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When discussing a certain discipline, be it mathematics or any other, the issue of rigor and foundation are completely separate. Rigor is established if you are

by pron 8y ago

When discussing a certain discipline, be it mathematics or any other, the issue of rigor and foundation are completely separate. Rigor is established if you are careful to always follow the axioms determined by the foundation. But the foundation, i.e. the choice of the axioms can only be established in another, lower level, discipline, which, in the case of mathematics is philosophy. In the early decades of the 20th century there were furious philosophical debates about the philosophy of the foundation of mathematics (between Brouwer's Intuitionism and Russell's Logicism, and also Hilbert's compromise of Formalism), but they all stemmed from an underlying assumption that mathematics gains its validity from some notion of philosophical truth. One could argue to no end on how the truth of mathematics is established, but a different perspective later emerged that avoids this debate altogether: that mathematics takes its validity not from truth but from utility (which is always relative to a specific task). We cannot say that one of those views is unacceptable, and if your position is that mathematical validity stems from utility, we cannot tell you that your foundation is shaky if its utility is established.

MaxBarraclough 8y ago
But axioms aren't assertions of truth. They're not declarations from on high about universal truths, they're how mathematicians ensure they're talking about the same things. Mathematics is about exploring the consequences of your axioms. Axioms are chosen, not dictated by the universe. If your axioms turn out to be inconsistent, well, congratulations on successfully showing that consequence of those axioms. Case in point: the axiom of choice. There is no objective truth value to it. The physical world gives us no 'answer'. Mathematicians can choose to adopt it, or not, and then explore the consequences. ( https://en.wikipedia.org/wiki/Axiom_of_choice#Statements_consistent_with_the_negation_of_AC https://en.wikipedia.org/wiki/Axiom_of_choice#Statements_con... )
- 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.

by pron 8y ago

MaxBarraclough 8y ago

But axioms aren't assertions of truth. They're not declarations from on high about universal truths, they're how mathematicians ensure they're talking about the same things. Mathematics is about exploring the consequences of your axioms. Axioms are chosen, not dictated by the universe. If your axioms turn out to be inconsistent, well, congratulations on successfully showing that consequence of those axioms. Case in point: the axiom of choice. There is no objective truth value to it. The physical world gives us no 'answer'. Mathematicians can choose to adopt it, or not, and then explore the consequences. ( https://en.wikipedia.org/wiki/Axiom_of_choice#Statements_consistent_with_the_negation_of_AC https://en.wikipedia.org/wiki/Axiom_of_choice#Statements_con... )

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.