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> Ok, but those pre-Gödel mathematicians were profoundly mistaken about the nature of mathematics. To put it bluntly: they were wrong, so why should I care what

by pron 8y ago

> Ok, but those pre-Gödel mathematicians were profoundly mistaken about the nature of mathematics. To put it bluntly: they were wrong, so why should I care what they thought? How do you know they were wrong? Has there been some breakthrough discovery in philosophy? And if they were, Gödel was wrongest of them all. He was the most extreme Platonist of them all. > If some set of axioms gives rise to interesting mathematical consequences, does that mean these axioms must have philosophical underpinnings? No. Does the 'validity' of the mathematics depend on the philosophical underpinnings of the axioms? No: it counts as mathematics either way. Again, the points you are raising have been discussed by logicians as part of the philosophy of mathematics. As usual, things are not as simple as you may think. > Perhaps there's a gap somewhere in my account of things, but I'm not seeing one so far. Perhaps that's because you've been thinking about this issue for a little bit, while logicians have been grappling with it for over a century. I may have confused you with the reference to physics. It is not the reason for the problem of foundation, but I used it as an example for a situation where something external to mathematics determines our choice of axioms. The Wikipedia entry is also a good overview, but as usual, contains a few (sometimes serious) inaccuracies: https://en.wikipedia.org/wiki/Philosophy_of_mathematics https://en.wikipedia.org/wiki/Philosophy_of_mathematics

pron 8y ago
BTW, if you want just a teaser to what it is that you may be missing, here's one: That mathematics explores the theorems derivable from axioms tells us little. Is mathematics about the formal manipulation of the symbolic expression of those axioms, some (nondeterministic) Turing machine, chewing up and spitting out strings of symbols? Nearly all mathematicians would say not (including Hilbert's Formalism, which some gravely mistake for saying mathematics is just what I described). But if not, this means that the linguistic symbols mean something, and in analytical philosophy, meaning is a sort of a mapping from linguistic terms to something outside the language. But if mathematical formulas mean something -- i.e. refer to something outside them -- what is the nature of that thing, and, more importantly, how do we learn about it?
- MaxBarraclough 8y ago
  > How do you know they were wrong? Has there been some breakthrough discovery in philosophy? Of course there has! Gödel! As you said, Russell believed that all of mathematics could be deduced from laws of logic that are true in the most absolute sense. Gödel showed that this is impossible, and his discovery ended Russell's project overnight. Russell was fundamentally mistaken about the nature of mathematics. No two ways about it. I already linked to the Wikipedia discussion of why Brouwer's idea about the law of the excluded middle, seems no longer to be taken seriously. > Gödel was wrongest of them all. He was the most extreme Platonist of them all. Maybe so, I'm afraid I really don't know his philosophy. > if mathematical formulas mean something -- i.e. refer to something outside them -- what is the nature of that thing, and, more importantly, how do we learn about it? But the mathematics derives the same with or without associating real-world meaning. Seems to me that this is enough to demonstrate that the mapping to something beyond the mathematics, is something that should be treated as quite separate from the mathematics itself (which is 'merely' axioms + consequences). Indeed, we see this in action with fields of mathematics that find real-world applications only years after the discovery of the mathematics itself. Suppose a field of mathematics is found to have several applications. Again, the mathematics is unchanged: the discovered applications are a different beast entirely. Doubtless I'm not the first to take this line of thought. > including Hilbert's Formalism, which some gravely mistake for saying mathematics is just what I described I'm afraid you're ahead of me again here -- that's still on my reading list. Also, perhaps just a nitpick: nondeterministic Turing machines might differ from deterministic ones in terms of space/time complexity (that's an open problem), but they're just as impotent against non-computable problems. Were you thinking of some kind of probabilistic machine?
  - pron 8y ago
    You seem to think that philosophy is about settling issues. It's more about understanding the question better. I also think you're confusing Russell with Hilbert, and it is not accurate that Gödel destroyed Hilbert's project, just changed its nature (or, rather, destroyed the original understanding but not subsequent ones). But Hilbert's idea was actually much more in line with yours -- he also believed that mathematics is about the process rather than some absolute truth, but in order to show that the process can lead anywhere it was important for him to prove consistency. After all, if our mathematics can prove any proposition (including, say, 1=0) then it's rather useless, whether or not it's "valid" in some sense. > I already linked to the Wikipedia discussion of why Brouwer's idea about the law of the excluded middle, seems no longer to be taken seriously. I think that neither Brouwer nor that Wikipedia article says what you think they do. :) There's a whole branch of logic, intuitionistic logic, and mathematics, constructive mathematics, that are largely based on Brouwer's ideas and rejection of LEM. They are particularly popular among some kind of computer scientists, BTW. > But the mathematics derives the same with or without associating real-world meaning. Seems to me that this is enough to demonstrate that the mapping to something beyond the mathematics, is something that should be treated as quite separate from the mathematics itself I'm not talking about real world meaning, nor "beyond mathematics", but beyond the formula (as a string). If the strings refer to anything, what is the nature of the things they refer to? > Also, perhaps just a nitpick: nondeterministic Turing machines might differ from deterministic ones in terms of space/time complexity (that's an open problem), but they're just as impotent against non-computable problems. Were you thinking of some kind of probabilistic machine? No, I simply meant that deduction in a formal system (like first-order logic) is a nondeterministic computation, and to emphasize the perspective that math is just a manipulation of symbols without meaning I referred to Turing machines. That they cannot settle non-computable problems is irrelevant, because neither can mathematicians, and whatever mathematicians can do, so can a TM.

by pron 8y ago

pron 8y ago

BTW, if you want just a teaser to what it is that you may be missing, here's one: That mathematics explores the theorems derivable from axioms tells us little. Is mathematics about the formal manipulation of the symbolic expression of those axioms, some (nondeterministic) Turing machine, chewing up and spitting out strings of symbols? Nearly all mathematicians would say not (including Hilbert's Formalism, which some gravely mistake for saying mathematics is just what I described). But if not, this means that the linguistic symbols mean something, and in analytical philosophy, meaning is a sort of a mapping from linguistic terms to something outside the language. But if mathematical formulas mean something -- i.e. refer to something outside them -- what is the nature of that thing, and, more importantly, how do we learn about it?

MaxBarraclough 8y ago
> How do you know they were wrong? Has there been some breakthrough discovery in philosophy? Of course there has! Gödel! As you said, Russell believed that all of mathematics could be deduced from laws of logic that are true in the most absolute sense. Gödel showed that this is impossible, and his discovery ended Russell's project overnight. Russell was fundamentally mistaken about the nature of mathematics. No two ways about it. I already linked to the Wikipedia discussion of why Brouwer's idea about the law of the excluded middle, seems no longer to be taken seriously. > Gödel was wrongest of them all. He was the most extreme Platonist of them all. Maybe so, I'm afraid I really don't know his philosophy. > if mathematical formulas mean something -- i.e. refer to something outside them -- what is the nature of that thing, and, more importantly, how do we learn about it? But the mathematics derives the same with or without associating real-world meaning. Seems to me that this is enough to demonstrate that the mapping to something beyond the mathematics, is something that should be treated as quite separate from the mathematics itself (which is 'merely' axioms + consequences). Indeed, we see this in action with fields of mathematics that find real-world applications only years after the discovery of the mathematics itself. Suppose a field of mathematics is found to have several applications. Again, the mathematics is unchanged: the discovered applications are a different beast entirely. Doubtless I'm not the first to take this line of thought. > including Hilbert's Formalism, which some gravely mistake for saying mathematics is just what I described I'm afraid you're ahead of me again here -- that's still on my reading list. Also, perhaps just a nitpick: nondeterministic Turing machines might differ from deterministic ones in terms of space/time complexity (that's an open problem), but they're just as impotent against non-computable problems. Were you thinking of some kind of probabilistic machine?
- pron 8y ago
  You seem to think that philosophy is about settling issues. It's more about understanding the question better. I also think you're confusing Russell with Hilbert, and it is not accurate that Gödel destroyed Hilbert's project, just changed its nature (or, rather, destroyed the original understanding but not subsequent ones). But Hilbert's idea was actually much more in line with yours -- he also believed that mathematics is about the process rather than some absolute truth, but in order to show that the process can lead anywhere it was important for him to prove consistency. After all, if our mathematics can prove any proposition (including, say, 1=0) then it's rather useless, whether or not it's "valid" in some sense. > I already linked to the Wikipedia discussion of why Brouwer's idea about the law of the excluded middle, seems no longer to be taken seriously. I think that neither Brouwer nor that Wikipedia article says what you think they do. :) There's a whole branch of logic, intuitionistic logic, and mathematics, constructive mathematics, that are largely based on Brouwer's ideas and rejection of LEM. They are particularly popular among some kind of computer scientists, BTW. > But the mathematics derives the same with or without associating real-world meaning. Seems to me that this is enough to demonstrate that the mapping to something beyond the mathematics, is something that should be treated as quite separate from the mathematics itself I'm not talking about real world meaning, nor "beyond mathematics", but beyond the formula (as a string). If the strings refer to anything, what is the nature of the things they refer to? > Also, perhaps just a nitpick: nondeterministic Turing machines might differ from deterministic ones in terms of space/time complexity (that's an open problem), but they're just as impotent against non-computable problems. Were you thinking of some kind of probabilistic machine? No, I simply meant that deduction in a formal system (like first-order logic) is a nondeterministic computation, and to emphasize the perspective that math is just a manipulation of symbols without meaning I referred to Turing machines. That they cannot settle non-computable problems is irrelevant, because neither can mathematicians, and whatever mathematicians can do, so can a TM.