Comments on: Mathematics and Reality http://www.airs.com/blog/archives/13 Ian Lance Taylor Sun, 12 Oct 2008 15:20:59 +0000 http://wordpress.org/?v=1.5.2 Comment on Mathematics and Reality by: Deep Mathematics I: “Typical” Theorems « The Twenty Eighth Line http://www.airs.com/blog/archives/13#comment-14703 Mon, 18 Aug 2008 11:18:11 +0000 http://www.airs.com/blog/archives/13#comment-14703 [...] Deep Mathematics I: “Typical” Theorems This is the first of what I hope will be a series of (not chronolgically consecutive) posts discussing what makes “deep” mathematics deep. This is, of course, not a new issue and has been dicussed many times before. I suppose this is more an issue of semantics than philosophy, but this is certainly one place the fact that mathematics is done by human beings rather just being being the “cold and austere” * discipline that it is so often viewed as being becomes apparant. [...] […] Deep Mathematics I: “Typical” Theorems This is the first of what I hope will be a series of (not chronolgically consecutive) posts discussing what makes “deep” mathematics deep. This is, of course, not a new issue and has been dicussed many times before. I suppose this is more an issue of semantics than philosophy, but this is certainly one place the fact that mathematics is done by human beings rather just being being the “cold and austere” * discipline that it is so often viewed as being becomes apparant. […]

]]> Comment on Mathematics and Reality by: Ian Lance Taylor http://www.airs.com/blog/archives/13#comment-10 Mon, 02 Jan 2006 06:58:26 +0000 http://www.airs.com/blog/archives/13#comment-10 To me, an example of a deep connection would be something like Newton's universal theory of gravitation, F = -G * (m1 * m2) / r**2. If that were a complete description of reality, not created out of underlying elements, then reality would be inherently mathematical. As to the objective reality of mathematics, that has to be interpreted in a Platonic sense; Plato argued that the ideal chair must really exist, because otherwise it would not be possible to speak of chairs as a concept at all. Most people today do not believe that, although I have spoken with people who do. I think Penrose was arguing that mathematics really does exist in the Platonic sense, so that mathematicians are not studying an abtract field of thought, they are studying a real object, or exploring a real universe, accessible through thought. As far as I can see, these distinctions can not make any practical difference. It's all just philosophical fun and games. To me, an example of a deep connection would be something like Newton’s universal theory of gravitation, F = -G * (m1 * m2) / r**2. If that were a complete description of reality, not created out of underlying elements, then reality would be inherently mathematical.

As to the objective reality of mathematics, that has to be interpreted in a Platonic sense; Plato argued that the ideal chair must really exist, because otherwise it would not be possible to speak of chairs as a concept at all. Most people today do not believe that, although I have spoken with people who do. I think Penrose was arguing that mathematics really does exist in the Platonic sense, so that mathematicians are not studying an abtract field of thought, they are studying a real object, or exploring a real universe, accessible through thought.

As far as I can see, these distinctions can not make any practical difference. It’s all just philosophical fun and games.

]]> Comment on Mathematics and Reality by: fche http://www.airs.com/blog/archives/13#comment-9 Mon, 02 Jan 2006 01:38:05 +0000 http://www.airs.com/blog/archives/13#comment-9 It makes one wonder what nature a "deep connection" between mathematics and reality could possibly be in order to satisfy the inquiry. I am satisfied to call something as "objective" or "real" if it meets some standard of proof, sort of like in the criminal justice system, or in the "consensus theory of reality". If one seeks a more formal isomorphism between the real and abstract, one would have to define what mappings would be acceptable. It makes one wonder what nature a “deep connection” between mathematics and reality could possibly be in order to satisfy the inquiry. I am satisfied to call something as “objective” or “real” if it meets some standard of proof, sort of like in the criminal justice system, or in the “consensus theory of reality”. If one seeks a more formal isomorphism between the real and abstract, one would have to define what mappings would be acceptable.

]]> Comment on Mathematics and Reality by: Ian Lance Taylor http://www.airs.com/blog/archives/13#comment-7 Sat, 31 Dec 2005 21:00:28 +0000 http://www.airs.com/blog/archives/13#comment-7 I see that I didn't give a good explanation of what I was thinking about. The question is not whether mathematics can describe reality; that is given. The question is whether there is a deep connection between mathematics and reality. Is reality inherently mathematical? When we study mathematics, are we studying something which is somehow tied deeply into reality? Is there something privileged about mathematics, beyond other types of human study? I think Penrose would say yes. I think Kant might say that mathematics is inherent in the way we view reality, though it may not be inherent in reality itself. I'm trying to say that there may not be a deep connection at all, using the physics of gases as an example of how mathematics can successfully describe reality without being deeply connected to it. I see that I didn’t give a good explanation of what I was thinking about. The question is not whether mathematics can describe reality; that is given. The question is whether there is a deep connection between mathematics and reality. Is reality inherently mathematical? When we study mathematics, are we studying something which is somehow tied deeply into reality? Is there something privileged about mathematics, beyond other types of human study? I think Penrose would say yes. I think Kant might say that mathematics is inherent in the way we view reality, though it may not be inherent in reality itself. I’m trying to say that there may not be a deep connection at all, using the physics of gases as an example of how mathematics can successfully describe reality without being deeply connected to it.

]]> Comment on Mathematics and Reality by: fche http://www.airs.com/blog/archives/13#comment-5 Sat, 31 Dec 2005 14:16:32 +0000 http://www.airs.com/blog/archives/13#comment-5 Can you explain why you believe that the existence of an underlying complex system somehow negates the claim that mathematics can describe a phenomenon? That underlying system can be made the target of subsequent mathematical study (as in statistical mechanics for gases). Can you explain why you believe that the existence of an underlying complex system somehow negates the claim that mathematics can describe a phenomenon? That underlying system can be made the target of subsequent mathematical study (as in statistical mechanics for gases).

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