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Well, that certainly was not the common perception in the philosophy of mathematics in the first half of the 20th century. Sure, they believed mathematics is an

by pron 8y ago

Well, that certainly was not the common perception in the philosophy of mathematics in the first half of the 20th century. Sure, they believed mathematics is an exploration of the axioms, but also that the axioms must be true in some deep philosophical sense. Brouwer wrote: [A]n incorrect theory, even if it cannot be inhibited by any contradiction that would refute it, is none the less incorrect,[1] and Russell believed that all of mathematics could be deduced from laws of logic that are true in the most absolute sense. It is also doubtful that this is the perception today. If we want to use mathematics to derive any result about the physical world -- and we most certainly do -- our axioms must be consistent with it. How do you ensure that that is the case? But such questions are not part of mathematics itself, just as the scientific method is not part of physics, but rather belong in a more fundamental discipline, called foundations of mathematics, or the philosophy of mathematics, which is usually studied by philosophers/logicians.[2] [1]: On the Significance of the Principle of Excluded Middle in Mathematics, 1923 [2]: https://plato.stanford.edu/entries/philosophy-mathematics/ https://plato.stanford.edu/entries/philosophy-mathematics/

MaxBarraclough 8y ago
> they believed mathematics is an exploration of the axioms, but also that the axioms must be true in some deep philosophical sense That's a contradiction, and it's easy enough to show it: a mathematician can research the consequences of the axiom of choice, and can research the consequences of its negation. It would be silly to deny that such research is legitimate mathematics. Brouwer's statement strikes me as circular. Beyond that, it seems to me that the law of excluded middle, is patently false. I already gave a counterexample: the axiom of choice. Neither the axiom, nor its negation, has a derivable truth value to settle upon. Ah, I see I'm behind the times [0] As to Russell, he of course turned out to be wrong. I refer of course to Gödel. I'm not sure I see your point in mentioning him, or for that matter Brouwer - do explain. > our axioms must be consistent with it They don't. We hope that we have the mathematics to advance physics, but we needn't hope that all mathematics is applicable to physics. It's perfectly legitimate for a mathematician to explore the consequences of denying the axiom of choice. (I already linked to such.) Such work would presumably never have application in physics, but it's still legitimate mathematics. > How do you ensure that that is the case? As you later allude to, that isn't a question of mathematics, it's a question of how and why mathematical discoveries map onto observable realities like physics and statistics. This is a completely legitimate line of question. The canonical article about how remarkable this is: Wigner's "The Unreasonable Effectiveness of Mathematics in the Natural Sciences". > such questions are not part of mathematics itself Agreed. [0] https://en.wikipedia.org/wiki/Law_of_excluded_middle#In_mathematical_logic https://en.wikipedia.org/wiki/Law_of_excluded_middle#In_math...
- 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.

by pron 8y ago

MaxBarraclough 8y ago

> they believed mathematics is an exploration of the axioms, but also that the axioms must be true in some deep philosophical sense That's a contradiction, and it's easy enough to show it: a mathematician can research the consequences of the axiom of choice, and can research the consequences of its negation. It would be silly to deny that such research is legitimate mathematics. Brouwer's statement strikes me as circular. Beyond that, it seems to me that the law of excluded middle, is patently false. I already gave a counterexample: the axiom of choice. Neither the axiom, nor its negation, has a derivable truth value to settle upon. Ah, I see I'm behind the times [0] As to Russell, he of course turned out to be wrong. I refer of course to Gödel. I'm not sure I see your point in mentioning him, or for that matter Brouwer - do explain. > our axioms must be consistent with it They don't. We hope that we have the mathematics to advance physics, but we needn't hope that all mathematics is applicable to physics. It's perfectly legitimate for a mathematician to explore the consequences of denying the axiom of choice. (I already linked to such.) Such work would presumably never have application in physics, but it's still legitimate mathematics. > How do you ensure that that is the case? As you later allude to, that isn't a question of mathematics, it's a question of how and why mathematical discoveries map onto observable realities like physics and statistics. This is a completely legitimate line of question. The canonical article about how remarkable this is: Wigner's "The Unreasonable Effectiveness of Mathematics in the Natural Sciences". > such questions are not part of mathematics itself Agreed. [0] https://en.wikipedia.org/wiki/Law_of_excluded_middle#In_mathematical_logic https://en.wikipedia.org/wiki/Law_of_excluded_middle#In_math...

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.