The Unreasonableness of Math Is Context Independence • Hacker News Board

Re: The Unreasonableness of Math Is Context Independence

#11

Post by giomasce » Sat, Feb 12, 2022, 9:04 AM UTC

ukj wrote: post #3

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) 2. http://math.andrej.com/2012/10/03/am-i-a-constructive-mathem...

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)

Re: The Unreasonableness of Math Is Context Independence

#12

Post by jonsen » Sat, Feb 12, 2022, 9:05 AM UTC

_xnmw wrote: post #8

lordnacho wrote: post #6

IMO math seems effective because everything that works is called math. So yeah, that quote from the beginning is right, it's selection. There are many different math concepts used to describe the world, everything from calculus to graph theory, geometry, and so on. These things have a two way relationship with the real world: they don't necessarily have to correspond with anything real, like Hardy's quote about his n…

What? I'm sorry but this is utter bullshit. Math is not just anything that works. Every hot new theory is assumed to "work" in the era which it is produced, and not everything is called "math". There has been no significant "wrong" result in the entire history of math since ancient times, nor has any significant result been jettisoned from the field of math, whereas every other field or discipline of study has been w…

I think you got the parent comment bacwards.

Re: The Unreasonableness of Math Is Context Independence

#13

Post by jacobolus » Sat, Feb 12, 2022, 9:08 AM UTC

_xnmw wrote: post #8

lordnacho wrote: post #6

IMO math seems effective because everything that works is called math. So yeah, that quote from the beginning is right, it's selection. There are many different math concepts used to describe the world, everything from calculus to graph theory, geometry, and so on. These things have a two way relationship with the real world: they don't necessarily have to correspond with anything real, like Hardy's quote about his n…

What? I'm sorry but this is utter bullshit. Math is not just anything that works. Every hot new theory is assumed to "work" in the era which it is produced, and not everything is called "math". There has been no significant "wrong" result in the entire history of math since ancient times, nor has any significant result been jettisoned from the field of math, whereas every other field or discipline of study has been w…

> no significant "wrong" result in the entire history of math since ancient times, nor has any significant result been jettisoned

This is a bit of an exaggeration. If you search around you can e.g. find https://mathoverflow.net/questions/35468/widely-accepted-mat... https://math.stackexchange.com/questions/139503/in-the-histo... https://mathoverflow.net/questions/27749/what-are-some-corre... https://mathoverflow.net/questions/879/most-interesting-math...

Re: The Unreasonableness of Math Is Context Independence

#14

Post by _xnmw » Sat, Feb 12, 2022, 9:12 AM UTC

jacobolus wrote: post #13

_xnmw wrote: post #8

Earlier quoted context omitted.

What? I'm sorry but this is utter bullshit. Math is not just anything that works. Every hot new theory is assumed to "work" in the era which it is produced, and not everything is called "math". There has been no significant "wrong" result in the entire history of math since ancient times, nor has any significant result been jettisoned from the field of math, whereas every other field or discipline of study has been w…

> no significant "wrong" result in the entire history of math since ancient times, nor has any significant result been jettisoned This is a bit of an exaggeration. If you search around you can e.g. find https://mathoverflow.net/questions/35468/widely-accepted-mat... https://math.stackexchange.com/questions/139503/in-the-histo... https://mathoverflow.net/questions/27749/what-are-some-corre... https://mathoverflow.net/…

Reading just the top answers from those threads, I see no significant results that have been disproved. Only the "intuitions" and "footnotes" and some "trivial assumptions" of mathematicians, but not an actual published result that was cited by other results and had significant consequences by invalidating other results.

Re: The Unreasonableness of Math Is Context Independence

#15

Post by [deleted] » Sat, Feb 12, 2022, 9:25 AM UTC

lordnacho wrote: post #6

IMO math seems effective because everything that works is called math. So yeah, that quote from the beginning is right, it's selection. There are many different math concepts used to describe the world, everything from calculus to graph theory, geometry, and so on. These things have a two way relationship with the real world: they don't necessarily have to correspond with anything real, like Hardy's quote about his n…

[deleted]

Re: The Unreasonableness of Math Is Context Independence

#16

Post by danielmorozoff » Sat, Feb 12, 2022, 9:29 AM UTC

>So perhaps the best way to build efficient abstractions in systems is to think about the flow of the system in terms of axioms and conditionals. The abstractions are axioms that can be grouped together and the conditionals are the boundaries between them.

I wonder how you square this idea of generalization with Godel's incompleteness theorems?

https://plato.stanford.edu/entries/goedel-incompleteness/

Re: The Unreasonableness of Math Is Context Independence

#17

Post by ggm » Sat, Feb 12, 2022, 9:47 AM UTC

giomasce wrote: post #10

ggm wrote: post #2

Except for the corner cases. So the trivial one is "angles in a triangle add up to 180" which works in a plane but not on the surface of a sphere so navigation has to use more than trivial trigonometry functions for accuracy at scale. The context is sometimes everything.

That's not a corner case. Either you defined "triangle" and "angles" to mean a plane triangle and angles, or that is not a theorem. Within the theory you're looking at, the definitions are not part of the context, they are part of the theory itself. So you're not depending on the context.

Like the other response you basically said context is everything. You just prefer to call the context axioms. What is context free here is the arithmetic.

Re: The Unreasonableness of Math Is Context Independence

#18

Post by ggm » Sat, Feb 12, 2022, 9:47 AM UTC

_xnmw wrote: post #9

ggm wrote: post #2

Except for the corner cases. So the trivial one is "angles in a triangle add up to 180" which works in a plane but not on the surface of a sphere so navigation has to use more than trivial trigonometry functions for accuracy at scale. The context is sometimes everything.

That's not a corner case, that's just more advanced math. No one ever claimed that triangulation on a plane is the same as triangulation on the surface of a sphere.

Like the other response you basically said context is everything. You just prefer to call the context axioms. What is context free here is the arithmetic.

Re: The Unreasonableness of Math Is Context Independence

#19

Post by lordnacho » Sat, Feb 12, 2022, 9:50 AM UTC

_xnmw wrote: post #8

lordnacho wrote: post #6

IMO math seems effective because everything that works is called math. So yeah, that quote from the beginning is right, it's selection. There are many different math concepts used to describe the world, everything from calculus to graph theory, geometry, and so on. These things have a two way relationship with the real world: they don't necessarily have to correspond with anything real, like Hardy's quote about his n…

What? I'm sorry but this is utter bullshit. Math is not just anything that works. Every hot new theory is assumed to "work" in the era which it is produced, and not everything is called "math". There has been no significant "wrong" result in the entire history of math since ancient times, nor has any significant result been jettisoned from the field of math, whereas every other field or discipline of study has been w…

Compare and contrast:

> everything that works is called math

> Math is not just anything that works

Can you see how you have read my comment wrong? "An A is a B" is not the same as "A B is and A", you're arguing against something that wasn't claimed.

If you wanted to come up with something sensible to say, you could bring up a theory that is backed up by something that isn't called math.

Re: The Unreasonableness of Math Is Context Independence

#20

Post by TheOtherHobbes » Sat, Feb 12, 2022, 10:37 AM UTC

_xnmw wrote: post #8

lordnacho wrote: post #6

IMO math seems effective because everything that works is called math. So yeah, that quote from the beginning is right, it's selection. There are many different math concepts used to describe the world, everything from calculus to graph theory, geometry, and so on. These things have a two way relationship with the real world: they don't necessarily have to correspond with anything real, like Hardy's quote about his n…

What? I'm sorry but this is utter bullshit. Math is not just anything that works. Every hot new theory is assumed to "work" in the era which it is produced, and not everything is called "math". There has been no significant "wrong" result in the entire history of math since ancient times, nor has any significant result been jettisoned from the field of math, whereas every other field or discipline of study has been w…

No. https://en.wikipedia.org/wiki/List_of_incomplete_proofs

You seem to be arguing from a personally idealised view of math, which doesn't match reality.

Real math is full of full of mis-starts, dead ends, and established mistakes which are later corrected.

Math is exactly like science. There's a cumulative core we can be very confident about, and more exploratory edges where results are more tentative and subject to review, correction, and expansion.