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Well, show me a mathematician who does not believe in the natural numbers. And yes, I am a Platonist, just as Gödel is. If you are not a Platonist, your view ca

Well, show me a mathematician who does not believe in the natural numbers. And yes, I am a Platonist, just as Gödel is. If you are not a Platonist, your view cannot be defended. It is as simple as that. I don't really care to convince non-Platonists, except for them to become Platonists. What is your definition of mathematical reasoning then? If it is just formal logic, we could not even work with natural numbers. Even better, we could not even justify formal logic. I am a big fan of formal logic, so much that I got myself a PhD in implementing formal logic on a computer. I want to do as much mathematical reasoning using formal logic as I can. But it is clear that formal logic has its limits, but that doesn't mean that mathematical reasoning has the same limits. The natural numbers show us otherwise.

kragen 3y ago
mathematics probably began as platonism, but eudoxos put it on a purely hypothetico-deductive basis before platon's corpse was cold, so this debate has been going on for a while already. however, from the end of the hellenistic age until the 19th century, mathematics was utterly dominated by platonism the social situation is quite different today. as you and i both know, there are plenty of formalists in mathematics, and quite a few constructivists as well, and none of them are platonists; we might even argue that at this point formalism is more popular than platonism among people working on foundations of mathematics. wildberger, a constructivist, is probably the most notorious living anti-platonist extremist who's still within the mainstream (i.e., he's a real mathematician, not just some bozo like me): https://njwildberger.com/2021/10/07/finite-versus-infinite-real-numbers/ https://njwildberger.com/2021/10/07/finite-versus-infinite-r... is a sample of his anti-platonist rhetoric. and, as you are certainly aware, voevodsky's extremely influential hott is fundamentally constructivist. https://en.wikipedia.org/wiki/Ultrafinitism https://en.wikipedia.org/wiki/Ultrafinitism is (or was) full of mathematicians who do (or did) not believe in the natural numbers. and it's easy to find working mathematicians saying things like https://old.reddit.com/r/math/comments/fx6bqv/why_did_brouwers_intuitionist_mathematics_not/fmsemke/ https://old.reddit.com/r/math/comments/fx6bqv/why_did_brouwe...: > I don't even know if 7 "exists," let alone pi. Such questions may be interesting, but they're philosophy, not math. Meanwhile, it's useful and mathematically interesting to count to 7, so we do it. from a sociological perspective, a reasonable definition of 'mathematical reasoning' might be 'whatever arguments platonists can use to convince formalists and constructivists, or vice versa, without resorting to empirical evidence'. (this obviously isn't a logical definition, but i would argue that definitions of commonly used words and phrases are always sociological rather than logical in nature, though we can certainly stipulate a logical definition within the context of a given discussion.) this excludes, for example, which territory rightfully belongs to belgium, whether the earth orbits in an ellipse, or whether the natural numbers exist or not; but it includes the area inside a particular proposed national border for belgium, the properties of ellipses, and the consequences of the peano axioms i think it is correct that formalists do not believe that formal logic is justified in any fundamental sense, but i haven't actually asked them for those who don't have a phd in logic but might still be reading this thread, a good short introduction to the late modern schism in mathematics (from an ardently platonist perspective!) is https://plus.maths.org/content/philosophy-applied-mathematics https://plus.maths.org/content/philosophy-applied-mathematic...; another introduction is https://web.archive.org/web/20170611144646/http://math.stanford.edu/~ebwarner/SplashTalk.pdf https://web.archive.org/web/20170611144646/http://math.stanf... and a much deeper introduction is https://plato.stanford.edu/entries/mathematics-constructive/ https://plato.stanford.edu/entries/mathematics-constructive/
- auggierose 3y ago
  Here's the thing: any definition of mathematical reasoning that excludes the natural numbers, isn't a definition of mathematical reasoning. For example, I cannot convince some constructivists that A or not A is always true when it comes to mathematical reasoning. But it is always true, anything else just doesn't make sense. So a constructivist who doesn't acknowledge that, I cannot take anything serious they say. Note that this is different from coming up with interpretations of ∨ and ¬ such that A ∨ ¬A is not always ⊤, such as Heyting Algebras, for example. And it is different from doing constructive mathematics, which is a very interesting subject. Voevodsky was not a constructivist. He was looking for a way to do the kind of mathematics he likes on a computer to ensure correctness. He first visited the Isabelle development group to find about that, but simple type theory was not flexible enough for his purposes. So afterwards he fell into the hands of the dependent type theorists, who are all constructivists. He tried to do classical mathematics in their framework, that's where his univalence axiom comes from, but that didn't really work out well so far, and I doubt it ever will (but I don't really understand it, so that's just my opinion).
  - kragen 3y ago
    certainly you have to be able to do mathematical reasoning about natural numbers, but that doesn't necessarily imply determining their ontological status another useful definition of mathematical reasoning is 'arguments strong enough to convince someone who cannot take seriously anything you are saying and does not trust you at all' i appreciate the correction about voevodsky; i don't have a phd in the field myself, so that may not be all i got wrong
    - auggierose 3y ago
      > certainly you have to be able to do mathematical reasoning about natural numbers, but that doesn't necessarily imply determining their ontological status That's true, of course you can use something without understanding it, but can you use something, and then deny its existence? I want to correct what I said earlier about the natural numbers and infinity. Of course you could add infinity to the natural numbers, if you wanted to. This can be done in different ways. It's just that natural numbers without infinity are a simpler starting point.

kragen 3y ago

mathematics probably began as platonism, but eudoxos put it on a purely hypothetico-deductive basis before platon's corpse was cold, so this debate has been going on for a while already. however, from the end of the hellenistic age until the 19th century, mathematics was utterly dominated by platonism the social situation is quite different today. as you and i both know, there are plenty of formalists in mathematics, and quite a few constructivists as well, and none of them are platonists; we might even argue that at this point formalism is more popular than platonism among people working on foundations of mathematics. wildberger, a constructivist, is probably the most notorious living anti-platonist extremist who's still within the mainstream (i.e., he's a real mathematician, not just some bozo like me): https://njwildberger.com/2021/10/07/finite-versus-infinite-real-numbers/ https://njwildberger.com/2021/10/07/finite-versus-infinite-r... is a sample of his anti-platonist rhetoric. and, as you are certainly aware, voevodsky's extremely influential hott is fundamentally constructivist. https://en.wikipedia.org/wiki/Ultrafinitism https://en.wikipedia.org/wiki/Ultrafinitism is (or was) full of mathematicians who do (or did) not believe in the natural numbers. and it's easy to find working mathematicians saying things like https://old.reddit.com/r/math/comments/fx6bqv/why_did_brouwers_intuitionist_mathematics_not/fmsemke/ https://old.reddit.com/r/math/comments/fx6bqv/why_did_brouwe...: > I don't even know if 7 "exists," let alone pi. Such questions may be interesting, but they're philosophy, not math. Meanwhile, it's useful and mathematically interesting to count to 7, so we do it. from a sociological perspective, a reasonable definition of 'mathematical reasoning' might be 'whatever arguments platonists can use to convince formalists and constructivists, or vice versa, without resorting to empirical evidence'. (this obviously isn't a logical definition, but i would argue that definitions of commonly used words and phrases are always sociological rather than logical in nature, though we can certainly stipulate a logical definition within the context of a given discussion.) this excludes, for example, which territory rightfully belongs to belgium, whether the earth orbits in an ellipse, or whether the natural numbers exist or not; but it includes the area inside a particular proposed national border for belgium, the properties of ellipses, and the consequences of the peano axioms i think it is correct that formalists do not believe that formal logic is justified in any fundamental sense, but i haven't actually asked them for those who don't have a phd in logic but might still be reading this thread, a good short introduction to the late modern schism in mathematics (from an ardently platonist perspective!) is https://plus.maths.org/content/philosophy-applied-mathematics https://plus.maths.org/content/philosophy-applied-mathematic...; another introduction is https://web.archive.org/web/20170611144646/http://math.stanford.edu/~ebwarner/SplashTalk.pdf https://web.archive.org/web/20170611144646/http://math.stanf... and a much deeper introduction is https://plato.stanford.edu/entries/mathematics-constructive/ https://plato.stanford.edu/entries/mathematics-constructive/

auggierose 3y ago
Here's the thing: any definition of mathematical reasoning that excludes the natural numbers, isn't a definition of mathematical reasoning. For example, I cannot convince some constructivists that A or not A is always true when it comes to mathematical reasoning. But it is always true, anything else just doesn't make sense. So a constructivist who doesn't acknowledge that, I cannot take anything serious they say. Note that this is different from coming up with interpretations of ∨ and ¬ such that A ∨ ¬A is not always ⊤, such as Heyting Algebras, for example. And it is different from doing constructive mathematics, which is a very interesting subject. Voevodsky was not a constructivist. He was looking for a way to do the kind of mathematics he likes on a computer to ensure correctness. He first visited the Isabelle development group to find about that, but simple type theory was not flexible enough for his purposes. So afterwards he fell into the hands of the dependent type theorists, who are all constructivists. He tried to do classical mathematics in their framework, that's where his univalence axiom comes from, but that didn't really work out well so far, and I doubt it ever will (but I don't really understand it, so that's just my opinion).
- kragen 3y ago
  certainly you have to be able to do mathematical reasoning about natural numbers, but that doesn't necessarily imply determining their ontological status another useful definition of mathematical reasoning is 'arguments strong enough to convince someone who cannot take seriously anything you are saying and does not trust you at all' i appreciate the correction about voevodsky; i don't have a phd in the field myself, so that may not be all i got wrong
  - auggierose 3y ago
    > certainly you have to be able to do mathematical reasoning about natural numbers, but that doesn't necessarily imply determining their ontological status That's true, of course you can use something without understanding it, but can you use something, and then deny its existence? I want to correct what I said earlier about the natural numbers and infinity. Of course you could add infinity to the natural numbers, if you wanted to. This can be done in different ways. It's just that natural numbers without infinity are a simpler starting point.