to steam engine and electricity. Guericke did not like mere speculation. You wrote: "The universe as a black box". Did you not understand that we are living within this box and have no possibility to look at it from outside? The very notion universe is in this sense a prison we cannot escape. This is however no relevant limit to physics if we accept that within the notion universe there is enough to investigate. It is perhaps possible to find almost any causes of something tangible but definitely impossible and therefore useless to deal with "the" primary cause. Causality and time are nearly synonymous in that they structure all events that already happened (probability one), to be seen from the chosen point of view (probability of correct observation smaller than one). Future causality and future time are not yet observable and more or less uncertain. Do you object?
Eckard
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Spanner?..just where does infinity begin?..somewhere between 0 + 1?
Infinity cannot possibly start from Zero, so it has to be One. The fractional reduction from 1 to 0 cannot produce the same "infinity" as the continueous whole number "infinity"? There can never be a total discrete zero, a very finite part "fraction" will exist for all Zero's.
In the opposite continuum, there can never be a viable "total" infinity, a continuum of whole consecutive numbers. One can class the Prime number process, when a new number is reached infinity is falsefied, but as infinity is a function of "time", you will eventually come across a prime number that will take a certain time to discover, or check, tou will need the collective time of at least TWO Universe's!
It is not where Infinity ends that is important, it is where one defines its source?
Ian your succint discrete point_infinity does not equal your total continuous infinity by fact of functioning time constraints?
Best p.v
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Infinity cannot possibly start from Zero, so it has to be One. The fractional reduction from 1 to 0 cannot produce the same "infinity" as the continueous whole number "infinity"? There can never be a total discrete zero, a very finite part "fraction" will exist for all Zero's.
In the opposite continuum, there can never be a viable "total" infinity, a continuum of whole consecutive numbers. One can class the Prime number process, when a new number is reached infinity is falsefied, but as infinity is a function of "time", you will eventually come across a prime number that will take a certain time to discover, or check, tou will need the collective time of at least TWO Universe's!
It is not where Infinity ends that is important, it is where one defines its source?
Ian your succint discrete point_infinity does not equal your total continuous infinity by fact of functioning time constraints?
Best p.v
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Paul,
"Ian your succint discrete point_infinity does not equal your total continuous infinity by fact of functioning time constraints?"
I'm not sure I follow you here. Can you clarify?
Ian
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"Ian your succint discrete point_infinity does not equal your total continuous infinity by fact of functioning time constraints?"
I'm not sure I follow you here. Can you clarify?
Ian
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Paul,
So called Rhind Papyrus shows how the old Egyptians in 1650 B.C. calculated with fractions. When Thales (624-545) stated that everything is made of water, he meant the physical universe should not be understood in terms of unconnected fragments but in terms of something continuous. His compatriot Anaximander (610-540) imagined infinitely many worlds, all made out of something infinitely extensed that has always existed ans will always exist. Pythagoras (570-500) claimed the opposite: All is number. This was proven wrong by irrational numbers. Parmenides (480) evaded infinity by rejecting it which in connection with assumed continuity led to the paradoxes of Zeno (450)
Aristotele (384-322) pointed out that infinity is the possibility to count endlessly. He was definitely correct in that there is no largest number infinity: Infinity actu non datur (There is no actual infinity).
Let me skip the further history and ask how for instance Dirichlet and Dedekind were misled by the obvious fact that infinite Fourier series must be complete. They concluded that "infinity" must be reached. Constructivists like Kronecker and Brouwer objected that infinity cannot be reached within finite time. This might be behind your "time constraints".
I do not share this point of view, and mainstream mathematics did also reject this idea of becoming.
My suggestion is slightly different. Well, we may operate with the fictions of the infinitely small as well as the infinitely large. We just have to avoid the mistake to believe that they can be reached by splitting or adding, respectively. They are no quantities but describe ideal qualities. In contrast to the mandatory tenet, infinity is not exhaustible.
Are numbers infinitely exact? This is an indispensable assumption. However, I maintain that we have to distinguish between rational and real numbers. While the sequence of natural numbers is changed if we add or remove a singular number we must not infer that for instance exclusion of pi from the continuum of the real numbers has any effect on this continuum. It simply does not matter because the distance between pi and the "next" larger number has the measure zero. That is, as some most excellent mathematicians understood: The TND is invalid in this case. Brutal mathematicians tried to remove the difference between 0.999... and one in Hilbert's manner by means of definition. Why not admitting that continuum and discrete numbers are mutually complementing but also mutually excluding qualities? This would not harm reasonable mathematics but show where Dirichlet and Dedekind went wrong.
Eckard
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So called Rhind Papyrus shows how the old Egyptians in 1650 B.C. calculated with fractions. When Thales (624-545) stated that everything is made of water, he meant the physical universe should not be understood in terms of unconnected fragments but in terms of something continuous. His compatriot Anaximander (610-540) imagined infinitely many worlds, all made out of something infinitely extensed that has always existed ans will always exist. Pythagoras (570-500) claimed the opposite: All is number. This was proven wrong by irrational numbers. Parmenides (480) evaded infinity by rejecting it which in connection with assumed continuity led to the paradoxes of Zeno (450)
Aristotele (384-322) pointed out that infinity is the possibility to count endlessly. He was definitely correct in that there is no largest number infinity: Infinity actu non datur (There is no actual infinity).
Let me skip the further history and ask how for instance Dirichlet and Dedekind were misled by the obvious fact that infinite Fourier series must be complete. They concluded that "infinity" must be reached. Constructivists like Kronecker and Brouwer objected that infinity cannot be reached within finite time. This might be behind your "time constraints".
I do not share this point of view, and mainstream mathematics did also reject this idea of becoming.
My suggestion is slightly different. Well, we may operate with the fictions of the infinitely small as well as the infinitely large. We just have to avoid the mistake to believe that they can be reached by splitting or adding, respectively. They are no quantities but describe ideal qualities. In contrast to the mandatory tenet, infinity is not exhaustible.
Are numbers infinitely exact? This is an indispensable assumption. However, I maintain that we have to distinguish between rational and real numbers. While the sequence of natural numbers is changed if we add or remove a singular number we must not infer that for instance exclusion of pi from the continuum of the real numbers has any effect on this continuum. It simply does not matter because the distance between pi and the "next" larger number has the measure zero. That is, as some most excellent mathematicians understood: The TND is invalid in this case. Brutal mathematicians tried to remove the difference between 0.999... and one in Hilbert's manner by means of definition. Why not admitting that continuum and discrete numbers are mutually complementing but also mutually excluding qualities? This would not harm reasonable mathematics but show where Dirichlet and Dedekind went wrong.
Eckard
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I blamed Dirichlet and Dedekind wrong. Let me start explaining. There are several problems concerning definitions that were made for R if we try to follow physical restriction to R+, cf. e.g. criticism by Terhardt of integration for unilateral Laplace transform from a small negative value to +oo or by Aseltine §3-6. Likewise, the usual illustration of Dirac impulses fails with restriction to R+. Common sense tells us that each single rational number must be countable in the sense it is different from any other one by a non vanishing value while genuine continuity as defined by Peirce demands that real numbers behave differently.
Dirichlet ignored this different properties when he gave the well known example of a function f(x) that is 0 for rational x and 1 for irrational x. While it has been called nowhere dense, I would rather see this the other way round.
Furthermore, Dirichlet attributed the mean value [f(x-0)+f(x+0)]/2 to discontinuous functions. Using pertaining integral tables I found out that the mean value is misleading. It does not make sense to exclude or include a "single" real number from the continuum of reals. It does not matter whether or not zero, infinity or any rational number exist within the continuum. There are infinitely many possible substitutes that cannot be distinguished from it. Accordingly I consider |sign(0)|=1 justified and a separate treatment for the very point zero between R+ and R- pointless.
Dedekind continued the strictly speaking inadmissible while pragmatic neglect of the fundamental difference between rational numbers and the uncountable manifold of potential points of continuum. He caused additional confusion when he referred to numbers as to points instead of measured that relate to a common unity.
Cantor caused a tempest in a teapot and ongoing trouble when added some useless speculations and the analogy based notion of cardinality. Isn't it sufficient to distinguish between in principle countable, discrete, rational numbers on one side and fictitious real numbers of continuum on the other side? I never heard of any application for aleph_2. Of course, I distinguish between let's say the number five as a rational number and an equivalent real number five.
Eckard
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Dirichlet ignored this different properties when he gave the well known example of a function f(x) that is 0 for rational x and 1 for irrational x. While it has been called nowhere dense, I would rather see this the other way round.
Furthermore, Dirichlet attributed the mean value [f(x-0)+f(x+0)]/2 to discontinuous functions. Using pertaining integral tables I found out that the mean value is misleading. It does not make sense to exclude or include a "single" real number from the continuum of reals. It does not matter whether or not zero, infinity or any rational number exist within the continuum. There are infinitely many possible substitutes that cannot be distinguished from it. Accordingly I consider |sign(0)|=1 justified and a separate treatment for the very point zero between R+ and R- pointless.
Dedekind continued the strictly speaking inadmissible while pragmatic neglect of the fundamental difference between rational numbers and the uncountable manifold of potential points of continuum. He caused additional confusion when he referred to numbers as to points instead of measured that relate to a common unity.
Cantor caused a tempest in a teapot and ongoing trouble when added some useless speculations and the analogy based notion of cardinality. Isn't it sufficient to distinguish between in principle countable, discrete, rational numbers on one side and fictitious real numbers of continuum on the other side? I never heard of any application for aleph_2. Of course, I distinguish between let's say the number five as a rational number and an equivalent real number five.
Eckard
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Eckard,
What I understand is that we dropped the ball some 300 years ago and I picked it up. My essay is my answer to the best of my abilities. So, it is possible/possible to ask and answer the question. What I need are people who can wrap their mind around this concept and back engineer it into our present knowledge to make it complete.
(Complete knowledge = physical description + logical understanding.)
"..mere speculation…. have no possibility… of something tangible… definitely impossible and therefore useless.."
These words tell me that you are not one of those people. (Maybe I’m not in the right forum ..)
All the bests,
Thanks,
Marcel,
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What I understand is that we dropped the ball some 300 years ago and I picked it up. My essay is my answer to the best of my abilities. So, it is possible/possible to ask and answer the question. What I need are people who can wrap their mind around this concept and back engineer it into our present knowledge to make it complete.
(Complete knowledge = physical description + logical understanding.)
"..mere speculation…. have no possibility… of something tangible… definitely impossible and therefore useless.."
These words tell me that you are not one of those people. (Maybe I’m not in the right forum ..)
All the bests,
Thanks,
Marcel,
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Marcel,
Yesterday LHC has been switched on again. I will appreciate a negative outcome that could possibly give rise for taking R+ more seriously and use mathematics no longer without knowing what is really behind it.
I will quote you because you are still considering yourself a monist:
“Understanding is about knowing the logic behind the existence and evolution of the universe.”
--- Logic behind the existence of the universe? As it is meant, this sounds unrealistic to me, because I understand the universe as something unlimited and ergo comprehensive but incomprehensible.
I rather see the logic behind the existence of the notion universe.
“The metaphysical and logical understanding of the universe is accessible, understandable and necessary for us to progress beyond the limits of physics. It is certain that this metaphysics still has to be engineered back into physics in order to produce something new and practical in our reality.
--- Understandable understanding. Hm.
“The universe has existed and evolved by itself for about 14 billion years.”
--- Isn’t this a hypothesis that is based on results from the experimental method?
--- Doesn’t it contradict to Monist Parmenides: “The universe consists of only one object, it exists timelessly and changelessly.” “There is no motion, since motion implies the existence of more than one thing, namely, a finishing place and a starting place.”
Zeno tried to demonstrate something I do not deny: “True existence evades measurement.”
--- No further comment,
all the best for you and respect for your effort,
Eckard
Let’s have you Marcel LeBel the last word: The box contains everything physicists ever wanted to know about the universe. The content of the box is the only thing that will allow us to make sense of all the theories and equations (unification). But they can’t get it with the empirical approach. The empirical approach is about finding things by experience, trial and error. It is a choice we made long ago between knowing and doing. The empirical test is not just the proof of a theory. It is before all a practical demonstration of control over some segment of the universe and this control provides the illusion that we understand what we are doing. We don’t
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Yesterday LHC has been switched on again. I will appreciate a negative outcome that could possibly give rise for taking R+ more seriously and use mathematics no longer without knowing what is really behind it.
I will quote you because you are still considering yourself a monist:
“Understanding is about knowing the logic behind the existence and evolution of the universe.”
--- Logic behind the existence of the universe? As it is meant, this sounds unrealistic to me, because I understand the universe as something unlimited and ergo comprehensive but incomprehensible.
I rather see the logic behind the existence of the notion universe.
“The metaphysical and logical understanding of the universe is accessible, understandable and necessary for us to progress beyond the limits of physics. It is certain that this metaphysics still has to be engineered back into physics in order to produce something new and practical in our reality.
--- Understandable understanding. Hm.
“The universe has existed and evolved by itself for about 14 billion years.”
--- Isn’t this a hypothesis that is based on results from the experimental method?
--- Doesn’t it contradict to Monist Parmenides: “The universe consists of only one object, it exists timelessly and changelessly.” “There is no motion, since motion implies the existence of more than one thing, namely, a finishing place and a starting place.”
Zeno tried to demonstrate something I do not deny: “True existence evades measurement.”
--- No further comment,
all the best for you and respect for your effort,
Eckard
Let’s have you Marcel LeBel the last word: The box contains everything physicists ever wanted to know about the universe. The content of the box is the only thing that will allow us to make sense of all the theories and equations (unification). But they can’t get it with the empirical approach. The empirical approach is about finding things by experience, trial and error. It is a choice we made long ago between knowing and doing. The empirical test is not just the proof of a theory. It is before all a practical demonstration of control over some segment of the universe and this control provides the illusion that we understand what we are doing. We don’t
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Pankaj,
You wrote: "How do we get back to knowing, rather than doing."
-- As someone whose mother tongue is not English, I have to admit being not sure I understood what you meant in your rather cryptic question without question mark.
Some experts argue not just the future but also the past is unknown to us. I do not think so. Well, it is imaginable that somebody lost his memory while he can be told what will most likely happen. My reasoning refers to physical reality which is exclusively determined by influences that originate in the past. So it is reality that does not know the future.
Does this answer your question?
Eckard
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You wrote: "How do we get back to knowing, rather than doing."
-- As someone whose mother tongue is not English, I have to admit being not sure I understood what you meant in your rather cryptic question without question mark.
Some experts argue not just the future but also the past is unknown to us. I do not think so. Well, it is imaginable that somebody lost his memory while he can be told what will most likely happen. My reasoning refers to physical reality which is exclusively determined by influences that originate in the past. So it is reality that does not know the future.
Does this answer your question?
Eckard
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Eckard,
I didn't even realize that I had placed a . instead of ? until I read your comment… nothing cryptic meant by that. I had read through this thread and found your point of view quite compelling, I found myself very often agreeing with you. I was curious about what you meant by "It is a choice we made long ago between knowing and doing.". I thought that I could learn something more from you, so I asked what you meant by that.
Something more specific then… when and how did we make the decision towards doing, rather than knowing… and who made this decision ( I mean of course in historical terms, and not necessarily focused on one individual)? I agree with you that prediction is not the same as understanding. What then would constitute understanding? How could we move towards understanding? Perhaps you will write something on these matters and I will understand some more of this world.
Thanks,
Pankaj
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I didn't even realize that I had placed a . instead of ? until I read your comment… nothing cryptic meant by that. I had read through this thread and found your point of view quite compelling, I found myself very often agreeing with you. I was curious about what you meant by "It is a choice we made long ago between knowing and doing.". I thought that I could learn something more from you, so I asked what you meant by that.
Something more specific then… when and how did we make the decision towards doing, rather than knowing… and who made this decision ( I mean of course in historical terms, and not necessarily focused on one individual)? I agree with you that prediction is not the same as understanding. What then would constitute understanding? How could we move towards understanding? Perhaps you will write something on these matters and I will understand some more of this world.
Thanks,
Pankaj
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Pankay.
One cannot predict the past but try to understand more or less what happened and what will happen. Animals are often steered by their innate instinct. I understand understanding as a variety of learned memory-based processes in the human brain. While I used to attend some seminars of an Institute for Neurobiology, I regret being unable and not ready to speculate. The more I get familiar with the function of brain, the more I realize understanding almost nothing.
Eckard
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One cannot predict the past but try to understand more or less what happened and what will happen. Animals are often steered by their innate instinct. I understand understanding as a variety of learned memory-based processes in the human brain. While I used to attend some seminars of an Institute for Neurobiology, I regret being unable and not ready to speculate. The more I get familiar with the function of brain, the more I realize understanding almost nothing.
Eckard
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Pankaj,
I am actually the one who wrote this:"It is a choice we made long ago between knowing and doing."
Knowing comes having a logical explanation to support a mathematical description. A logical description is about really understanding the physical laws we use to describe the various behaviors of the universe.
How do we get back to "understanding"? We may do that by admitting that something does really exist by itself in a metaphysical way and by deducing the logical requirements of such existence. Nothing in empirical science admits that something exists by itself. The empirical approach only deal with our experience ....
Hope this helps,
Marcel, ( I am still around!)
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I am actually the one who wrote this:"It is a choice we made long ago between knowing and doing."
Knowing comes having a logical explanation to support a mathematical description. A logical description is about really understanding the physical laws we use to describe the various behaviors of the universe.
How do we get back to "understanding"? We may do that by admitting that something does really exist by itself in a metaphysical way and by deducing the logical requirements of such existence. Nothing in empirical science admits that something exists by itself. The empirical approach only deal with our experience ....
Hope this helps,
Marcel, ( I am still around!)
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Thanks, Marcel... and apologies to you and also to Eckard for my mistake in attribution. I enjoyed reading the conversation and learned from both of you.
Marcel, what you say reminds me of Bernard D'espagnat's position when he speaks of a 'veiled reality'. He speaks of an "empirical reality" and an "ultimate reality", and further says that we may say nothing about the ultimate reality. I take it that you think that we ought to be able to say something about it. I've thought, perhaps as you, that what we call our ontology is really a phenomenology, due to the empirical nature of our investigations. I think that what we call "objective reality" is really a consensual, subjective reality.
One problem in saying that something exists in a metaphysical way is the lack of applicability of even the word 'existence'. Heisenberg wrote that even concepts like 'space', 'time' and 'existence" have only limited applicability. How can we say anything at all about this metaphysical reality? Its a tough task, to be sure.
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Marcel, what you say reminds me of Bernard D'espagnat's position when he speaks of a 'veiled reality'. He speaks of an "empirical reality" and an "ultimate reality", and further says that we may say nothing about the ultimate reality. I take it that you think that we ought to be able to say something about it. I've thought, perhaps as you, that what we call our ontology is really a phenomenology, due to the empirical nature of our investigations. I think that what we call "objective reality" is really a consensual, subjective reality.
One problem in saying that something exists in a metaphysical way is the lack of applicability of even the word 'existence'. Heisenberg wrote that even concepts like 'space', 'time' and 'existence" have only limited applicability. How can we say anything at all about this metaphysical reality? Its a tough task, to be sure.
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Marcel,
Anton Zeilinger seems to be saying the same as D'Espagnat in http://www.signandsight.com/features/614.html
Q
:So there is in fact something that exists independently of us. And the moon is also there when I'm not looking at it.
AZ: Something exists, but it is not directly accessible to us. Only indirectly. And whether this thing must really be called the "moon" is another question. That is also a construct.
Q: But there is something up there...
AZ: ... the word "there" is yet another construct. Space and time are concepts aimed at giving meaning to our world of appearances. So they are entirely reasonable constructs. By no means do I want to give the impression that I believe everything is just our imagination.
Q: The world as a huge theatre that only plays in our heads. AZ: That is certainly not my view of things. Q: Then what would you call it, this something that you can't call moon or space or time – this something that exists independently of us?
AZ: Wouldn't I be making another qualification if I tried to give it a name? Isn't it enough if I just say it exists? As soon as you use words like "world" or "universe", you start lugging about all that conceptual ballast again.
Q: But you defend the thesis that there is an "original matter of the universe": information.
AZ: Yes. For me the concept of "information" is at the basis of everything we call "nature". The moon, the chair, the equation of states, anything and everything, because we can't talk about anything without de facto speaking about the information we have of these things. In this sense the information is the basic building block of our world.
Q: But just now you spoke of a world that exists independently of us.
AZ: That's right. But this world is not directly ascertainable or describable. Because every description must be done in terms of the information, and so you inevitably get into circular reasoning. There's a limit we can't cross. And even a civilisation on Alpha Centauri can't cross it. For me that's something almost mystical.
How will we get past this barrier?
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Anton Zeilinger seems to be saying the same as D'Espagnat in http://www.signandsight.com/features/614.html
Q
:So there is in fact something that exists independently of us. And the moon is also there when I'm not looking at it.
AZ: Something exists, but it is not directly accessible to us. Only indirectly. And whether this thing must really be called the "moon" is another question. That is also a construct.
Q: But there is something up there...
AZ: ... the word "there" is yet another construct. Space and time are concepts aimed at giving meaning to our world of appearances. So they are entirely reasonable constructs. By no means do I want to give the impression that I believe everything is just our imagination.
Q: The world as a huge theatre that only plays in our heads. AZ: That is certainly not my view of things. Q: Then what would you call it, this something that you can't call moon or space or time – this something that exists independently of us?
AZ: Wouldn't I be making another qualification if I tried to give it a name? Isn't it enough if I just say it exists? As soon as you use words like "world" or "universe", you start lugging about all that conceptual ballast again.
Q: But you defend the thesis that there is an "original matter of the universe": information.
AZ: Yes. For me the concept of "information" is at the basis of everything we call "nature". The moon, the chair, the equation of states, anything and everything, because we can't talk about anything without de facto speaking about the information we have of these things. In this sense the information is the basic building block of our world.
Q: But just now you spoke of a world that exists independently of us.
AZ: That's right. But this world is not directly ascertainable or describable. Because every description must be done in terms of the information, and so you inevitably get into circular reasoning. There's a limit we can't cross. And even a civilisation on Alpha Centauri can't cross it. For me that's something almost mystical.
How will we get past this barrier?
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I think this is a great place to start a new discussion (though the nature of mathematics discussion can go on here). "Understanding" is a hugely interesting and deep topic that deserves some discussion on its own.
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Ian,
I did not pay too much attention to FQXi recent, but you did get a very interesting topic of discussion here.
Before answering your questions head on, I want to have a little digression. How does math compare with heuristic arguments? I contend that math proofs and results are like limiting cycles in phase space. A correct proof today is correct forever, while a heuristic argument can either become invalidated over time, or become part of a proof. It is therefore clear that math is timeless and has an independent existence.
But do math and reality have a dual relationship as Gordon McCabe contends? To answer this one has to answer what is reality? Some new mathematics may not have any obvious usefulness in the real world, but this does not mean they are not part of reality. The very paper and ink marks comprising the new math is part of reality. The new question is: do all maths play a role in nature? To answer this one needs to look at examples. What role did crystallographic groups play in the first seconds after Bib Bang? None (or a very minor one). But they become important after first crystals began forming. The moral of the story is that emergence can make any math useful under appropriate conditions. The only question remaining is: is all math playing a UNIVERSAL role in nature? Obviously not. A Minkowski space with 45 spatial and 1 time dimension is not physical.
And now back to Gordon McCabe: reality can describe all maths, and maths can model all reality. I touched above on the first implication. The second one is true if there are not supernatural explanations of reality. Only in that case reality is non-contradictory and can be modeled by non-contradictory mathematical structures.
Florin
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I did not pay too much attention to FQXi recent, but you did get a very interesting topic of discussion here.
Before answering your questions head on, I want to have a little digression. How does math compare with heuristic arguments? I contend that math proofs and results are like limiting cycles in phase space. A correct proof today is correct forever, while a heuristic argument can either become invalidated over time, or become part of a proof. It is therefore clear that math is timeless and has an independent existence.
But do math and reality have a dual relationship as Gordon McCabe contends? To answer this one has to answer what is reality? Some new mathematics may not have any obvious usefulness in the real world, but this does not mean they are not part of reality. The very paper and ink marks comprising the new math is part of reality. The new question is: do all maths play a role in nature? To answer this one needs to look at examples. What role did crystallographic groups play in the first seconds after Bib Bang? None (or a very minor one). But they become important after first crystals began forming. The moral of the story is that emergence can make any math useful under appropriate conditions. The only question remaining is: is all math playing a UNIVERSAL role in nature? Obviously not. A Minkowski space with 45 spatial and 1 time dimension is not physical.
And now back to Gordon McCabe: reality can describe all maths, and maths can model all reality. I touched above on the first implication. The second one is true if there are not supernatural explanations of reality. Only in that case reality is non-contradictory and can be modeled by non-contradictory mathematical structures.
Florin
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" ... maths can model all reality. ... true if there are not supernatural explanations of reality. Only in that case reality is non-contradictory and can be modeled by non-contradictory mathematical structures."
"contradictory reality"? May we suspect reality to be wrong? If a Gordon McCabe contends duality between theory and reality I will never take him seriously.
Laplace admitted to Napoleon not to need the hypothesis god. I feel neither in position nor obliged to have a demon at hand and model anything by means of mathematics. What about hearing I rather trust in physiology. Except for a theorem by Wiener/Chintchine all mathematical approaches so far were rather misleading in this field of science.
Eckard
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"contradictory reality"? May we suspect reality to be wrong? If a Gordon McCabe contends duality between theory and reality I will never take him seriously.
Laplace admitted to Napoleon not to need the hypothesis god. I feel neither in position nor obliged to have a demon at hand and model anything by means of mathematics. What about hearing I rather trust in physiology. Except for a theorem by Wiener/Chintchine all mathematical approaches so far were rather misleading in this field of science.
Eckard
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Florin,
That is an excellent point regarding math v. heuristic arguments. And emergence is clearly an issue here. Perhaps mathematics, in some sense, is a vision of the future or what is ultimately possible. Hmmm. This requires more thought on my part when I'm not utterly exhausted.
Ian
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That is an excellent point regarding math v. heuristic arguments. And emergence is clearly an issue here. Perhaps mathematics, in some sense, is a vision of the future or what is ultimately possible. Hmmm. This requires more thought on my part when I'm not utterly exhausted.
Ian
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Ian,
You are right. And the future possibility is unrevealed today because mathematics in inexhaustible. Thinking outside the box about emergence, I believe no where but in humor we see this in its extreme form. Humor is very contextual and extremely sensitive to the natural selection rules of the sociological “landscape” (to paraphrase Susskind). For example, to people 100 years ago, a Seinfeld episode would looked completely strange. We laugh at different things because we constantly find new patterns in our daily life. And those patterns are just new emergent mathematical axioms. But this is maybe all too fuzzy. We can pick another area: morality and law. We can understand morality as an ever evolving landscape of fuzzy heuristic possibilities and the new laws as quasi-formalized axiomatic systems that passed at least some form of consistency check.
Florin
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You are right. And the future possibility is unrevealed today because mathematics in inexhaustible. Thinking outside the box about emergence, I believe no where but in humor we see this in its extreme form. Humor is very contextual and extremely sensitive to the natural selection rules of the sociological “landscape” (to paraphrase Susskind). For example, to people 100 years ago, a Seinfeld episode would looked completely strange. We laugh at different things because we constantly find new patterns in our daily life. And those patterns are just new emergent mathematical axioms. But this is maybe all too fuzzy. We can pick another area: morality and law. We can understand morality as an ever evolving landscape of fuzzy heuristic possibilities and the new laws as quasi-formalized axiomatic systems that passed at least some form of consistency check.
Florin
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Rather than talking about an "underlying" order and thus elevating Mathematics to almost an ontological status, couldn't we just say that there is regularity, and it should obviously be possible to derive quantitative statements that reflect the fact there are stable "ratio" relationships. Occam's razor would seem to demand this view, but Platonic ideas are deeply rooted in Western culture, which...
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I have a strong evolutionary standpoint on mathematics. I consider mathematics to be nothing more than a communication product of the evolution of human brains, which means that understanding mathematics first boils down to understanding human brains.
I speculate that brains started with the macroscopic multicellular organism's need to move around its environment to reach better living...
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I speculate that brains started with the macroscopic multicellular organism's need to move around its environment to reach better living...
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I think there is a logical evolution to repetetive_events, that would instigate repetative solution? Think of how the humans grasped shaping the end of a blunt piece of wood. I can imagine our far distant ancestors throwing objects at other intruding wild species, say a prairie pack of wolves trying to snatch a tribes stored food supply.
Blunt objects would have maybe scattered one or two wild animals, but eventually all the objects thrown would have contained an accidental "sharp" object, which would have injured animal to a degree of killing it. Then the ratio of blunt to sharp objects being thrown would have evolved in favour of only "sharpened" objects being thrown? Once tribes had taken this repetative action of choosing sharp objects, shaping "weapons" would be the next action? the tribes would have honed their skill of repetative throwing and hitting objects as "practice", what we would today deem as field sports events?
Once the brain has experienced a certain amount of repetative actions, I think there is a process that kicks in, and the activity is logically accepted. Mathematics is the process of "honing" repetative events,as a post_logic activity ?
mathematics therfore, I think are a construct of Human evolution.
best p.v
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Blunt objects would have maybe scattered one or two wild animals, but eventually all the objects thrown would have contained an accidental "sharp" object, which would have injured animal to a degree of killing it. Then the ratio of blunt to sharp objects being thrown would have evolved in favour of only "sharpened" objects being thrown? Once tribes had taken this repetative action of choosing sharp objects, shaping "weapons" would be the next action? the tribes would have honed their skill of repetative throwing and hitting objects as "practice", what we would today deem as field sports events?
Once the brain has experienced a certain amount of repetative actions, I think there is a process that kicks in, and the activity is logically accepted. Mathematics is the process of "honing" repetative events,as a post_logic activity ?
mathematics therfore, I think are a construct of Human evolution.
best p.v
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Dear Eckard,
I think I should say more since your interest in the audible spectrum is very high:
"...You finally lost me when you claimed that my_zero is not 4 pi 10^-7 Vs/Am but the relation between the velocity of light (obviously in vacuum because there is no known value for the velocity of light in copper) and the velocity of sound. This is perhaps easily falsifiable. Sorry."
The velocity of sound is a rate of interaction between particles of matter. Similarly to that of light, sound as we hear it, is a limited part of the sound spectrum. In other words, particles interact between one another, theoretically speeking, over any distance. It will usually be the case that most of the interaction is undetectible by the hearing sense. Even in free space there are particles and they do interact with one another in the same manner in which more dense particles create the sensation we call sound. The difference is not of kind but rather of intensity. Therefore, in the strictest sense, if the effect is called sound in its most general form, then there is an effect of sound even in free space. We do not need to hear it in order to anticipate that it must be occurring even there. What do you think? By the way, I do not recall ever saying that the magnitude of mu_zero is not 4 pi 10^-7 (appropriate units can be added here, either yours or mine. They are different).
James
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I think I should say more since your interest in the audible spectrum is very high:
"...You finally lost me when you claimed that my_zero is not 4 pi 10^-7 Vs/Am but the relation between the velocity of light (obviously in vacuum because there is no known value for the velocity of light in copper) and the velocity of sound. This is perhaps easily falsifiable. Sorry."
The velocity of sound is a rate of interaction between particles of matter. Similarly to that of light, sound as we hear it, is a limited part of the sound spectrum. In other words, particles interact between one another, theoretically speeking, over any distance. It will usually be the case that most of the interaction is undetectible by the hearing sense. Even in free space there are particles and they do interact with one another in the same manner in which more dense particles create the sensation we call sound. The difference is not of kind but rather of intensity. Therefore, in the strictest sense, if the effect is called sound in its most general form, then there is an effect of sound even in free space. We do not need to hear it in order to anticipate that it must be occurring even there. What do you think? By the way, I do not recall ever saying that the magnitude of mu_zero is not 4 pi 10^-7 (appropriate units can be added here, either yours or mine. They are different).
James
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Sorry everyone, I thought I was in a different forum speaking to Eckard. I do not know how I managed to cause my message to appear here. Anyway, please carry on and disregard it. Thank you.
James
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James
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Genevieve,
I have a simple question/challenge for your evolutionary standpoint on mathematics. Was 1+1=2 true before humans (or any other intelligent beings) realized it?
On the other hand, you are right about the different logics operating in different domains. In quantum mechanics, the Boolean logic of set theory does not apply, but the logic of subspaces (where de distributive property is replaced by the modularity property for lattices). Still, there are so many different logics possible, and yet Nature seems to favor only one. Why is that?
Florin
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I have a simple question/challenge for your evolutionary standpoint on mathematics. Was 1+1=2 true before humans (or any other intelligent beings) realized it?
On the other hand, you are right about the different logics operating in different domains. In quantum mechanics, the Boolean logic of set theory does not apply, but the logic of subspaces (where de distributive property is replaced by the modularity property for lattices). Still, there are so many different logics possible, and yet Nature seems to favor only one. Why is that?
Florin
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Florin,
you wrote:"there are so many different logics possible, and yet Nature seems to favor only one. Why is that?"
Do not blame nature for being wrong. Theory is patient, nonsense has no limits, and it provides honor as well as money. Mutually contradicting theories cannot likewise fit simultaneously to the same object. That's why I vote for ultimate realism.
The Greeks established the logical connections among their results, deducing the theorems from starting assumptions (axioms). For Aristotle, the axioms are truth, and hence the theorems are also truth. Leibniz conceived the idea of symbolic logic, a universal language in which all rational thinking could be expressed. Intuitionists can, and do, deny that, for any mathematical statement p, it is a logical truth that 'either p or not p'. Equideductive logic is offered instead of set theory. I fear, it does not solve the problems. Sound common sense is still necessary.
Anyway, your question seems to be worth a very radical consideration.
Eckard
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you wrote:"there are so many different logics possible, and yet Nature seems to favor only one. Why is that?"
Do not blame nature for being wrong. Theory is patient, nonsense has no limits, and it provides honor as well as money. Mutually contradicting theories cannot likewise fit simultaneously to the same object. That's why I vote for ultimate realism.
The Greeks established the logical connections among their results, deducing the theorems from starting assumptions (axioms). For Aristotle, the axioms are truth, and hence the theorems are also truth. Leibniz conceived the idea of symbolic logic, a universal language in which all rational thinking could be expressed. Intuitionists can, and do, deny that, for any mathematical statement p, it is a logical truth that 'either p or not p'. Equideductive logic is offered instead of set theory. I fear, it does not solve the problems. Sound common sense is still necessary.
Anyway, your question seems to be worth a very radical consideration.
Eckard
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Eckard,
I do not “blame” nature for anything. I was just restating Wheeler’s question: “why those equations?” into “why this logic?”
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I do not “blame” nature for anything. I was just restating Wheeler’s question: “why those equations?” into “why this logic?”
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Florin,
Yes, I read your essay and wonder about your references 3-6 to a bit too Platonic ideas. Let me refer to an undergraduate text by Anglin who compares in detail Platonists like Goedel with Formalists like Hilbert and intuitionists like Brouwer. You repeatedly mentioned "the Platonic world of abstract mathematics".
I found an utterance by von Neumann:
"As a mathematical discipline travels far from its empirical source, or still more, if it is a second or third generation only indirectly inspired by ideas coming from 'reality,' it is beset with very grave dangers. It becomes more and more purely aestheticizing , more and more purely l'art pour le'art. ... In other words, at a great distance from its empirical source, or after much 'abstract' inbreeding, a mathematical subject is in danger of degeneration." [The Mathematician, in R.B. Heywood, ed. The Works of the Mind, Univ. of Chicago Press. References to Collected Works, Vol.I: Logic, theory of sets, and quantum mechanics, ed. A. H. Taub, Oxford, Pergamon.]
Do you know Weyl's diagram?
Weyl Russell Zermelo
Brouwer Hilbert
More constructive tendencies are located to the left, more axiomatic ones to the right, more evident and deeper founded on the bottom, more customary on top.
more
Eckard
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Yes, I read your essay and wonder about your references 3-6 to a bit too Platonic ideas. Let me refer to an undergraduate text by Anglin who compares in detail Platonists like Goedel with Formalists like Hilbert and intuitionists like Brouwer. You repeatedly mentioned "the Platonic world of abstract mathematics".
I found an utterance by von Neumann:
"As a mathematical discipline travels far from its empirical source, or still more, if it is a second or third generation only indirectly inspired by ideas coming from 'reality,' it is beset with very grave dangers. It becomes more and more purely aestheticizing , more and more purely l'art pour le'art. ... In other words, at a great distance from its empirical source, or after much 'abstract' inbreeding, a mathematical subject is in danger of degeneration." [The Mathematician, in R.B. Heywood, ed. The Works of the Mind, Univ. of Chicago Press. References to Collected Works, Vol.I: Logic, theory of sets, and quantum mechanics, ed. A. H. Taub, Oxford, Pergamon.]
Do you know Weyl's diagram?
Weyl Russell Zermelo
Brouwer Hilbert
More constructive tendencies are located to the left, more axiomatic ones to the right, more evident and deeper founded on the bottom, more customary on top.
more
Eckard
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Genevieve,
That is an intriguing idea, but I think I'm with Florin on this, particularly since numerous species can do simple math.
I think it is important to distinguish between the formalism and the intrinsic. So, certainly the formalism is a product of biological, social, and cultural evolution since it is, at its core, symbology. But what lies beneath that has to be more than simply evolutionary since that would imply mathematics doesn't exist in places where humans don't exist which makes no logical sense.
Ian
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That is an intriguing idea, but I think I'm with Florin on this, particularly since numerous species can do simple math.
I think it is important to distinguish between the formalism and the intrinsic. So, certainly the formalism is a product of biological, social, and cultural evolution since it is, at its core, symbology. But what lies beneath that has to be more than simply evolutionary since that would imply mathematics doesn't exist in places where humans don't exist which makes no logical sense.
Ian
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Stop the Weirdness!
It all started with Planck's 'energy quanta' and Einstein's 'photons'. The first explained 'blackbody radiation' while the second explained the 'photoelectric effect'. From these beginnings we have a view of the world evolve that has become increasingly mathematically abstract and counter-intuitive. But what if we could turn back the clock some 100 years and consider a...
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It all started with Planck's 'energy quanta' and Einstein's 'photons'. The first explained 'blackbody radiation' while the second explained the 'photoelectric effect'. From these beginnings we have a view of the world evolve that has become increasingly mathematically abstract and counter-intuitive. But what if we could turn back the clock some 100 years and consider a...
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Dear Constantinos Ragazas,
While I need time for digesting your arguments, I already would like asking you to comment on my possibly related ones to be found at fqxi topics 369 and 527 and my IEEE paper 'Adaptation of Spectral Analysis to Reality' and a manuscript 'A Still Valid Argument by Ritz'.
I just wonder why you wrote 'energy quanta' instead of 'quanta of action' and you did not reveal your scientific background and affiliation. At least, you should indicate what part of currently accepted theory you are considering ill-founded or possibly flawed, why, and with which testable consequences. We should be careful before declaring the many many experts wrong.
Eckard Blumschein
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While I need time for digesting your arguments, I already would like asking you to comment on my possibly related ones to be found at fqxi topics 369 and 527 and my IEEE paper 'Adaptation of Spectral Analysis to Reality' and a manuscript 'A Still Valid Argument by Ritz'.
I just wonder why you wrote 'energy quanta' instead of 'quanta of action' and you did not reveal your scientific background and affiliation. At least, you should indicate what part of currently accepted theory you are considering ill-founded or possibly flawed, why, and with which testable consequences. We should be careful before declaring the many many experts wrong.
Eckard Blumschein
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Dear Eckard Blumschein.
Since you asked, I am a retired math teacher having taught at The Lawrenceville School in NJ for some thirty years. But this is not about me. It's about a result that I stumbled upon by accident some ten years ago and more recently developed into a series of short notes. Foremost of these is the mathematical derivation of Planck's Law (or variation of it) showing that it is an exact mathematical identity (a tautology) that describes the interaction of energy.
I cannot comment on your many referenced works since I don't have the background to do so. But I have no doubt you will be able to follow my simple mathematical derivations and arguments. Since ultimately we are all seeking the same thing, an understanding of our world that 'makes sense', I trust that you will consider these independent of me. I humbly submit these results for your consideration and welcome a sustained dialog on them.
Best regards,
Constantinos
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Since you asked, I am a retired math teacher having taught at The Lawrenceville School in NJ for some thirty years. But this is not about me. It's about a result that I stumbled upon by accident some ten years ago and more recently developed into a series of short notes. Foremost of these is the mathematical derivation of Planck's Law (or variation of it) showing that it is an exact mathematical identity (a tautology) that describes the interaction of energy.
I cannot comment on your many referenced works since I don't have the background to do so. But I have no doubt you will be able to follow my simple mathematical derivations and arguments. Since ultimately we are all seeking the same thing, an understanding of our world that 'makes sense', I trust that you will consider these independent of me. I humbly submit these results for your consideration and welcome a sustained dialog on them.
Best regards,
Constantinos
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Hi Constantinos,
"Foremost of these is the mathematical derivation of Planck's Law (or variation of it) showing that it is an exact mathematical identity (a tautology) that describes the interaction of energy."
I thought your first message was quite general. I think this one is more specific. In my opinion, you have enterred a forum that is especially thoughtful and patient. However, there is more action to be found in the blogs section. I think that posting in both locations will help you receive more varied results. You will find, because of the convenient listing of recent messages, that the blogs section attracts more attention and also the risk of strongly opposite, even possibly offensively posed, views.
Most of which gets posted in the blogs section does not pertain to the subject of any particular blog. FQXi is very patient with new postings. Choose one that is not currently very active and hopefully fits at least minimally with your ideas. Your post will stay visual longer. I do not know what Ian might say about your ideas; but, judging from what you have said thus far, I think that you may have something worthwhile to add to these discussions. I suggest that you post simultaneously in both locations. As, I mentioned above, you have a chance of receiving a better variety of responses. You need to be prepared to explain, in relatively short form, that which you want to communicate.
James
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"Foremost of these is the mathematical derivation of Planck's Law (or variation of it) showing that it is an exact mathematical identity (a tautology) that describes the interaction of energy."
I thought your first message was quite general. I think this one is more specific. In my opinion, you have enterred a forum that is especially thoughtful and patient. However, there is more action to be found in the blogs section. I think that posting in both locations will help you receive more varied results. You will find, because of the convenient listing of recent messages, that the blogs section attracts more attention and also the risk of strongly opposite, even possibly offensively posed, views.
Most of which gets posted in the blogs section does not pertain to the subject of any particular blog. FQXi is very patient with new postings. Choose one that is not currently very active and hopefully fits at least minimally with your ideas. Your post will stay visual longer. I do not know what Ian might say about your ideas; but, judging from what you have said thus far, I think that you may have something worthwhile to add to these discussions. I suggest that you post simultaneously in both locations. As, I mentioned above, you have a chance of receiving a better variety of responses. You need to be prepared to explain, in relatively short form, that which you want to communicate.
James
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James,
. . . as bad as the comment directed to you is, believe me when I say I've had worst! Really vicious! But I wont dwell on that. By nature I am a very positive and optimistic person. My love of ideas would not let me be anything else. Nothing can dissuade me from reasoning other than reasoning itself. Hard to abandon what you know to be True. So feel assured that I will persist, since really this is not about me!
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. . . as bad as the comment directed to you is, believe me when I say I've had worst! Really vicious! But I wont dwell on that. By nature I am a very positive and optimistic person. My love of ideas would not let me be anything else. Nothing can dissuade me from reasoning other than reasoning itself. Hard to abandon what you know to be True. So feel assured that I will persist, since really this is not about me!
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Why should physical quantities follow mathematical formalisms? Without getting into long and protracted philosophical discussions, I just want to say that the essence of Physics is 'measurement' while the essence of Math is 'logical consistency'. So why should mathematical derivations be reflected in physical experiments? I like to propose that the way this would be possible is if all Basic Universal Law are mathematical identities (tautologies) that describe the interactions of measurement.
In the short paper “Plancks' Law is an Exact Mathematical Identity” we show this to be true for measurements of energy. In the equally short paper “The Interaction of Measurement” we show that a physical quantity cannot be 'known' by measurements of it. Thus, there is an underlying 'hidden reality' that we can never know but can only 'measure' and all our derivations of this 'hidden reality' are mathematical certainties that only relate to our measurements.
If we seek to describe such underlying 'hidden nature' directly through our mathematical formalisms and physical theories, if we try to theoretically model it as something separate and beyond our measurements of it, we risk falling into the same 'rabbit's hole' that medieval intellectuals had fallen seeking to count the number of angles that can sit at the head of a pin!
We can only know what we can 'observe' and 'measure'. Thus it all comes back to us, our human Understanding of what Is to Us, but not what Is in Itself! Our very Understanding is a form of 'measurement'. And just as you cannot truly know someone by taking 'measures' of their behavior, equally we cannot know a physical quantity by taking 'measurements' of it.
There is nothing 'absolutely real' about our theories! There are just theories that 'make sense' and theories that 'make none-sense'. When a theory has become so abstract and so removed from the 'sensible experience' of our lives, when it creates 'mathematical certainties' that are counter-intuitive and 'weird', it's time to 'stop the weirdness' and to re-think our Universe and re-Create it anew.
Constantinos
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In the short paper “Plancks' Law is an Exact Mathematical Identity” we show this to be true for measurements of energy. In the equally short paper “The Interaction of Measurement” we show that a physical quantity cannot be 'known' by measurements of it. Thus, there is an underlying 'hidden reality' that we can never know but can only 'measure' and all our derivations of this 'hidden reality' are mathematical certainties that only relate to our measurements.
If we seek to describe such underlying 'hidden nature' directly through our mathematical formalisms and physical theories, if we try to theoretically model it as something separate and beyond our measurements of it, we risk falling into the same 'rabbit's hole' that medieval intellectuals had fallen seeking to count the number of angles that can sit at the head of a pin!
We can only know what we can 'observe' and 'measure'. Thus it all comes back to us, our human Understanding of what Is to Us, but not what Is in Itself! Our very Understanding is a form of 'measurement'. And just as you cannot truly know someone by taking 'measures' of their behavior, equally we cannot know a physical quantity by taking 'measurements' of it.
There is nothing 'absolutely real' about our theories! There are just theories that 'make sense' and theories that 'make none-sense'. When a theory has become so abstract and so removed from the 'sensible experience' of our lives, when it creates 'mathematical certainties' that are counter-intuitive and 'weird', it's time to 'stop the weirdness' and to re-think our Universe and re-Create it anew.
Constantinos
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Constantinos,
You said:
"We can only know what we can 'observe' and 'measure'."
I absolutely agree. The "overabstractification" of physics drives me nuts.
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You said:
"We can only know what we can 'observe' and 'measure'."
I absolutely agree. The "overabstractification" of physics drives me nuts.
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Greetings Ian, thank you for your response.
As careful as I am in crafting my ideas and arguments, there is always some significant aspect that is left out or some misreading that is left in the written words. For example, if I were to re-write the quoted statement in your post I would write
"We can only know our 'observations' and our 'measurements' ".
We really cannot know 'what Is' but only 'what we observe', 'what we measure' and 'what we understand'. These are the essences upon which we create our World (whether it be our physical universe or the world of our personal lives). Any attempt to model the World of 'what Is' in my view leads to Metaphysics and to Modern Physics. In my paper The Interaction of Measurement I show that we cannot know a physical quantity (thought of as a function of time) through our direct measurements of it.
The essence of Physics in my view is 'measurement', while mathematics only provides 'logical certainties'. If we were to combine the two, it seems to me that a new Mathematical Foundation of Physics may be needed. One that establishes Basic Law of Physics as mathematical identities (tautologies) that describe the interaction of measurement. This I show is possible in the case of energy in my short paper, Planck's Formula is an Exact Mathematical Identity.
Though I argue that we cannot know 'what Is', I do not go as far as saying that 'nothing Is'. I simply acknowledge that the very asking of the question (let alone answering it) is meaningless.
Constantinos
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As careful as I am in crafting my ideas and arguments, there is always some significant aspect that is left out or some misreading that is left in the written words. For example, if I were to re-write the quoted statement in your post I would write
"We can only know our 'observations' and our 'measurements' ".
We really cannot know 'what Is' but only 'what we observe', 'what we measure' and 'what we understand'. These are the essences upon which we create our World (whether it be our physical universe or the world of our personal lives). Any attempt to model the World of 'what Is' in my view leads to Metaphysics and to Modern Physics. In my paper The Interaction of Measurement I show that we cannot know a physical quantity (thought of as a function of time) through our direct measurements of it.
The essence of Physics in my view is 'measurement', while mathematics only provides 'logical certainties'. If we were to combine the two, it seems to me that a new Mathematical Foundation of Physics may be needed. One that establishes Basic Law of Physics as mathematical identities (tautologies) that describe the interaction of measurement. This I show is possible in the case of energy in my short paper, Planck's Formula is an Exact Mathematical Identity.
Though I argue that we cannot know 'what Is', I do not go as far as saying that 'nothing Is'. I simply acknowledge that the very asking of the question (let alone answering it) is meaningless.
Constantinos
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Ian and Constantinos,
While I share your attitude, I am an engineer who trusts in sound prediction and benefits from complex calculus.
I would like to convey the insight that the key question is not knowledge but a clear distinction between influences that are merely to be expected and those that already lead to something which can now be observed and measured in principle.
In terms of physical theory, that now is not necessarily the now of reality. It rather corresponds to the moment under consideration, i. e., to the latest point of time at which something can influence the process.
Eckard
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While I share your attitude, I am an engineer who trusts in sound prediction and benefits from complex calculus.
I would like to convey the insight that the key question is not knowledge but a clear distinction between influences that are merely to be expected and those that already lead to something which can now be observed and measured in principle.
In terms of physical theory, that now is not necessarily the now of reality. It rather corresponds to the moment under consideration, i. e., to the latest point of time at which something can influence the process.
Eckard
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Dear Eckard,
… your call for experimental confirmation and predictability is well taken and appreciated.
There are some new posts re: double-slit experiment explanation in the blog section of Edge of Physics. Just in case you were not aware of this …
Best,
Constantinos
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… your call for experimental confirmation and predictability is well taken and appreciated.
There are some new posts re: double-slit experiment explanation in the blog section of Edge of Physics. Just in case you were not aware of this …
Best,
Constantinos
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Incidentally, I have completely lost track of the discussion here. I find this software annoying and think FQXi ought to switch to software like they use over at places like nForum and Meta.MathOverflow.
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Hi Constantinos,
i read your explanation of the double-slit-experiment and i have a question to it.
If the emission of an electron (by passing the double-slit) has to be thought as a continous field of energy - consisting of the energy of exactly one electron - and distributed in its strengthness due to the probability-interpretation of QM over the detection-screen in the well known manner of an interference-pattern - then, due to the laws of addition the area exactly in the middle behind the two slits has the highest probability to pop up as a light-spot *firstly*. Therefore the first light-spot after accumulation of enough energy must be in exactly this area. Could this have been really the case in all the experiments? And why are there light-spots in areas where there had to be exactly destructive interference - means, a probability of zero?
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i read your explanation of the double-slit-experiment and i have a question to it.
If the emission of an electron (by passing the double-slit) has to be thought as a continous field of energy - consisting of the energy of exactly one electron - and distributed in its strengthness due to the probability-interpretation of QM over the detection-screen in the well known manner of an interference-pattern - then, due to the laws of addition the area exactly in the middle behind the two slits has the highest probability to pop up as a light-spot *firstly*. Therefore the first light-spot after accumulation of enough energy must be in exactly this area. Could this have been really the case in all the experiments? And why are there light-spots in areas where there had to be exactly destructive interference - means, a probability of zero?
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Hello again Stefan. I just posted a reply to your other post addressed to me. Thank you for your very good questions. Following up on my reply to your other question, let me just say that there are local conditions on the detection screen (beyond the radiated pattern from the emitted electrons) that we just don't know, making any precise prediction just impossible (even if we were to predict the 'area' where the first flash will occur). We may agree that QM does not 'lie' here. But QM does not provide a 'physical explanation' either. What I am hoping with this is to help provide some 'Physical Realism' to what now has become 'quantum weirdness'.
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I just found your site. I am in love. I just forwarded the link to all I know. Good luck on future growth. I look forward to it!!
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Ian,
I was involved in a discussion at another website. My involvment was not as a main participant. I have copied edited parts of my few messages and include them below. I have removed all texts except for mine. The point I was making is that thermodynamic entropy is not yet explained. Skipping past it to expressions of statistical mechanics is not satisfying to me. I think the...
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I was involved in a discussion at another website. My involvment was not as a main participant. I have copied edited parts of my few messages and include them below. I have removed all texts except for mine. The point I was making is that thermodynamic entropy is not yet explained. Skipping past it to expressions of statistical mechanics is not satisfying to me. I think the...
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James,
I highly recommend reading Dan Schroeder's somewhat unusual textbook An Introduction to Thermal Physics, specifically chapter 3 (though the basic ideas presented in that chapter also appear in a paper he co-authored with Tom Moore in the American Journal of Physics in the late 1990s). Based on Dan's and Tom's work, I think thermodynamic entropy (and temperature) make sense, but its not always presented in the form given by Schroeder and Moore which makes it confusing. It's really the honing of Shannon's ideas.
Ian
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I highly recommend reading Dan Schroeder's somewhat unusual textbook An Introduction to Thermal Physics, specifically chapter 3 (though the basic ideas presented in that chapter also appear in a paper he co-authored with Tom Moore in the American Journal of Physics in the late 1990s). Based on Dan's and Tom's work, I think thermodynamic entropy (and temperature) make sense, but its not always presented in the form given by Schroeder and Moore which makes it confusing. It's really the honing of Shannon's ideas.
Ian
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Ian:
I like the idea of attempting to identify the limits of mathematics because insight into this issue would help to flag the moments when physics or other subjects, such as economics, might be going off the deep end. It is a profoundly challenging question, though, and it sits at the intersection of philosophy, mathematics and science. This is one of those situations where insight into a large question is gained by asking an even larger question.
So let's ask instead: What is the difference between abstract mathematics - with no applications - and real mathematics? A good argument can be raised that there is no difference for one simple reason - all mathematics is abstract. There is a profound difference between useful and real. Abstractions from reality can be exceedingly useful, but they aren't real - they are idealized mental forms that are intentional simplifications of reality. The number 3 seems very real, and so does pi, but there is a reason why you can't see either number written in the sky or on the waves of the sea - these numbers exist only in your mind. They spring from the reality that defines them and makes them meaningful in more than one situation. We shouldn't be surprised that they are useful in multiple situations - we chose them as useful abstractions from reality for exactly that reason.
Long ago, I browsed through a book on the Tao in a library, and didn't find much useful, but one idea resonated strongly with me. It was stated something like - the abstraction of the thing is not the thing itself. This is a powerful and useful observation. It turns out that the Tao has many translations and I have never again seen this quote in exactly the same way, but the idea is eternal. The word cow is not a cow, A picture of a cow is not a cow. Your thought of a cow when you see the word cow or smell cows is not a cow. Our understanding that a cow herd contains individual cows does not create any cows. Only that individual creature on Farmer Jones' farm, named Bessy, calmly chewing grass with a bell around her neck as we watch her with our very eyes is a cow. She is unlike all other cows - a reality we forget when we abstract her.
For every cow there are many abstractions of it. Likewise, it is possible to abstract reality into mathematics in a infinite number of ways. But if our goal is to understand reality we should never confuse it with mathematics or with any of our abstractions. Physics and other sciences begin to diverge from reality the instant this confusion is made.
Stan
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I like the idea of attempting to identify the limits of mathematics because insight into this issue would help to flag the moments when physics or other subjects, such as economics, might be going off the deep end. It is a profoundly challenging question, though, and it sits at the intersection of philosophy, mathematics and science. This is one of those situations where insight into a large question is gained by asking an even larger question.
So let's ask instead: What is the difference between abstract mathematics - with no applications - and real mathematics? A good argument can be raised that there is no difference for one simple reason - all mathematics is abstract. There is a profound difference between useful and real. Abstractions from reality can be exceedingly useful, but they aren't real - they are idealized mental forms that are intentional simplifications of reality. The number 3 seems very real, and so does pi, but there is a reason why you can't see either number written in the sky or on the waves of the sea - these numbers exist only in your mind. They spring from the reality that defines them and makes them meaningful in more than one situation. We shouldn't be surprised that they are useful in multiple situations - we chose them as useful abstractions from reality for exactly that reason.
Long ago, I browsed through a book on the Tao in a library, and didn't find much useful, but one idea resonated strongly with me. It was stated something like - the abstraction of the thing is not the thing itself. This is a powerful and useful observation. It turns out that the Tao has many translations and I have never again seen this quote in exactly the same way, but the idea is eternal. The word cow is not a cow, A picture of a cow is not a cow. Your thought of a cow when you see the word cow or smell cows is not a cow. Our understanding that a cow herd contains individual cows does not create any cows. Only that individual creature on Farmer Jones' farm, named Bessy, calmly chewing grass with a bell around her neck as we watch her with our very eyes is a cow. She is unlike all other cows - a reality we forget when we abstract her.
For every cow there are many abstractions of it. Likewise, it is possible to abstract reality into mathematics in a infinite number of ways. But if our goal is to understand reality we should never confuse it with mathematics or with any of our abstractions. Physics and other sciences begin to diverge from reality the instant this confusion is made.
Stan
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Dear Stan,
Well, applied mathematics is applied abstraction. Someone said: "The map is not the territory".
Hopefully, the community of physicists will not feel too offended if I reiterate that predominant physics is ignorant of something that is well understood by common sense: abstracted time is different from real time.
In particular, W. Ritz was correct in 1909 when he insisted that future events cannot influence the past while Einstein preferred to stick on the traditional belief in an a priori given time from minus infinity to plus infinity. Ritz died soon.
Maybe, Minkowski got ill and died because he felt being wrong with his exciting idea of a belonging complex spacetime.
If I recall correctly, Schwarzschild died in 1916 before his complete solution for the metric around a point mass was interpreted as reality even for past and future singularity and beyond.
If I compare the about 4 000 Mio € expense for LHC with the 750 000 Mio € agreed in order to finance some almost bankrupt generous European states, I see the LHC worth his money if one will draw the due consequences from the outcome.
Regards,
Eckard
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Well, applied mathematics is applied abstraction. Someone said: "The map is not the territory".
Hopefully, the community of physicists will not feel too offended if I reiterate that predominant physics is ignorant of something that is well understood by common sense: abstracted time is different from real time.
In particular, W. Ritz was correct in 1909 when he insisted that future events cannot influence the past while Einstein preferred to stick on the traditional belief in an a priori given time from minus infinity to plus infinity. Ritz died soon.
Maybe, Minkowski got ill and died because he felt being wrong with his exciting idea of a belonging complex spacetime.
If I recall correctly, Schwarzschild died in 1916 before his complete solution for the metric around a point mass was interpreted as reality even for past and future singularity and beyond.
If I compare the about 4 000 Mio € expense for LHC with the 750 000 Mio € agreed in order to finance some almost bankrupt generous European states, I see the LHC worth his money if one will draw the due consequences from the outcome.
Regards,
Eckard
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Eckard:
I think Ian senses, along with some other physicists, that physics may have gotten lost somewhere along the way. He is looking for a method that will flag an abstraction as having no parallel in the real world. There are many ways to approach a problem like this, and if you have an open mind insights can be obtained that give you the ability to tell the difference even when the rule...
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I think Ian senses, along with some other physicists, that physics may have gotten lost somewhere along the way. He is looking for a method that will flag an abstraction as having no parallel in the real world. There are many ways to approach a problem like this, and if you have an open mind insights can be obtained that give you the ability to tell the difference even when the rule...
view entire post
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Thanks for your reply, Stan. My apologies on not responding sooner. I have been insanely busy lately and haven't had a chance to check on this board recently.
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Just as the "logarithmic integral" Li(x) can be used to approximate
how many prime numbers there are under a given number x,
the "non-trivial polygonal number counting function"
which you can find here:
on_polygonal_numbers.pdf
can be used to approximate, (with even greater accuracy)
how many "polygonal numbers of rank greater than 2"
there are under a given number x.
What makes this "counting function" so unusual
(and of possible interest to the physics community)
is that it seems to require the two most important
"physical constants", alpha and mu.
I personally have no theories as to why this should be so,
and am presenting this as a mathematical function only.
I welcome your comments.
Don
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how many prime numbers there are under a given number x,
the "non-trivial polygonal number counting function"
which you can find here:
on_polygonal_numbers.pdf
can be used to approximate, (with even greater accuracy)
how many "polygonal numbers of rank greater than 2"
there are under a given number x.
What makes this "counting function" so unusual
(and of possible interest to the physics community)
is that it seems to require the two most important
"physical constants", alpha and mu.
I personally have no theories as to why this should be so,
and am presenting this as a mathematical function only.
I welcome your comments.
Don
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Ian,
Thank you for this forum. The linkage between mathematics and physics and its limits is the most important foundational question. My work developing www.zenophysics.com indicates that much of the trouble with physics can be traced to the failure of Newton's calculus to counter Zeno's paradox of motion. Here is the story:
1. Zeno's paradox of motion: If you assume that points are...
view entire post
Thank you for this forum. The linkage between mathematics and physics and its limits is the most important foundational question. My work developing www.zenophysics.com indicates that much of the trouble with physics can be traced to the failure of Newton's calculus to counter Zeno's paradox of motion. Here is the story:
1. Zeno's paradox of motion: If you assume that points are...
view entire post
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Dear Don Limuti,
I also consider the basis of mathematic most fundamental to physics.
If you agree with Euclid that a point has no parts and with Peirce that a each part of continuum has parts then a continuum cannot be resolved in a finite amount of single points and there is no closest point to any point. This is not yet a physical but a mathematical explanation.
What "trouble in physics" did you resolve?
Given I am correct then quantum computing will not work as promised, several paradoxes will vanish, the LHC will not fulfill expectations, and several so far mandatory tenets will turn out at least questionable.
I found out:
1) Pupils of Gauss changed the good old notion of number as a measure to the rather questionable older understanding as a pebble.
2) IR+ is equivalent to IR and fits better to virtually all original physical quantities with natural zero including volume, area, distance, elapsed time, radius, temperature, pressure, number of electrons and other items ... , probability, ...
3) According to 2) Analysis of measured data does not require a complex valued Fourier transformation but just a real-valued Cosine transformation. The FT is redundant. In addition to the information available with the CT it only provides information on the arbitrarily chosen point of reference.
4) The fathers of QM introduced unilateral functions of frequency directly in a complex domain in a manner they borrowed from common practice where unilateral functions of time are considered the original ones. However they changed to the Hamiltonian point of view. This led to misinterpretation.
5) Perhaps, length contraction does not likewise belong to decreasing and increasing distance.
6) Space-time seems to be ill-founded because it would be anticipatory, and the sign of rotation clockwise or anticlockwise is arbitrary.
7) G. Cantor and his proponents did not refute Galilei's argument: The relations smaller, equal to, and larger are not valid for infinite quantities. As Fraenkel admitted: There is a 4th logical possibility: incomparability, and Cantor's naive definition of a set is untenable.
Regards,
Eckard
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I also consider the basis of mathematic most fundamental to physics.
If you agree with Euclid that a point has no parts and with Peirce that a each part of continuum has parts then a continuum cannot be resolved in a finite amount of single points and there is no closest point to any point. This is not yet a physical but a mathematical explanation.
What "trouble in physics" did you resolve?
Given I am correct then quantum computing will not work as promised, several paradoxes will vanish, the LHC will not fulfill expectations, and several so far mandatory tenets will turn out at least questionable.
I found out:
1) Pupils of Gauss changed the good old notion of number as a measure to the rather questionable older understanding as a pebble.
2) IR+ is equivalent to IR and fits better to virtually all original physical quantities with natural zero including volume, area, distance, elapsed time, radius, temperature, pressure, number of electrons and other items ... , probability, ...
3) According to 2) Analysis of measured data does not require a complex valued Fourier transformation but just a real-valued Cosine transformation. The FT is redundant. In addition to the information available with the CT it only provides information on the arbitrarily chosen point of reference.
4) The fathers of QM introduced unilateral functions of frequency directly in a complex domain in a manner they borrowed from common practice where unilateral functions of time are considered the original ones. However they changed to the Hamiltonian point of view. This led to misinterpretation.
5) Perhaps, length contraction does not likewise belong to decreasing and increasing distance.
6) Space-time seems to be ill-founded because it would be anticipatory, and the sign of rotation clockwise or anticlockwise is arbitrary.
7) G. Cantor and his proponents did not refute Galilei's argument: The relations smaller, equal to, and larger are not valid for infinite quantities. As Fraenkel admitted: There is a 4th logical possibility: incomparability, and Cantor's naive definition of a set is untenable.
Regards,
Eckard
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Dear Eckard,
The trouble with physics: The axioms of geometry* do not allow quantum particles to move.
I propose to makes the axioms of geometry legitimate for quantum particles by making particles move in a special way. That way of moving is what I call wavelength-hopping. The details are on www.digitalwavetheory.com
Thanks,
Don L.
* 1. Dimensionless Points 2. A continuum of points on a line.
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The trouble with physics: The axioms of geometry* do not allow quantum particles to move.
I propose to makes the axioms of geometry legitimate for quantum particles by making particles move in a special way. That way of moving is what I call wavelength-hopping. The details are on www.digitalwavetheory.com
Thanks,
Don L.
* 1. Dimensionless Points 2. A continuum of points on a line.
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The origin of special relativity has much to do with the origin of dynamics.
Useful axioms or theorems that could act as fundaments for physics exist, but they treat only the static relational structure of physical items and physical fields. These law sets consist for a part of traditional quantum logic as it is defined by Birkhoff, von Neumann, Jauch, Piron and others. Quantum logic says...
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Useful axioms or theorems that could act as fundaments for physics exist, but they treat only the static relational structure of physical items and physical fields. These law sets consist for a part of traditional quantum logic as it is defined by Birkhoff, von Neumann, Jauch, Piron and others. Quantum logic says...
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I am new to this forum, so I forgot to add my name and I did not introduce myself. I am a retired physist. I spend my retirement on thinking what is good and what is bad about contempory physics. By browsing around I encountered a series of important inconsistencies. All have to to with the incorrect implementation of mathematics. An example is the fact that according to the isomorphism between the lattice of quantum logical propositions and the lattice of closed subspaces of an inifinite separatable Hilbert space the quantum propositions can be represented by the corresponding closed subspaces. If everything that can be said about a physical item is formulated as a proposition, then that proposition represents the physical item and the corresponding subspace also represents that item. Why then will most physicists represent physical items by wave functions or by density operators. The wave function corresponds with a single Hilbert vector instead of with a multidimensional subspace. How can a single Hilbert vector carry all the properties of the DNA molecule that represents on its turn the characteristics of Napoleon? A multdimensional subspace would fit far better.
There are not so much divergences between physics and mathematics, but instead there exist many inconsistencies in the way mathematics is applied to physics. This is not the failure of physics. It is the failure of the physicists.
Greathings, Hans
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There are not so much divergences between physics and mathematics, but instead there exist many inconsistencies in the way mathematics is applied to physics. This is not the failure of physics. It is the failure of the physicists.
Greathings, Hans
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Dear All,
Mathematical representation of absolute truth, zero = i = infinity can be deduced as follows
If 0 x 0 = 0 is true, then 0 / 0 = 0 is also true
If 0 x 1 = 0 is true, then 0 / 0 = 1 is also true
If 0 x 2 = 0 is true, then 0 / 0 = 2 is also true
If 0 x i = 0 is true, then 0 / 0 = i is also true
If 0 x ~ = 0 is true, then 0 / 0 = ~ is also true
It seems that mathematics, the universal language, is also pointing to the absolute truth that 0 = 1 = 2 = i = ~, where "i" can be any number from zero to infinity. We have been looking at only first half of the if true statements in the relative world. As we can see it is not complete with out the then true statements whic are equally true. As all numbers are equal mathematically, so is all creation equal "absolutely".
This proves that 0 = i = ~ or in words "absolutely" nothing = "relatively" everything or everything is absolutely equal. Singularity is not only relative infinity but also absolute equality. There is only one singularity or infinity in the relativistic universe and there is only singularity or equality in the absolute universe and we are all in it.
Love,
Sridattadev.
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Mathematical representation of absolute truth, zero = i = infinity can be deduced as follows
If 0 x 0 = 0 is true, then 0 / 0 = 0 is also true
If 0 x 1 = 0 is true, then 0 / 0 = 1 is also true
If 0 x 2 = 0 is true, then 0 / 0 = 2 is also true
If 0 x i = 0 is true, then 0 / 0 = i is also true
If 0 x ~ = 0 is true, then 0 / 0 = ~ is also true
It seems that mathematics, the universal language, is also pointing to the absolute truth that 0 = 1 = 2 = i = ~, where "i" can be any number from zero to infinity. We have been looking at only first half of the if true statements in the relative world. As we can see it is not complete with out the then true statements whic are equally true. As all numbers are equal mathematically, so is all creation equal "absolutely".
This proves that 0 = i = ~ or in words "absolutely" nothing = "relatively" everything or everything is absolutely equal. Singularity is not only relative infinity but also absolute equality. There is only one singularity or infinity in the relativistic universe and there is only singularity or equality in the absolute universe and we are all in it.
Love,
Sridattadev.
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This is an elementary slapdash answer before I lose my cool and run away from this conversation.
You state above: "What I am interested in from a non-speculative standpoint are those places that mathematics *diverges* from the physical world, i.e. those places in which mathematics describes something that is very clearly not physically possible. Now, this obviously does not mean any instance of infinity since, as an example, infinity is necessary to resolve such physical paradoxes as those conjectured by Zeno. But there *are* instances where mathematics in some infinite limit ends up describing something that is not physically possible. There are probably plenty of examples that don't involve infinity."
My intuition and mathematical researches lead me to feel strongly that insight into the above problem can be gained by a fuller understanding of the relationships between Thought, Mathematics, and Physical reality. This can be accomplished by comparing the math describing both thought and reality. The mathematics (symbology) is the same in each case, even though the two Subjects: Thought and Reality are different Kinds of subjects. So the math contains the clues.
Two specifics come to mind: 1)George Boole's law of thought: X^2=X, and 2)the Quantum Mechanical linear operator |Y>
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You state above: "What I am interested in from a non-speculative standpoint are those places that mathematics *diverges* from the physical world, i.e. those places in which mathematics describes something that is very clearly not physically possible. Now, this obviously does not mean any instance of infinity since, as an example, infinity is necessary to resolve such physical paradoxes as those conjectured by Zeno. But there *are* instances where mathematics in some infinite limit ends up describing something that is not physically possible. There are probably plenty of examples that don't involve infinity."
My intuition and mathematical researches lead me to feel strongly that insight into the above problem can be gained by a fuller understanding of the relationships between Thought, Mathematics, and Physical reality. This can be accomplished by comparing the math describing both thought and reality. The mathematics (symbology) is the same in each case, even though the two Subjects: Thought and Reality are different Kinds of subjects. So the math contains the clues.
Two specifics come to mind: 1)George Boole's law of thought: X^2=X, and 2)the Quantum Mechanical linear operator |Y>
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Dang, I had a totally brilliant thread there for you to contemplate, and then deleted most of it with an accidental keystroke. But because of the lack of feedback I get in these threads, that should be no problem for anyone. So, I ain't worth retyping. Here I'll just finish the last cut off sentence above:
2) the quantum mechanical linear operator |Y>
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2) the quantum mechanical linear operator |Y>
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WTF! Warning, do not attempt to type the linear operator |Y> .
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The square of the product of a ket and a bra vector (with the ket on the left of the bra) is equal to the original bra and ket. This is impossible to represent in this thread symbollically, apparently, as it deletes all. Anyway, this is the same form of equation as in the Boolean law of thought.
See my essay for symbolical demonstration, and how the interpretation of these equations is accomplished. Typing all this a fifth time just ain't gonna happen...
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See my essay for symbolical demonstration, and how the interpretation of these equations is accomplished. Typing all this a fifth time just ain't gonna happen...
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The bra and kets have to be conjugate imaginaries of each other for this equation to hold, of course...
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