Comment by pron

Comment by 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/

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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  pron 5 days ago
  
  > Instead, at worst, he would simply move it from the category of being a real formula, to being an ideal formula.
  No. No result, either ideal or real, may contradict reality (it's just that since infinitary statements do not describe reality, they obviously cannot contradict it). You can think about it like this: According to Hilbert, a valid mathematical foundation is any logical theory that is a conservative extension of reality. ZFC, constructive foundations, and ultrafinitist foundations all satisfy that, so they would all be valid foundations according to that principle.
  > For example, Euclid's geometry doesn't agree with physical reality.
  It may not describe physical reality, but it doesn't contradict it.
  > 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.
  Not only does that disagree with Hilbert's formalism, it also disagrees with constructivism. The question of the philosophy of mathematics is precisely what, if anything, does mathematics describe beyond symbols.
  
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