Comments on: mathematics and universal truth http://www.evolution-nextstep.com/archives/2508 take the next step Fri, 05 Dec 2008 13:41:20 +0000 http://wordpress.org/?v=2.6.5 By: j.d. http://www.evolution-nextstep.com/archives/2508#comment-3550 j.d. Wed, 15 Feb 2006 16:43:30 +0000 http://www.evolution-nextstep.com/archives/2006/02/14/2508/#comment-3550 You wrote: <blockquote>Even if numbers exist ab initio, it need not imply that math as we conceive it must all be universally true. Even if it is a set of true statements about universal truths, that doesn’t mean that it’s the only way to discuss those truths.</blockquote> That is, of course, true in a sense. On some level mathematics is a creation of pure reason. On another, the question is what does a given mathematical system (i.e., the axioms it rests on) <em>express</em>? Does it express some fundamental property of the universe (a "universal truth"), or is it a Man-made word game? I believe it is the former. I don't think that <em>all</em> of mathematics expresses universal truth. (I now realize why you might think I did; I'll revise it later tonight.) Goedel's theorems seem to say that at least <em>some part</em> of it does; in fact, that's what he himself believed. The truth of any formal system rests on the axioms on which it's built, and Goedel's result seems to say to me that at least <em>some part</em> (not necessarily the totality) of mathematics rests upon its axioms, at least one of which ought to be one of Goedel's true but unprovable statements. What I'm saying here is that there is <em>some part</em> of mathematics which expresses a "universal truth", i.e., a fundamental property of the universe. You wrote:

Even if numbers exist ab initio, it need not imply that math as we conceive it must all be universally true. Even if it is a set of true statements about universal truths, that doesn’t mean that it’s the only way to discuss those truths.

That is, of course, true in a sense. On some level mathematics is a creation of pure reason.

On another, the question is what does a given mathematical system (i.e., the axioms it rests on) express? Does it express some fundamental property of the universe (a “universal truth”), or is it a Man-made word game? I believe it is the former.

I don’t think that all of mathematics expresses universal truth. (I now realize why you might think I did; I’ll revise it later tonight.) Goedel’s theorems seem to say that at least some part of it does; in fact, that’s what he himself believed.

The truth of any formal system rests on the axioms on which it’s built, and Goedel’s result seems to say to me that at least some part (not necessarily the totality) of mathematics rests upon its axioms, at least one of which ought to be one of Goedel’s true but unprovable statements. What I’m saying here is that there is some part of mathematics which expresses a “universal truth”, i.e., a fundamental property of the universe.

]]> By: Josh http://www.evolution-nextstep.com/archives/2508#comment-3544 Josh Tue, 14 Feb 2006 19:55:03 +0000 http://www.evolution-nextstep.com/archives/2006/02/14/2508/#comment-3544 The fact that babies (or monkeys or apes) have a concept of "2" or "3" doesn't necessarily imply that math, or even those concepts, are truly universal. It could be that these are concepts which the primates evolved and which we share at some deep developmental level. I think I once saw something about crows and counting, and that would be a better test. If species whose intelligence can be seen as independently evolved show similar mathematical abstractions, that suggests something about the nature of the concepts (corvids are super-smart, but not all birds are, nor are various mammals and reptiles which branch off in between primates and corvids). Of course, there's a huge gulf between counting and the rules of logic which govern math. It's always amazed me what you can derive given 0, 1, logic, and addition (maybe addition comes naturally with 0 and 1, I can't remember now). Even if numbers exist ab initio, it need not imply that math as we conceive it must all be universally true. Even if it is a set of true statements about universal truths, that doesn't mean that it's the only way to discuss those truths. Natural scientists get into this debate in deciding whether natural laws exist, and if so, whether they are unique, or whether one could express the concept of thermodynamic laws in only two laws, or segment them into three parts differently than we now study them. Fermat's last theorem probably represents some deeper truth about numbers, and we find particular arrangements interesting, so we express that truth in terms of a^n+b^n=c^n, but that need not be unique, and the expression of the truth may not be universal, even if the truth is. It's an interesting topic. The fact that babies (or monkeys or apes) have a concept of “2″ or “3″ doesn’t necessarily imply that math, or even those concepts, are truly universal. It could be that these are concepts which the primates evolved and which we share at some deep developmental level. I think I once saw something about crows and counting, and that would be a better test. If species whose intelligence can be seen as independently evolved show similar mathematical abstractions, that suggests something about the nature of the concepts (corvids are super-smart, but not all birds are, nor are various mammals and reptiles which branch off in between primates and corvids).

Of course, there’s a huge gulf between counting and the rules of logic which govern math. It’s always amazed me what you can derive given 0, 1, logic, and addition (maybe addition comes naturally with 0 and 1, I can’t remember now).

Even if numbers exist ab initio, it need not imply that math as we conceive it must all be universally true. Even if it is a set of true statements about universal truths, that doesn’t mean that it’s the only way to discuss those truths. Natural scientists get into this debate in deciding whether natural laws exist, and if so, whether they are unique, or whether one could express the concept of thermodynamic laws in only two laws, or segment them into three parts differently than we now study them. Fermat’s last theorem probably represents some deeper truth about numbers, and we find particular arrangements interesting, so we express that truth in terms of a^n+b^n=c^n, but that need not be unique, and the expression of the truth may not be universal, even if the truth is.

It’s an interesting topic.

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