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It's not a matter of legitimacy (which is a social construct) but of foundation. Russell, Brouwer and Hilbert believed that mathematics must be based on a solid

by pron 8y ago

It's not a matter of legitimacy (which is a social construct) but of foundation. Russell, Brouwer and Hilbert believed that mathematics must be based on a solid philosophical foundation so that it leads to some "Truth." I am not saying this is the only way to think about the philosophy of mathematics -- in fact, my original comment said just the opposite -- but it was very much at the center of the mathematical world in the early decades of the 20th century, and is still a matter of debate among logicians. The philosophy of mathematics is a deep and complex subject. It does not seek to mathematically derive theorems from mathematical axioms, but to derive mathematical axioms from philosophical underpinnings -- whether they are physical reality, some Platonic reality, or even common sense. Logicians don't think the subject is silly at all, and if you're interested in learning more about it, the SEP link I provided above is a great place to start.

MaxBarraclough 8y ago
> it was very much at the center of the mathematical world in the early decades of the 20th century 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? > It does not seek to mathematically derive theorems from mathematical axioms, but to derive mathematical axioms from philosophical underpinnings I don't agree. If you don't add new axioms, then you are 'merely' deriving theorems. If you do add new axioms, fine, you're just adopting a new set of axioms to explore through derivation. 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. It would be pure silliness to dismiss non-Euclidean geometry on the grounds that Hey, you just made that up! Perhaps there's a gap somewhere in my account of things, but I'm not seeing one so far. > to derive mathematical axioms from philosophical underpinnings -- whether they are physical reality, some Platonic reality, or even common sense Which of these underpins non-Euclidean geometry? How about fields with no practical applications? How about, as I've mentioned several times, research into what happens when you deny the axiom of choice? They're all still valid fields of mathematics. The 'underpinnings' of the chosen axioms are of no consequence: it's valid mathematics either way. The miracle is that we're able to be so successful with fields like physics and statistics. Picking the right special sets of axioms, we've been able to derive huge amounts. > the SEP link I provided above is a great place to start Looks it - will read when I get a moment.
- 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?
    - 8y ago

by pron 8y ago

MaxBarraclough 8y ago

> it was very much at the center of the mathematical world in the early decades of the 20th century 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? > It does not seek to mathematically derive theorems from mathematical axioms, but to derive mathematical axioms from philosophical underpinnings I don't agree. If you don't add new axioms, then you are 'merely' deriving theorems. If you do add new axioms, fine, you're just adopting a new set of axioms to explore through derivation. 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. It would be pure silliness to dismiss non-Euclidean geometry on the grounds that Hey, you just made that up! Perhaps there's a gap somewhere in my account of things, but I'm not seeing one so far. > to derive mathematical axioms from philosophical underpinnings -- whether they are physical reality, some Platonic reality, or even common sense Which of these underpins non-Euclidean geometry? How about fields with no practical applications? How about, as I've mentioned several times, research into what happens when you deny the axiom of choice? They're all still valid fields of mathematics. The 'underpinnings' of the chosen axioms are of no consequence: it's valid mathematics either way. The miracle is that we're able to be so successful with fields like physics and statistics. Picking the right special sets of axioms, we've been able to derive huge amounts. > the SEP link I provided above is a great place to start Looks it - will read when I get a moment.

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?
  - 8y ago