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Previous Contests
Questioning the Foundations
Which of Our Basic Physical Assumptions Are Wrong?
May 24 - August 31, 2012
Contest Partners: The Peter and Patricia Gruber Foundation, SubMeta, and Scientific American
read/discuss • winners
Is Reality Digital or Analog?
November 2010 - February 2011
Contest Partners: The Peter and Patricia Gruber Foundation and Scientific American
read/discuss • winners
What's Ultimately Possible in Physics?
May - October 2009
Contest Partners: Astrid and Bruce McWilliams
read/discuss • winners
The Nature of Time
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FQXi FORUM
May 24, 2013
CATEGORY:
FQXi Essay Contest - Spring, 2012
[back]
TOPIC: Of Mathematics and Radical Change: Alain Badiou’s Set-Theoretical Ontology by Glenn Gomes [refresh]
TOPIC: Of Mathematics and Radical Change: Alain Badiou’s Set-Theoretical Ontology by Glenn Gomes [refresh]
Essay Abstract
This paper endeavors to describe an ability, right in the Being of mathematics, to sustain a dogmatic set of scientific theories along with the very mechanism for their eventual questioning and upheaval by committed individuals. Mathematics is the indisputable foundation upon which our thought and discourse of the physical universe stands. But what if this foundation is inherently cracked? And what if our well-intentioned attempts to create complete and consistent theories of the world merely act to cover up and repress these congenital cracks, only to have them manifest in the form of anomalies that can rupture our closely-held scientific beliefs and worldviews if we choose to interrogate them? This paper brings these possibilities to light by reviewing the development of the axiomatic set-theory that grounds our modern mathematics, and the resulting opportunity for scientific revolution and theory change that it allows for. Our project proceeds through the lens of philosopher Alain Badiou’s work on a set-theoretical ontology, and his attempt to use the discourse of set theory to describe the very mechanism of revolution within such afar fields as mathematics, politics, and art. We conclude with a case study of the Einsteinian revolution from within the Newtonian world of physical theories, through crucial innovations in non-Euclidian geometry, as an example of how unquestioned and dogmatically reinforced cracks in a quasi-complete scientific system can bear the fruits of radical change and transformation, and how approaches to set-theory provide the ontological description of these important phenomena.
Author Bio
Glenn Gomes is a graduate of Columbia University with a degree in Biomedical Engineering and minors in Philosophy and Religion. He has worked at UC Davis Medical Center as a research specialist, researching schizophrenia and bipolar disorder through EEG and fMRI imaging. He is currently pursuing both his MD and MPH at Tufts University School of Medicine.
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This paper endeavors to describe an ability, right in the Being of mathematics, to sustain a dogmatic set of scientific theories along with the very mechanism for their eventual questioning and upheaval by committed individuals. Mathematics is the indisputable foundation upon which our thought and discourse of the physical universe stands. But what if this foundation is inherently cracked? And what if our well-intentioned attempts to create complete and consistent theories of the world merely act to cover up and repress these congenital cracks, only to have them manifest in the form of anomalies that can rupture our closely-held scientific beliefs and worldviews if we choose to interrogate them? This paper brings these possibilities to light by reviewing the development of the axiomatic set-theory that grounds our modern mathematics, and the resulting opportunity for scientific revolution and theory change that it allows for. Our project proceeds through the lens of philosopher Alain Badiou’s work on a set-theoretical ontology, and his attempt to use the discourse of set theory to describe the very mechanism of revolution within such afar fields as mathematics, politics, and art. We conclude with a case study of the Einsteinian revolution from within the Newtonian world of physical theories, through crucial innovations in non-Euclidian geometry, as an example of how unquestioned and dogmatically reinforced cracks in a quasi-complete scientific system can bear the fruits of radical change and transformation, and how approaches to set-theory provide the ontological description of these important phenomena.
Author Bio
Glenn Gomes is a graduate of Columbia University with a degree in Biomedical Engineering and minors in Philosophy and Religion. He has worked at UC Davis Medical Center as a research specialist, researching schizophrenia and bipolar disorder through EEG and fMRI imaging. He is currently pursuing both his MD and MPH at Tufts University School of Medicine.
Download Essay PDF File
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Hi Glenn Gomes,
I looked in vain for the radical change promised in the title. Except for who commented on set theory, Riemannian geometry and the like in which fashion, I did not find anything I was not yet aware of. Well, some quoted sentences correspond to my own attitude. However, where are concrete applications?
Some radical comments in my essay will definitely shock you. Please be nonetheless fair, try to understand and possibly refute my factual arguments. If you are an independent thinker I may even hope you will overcome what you learned.
You quoted someone unknown to me who did not mention Euclid: "The axiomatization of set theory as the foundation for mathematics completed the process begun by Descartes and the arithmetization of geometry, namely, the liberation of mathematics from all spatial or sensory intuition."
Descartes resumed ancient geometry which culminated in Euclid's axioms and his anything but intuitive definition of number, I revealed the issue of intuition mainly the other was round than set theorists, and I consider arithmetization a method that does not need illusory intuitively based (set theoretic) rigor. Admittedly there were and still are only a few opponents to set theory.
In my essay I quoted one out of 16 more of less intriguing contributions by Detlef D. Spalt. Perhaps you do not even know his name.
You wrote "... universe of Cantor’s that was properly axiomatized by Ersnt Zermelo and Abaraham Fraenkel ...". Sorry, I see four bewildering details.
Eckard
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I looked in vain for the radical change promised in the title. Except for who commented on set theory, Riemannian geometry and the like in which fashion, I did not find anything I was not yet aware of. Well, some quoted sentences correspond to my own attitude. However, where are concrete applications?
Some radical comments in my essay will definitely shock you. Please be nonetheless fair, try to understand and possibly refute my factual arguments. If you are an independent thinker I may even hope you will overcome what you learned.
You quoted someone unknown to me who did not mention Euclid: "The axiomatization of set theory as the foundation for mathematics completed the process begun by Descartes and the arithmetization of geometry, namely, the liberation of mathematics from all spatial or sensory intuition."
Descartes resumed ancient geometry which culminated in Euclid's axioms and his anything but intuitive definition of number, I revealed the issue of intuition mainly the other was round than set theorists, and I consider arithmetization a method that does not need illusory intuitively based (set theoretic) rigor. Admittedly there were and still are only a few opponents to set theory.
In my essay I quoted one out of 16 more of less intriguing contributions by Detlef D. Spalt. Perhaps you do not even know his name.
You wrote "... universe of Cantor’s that was properly axiomatized by Ersnt Zermelo and Abaraham Fraenkel ...". Sorry, I see four bewildering details.
Eckard
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Hello Eckard,
Thank you for your comments! I'm impressed that none of this was new to you, but I imagine that much of it will be for others. Let me see if I can clarify some of your specific concerns:
1) What is "radical change"? In Badiou's language, radical change would correspond to a complete rupture in the prevailing bodies (sets) and languages (predicates) of a situation through the forcing and incorporation of a generic set. In socio-scientific language, this would correspond to a traditional Kuhnian "paradigm shift" where a new discovery, or change in perspective, in response to theoretical stress may cause a complete break with the current paradigm and create a new paradigm incommensurable to the initial one. (I had wanted to include a section on how I believe Badiou to have, in a way, described a mathematical formalization of Khunian paradigms and paradigm shifts, but there was no space). The key here is a "complete" rupture with the current state of affairs, not due to some mystical intervention, but through the work of committed individuals.
I tried to argue the Einstienian revolution as an example of such a radical change/paradigm shift that may be formally descibed using the set-theoretical ontology outlined in the paper. The beauty of Badiou's philosophy is its being a philosophy of "praxis": a philosophy that claims to tie this mathemtical ideology to actual practice. I will refer you to Badiou's works for how he applies this set-theory (as well as category theory) to actual/historical mathematical, political, and artistic situations. So as not to take up too much space here, I can give more examples myself if you (or others) request them. I personally find these practical applications fascinating.
Additionally, radical change may be contrasted with what Badiou calls mere "modification" or "simple becoming" of a situation; akin to a Khunian "reinforcing" of the dominant paradigm, where a scientific innovation may occur, but still falls under the language of that paradigm.
2) The quote you mention is from Peter Hallward, a philosopher and foremost commentor of Badiou's works. I chose the quote because I felt that it perfectly expressed both the crucial philisophical stakes and mathematical stakes involved in the axiomatization of set theory. I had no intention of mentioning Euclid at that point.
3) I have not heard of Mr. Spalt , but will look him up. I am curently travelling, but will read your essay as soon as I get the chance!
4) While it was Cantor that formulated the basics of set-theory, it was Zermelo and Frankel (amongst others) who axiomatized Cantor's principles. I do not believe this to be a point of contention, unless I am missing something?
Please let me know if you have any more questions, and thank you for the comments!
- Glenn
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Thank you for your comments! I'm impressed that none of this was new to you, but I imagine that much of it will be for others. Let me see if I can clarify some of your specific concerns:
1) What is "radical change"? In Badiou's language, radical change would correspond to a complete rupture in the prevailing bodies (sets) and languages (predicates) of a situation through the forcing and incorporation of a generic set. In socio-scientific language, this would correspond to a traditional Kuhnian "paradigm shift" where a new discovery, or change in perspective, in response to theoretical stress may cause a complete break with the current paradigm and create a new paradigm incommensurable to the initial one. (I had wanted to include a section on how I believe Badiou to have, in a way, described a mathematical formalization of Khunian paradigms and paradigm shifts, but there was no space). The key here is a "complete" rupture with the current state of affairs, not due to some mystical intervention, but through the work of committed individuals.
I tried to argue the Einstienian revolution as an example of such a radical change/paradigm shift that may be formally descibed using the set-theoretical ontology outlined in the paper. The beauty of Badiou's philosophy is its being a philosophy of "praxis": a philosophy that claims to tie this mathemtical ideology to actual practice. I will refer you to Badiou's works for how he applies this set-theory (as well as category theory) to actual/historical mathematical, political, and artistic situations. So as not to take up too much space here, I can give more examples myself if you (or others) request them. I personally find these practical applications fascinating.
Additionally, radical change may be contrasted with what Badiou calls mere "modification" or "simple becoming" of a situation; akin to a Khunian "reinforcing" of the dominant paradigm, where a scientific innovation may occur, but still falls under the language of that paradigm.
2) The quote you mention is from Peter Hallward, a philosopher and foremost commentor of Badiou's works. I chose the quote because I felt that it perfectly expressed both the crucial philisophical stakes and mathematical stakes involved in the axiomatization of set theory. I had no intention of mentioning Euclid at that point.
3) I have not heard of Mr. Spalt , but will look him up. I am curently travelling, but will read your essay as soon as I get the chance!
4) While it was Cantor that formulated the basics of set-theory, it was Zermelo and Frankel (amongst others) who axiomatized Cantor's principles. I do not believe this to be a point of contention, unless I am missing something?
Please let me know if you have any more questions, and thank you for the comments!
- Glenn
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Hi Glen,
If I was young and educated as were you and many others, I too would definitely have no chance to question that Kuhnian (you wrote twice Khunian) paradigm shifts towards set theory and Einstein's theory of relativity are necessary steps in the correct direction.
Please do not take it amiss if I suspect your knowledge of German language is at best limited. And since you wrote "Cantor which" instead of "Cantor who" and your name is Gomez, I guess your background could be Spanish. Profound command of German is a must for anybody who intends to read Spalt because his books were not yet translated into English. It is also highly recommendable reading the original papers by Georg Cantor, Zermelo, Hilbert, Fraenkel, Einstein (you wrote Einstien), Hermann Weyl in German.
Is Badiou's philosophy "that claims to tie this mathemtical ideology to actual practice" really practical? The topic of this contest is: Which of Our Basic Physical Assumptions is Wrong?".
Do you know that Fraenkel (you wrote Frankel) in [7] of my essay could not deny that Cantor's definition of an infinite set was untenable? Zermelo in 1904 and 1908 seemingly rescued Cantor's paradise by fabricating the axiom of choice based on the exhaustion of the inexhaustible. Cantor himself who had driven Kronecker in desperation got insane and died in a madhouse.
Eckard
post approved
If I was young and educated as were you and many others, I too would definitely have no chance to question that Kuhnian (you wrote twice Khunian) paradigm shifts towards set theory and Einstein's theory of relativity are necessary steps in the correct direction.
Please do not take it amiss if I suspect your knowledge of German language is at best limited. And since you wrote "Cantor which" instead of "Cantor who" and your name is Gomez, I guess your background could be Spanish. Profound command of German is a must for anybody who intends to read Spalt because his books were not yet translated into English. It is also highly recommendable reading the original papers by Georg Cantor, Zermelo, Hilbert, Fraenkel, Einstein (you wrote Einstien), Hermann Weyl in German.
Is Badiou's philosophy "that claims to tie this mathemtical ideology to actual practice" really practical? The topic of this contest is: Which of Our Basic Physical Assumptions is Wrong?".
Do you know that Fraenkel (you wrote Frankel) in [7] of my essay could not deny that Cantor's definition of an infinite set was untenable? Zermelo in 1904 and 1908 seemingly rescued Cantor's paradise by fabricating the axiom of choice based on the exhaustion of the inexhaustible. Cantor himself who had driven Kronecker in desperation got insane and died in a madhouse.
Eckard
post approved
Well now, if this isn't precisely the point.
We are loathe to question the characteristics of our own shadow -- let alone even enumerate them in the first place -- because we're just too busy forcing the rest of the universe to conform to our strict beliefs.
Bravo!
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We are loathe to question the characteristics of our own shadow -- let alone even enumerate them in the first place -- because we're just too busy forcing the rest of the universe to conform to our strict beliefs.
Bravo!
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Dear Edward and Glenn,
You are invited see the result of radical thinking in mathematics in the essay on Five Dimensions of universe .
You will see,w e question validity of arithmatic numbers to express quantity and reach a subset of transfinite numbers that can express the quantity.
Look forward to your evaluation on departure from conventional approach to mathematics.
thanks & Regards,
Vijay Gupta
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You are invited see the result of radical thinking in mathematics in the essay on Five Dimensions of universe .
You will see,w e question validity of arithmatic numbers to express quantity and reach a subset of transfinite numbers that can express the quantity.
Look forward to your evaluation on departure from conventional approach to mathematics.
thanks & Regards,
Vijay Gupta
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Dear Glenn Gomes,
Law of sines for tetrahedral, may be applicable to describe the string dynamics of the infinite universe for quantizing in finite conditions, in that its consistency in continuum is expressional.
With best wishes,
Jayakar
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Law of sines for tetrahedral, may be applicable to describe the string dynamics of the infinite universe for quantizing in finite conditions, in that its consistency in continuum is expressional.
With best wishes,
Jayakar
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Hello Jayakar,
Thank you for directing me to your essay.
In terms of mine, I would like to point out that I address Mathematics, not in its algebraic, geometric, or calculus based characteristics, but rather in its foundation as a set-theoretical construct.
- Glenn
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Thank you for directing me to your essay.
In terms of mine, I would like to point out that I address Mathematics, not in its algebraic, geometric, or calculus based characteristics, but rather in its foundation as a set-theoretical construct.
- Glenn
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Hello Glenn,
Very interesting and unusual essay for the contest. It makes us look for new ways to solve old problems in the foundations of physics and mathematics. I constantly read and ponder in this regard the paper by Zenkin «Scientific Counter-Revolution in Mathematics» and contemplate his idea: "the truth should be drawn and should be presented to" an unlimited circle "of spectators".
Sincerely, Vladimir Rogozhin
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Very interesting and unusual essay for the contest. It makes us look for new ways to solve old problems in the foundations of physics and mathematics. I constantly read and ponder in this regard the paper by Zenkin «Scientific Counter-Revolution in Mathematics» and contemplate his idea: "the truth should be drawn and should be presented to" an unlimited circle "of spectators".
Sincerely, Vladimir Rogozhin
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Vladimir,
Thank you for looking at my paper! I shall have to read this paper by Zenkin. That is a very pertinent quote with regard to our modern mathematics.
Regards,
Glenn
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Thank you for looking at my paper! I shall have to read this paper by Zenkin. That is a very pertinent quote with regard to our modern mathematics.
Regards,
Glenn
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A very good essay. In my opinion, the essay fits the topic perfectly.
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Thank you very much for the kind words and reading my essay! I was having a hard time while writing the paper to make sure it focused on the essay topic, even though I felt that it was very relavent. It's nice to see people agreeing that it fits the topic.
Regards,
Glenn
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Regards,
Glenn
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Glenn
I noted your attitude to non- euclidean geometry
In the last essay competition my essay
http://www.fqxi.org/community/forum/topic/946
dedicated ANALOGY BETWEEN PHYSICS AND MATH.
Can you comment?
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I noted your attitude to non- euclidean geometry
In the last essay competition my essay
http://www.fqxi.org/community/forum/topic/946
dedicated ANALOGY BETWEEN PHYSICS AND MATH.
Can you comment?
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Dear Glenn Gomes
I think so :
Mathematics is just a tool to do the job is directed by the thinking of intellectual.
It only accepted to manifest when a result of the thinking of intellectual was accepted.
Mathematics is a great tool, the errors can not be caused by a tool.
In my opinion: we be must to fix for people who have used the tool.
How do you think ?
Kind Regards !
Hải.Caohoàng of THE INCORRECT ASSUMPTIONS AND A CORRECT THEORY
August 23, 2012 - 11:51 GMT on this essay contest.
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I think so :
Mathematics is just a tool to do the job is directed by the thinking of intellectual.
It only accepted to manifest when a result of the thinking of intellectual was accepted.
Mathematics is a great tool, the errors can not be caused by a tool.
In my opinion: we be must to fix for people who have used the tool.
How do you think ?
Kind Regards !
Hải.Caohoàng of THE INCORRECT ASSUMPTIONS AND A CORRECT THEORY
August 23, 2012 - 11:51 GMT on this essay contest.
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If you do not understand why your rating dropped down. As I found ratings in the contest are calculated in the next way. Suppose your rating is 
and 
was the quantity of people which gave you ratings. Then you have 
of points. After it anyone give you 
of points so you have 
of points and 
is the common quantity of the people which gave you ratings. At the same time you will have 
of points. From here, if you want to be R2 > R1 there must be: 
or  / (N_1+1) >S_1/ N_1)
or 
In other words if you want to increase rating of anyone you must give him more points then the participant`s rating was at the moment you rated him. From here it is seen that in the contest are special rules for ratings. And from here there are misunderstanding of some participants what is happened with their ratings. Moreover since community ratings are hided some participants do not sure how increase ratings of others and gives them maximum 10 points. But in the case the scale from 1 to 10 of points do not work, and some essays are overestimated and some essays are drop down. In my opinion it is a bad problem with this Contest rating process. I hope the FQXI community will change the rating process.
Sergey Fedosin
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Sergey Fedosin
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Dear Glenn,
Thanks for your important contribution! After reading your essay, I looked at your bio expecting to find you an eminent mathematician in one of the world's great research institutions and was surprised to see that you are on another path entirely! Regarding your essay, some thoughts come to mind:
1. Nice discussion of Cantor, set theory, and ordinals. In my own efforts to understand quantum gravity, I have found it necessary to deal with something similar, though somewhat more general (which I call a "semiordinal;" the object in footnote 14 of my essay is an example of one). All this arises from the most naive physical ideas of cause and effect and reasonable local conditions!
2. I've run into the continuum hypothesis in thinking about physics too! It's funny how most people regard these things (and other "purely mathematical issues" like undecidability and the uncountable axiom of choice) as physically irrelevant when they are anything but.
3. Regarding ZFC, Russell's paradox, etc... as you know the ordinals form a proper class, not a set. Hence, again these issues become physically relevant from the most primitive physical considerations.
4. Regarding Godel's work and model theory... Torsten Asselmeyer-Maluga, Jerzy Krol, and Michael Goodband all have excellent essays here that would interest you.
5. Regarding geometry, as an aspiring algebraic geometer, I strongly suspect that the geometry of smooth manifolds may ultimately prove "too good to be true" in regard to spacetime structure. But it's been fantastically useful and astonishingly accurate so far.
6. Excellent endnotes!
Thanks again for the interesting read! Your work rates very highly in my opinion. Take care,
Ben Dribus
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Thanks for your important contribution! After reading your essay, I looked at your bio expecting to find you an eminent mathematician in one of the world's great research institutions and was surprised to see that you are on another path entirely! Regarding your essay, some thoughts come to mind:
1. Nice discussion of Cantor, set theory, and ordinals. In my own efforts to understand quantum gravity, I have found it necessary to deal with something similar, though somewhat more general (which I call a "semiordinal;" the object in footnote 14 of my essay is an example of one). All this arises from the most naive physical ideas of cause and effect and reasonable local conditions!
2. I've run into the continuum hypothesis in thinking about physics too! It's funny how most people regard these things (and other "purely mathematical issues" like undecidability and the uncountable axiom of choice) as physically irrelevant when they are anything but.
3. Regarding ZFC, Russell's paradox, etc... as you know the ordinals form a proper class, not a set. Hence, again these issues become physically relevant from the most primitive physical considerations.
4. Regarding Godel's work and model theory... Torsten Asselmeyer-Maluga, Jerzy Krol, and Michael Goodband all have excellent essays here that would interest you.
5. Regarding geometry, as an aspiring algebraic geometer, I strongly suspect that the geometry of smooth manifolds may ultimately prove "too good to be true" in regard to spacetime structure. But it's been fantastically useful and astonishingly accurate so far.
6. Excellent endnotes!
Thanks again for the interesting read! Your work rates very highly in my opinion. Take care,
Ben Dribus
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Hello Ben,
Please forgive me for the lateness of my response! Having just started medical school, it has been both exciting and overwhelming. I must say though, reading your comments absolutely made my day, and I very much appreciate them and take them to heart.
I actually had a very hard time deciding whether to go into graduate school for philosophy of science/mathematics, which is my passion. However, I also saw myself doing good community work as a physician. This paper was my attempt to contribute an introduction on Alain Badiou's philosophical work to mathematicians and scientists, before leaving these topics for medicine; at least until the future. I was a bit hesitant to submit it, unsure of how relevant it was to the contest topic, and in comparison to the high level physics papers also submitted by experts in those fields. But it is comments like yours that make it worth it to me, regardless of the contest outcome.
I'm happy that you enjoyed the paper. I will try my best to read the papers you have recommended, and particularly your submission. You will forgive me again if I am delayed with this.
This tension between how we conceptualize mathematics and their relation to physical entities is, I agree, absolutely fascinating. I must read more on the attempts to bring algebra and geometry together, and I wish you all the best in your endeavors. Thank you again for reading my paper, and for your generous comments.
All the best,
Glenn
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Please forgive me for the lateness of my response! Having just started medical school, it has been both exciting and overwhelming. I must say though, reading your comments absolutely made my day, and I very much appreciate them and take them to heart.
I actually had a very hard time deciding whether to go into graduate school for philosophy of science/mathematics, which is my passion. However, I also saw myself doing good community work as a physician. This paper was my attempt to contribute an introduction on Alain Badiou's philosophical work to mathematicians and scientists, before leaving these topics for medicine; at least until the future. I was a bit hesitant to submit it, unsure of how relevant it was to the contest topic, and in comparison to the high level physics papers also submitted by experts in those fields. But it is comments like yours that make it worth it to me, regardless of the contest outcome.
I'm happy that you enjoyed the paper. I will try my best to read the papers you have recommended, and particularly your submission. You will forgive me again if I am delayed with this.
This tension between how we conceptualize mathematics and their relation to physical entities is, I agree, absolutely fascinating. I must read more on the attempts to bring algebra and geometry together, and I wish you all the best in your endeavors. Thank you again for reading my paper, and for your generous comments.
All the best,
Glenn
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