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Math is not context-independent. The meta-mathematical assumptions (axioms) are the context. Different axioms produce different truths; or if you want - they p

by ukj 5y ago

Math is not context-independent. The meta-mathematical assumptions (axioms) are the context. Different axioms produce different truths; or if you want - they produce different Mathematical universes [1]. Maths is relative like Physics is relative - it depends on your frame of reference [2]. 1. https://en.wikipedia.org/wiki/Universe_(mathematics) https://en.wikipedia.org/wiki/Universe_(mathematics) 2. http://math.andrej.com/2012/10/03/am-i-a-constructive-mathematician/ http://math.andrej.com/2012/10/03/am-i-a-constructive-mathem...

giomasce 5y ago
I'd put that in a different way: the point of maths is not really being context-independent, but to make it very clear what is the context. So, let us consider a statement A which is true provided that a certain set of hypotheses B is true. You might either consider that "A is true in the context of B" (what is commonly written as "B |- A"), so you have a context (but it is very clearly stated what it is). Or you can (often) write it as an implication: "B -> A". The whole sentence "B -> A" is an absolute, it has no context any more, because the context has been absorbed in the antecedent. (yes, I know I am oversimplifying something, take this at the "philosophical" level)
- ukj 5y ago
  From the lens of the Curry-Howard isomorphism where logic, category theory and type theory are just different perspectives on the same sort of mental human activity... Implication (logic) is the same thing as internal hom (Category theory); or Function type (type theory). It is just syntax. B |- A in logic translates to f::B -> A in Haskell. https://ncatlab.org/nlab/show/computational%20trilogy#rosetta_stone https://ncatlab.org/nlab/show/computational%20trilogy#rosett... And then the context left implicit is the transformation of B to A e.g the concrete steps for tranforming B to A. The implementation of f. There is no escaping The Hierarchy. https://en.wikipedia.org/wiki/Chomsky_hierarchy#The_hierarchy https://en.wikipedia.org/wiki/Chomsky_hierarchy#The_hierarch...
andrewflnr 5y ago
Yeah, context-dependence is a matter of degree. However, if you rephrase the article in terms of drastically reducing context dependence, particularly eliminating physical circumstance from the context, it still says something mostly true and important.
- ukj 5y ago
  There are, indeed, a lot of trivial truisms in the article. >once you’ve matched the solution to the scenario the solution is able to produce the desired result without modification Translation: once you have identified the context in which the "solution" is applicable then the "solution" works. Would it even be called a "solution" if it wasn't applicable in context?
  - andrewflnr 5y ago
    Yeah, but it's genuinely remarkable that a solution should ever be applicable to more than one problem, other than where it was first devised. It's easy to be numb to that if you've grown up with math showing up all over the place, but that's what the "unreasonable effectiveness" thing is all about.
    - ukj 5y ago
      shrug it is no more; or less remarkable to me than multiple solutions existing for the same problem. Multiple algorithms (proofs) of theorems. Multiple different theories explaining one and the same phenomenon. The principle of equifinality is everywhere in open systems. Only exemplified further that Logic, Programming and Mathematics are essentially three different tools for the same job.

by ukj 5y ago

giomasce 5y ago

I'd put that in a different way: the point of maths is not really being context-independent, but to make it very clear what is the context. So, let us consider a statement A which is true provided that a certain set of hypotheses B is true. You might either consider that "A is true in the context of B" (what is commonly written as "B |- A"), so you have a context (but it is very clearly stated what it is). Or you can (often) write it as an implication: "B -> A". The whole sentence "B -> A" is an absolute, it has no context any more, because the context has been absorbed in the antecedent. (yes, I know I am oversimplifying something, take this at the "philosophical" level)

ukj 5y ago
From the lens of the Curry-Howard isomorphism where logic, category theory and type theory are just different perspectives on the same sort of mental human activity... Implication (logic) is the same thing as internal hom (Category theory); or Function type (type theory). It is just syntax. B |- A in logic translates to f::B -> A in Haskell. https://ncatlab.org/nlab/show/computational%20trilogy#rosetta_stone https://ncatlab.org/nlab/show/computational%20trilogy#rosett... And then the context left implicit is the transformation of B to A e.g the concrete steps for tranforming B to A. The implementation of f. There is no escaping The Hierarchy. https://en.wikipedia.org/wiki/Chomsky_hierarchy#The_hierarchy https://en.wikipedia.org/wiki/Chomsky_hierarchy#The_hierarch...

andrewflnr 5y ago

Yeah, context-dependence is a matter of degree. However, if you rephrase the article in terms of drastically reducing context dependence, particularly eliminating physical circumstance from the context, it still says something mostly true and important.

ukj 5y ago
There are, indeed, a lot of trivial truisms in the article. >once you’ve matched the solution to the scenario the solution is able to produce the desired result without modification Translation: once you have identified the context in which the "solution" is applicable then the "solution" works. Would it even be called a "solution" if it wasn't applicable in context?
- andrewflnr 5y ago
  Yeah, but it's genuinely remarkable that a solution should ever be applicable to more than one problem, other than where it was first devised. It's easy to be numb to that if you've grown up with math showing up all over the place, but that's what the "unreasonable effectiveness" thing is all about.
  - ukj 5y ago
    shrug it is no more; or less remarkable to me than multiple solutions existing for the same problem. Multiple algorithms (proofs) of theorems. Multiple different theories explaining one and the same phenomenon. The principle of equifinality is everywhere in open systems. Only exemplified further that Logic, Programming and Mathematics are essentially three different tools for the same job.