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Mathematical Platonism

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Posted by Paul R. Martin on November 18, 2003 17:26:56 UTC

Hi Kyle,

It is delightfully refreshing to be reading your insightful and witty comments again.

"So the gazillion dollar question regards the statement "Either P or -P" --> is the statement itself true or false? Is it true or false that P is either true or false?

Most people (I think) would say 'true' -- that is, 'Either P or -P' must be true. I did an informal survey last year and Paul Martin and Yanniru both effectively said 'true' i.e. both took mathematical Platonic positions that consider mathematical objects to exist 'out there' independent of human consciousness -- (that is, assuming that I understood them correctly...) and I counted you as a 'true' as well, Alan, since you once stated "Physics = Math + Consciousness" -- which I take to be a Platonic statement regarding mathematics."


I agree with the price you put on that question. It's worth at least a gazillion dollars. I think the question and its answer holds a major key to the furthering of our understanding of the universe. You are correct in characterizing my Platonic stance that "mathematical objects ... exist 'out there' independent of human consciousness". I also agree with Alan's formula that "Physics = Math + Consciousness".

But I don't think you understood me quite correctly. My position is that mathematical objects do indeed exist in this way, but I also have a strong conviction that none of those objects is infinite in any way.

In particular, I don't believe that an infinite decimal expansion of pi exists anywhere or ever will. Furthermore, I believe that the eventual abandonment of the ridiculous (IMHO) notion of infinity will allow for a great leap forward in human understanding. In the context of religion, I believe that the sooner we give up on the notion that God is infinite (and while we're at it, throw out "perfect", "omnipotent", etc.) the more effectively and truthfully we may be able to approach God. In the context of mathematics, I have explained in many earlier posts why I think a more useful and consistent mathematics would result if we denied the Axiom of Choice, and thus disallowed the notion of infinity in mathematics. Physics already eschews the notion of infinity and takes great pains to keep it from appearing in any of their theories.

Let me summarize my position on the question of infinity. If you allow it in mathematics, then you assume, in particular, that there are an infinite number of integers. Cantor first showed the logical consequences of infinite sets and immediately discovered paradoxes (i.e. nonsense) as a result. Russell and others tried various ways of sweeping the resulting problems under the rug. Goedel finally showed, with his famous theorem, the general consequence that any mathematical system which contains the notion of infinity is either inconsistent (i.e. contains nonsense) or it is incomplete, and you can't tell which. IMHO this should be sufficient cause to abandon the notion by denying the AC. But, for the past 100 years, mathematicians have seen fit to do otherwise.

In my view, the philosophical implications are that 1) God is finite and has created a finite world, and, 2) the consistencies we see in the world are the result of the consistent (i.e. finite) mathematics which God worked out and placed into Platoland far in advance of the Big Bang.

But, I digress. Back to your interesting gazillion dollar question. We could talk about P, but I think it is unduly complicated by bringing in pi and the dubious question of infinity. Instead, let's talk about the question (if I remember right) which you posed in your survey. (If I remember wrong, let's talk about this question anyway.)

Let me paraphrase your statement of your question about P: The question regards the statement "This statement is false". Let me call "This statement is false", Q. So the new question at hand is, "Is it true or false that Q is either true or false?"

You wrote that in your earlier survey I had "effectively said 'true'" to this question. That needs to be corrected because that is not my answer. My answer is 'false'. That is, I claim that the statements P and Q are neither true nor false. Instead, they belong in a third category I would call 'nonsense'. P is nonsense, in my view, because it depends on the existence of an infinite decimal expansion of pi, which I say is non-existent, and furthermore, Goedel has proved that if such an expansion did exist, it would necessarily contain inconsistencies if the system were completely developed. Q is nonsense, in my view, because it is self-contradictory: if it is true, then it is false and if it is false, then it is true.

At this point, let me make another digression that I have been itching to write for a long time, but the opportunity has not been quite as ripe as it is now. The question I labeled 'Q' is, of course, very familiar to all of us. But as interesting as it is, I think it is even more revealing to consider the question, "This statement is true." Let me call that one 'B'.

So now we ask, is B either true or false? Is B true? Is B false?

It turns out that the answer to all three questions is 'yes'. So is B useful? Well, if you think about it, there are a great many people who use it as a fundamental basis of their beliefs. It is no coincidence that I chose the symbol 'B' for the statement. What I had in mind was the fact that if you ask a typical Christian for the fundamental basis for their beliefs, they will claim that the Bible provides it. And if you ask them why they think the Bible is so trustworthy, they will quote you passages from the Bible which declare that it is.

Returning, again, from my digression, I need to get to the main reason I am writing this response to you, Kyle. I have missed talking to you since I returned your tapes but I have been thinking about you. In particular, I thought of you when I read a book titled, "Symmetry and the End of Probability", by Thinh Tran (AKA Dangson). I have an extra copy of the book which I plan to mail to you this morning. It is a gift I would like you to keep whether you read it or not.

In this book, Tran convincingly (to me at least, and you know how gullible I am) demonstrates that even though Probability Theory might be good math, it does not match reality. In my view, this is true because (yawn) math has included the non-sensical notion of infinity where reality has not. I think Tran agrees with me on that but I think he sees a difference beyond that.

I am not really competent to represent Tran's views, but for people who might read this and not read Tran's book, let me try anyway. He says that Probability Theory is based on the notion of infinite randomness. That is, that probabilistic behavior is consistent with the results of the limit as the sample size approaches infinity. So, for example, if you were to ask your question, P, about the possible appearance of 100 consecutive zeros in the expansion of pi, you would have to consider an infinite expansion. Tran says that nothing real is infinite and so there is really no infinite randomness. Instead, he introduces the conjugate notion of symmetry to offset the randomness that might be present. Thus, he introduces the quantity, k, or the k-factor, which is sort of a ratio between the amount of randomness present and the amount of symmetry present in the system. Probability Theory, then, becomes a special case of Tran's more inclusive Distribution Theory where k = infinity. He claims that there is no real system in which k = infinity, so classical Probability Theory doesn't apply to any real cases. The easy way to think of the k-factor is that it is the number of standard deviations required to include all the data. If you think about any real system, you never need an infinite number of standard deviations to include all the data so his scheme makes sense to me.

Anyway, I think Tran's proposal is exactly along the lines I have proposed of dispensing with infinity and as a result, making a great leap forward, not only in mathematics and physics, but in philosophy and religion as well.

Now, I have to wrap up a book and get it in the mail.

Warm regards,

Paul

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