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I wish I understood how mathematicians conceptualize the notion of 'right' and 'wrong' axioms in light of Gödel. I would've thought that the incompleteness theo

I wish I understood how mathematicians conceptualize the notion of 'right' and 'wrong' axioms in light of Gödel. I would've thought that the incompleteness theorems would be taken as evidence against Platonism, but clearly this didn't happen. What do people think is the source of 'ground truth'? Is it more than aesthetic appeal? Are there any articles or books that convey it? I've talked to a friend with a doctorate in algebraic geometry about this several times, but all I've really gotten out of her is that if I studied a lot more math, I would gradually get the same sense. (For reference, I have about as much mathematics background as a US student who minored in maths.)

n4r9 2y ago
> the incompleteness theorems would be taken as evidence against Platonism I wonder if this is a map/territory thing. Incompleteness theorems apply to models (axiomatisations) of mathematics, e.g. ZFC set theory. Platonism applies to the actual act of doing mathematics, i.e. persuading other mathematicians that certain statements about abstract entities are true or false.
- dr_dshiv 2y ago
  Platonism is about the reality of math behind the notation and beyond the human mind — Like that spheres are real even though there is no perfect sphere in the material world. This is a common opinion among mathematicians and physicists even though it seems unscientific (as it poses reality beyond matter)
  - n4r9 2y ago
    Right. According to Platonism, a mathematician publishing a result about spherical geometry is in fact exploring an objective reality beyond human minds. Contrast that with, say, intuitionism, according to which they are simply describing the mental activity which arises as a result of considering an object of shared understanding known as a "sphere".
vlz 2y ago
This comment from a previous thread https://news.ycombinator.com/item?id=27846084 https://news.ycombinator.com/item?id=27846084 links to a two-part paper called "Believing the axioms", part 1: https://www.cs.umd.edu/~gasarch/BLOGPAPERS/belaxioms1.pdf https://www.cs.umd.edu/~gasarch/BLOGPAPERS/belaxioms1.pdf I have only taken a short look and am not qualified to summarize, but it seems this would have an answer if there is one.
skissane 2y ago
> I would've thought that the incompleteness theorems would be taken as evidence against Platonism, but clearly this didn't happen I don't think they really are evidence against it. The first incompleteness theorem says (to put it simply) there are truths about the natural numbers you can't prove, and (if we equate proof with knowledge) can't know. I don't know why a Platonist would find that objectionable. I mean, naïve materialism would imply there are lots of facts about the material world we are never going to be able to know (e.g. the particular arrangement of rocks on a lifeless planet in a distant galaxy right now, or as close to right now as relativity permits). If unknowable truths isn't evidence against materialism, why would it be evidence against Platonism? Really, Gödel's theorems were a much bigger problem for formalism than Platonism. Formalists wanted to identify mathematical truth with provability, and Gödel shattered that dream. Platonists never dreamt the dream, so its destruction didn't discourage them.

n4r9 2y ago

> the incompleteness theorems would be taken as evidence against Platonism I wonder if this is a map/territory thing. Incompleteness theorems apply to models (axiomatisations) of mathematics, e.g. ZFC set theory. Platonism applies to the actual act of doing mathematics, i.e. persuading other mathematicians that certain statements about abstract entities are true or false.

dr_dshiv 2y ago
Platonism is about the reality of math behind the notation and beyond the human mind — Like that spheres are real even though there is no perfect sphere in the material world. This is a common opinion among mathematicians and physicists even though it seems unscientific (as it poses reality beyond matter)
- n4r9 2y ago
  Right. According to Platonism, a mathematician publishing a result about spherical geometry is in fact exploring an objective reality beyond human minds. Contrast that with, say, intuitionism, according to which they are simply describing the mental activity which arises as a result of considering an object of shared understanding known as a "sphere".

vlz 2y ago

This comment from a previous thread https://news.ycombinator.com/item?id=27846084 https://news.ycombinator.com/item?id=27846084 links to a two-part paper called "Believing the axioms", part 1: https://www.cs.umd.edu/~gasarch/BLOGPAPERS/belaxioms1.pdf https://www.cs.umd.edu/~gasarch/BLOGPAPERS/belaxioms1.pdf I have only taken a short look and am not qualified to summarize, but it seems this would have an answer if there is one.

skissane 2y ago

> I would've thought that the incompleteness theorems would be taken as evidence against Platonism, but clearly this didn't happen I don't think they really are evidence against it. The first incompleteness theorem says (to put it simply) there are truths about the natural numbers you can't prove, and (if we equate proof with knowledge) can't know. I don't know why a Platonist would find that objectionable. I mean, naïve materialism would imply there are lots of facts about the material world we are never going to be able to know (e.g. the particular arrangement of rocks on a lifeless planet in a distant galaxy right now, or as close to right now as relativity permits). If unknowable truths isn't evidence against materialism, why would it be evidence against Platonism? Really, Gödel's theorems were a much bigger problem for formalism than Platonism. Formalists wanted to identify mathematical truth with provability, and Gödel shattered that dream. Platonists never dreamt the dream, so its destruction didn't discourage them.