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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 a

by btilly 1y 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.

pron 1y 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.
- btilly 1y 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.
  - pron 1y 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.
    - btilly 1y ago
      I have absolutely no idea how physical reality can contradict a statement about a formal symbol game. I am also convinced that what you are saying is not what Hilbert actually meant. But he died before I was born, so I can't ask him. Besides, I don't speak German.
      - tsimionescu 1y ago
        I think what the poster above means is that any formal system in which, say, 1 + 1 = 75, even if internally consistent as a set of symbols, would not be considered a valid logical foundation for mathematics by Hilbert, because it directly contradicts physical reality. Of course, this is under the assumption that 1 and + and 75 are meant to have their usual meanings, not that we've redefined the + operator to mean "the 73rd successor", nor that we're operating in some exotic numerical base where 75 is just a different set of symbols to describe the number 2.
      - pron 1y ago

by btilly 1y ago

pron 1y 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.

btilly 1y 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.
- pron 1y 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.
  - btilly 1y ago
    I have absolutely no idea how physical reality can contradict a statement about a formal symbol game. I am also convinced that what you are saying is not what Hilbert actually meant. But he died before I was born, so I can't ask him. Besides, I don't speak German.
    - tsimionescu 1y ago
      I think what the poster above means is that any formal system in which, say, 1 + 1 = 75, even if internally consistent as a set of symbols, would not be considered a valid logical foundation for mathematics by Hilbert, because it directly contradicts physical reality. Of course, this is under the assumption that 1 and + and 75 are meant to have their usual meanings, not that we've redefined the + operator to mean "the 73rd successor", nor that we're operating in some exotic numerical base where 75 is just a different set of symbols to describe the number 2.
    - pron 1y ago