Hacker News Board Hacker News, browsed like a classic message board
Live data from Hacker News

The Unreasonableness of Math Is Context Independence

bellmar.medium.com

Post reply on HN
Print view View original

41–50 of 52 posts

Re: The Unreasonableness of Math Is Context Independence

#41

Post by ukj »

giomasce wrote: post #11
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…

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#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_hierarch...

Re: The Unreasonableness of Math Is Context Independence

#42

Post by [deleted] »

_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…

[deleted]

Re: The Unreasonableness of Math Is Context Independence

#44

Post by agarsev »

qboltz wrote: post #34
agarsev wrote: post #28

Earlier quoted context omitted.

I think this is a very widespread idea, I used to believe in it too, perhaps due to our background. However, if you work on translating science to computers, you soon find it's not so true. There are some many ideas in science which are not mathematically encoded, but rather in human language, it's kind of frustrating. The "low-level" sciences like physics have spoilt us with their very math-like nature. But even in…

Could you cite some specific examples? I'm finding the deeper I study biology, the more certain I am that complex models with both classical and quantum parameters will eventually be able to predict the overwhelming majority of macromolecular behavior such as protein folding and DNA recombination. Once you start dealing with concepts bigger than that you get into another mathematical description with Markov chain sty…

In biology: morphology of organisms, evolution, ecology. And those deal with systems, so they use a lot of math. But interspersed with the math, you always find natural language descriptions, definitions, explanations, which are necessary for understanding and complete modelling of the theory. These make reference to the shared human experience of the world, and are not formalized in logic. Not that they cannot be, or at least so I hope. But we're very far from it today, that's what I mean.

Maybe relatedly, humans think of the world in fuzzy terms. At some point we're going to need a system for formalizing fuzzy thought, and no, fuzzy logic is not it, because that's just a continuous extension to boolean logic. Human thinking is fuzzy beyond that. But, as a computational linguist, I sometimes worry that we already have that system: natural languages!

Re: The Unreasonableness of Math Is Context Independence

#45

Post by andrewflnr »

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...

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.

Re: The Unreasonableness of Math Is Context Independence

#46

Post by andrewflnr »

ggm wrote: post #18
_xnmw wrote: post #9

Earlier quoted context omitted.

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.

Arithmetic still depends on which ring/field axioms you're using. :) I think you would have to agree, though, that axioms are a much more tractable kind of context than "a whole honking physical situation with atoms and entropy", which I think is the real kernel of the article.

Re: The Unreasonableness of Math Is Context Independence

#47

Post by giomasce »

ggm wrote: post #17
giomasce wrote: post #10

Earlier quoted context omitted.

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.

Why should arithmetic be more or less context free than anything else? It's defined by axioms and definitions as much as the rest.

Re: The Unreasonableness of Math Is Context Independence

#48

Post by xtiansimon »

red_admiral wrote: post #23

I really liked this, and it's also why it's so hard to teach mathematics, which is part of my current job. Most people think in a context-dependent way. If you ask, suppose Jane has three apples and John gives her two more apples, how many does she have - then most kids at the appropriate level will visualise apples and count to five. Give exactly the same problem but with "Jane has five McGuffins" and you'll get a c…

One of my favorite _rudimentary ideas about mathematics_ comes from philosopher Cathy Legg describing the work of Charles Sanders Perice:

_"Perice had a hypothetical interpretation of mathematics. So mathematics doesn’t talk about what’s actual at all. Mathematics makes no positive claims. Mathematics just tells you if you make this hypothesis, then this must follow. So mathematics is the science that draws necessary conclusions."_

If you get your head around that, then apples and McGuffins are both permissible.

Episode 81: Cathy Legg discusses what Peirce’s categories can do for you https://elucidations.hum.uchicago.edu/Legg_WhatPeircesCatego...

Re: The Unreasonableness of Math Is Context Independence

#49

Post by ukj »

andrewflnr wrote: post #45
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...

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.

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?

Re: The Unreasonableness of Math Is Context Independence

#50

Post by qboltz »

agarsev wrote: post #44
qboltz wrote: post #34

Earlier quoted context omitted.

Could you cite some specific examples? I'm finding the deeper I study biology, the more certain I am that complex models with both classical and quantum parameters will eventually be able to predict the overwhelming majority of macromolecular behavior such as protein folding and DNA recombination. Once you start dealing with concepts bigger than that you get into another mathematical description with Markov chain sty…

In biology: morphology of organisms, evolution, ecology. And those deal with systems, so they use a lot of math. But interspersed with the math, you always find natural language descriptions, definitions, explanations, which are necessary for understanding and complete modelling of the theory. These make reference to the shared human experience of the world, and are not formalized in logic. Not that they cannot be, o…

I don't think was specific enough, I guess what I'm looking for is something that we can describe with language that doesn't have at least some sort of parameterization in regards to physics.

So take the original reaction of DNA from just inorganics, I typed those words, but have no reference for what the model actually is. What I do however have, is words for each of those things, and a set of impossibilities for what it could "not" mean.

However, the reference is not born out in terms of nothing, each of those words has a set of things that we do have models for, we have models for atoms, reactions, DNA, etc.

So in reality the sentence describes something that we simply can't point to specifics on, but is in no way "unexplainable" in terms of its logic.

Another example would be dark matter, we use those words, but really they just stand for a set of observations, empirical measurements just operating outside of the patterns we are used to, but certainly not without something to point to.

If there's some shared experience that we can't express logically, I'm at least personally unfamiliar with it, I would need some further understanding of what you have in mind.

I could also be wildly misreading what you mean, semantics are not my favorite over text.

Post reply on HN