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Comment by hackandthink | Hacker News Reader

Mathematics has a philosophical bent.

Philosophy tries to make implicit understanding explicit and usually fails in interesting ways.

(A part of) Mathematics tries to get the essence of numbers, calculation or the continuum and formulates corresponding axioms.

So Mathematics tries to make the implicit understandig of for example natural numbers explicit.

Mathematics doesn't fail usually because the scope is much smaller.

I think some Mathematicians whose scope is very wide should be regarded as philosophers.

Per Martin Löf:

"Martin-Löf type theory is a formal logical system and philosophical foundation for constructive mathematics" (1)

William Lawvere: his topos theory amalgates Logic and Geometry and leads to a kind of natural philosophy (2)

(1) https://plato.stanford.edu/entries/type-theory-intuitionisti...

(2) https://ncatlab.org/nlab/show/William+Lawvere#RelationToPhil...

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> Philosophy tries to make implicit understanding explicit and usually fails in interesting ways.

> Mathematics doesn't fail usually because the scope is much smaller.

Another way to look at this: we try to make implicit understanding explicit in a large number of fields. The fields in which it succeeds we call mathematics, and the other fields in which it fails we call philosophy.

This hits to the core of CS, actually. Any definition of explicit to make your proposition work is equivalent to “computable”. any decent formalism of mathematics is computable.

Plug for implicit understanding though. Your very claim is not explicable; you can’t explicate the non existence of explicability. You can’t formally denote areas for which there’s no formal descriptions. This is the escape hatch out of logical realism (and yeah, that’s not an explicable claim, either; how fun!)

> Any definition of explicit to make your proposition work is equivalent to “computable”. any decent formalism of mathematics is computable.

I am not sure that this is the case actually. Many results in mathematics only relate to the (non) existence of certain objects, without explicit recipes for finding (i.e., computing) them. And obviously philosophers started arguing whether an object that cannot be found actually exists or not :)

Proving non existence uses some type of law of excluded middle. It’s a very concrete and computable style of argument, although its subjects may not be.

My favourite example this is Mark Braverman's demonstration that all Quadratic Julia sets are computable (as in computable real numbers: meaning that you can produce arbitrary fine approximations).

However, as Mark notes, this proof proceeds by deriving 5 alternative programs to do the computation. For any given Julia set parameter 'c' (given as a computable real number) one of the 5 programs will compute the Julia set. Which one? Who knows. It depends discontinuously on the properties of 'c'.

It has been striking to me how post-modern philosophy seems to have recentered itself around the questions of language... while «naming things» is considered one of the ~~two~~ three «hardest problems in CS» !

Not directly related to the above, but maybe also relevant : Wolfram's «new science» project centered around computability : https://writings.stephenwolfram.com/2017/05/a-new-kind-of-sc...

Words are not a perfect mapping to the reality they describe. Never understood why people spend so much time worrying about it.

Because this is such an easy and common trap to fall into (because it's a very economical shortcut ?), we need to be super vigilant about it ?

Speaking of which, might the feeling about the existence of numbers come from a similar shortcoming in our neurological structure ?

https://www.lesswrong.com/posts/mHNzpX38HkZQrdYn3/philosophy...

Not directly related, we're now surrounded by computers, which are even dumber (some forms of machine learning and analog computers aside ?) in that they have zero flexibility in treating symbols as something else...

I’m not really bickering about names here, just trying to succinctly describe ideas.

The irony of the GGP comment is that it claims mathematical ideas are successful ones and philosophical ones aren’t, but that’s not a mathematical claim. So either it’s right and it’s wrong, or it’s wrong.

This seems to weirdly ignore the halting problem and other parts of mathematics that can never be complete in our universe.

The halting problem itself is not computable, that doesn’t mean the proof that it’s not computable isn’t itself computable.

Saul Kripke is a nice example here. He did mathematics and philosophy.

His later philosophical work (Kripkenstein) had a wider scope and was less mathematical.

(maybe Wittgenstein fits as well: from the (simple minded) mathematical Tractatus to Language Games)

Wittgenstein's or Kripke's later "explicit understanding" did not fail, it's just not as well defined as mathematics.

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