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FQXi FORUM
CATEGORY: Ultimate Reality
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TOPIC: Tegmark's Mathematical Universe
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While he would be too modest to toot his own horn, there's nothing stopping me from pointing out that 'mad Max' has recently put out a very interesting paper on whether all possible formal systems (and the 'physical' universes that some subset of them describe) are equally 'real'.
I like lots of things about this paper. One part that I worry about -- or perhaps misunderstand -- is the connection with Godel's theorem. Max makes an extended argument that if we exclude all human-centric 'baggage' then the universe must be mathematical, and in particular isomorphic to some formal system. He then limits this to 'Godel-complete' formal systems. But why? My feeling is that exactly what Godel proved is that there are mathematical truths that are not decidable using the formal system within which they are formulated. So the *truth or falsity* of them ('truthiness'?) is (a) baggage-free, but (b) outside of any formal system. So in his scheme is it real or not? I don't know.
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Well, if we accept the Mathematical Universe Hypothesis (which is pretty neat) then ALL mathematics is "real" -- as that's what the universe, err, is. But that can't be the last word, for a couple of reasons. First, our universe seems fairly simple, at least compared to how complicated it could be if all mathematics were manifested. Second, we run up against G??del's incompleteness theorem. I suspect I know just enough about G??delian incompletenes to get into trouble (and even if I did truly know it, I wouldn't be able to prove I knew it), but I'll offer up some ideas anyway.
G??del's incompleteness theorem is the dagger through the heart of Hilbert's program to build all of mathematics from a simple logical foundation. Real mathematics (what real mathematicians (i.e. not me) do) is usually about proving things (theorems), and those things, including the proofs, can be encoded as numbers and operations between them. Hilbert hoped all true theorems in mathematics, with proofs, could be built up with logic. But G??del came up with this zinger: "This theorem is true but cannot be proven to be true." And there it sits, a theorem, which can be encoded by a G??del number, that is true but has no proof. So any sufficiently complicated formal system, in which these G??del numbers can be formulated, is incomplete since it has true statements that aren't provable.
This incompleteness wouldn't be much of a bother for physicists (who tend not to care much for proofs anyway) except the MUH proposes that mathematics, including proofs, is our long sought Theory Of Everything. So, as far as I can tell, Max Tegmark introduced the Computable Universe Hypothesis in order to dodge G??delian incompleteness. The CUH says maybe the universe consists of mathematics simple enough that G??del's theorem doesn't apply. It's a neat idea, since it also explains why the universe seems relatively comprehensible, but I sure hope it's wrong. If it's right, it implies the universe is finite, and anyone trying to build a TOE out of continuous pieces (like me or them) would have to toss the whole effort and (like him) consider something more like Legos. Now, I've always loved Legos, but they got kinda boring after age twelve or so, and I'd be sad now if that's all there was to play with. So, if we accept the MUH, how do we dodge the CUH? I can think of two options:
1) Live with G??delian incompleteness as part of our universe. This doesn't seem so bad to me, but as a physicist I'm more interested in truth than in proofs. I don't know why it's bad to have things about the universe that are true but not provable. (And it would be frustrating for mathematicians -- which is fun to see as a physicist often humbled by math papers.) I'm hoping someone else here (maybe Janna Levin?) will argue why a mathematical universe would vanish in a puff of logic if there were true things that weren't provable.
2) Limit the MUH another way (with minimal carry-on baggage). I don't know enough math to argue this well -- but I remember that naive set theory had problems with G??delish paradoxes which led to the formulation of axiomatic set theory. Maybe the universe is just a complex mathematical shape, or a geometric automata, that exists and acts in a way describable by a set of mathematical axioms that preclude G??delian self-reference? It's true that we can write down and think about G??del's theorem -- but that doesn't necessarily mean our substrate cares about the theorem, does it?
Anyway, fun stuff to think about.
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My take on why his central thesis is flawed:
http://aeolist.wordpress.com/2007/04/26/confusing-baggage/
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What I like most about such ideas is that it demistifies some couterintuitive aspects of arificial intelligence, the subjective aspects about our personal experiences etc. If indeed all mathematical systems are universes in their own right, then consider the (personal) universe that is defined by the formal rules that describes how someone's brain works.
Personal experiences are simply events that happen in such personal universes. We are such personal universes but we find ourselves living inside simulations performed by neural networks in a universe that is accurately decribed by the Standard Model and General Relativity. Presumably this is because very complex universes (that can onlty be specified using a large amount of data) have a small measure while simple universes that can be specified with a small amount of information have a large measure.
In case of our minds, the formal rules are not really how exactly the neurons are connected to each other, that's just the way the program is implemented in this universe. But there also exists a "source code" that describes who we are. The way one implements the program does not matter to the program itself.
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There's also another way to argue that ERH implies MUH than outlined in the paper.
If every mathematical structure exists, we can choose the mathematical structure which approximates our universe so well we could not distinguish it from our own, given that we can not distinguish it; our universe can be described by that mathematical structure and given that there is no way to distinguish it, it actually is this structure.
Now I must agree that ERH is not exactly the same as stating that every mathematical structure exists. It certainly is at least as general a proposition as ERH and it can also be considered as a form of ERH given that mathematical existence implies an external physical existence.
I also do not see how any of this is counterintuitive; given that we would simulate a given SAS (Self Aware Structure) two times in an exact way. It would be absurd for the SAS to state that he was the first or second being simulated, given there is no way to distinguish the different simulations. Only the probability measure can/will be affected. In a broader context I do not like the wording that we would be living in a simulated reality as this is again a absurd statement. Given that we accept MUH, we exist regardless of the fact if we are being simulated or not. Would it not be counterintuitive that only SAS being simulated exist and those who are not simulated do not exist ?
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William wrote: "given that mathematical existence implies an external physical existence."
That is an extremely strong "given", which if true would imply the [in my opinion absurd] consequence that every possible mathematical structure exists in some universe. I also can't think of any good arguments for it.
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Re. Ponder Stibbons, I don't see that it is absurd that every mathematical structure should exist. The hypothesis "all mathematical structures exist" is simpler than "only certain mathematical structures exist" and so is to be preferred by Occam. In fact, the MUH is the simplest possible metaphysics, having no free parameters at all.
Re. Garett Lisi: I don't see that the CUH implies a finite universe. If the computation never stops (halts), the universe will be infinite. One might posit existence as identical with proof. On this analysis the universe is a mathematical proof machine. Godel would imply that there must thus be true theorems (possible universes) that are not provable (never exist).
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Of course math can explain the universe. Even astrology has a math behind it that allows precise calculations of how the stars supposedly influence different people's lives, although they lack an adequate explanation of the physical process involved.
Understanding the universe will require looking at the issue of physical dimensions of reality in strict mathematical terms instead of relying on the physical senses.
In the strict mathematical sense a dimension is a characteristic or variable. The only information the three Euclidean dimensions provide is length, width and height. They provide the location of "points", but cannot provide any further information about the points. These dimensions cannot say if the point has some type of "spin" or "charm".
If reality consisted only of these three dimensions, we would see a world in black and white only. Our eyes could only determine that something was in a location or not. They could not determine other characteristics of objects because characteristics like color require additional information provided by other variables. The eyes can determine if an object is reflecting or emitting light or if light is merely passing through it, but this characteristic of objects doesn’t depend on length, width or height. One object could not be a different temperature than another in a 3 dimensional reality, because temperature differences require an additional characteristic or variable.
Our eyes cannot see gravity or a magnetic field. Yet, we know both exist because we can see or feel the effects of them. When we see something fall off the edge of a table we only see it fall. We do not see the invisible force that draws it to the floor.
Gravity and magnetic attraction are not defined by simple length, width and height. The sun occupies a greater 3-dimensional area than earth and has higher gravitational attraction, but a black hole the size of a ping pong ball could have higher gravitational attraction than the sun. The density of matter might determine gravity, but density is a different dimension from length, width and height.
Other dimensions would not necessarily have to be “higher” dimensions. There is no mathematical reason that would preclude the existence of various different “spaces” with different numbers of dimensions. For example, there could be an energy space and a matter space. One or more of the dimensions of these different spaces could be unique to that specific space. Other dimensions could be the same with the different spaces intersecting in the shared dimensions. We may perceive reality as 3-dimensional because the various spaces that make up reality intersect in the dimensions of length, width and height.
If other dimensions are actually higher dimensions than the space in which we exist, then it would be impossible for the eyes to see them. Higher dimensions could only be perceived indirectly through the projection of their influence on our space. A projection from a higher dimensional space into a lower dimensional space can only possess the dimensions of the lower space.
Physicists treat gravity as a force, but have some trouble explaining it. Perhaps gravity isn’t a force per se, but merely a dimension of matter. A given object may have a length, width, height and a gravity.
Albert Einstein suggested that gravity could warp space. Perhaps it is matter that warps space through impact on an elasticity dimension and it is the warping of space that causes the effect called gravity.
A dynamic intersection of spaces might explain some aspects of quantum physics such as tunneling or particles that seem to wink in or out or reality. The winking could result from the particle being in the space we perceive at one time, but not the other time. If this explanation is correct then time travel would be impossible because a moment of time would be only a temporary intersection rather than a location that could be traveled to in a science fiction story.
Perhaps the different flavors of quarks are actually different dimensions of quarks. An up quark would be a quark in the up dimension. A down quark would be in the down dimension. Each quark might have its own Euclidean space with a quark in a location or not in a location. The different quark spaces might then combine to form a matter space.
What we call human intelligence or consciousness seems to exist in a higher physical dimension. Human understanding of the physical world and ability to alter it implies knowledge that could only be obtained by viewing that world from “above”. Beings living in a 2-dimensional world would only be able to perceive locations as being so far forward then right, left or back from the present location. They could not recognize if that location was on the other side of a barrier from them because recognizing that fact would require being able to see the world from above which would be a higher dimension.
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Reason:"In the strict mathematical sense a dimension is a characteristic or variable. The only information the three Euclidean dimensions provide is length, width and height. They provide the location of "points", but cannot provide any further information about the points."
But then Height is relative to a width, and width is reletive to a length?
Thus:"If reality consisted only of these three dimensions, we would see a world in black and white only. Our eyes could only determine that something was in a location or not. They could not determine other characteristics of objects because characteristics like color require additional information provided by other variables." we see a "range" of colours..because each colour is relative to another colour, ie spectrum?
We do not see just black and white, not because there are MORE dimensions, but they are part of a combination, spectrum?
There cannot be more than 3-Dimensions for our physical Universe, at this moment in time, this is not to say that the Universe will always remain 3-D. I believe the Universe to be dynamic, it changes over time, and in the future it will develop "extra" dimensions.
A simple water molocule cannot be placed within a 5-Dimensional space and remain a water molocule?..the stucture of matter within space-time is 3-D. But all of 3-D structures have "edges", these can be viewed as 2-D boundaries, thus as you state, fields or forces help gather matter into 3-D forms.
It is interesting that all of forces/fields are hard to detect..visually that is, because edges can have linearrly finite form, from this perspective, 2-D fields always "surround" 3-D structures.
You cannot find 2-D fields within a 3-D structure, but you can find a 3-D structure within a 2-D field!
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I am not a mathematician and when I happen to read or hear some complicated mathematical issues, I always feel very ignorant, but before falling in a deep depression I react imagining to be in front of a mathematical formula and asking it to show me how it interact with all other formulas to construct and regulate the reality I am living in.
When I see that the formula remain still, I feel better, and realise that, contrary to the formula, my ignorance doesn’t prevent me to act in a free and constructing way.
Perhaps I am wrong, but my vision of mathematic is that of an instrument that serve its purpose only if is used by an operator, (man, animal and down to the single element of nature)
At this point a question arise: can the single element of nature that uses the mathematical instrument, be considered o represented as a mathematical function?
I don’t believe so, and I feel backed also by G??del theorem, the operator to correctly operate the instrument, must necessarily be positioned a step higher, (whatever this step higher means)
Am I right? Hope so.
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I agree with Jonathan Colby (may 23, 2007) that the hypothesis “all mathematical structure exist” is simpler than “only certain mathematical structure exist”, and therefore preferable, I am also convinced that the first would be in better position to represent our reality.
I recon also that my conviction in not a scientific proof, in fact there could be only two way of proving it:
¬? Verify one by one all structures,
¬? Introducing, according to G??del a meta-rule.
Not having a meta-rule, I would be stuck, if it weren’t for the awareness of disposing, as individual, a partial or limited meta-rule: the possibility of making a choice.
Having the possibility of a choice means that instead of a definite proof, I can propose an hypothesis, verify it, and, if not happy, propose a new and different hypothesis.
With any luck, even if I will never arrive to a definite proof, in meantime I will have proved myself to be real and alive.
Now the question: if all mathematical structure exists, would there also be a structure capable of making a choice like I do myself? In other word produce a partial or limited meta-rule to self-justify it existence? (And therefore partially satisfy the G??del theorem?)
The question is not an easy one, and might be left with many other unsolved theoretical questions, but there is a fact to consider: in this moment our scientific evolution is stuck to a similar problem: how many links can a simple element of nature install with others elements?
Take away the “one to one” possibility, there are only two left:
¬? One to many
¬? One to all
No need to say that the first possibility has been the choice of the traditional science and, at this time, most, if not all, mathematical models respect this choice.
A choice that, due to the fact that it is impossible to give a fixed amount for the “many”, it is not without incongruence, an incongruence that has been “partially” overcome by introducing average and probability calculus.
At this point, as there are still many phenomenon of our reality (most of them related to free choice, evolution and complexity) that are difficult/impossible to be represented with traditional mathematical models, the previous question becomes relevant, could the reason be of having chosen the “many” instead of the “all”?
I certainly realize that the choice of “all” would envisage a reality different from what we are used to, and at the same time it would need adequate mathematical structures (including its’ meta rule) to represent it, but if the proof between “all” and “many “ is only a matter of choice, and providing hypothesis to support it, be sure that the “all” would be my choice.
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Readers of crime fiction (or other interest-driven novels) know that a dazzling beginning ultimately leads to a disappointing end. Apparently the same happens in The MU. Staring With MUH which should dislodge ERH, it soon adds CUH. Then in VII.G Tegmark candidly admits
"that virtually all historically successful theories of physics violate the CUH, and that it is far from obvious whether a viable computable alternative exists. The main source of CUH violation comes from incorporating the continuum, usually in the form of real or complex numbers". Say goodbye to the CUH and start thinking about the continuum. It was the pride and joy of XIXth century mathematics but in the next century it produced mostly trouble, witness the 'annoying infinities' that Tegmark mentions, and of course it lead, albeit indirectly, to Godel's result.
It's been a long time since the Lowenheim-Skolem theorem told us that everything has a countable model, but nobody took it ontologically, except postmodernists who said that il n'y a pas de hors-texte, and perhaps a few others.
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Is there any theory out there that addresses either of my two main concerns about using mathematics to describe physical reality? Or are my concerns unfounded?
1) All physical theories seem to utilize the mathematical constant pi. Pi is the ratio of the circumference of a circle to its diameter, in Euclidean space. In Euclidean space it is constant and incommensurable. Since physical space is non-Euclidean, the ratio of the circumference of a circle to its diameter will depend on the local curvature of space-time. Is pi included in physical theories because there is some need to include a factor of roughly 3.14159 in the equations, thus making it a purely mathematical concept? However, if it is the geometric relation and not the factor of 3.14159 that is relevant, does it make sense to include a Euclidean geometric term in a non-Euclidean theory of space-time? If pi was not constant, but determined by General Relativity, then quantum theory would be affected by General Relativity as well?
2) Is there any reason to think that the physical continuum is the same as the mathematical continuum? Arguments for including incommensurable numbers in the mathematical continuum include the fact that the point where the in-circle of a square and the diagonal of the square intersect is incommensurable. So in order for there to be an actual intersection of this line and arc there must be an irrational number. But in physics, where such lines are merely abstractions, does there really have to be such an intersection point in space-time? Would the whole subject be simplified if these very possibly "unreal" points were excluded from the mathematical system used to describe the physical world?
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Especially having discovered quantum entanglement it could be thought that understanding a large psrt of reality is not about an explanation that describes mathematically quantified properties of a cause or its effects.
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