Comment by btilly

Comment by btilly 5 days ago

My understanding of how Hilbert meant it is summed up in this quote from him: "Mathematics is a game played according to certain simple rules with meaningless marks on paper." I think that in part because I read Constance Reid's excellent biography Hilbert! It traces in some detail his thinking over his life, and how he came to formalism. His thinking about the nature of existence was particularly interesting.

If you think that he meant something else, please find somewhere where he said something that didn't boil down to that.

As for what most people think about the philosophical implications, nobody should be expected to have any meaningful philosophical opinions about topics that they have not yet tried to think about. I know that I didn't.

After you've thought about it, you may well have a dramatically different opinion than I do. For example Gödel thought that mathematical existence was real, since mathematics exists in God's mind. This idea made it important to him to decide which set of axioms was right, where right means, "These are the axioms that God must have settled on, and that therefore exist in His mind." This lead to such ironies as the fact that after proving that the consistency of ZF implies the consistency of ZFC, he then concluded that that the construction was so unnatural that Choice couldn't be one of God's axioms!

I don't agree with Gödel. For a start, I don't believe that God exists. And after I thought about it more, I realized that what I want existence to mean, isn't what mathematicians mean when they say "exists". I'm willing to use language in their way when I'm talking to them. But I'm always aware that it doesn't mean what I want it to mean.

ConspiracyFact 2 days ago

I thought it was generally understood that Hilbert didn’t literally believe that. Do you seriously believe that he believed it?

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btilly 2 days ago

What I said about Hilbert's belief was a quote by him. So of course I believe it.
His full views were more complicated, and are already discussed elsewhere in this discussion.
My statement about Gödel is not an exact quote, but fits what I've heard from every source on the topic. He was a very deeply religious genius.

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pron 5 days ago

I can't locate my Heijenoort right now, but here's a description from the Stanford Encyclopedia of Philosophy [1] (which points to Heijenoort):

The analogy with physics is striking... In the second half of the 1920s, Hilbert replaced the consistency program with a conservativity program: Formalized mathematics was to be considered by analogy with theoretical physics. The ultimate justification for the theoretical part lies in its conservativity over “real” mathematics: whenever theoretical, “ideal” mathematics proves a “real” proposition, that proposition is also intuitively true. This justifies the use of transfinite mathematics: it is not only internally consistent, but it proves only true intuitive propositions (and indeed all, since a formalization of intuitive mathematics is part of the formalization of all mathematics).

In 1926, Hilbert introduced a distinction between real and ideal formulas. This distinction was not present in 1922b and only hinted at in 1923. In the latter, Hilbert presents first a formal system of quantifier-free number theory about which he says that “The provable formulae we acquire in this way all have the character of the finite”

In other words, Hilbert does not require assigning any sense of truth beyond the symbolic one to those mathematical statements that do not correspond to physical reality, but those statements that can correspond to physical reality (i.e. the "real formulas") must do so, and those "real formulas" are meaningfully true beyond the symbols.

The earlier formalism (mathematics is just symbols) could no longer be justified after Gödel, as consistency was its main justification.

If anything, I think it's constructivism that suffers from a philosophical issue in requiring meaning that isn't physically realisable -- unlike ultrafinitism, for example. Personally, I find both Hilbert's formalism and ultrafinitism more philosophically satisfying than constructivism, as they're both rooted in physical reality, whereas constructivism is based on "computation in principle" (but not in practice!).

> As for what most people think about the philosophical implications, nobody should be expected to have any meaningful philosophical opinions about topics that they have not yet tried to think about

I mean people who have thought about it.

[1]: https://plato.stanford.edu/entries/hilbert-program/

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btilly 5 days ago

I was responding to this statement of yours, "I don't think that humans philosophically reject non-constructive results."
Some of the humans who have thought about it do reject them. Some of the humans who have thought about it don't reject them.
Most humans, including most mathematicians, have never truly thought about it.

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- pron 5 days ago
  
  > Some of the humans who have thought about it do reject them.
  I think they reject them only if they misrepresent Hilbert's formalism, because formalism does not assign them any meaning of truth beyond the symbolic. It makes no statement that could be rejected: a mathematical theorem that proves a set "exists" does not necessarily make any claim about its "actual" existence (unlike, say, Platonism). You asked in what sense does such a set exist, and Hilbert would say, great question, which is why I don't claim there necessarily is any such sense.
  What those who reject Hilbert's formalism reject is the validity of a system of mathematics where only some but not all propositions are "externally" meaningful, but such a rejection, I think, can only be on aesthetic grounds, because, again, for Hilbert, all "valid" foundations must agree with physical reality when it comes to statements that could be assigned physical meaning. If ZFC led to any result that doesn't agree with physical reality, Hilbert would reject it, too. But it hasn't yet.
  
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  btilly 5 days ago
  
  I believe that you are misrepresenting Hilbert here.
  If ZFC led to a result that doesn't agree with physical reality, Hilbert would not reject that result. Instead, at worst, he would simply move it from the category of being a real formula, to being an ideal formula. For example, Euclid's geometry doesn't agree with physical reality. Therefore it is an ideal formula, not a real formula. And yet we do not reject this geometry from mathematics.
  But the distinction between real and ideal is a question for physics. It is not a question that mathematicians need worry about. The questions that mathematicians need worry about are entirely those which are internal to the formal game.
  
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