6 ms·

So you are saying Kepler and Galileo used formal axiom systems? Give me a break. 1. needs a citation. I never believed such a nonsense. Common sense with resp

So you are saying Kepler and Galileo used formal axiom systems? Give me a break. 1. needs a citation. I never believed such a nonsense. Common sense with respect to the physical world is different to common sense with respect to the world of mathematical objects. In fact I don't like the word "common sense" when applied to mathematics, I prefer the expression "mathematical reasoning". In fact we model the physical world by representing it via mathematical objects, because mathematical reasoning is so much more reliable than common sense with respect to the physical world. In fact the advantage of such a mathematical model which we can subject to mathematical reasoning are so big that we are OK with the fact that the model might only be approximating physical reality. In the mathematical world formal logic is a great tool for mathematical reasoning. Mathematical reasoning tells us that if we can represent our reasoning via formal logic, then we can be sure that it is correct. But formal logic has its limits, and in the end mathematical reasoning is the final arbiter, not formal logic: The peano axioms are consistent, because the natural numbers exist. Which is obvious and doesn't require a formal proof (which does not exist, by the way, and will never exist; but that we can know this, is a testament to the power of mathematical reasoning).

kragen 3y ago
no, kepler and galileo didn't understand formal axiom systems; even today you can read galileo's mystified remarks upon encountering that manner of thought in the surviving fragments of eudoxos when you say 'because the natural numbers exist' what you are really saying is 'because i am a platonist'. while that is a perfectly reasonable reason for you to believe things, it's not going to convince anyone else, in many cases not even other platonists. i reserve the term 'mathematical reasoning' for things that do, and i think that's a more widespread definition than the one you are using
- auggierose 3y ago
  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/

kragen 3y ago

no, kepler and galileo didn't understand formal axiom systems; even today you can read galileo's mystified remarks upon encountering that manner of thought in the surviving fragments of eudoxos when you say 'because the natural numbers exist' what you are really saying is 'because i am a platonist'. while that is a perfectly reasonable reason for you to believe things, it's not going to convince anyone else, in many cases not even other platonists. i reserve the term 'mathematical reasoning' for things that do, and i think that's a more widespread definition than the one you are using

auggierose 3y ago
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/