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The Unreasonableness of Math Is Context Independence

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Re: The Unreasonableness of Math Is Context Independence

#31

Post by asimpletune »

lordnacho wrote: post #19
_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…

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.

What about "experts"? Like, I would say there are people out there who are valuable because they know how to do stuff. It's not required for them to explain how they do it, and often times it's the case that they can't, otherwise they would simply explain and we'd all be experts. Rather, we need them precisely because the results they deliver are not able to be broken down into a sequence of steps that anyone could follow - since many of them can't explain. So it's something that "works" but isn't math.

I'll add that eventually experts are replaced, but then by that time there are new experts. The problem domain evolves and what used to require experts is replaced with math, and the new experts are working in the area where things can't be math.

Conceptually I think I'm on point for this, but I don't know if my examples are super good. I'd say business, human language, politics, medicine, and art are all examples of things that have experts. In each of these fields that are things that work, but it's not yet backed up by math.

Maybe it's more accurate to say, given an infinite amount of time and intelligence, everything becomes math? And I think that makes sense, but I'm sort of inclined to believing in an objective, yet logistically intractable reality.

Re: The Unreasonableness of Math Is Context Independence

#32

Post by Aerroon »

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…

But the math we teach in school is context-sensitive. It matters what set your inputs are from and what set the output is supposed to be in. We usually don't mention that we're doing math on real numbers, we assume that based on the context.

22 + 8 = 6

This would be incorrect in an average math class, but when you're dealing with the clock then it's something understands. 8 hours after 10 pm is 6 am.

ab = ba

We teach the commutative property as though it is universal, but it isn't. With real numbers? Sure! Swap two matrices though and you're in trouble.

I don't think kids should necessarily be taught differently, but there is definitely (implicit) context involved in math. Even in geometry: the inner angles of a triangle add up to 180 degrees, right? But in spherical geometry the sum of the inner angles of a triangle can be larger.

Re: The Unreasonableness of Math Is Context Independence

#33

Post by lordnacho »

asimpletune wrote: post #31
lordnacho wrote: post #19

Earlier quoted context omitted.

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.

What about "experts"? Like, I would say there are people out there who are valuable because they know how to do stuff. It's not required for them to explain how they do it, and often times it's the case that they can't, otherwise they would simply explain and we'd all be experts. Rather, we need them precisely because the results they deliver are not able to be broken down into a sequence of steps that anyone could f…

Sure, tacit knowledge is a real thing that people talk about. But I see expertise as a kind of navigation through murky waters rather than "theory" which tends to be an explicit thing.

One thing experts can do is tell you when a theory is applicable.

Re: The Unreasonableness of Math Is Context Independence

#34

Post by qboltz »

agarsev wrote: post #28
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…

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 style models for cellular proliferation, followed by network analysis for tissue growth.

You can take that up further and further, I'm sure you're somewhat familar.

My question is, even if you have some examples, what do you find to be some kind of theoretical limit to the modelling that would actually be accurate?

Not a limit to the accuracy, that must simply always exist, but a limit to what can be successfully modeled at least to "acceptably correct" for use in some application?

Re: The Unreasonableness of Math Is Context Independence

#35

Post by bannedbybros »

drewcoo wrote: post #4

I chased 2-3 linked articles deep am still wondering what is meant by reasonable here. Or "reasonably effective." Is that just an example of the ineffective reasonability of essays? My best guess at this point is that reasonable is what a person expects. And if that's so, it's subjective. And math abstracts realities into imperfect but objective simulacra. So I think the claim is that math is made of abstract rules.…

No post body was provided.

Re: The Unreasonableness of Math Is Context Independence

#36

Post by bannedbybros »

danielmorozoff wrote: post #16

>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/

No post body was provided.

Re: The Unreasonableness of Math Is Context Independence

#37

Post by pfdietz »

morelisp wrote: post #25
nicoburns wrote: post #24

Earlier quoted context omitted.

I have a good friend who works as a high school physics teacher, and this is apparently exactly how they teach: they teach the intuition behind the problems so that the kids can visualise them.

And this works up until about high school physics, but not too much further. Not only do many interesting mathematical objects lack some intuitive basis, some are interesting specifically because they behave counter-intuitively.

Mathematics builds abstractions, but at any level you first see the less abstract things that motivated the abstraction.

Re: The Unreasonableness of Math Is Context Independence

#38

Post by zozbot234 »

Aerroon wrote: post #32
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…

But the math we teach in school is context-sensitive. It matters what set your inputs are from and what set the output is supposed to be in. We usually don't mention that we're doing math on real numbers, we assume that based on the context. 22 + 8 = 6 This would be incorrect in an average math class, but when you're dealing with the clock then it's something understands. 8 hours after 10 pm is 6 am. ab = ba We teach…

Yes, abstracting away from the context only works if you can tell that the problem really is context-independent. This works well with apples, not so well with e.g. commutativity once you get to things like matrices. So abstraction (from "apples" to "numbers" to "matrices") can sometimes reintroduce context that had previosly been discarded.

Re: The Unreasonableness of Math Is Context Independence

#39

Post by pmichaud »

drewcoo wrote: post #4

I chased 2-3 linked articles deep am still wondering what is meant by reasonable here. Or "reasonably effective." Is that just an example of the ineffective reasonability of essays? My best guess at this point is that reasonable is what a person expects. And if that's so, it's subjective. And math abstracts realities into imperfect but objective simulacra. So I think the claim is that math is made of abstract rules.…

In the original essay, "reasonable" specifically meant "rational" in the sense of "able to be deduced from first principles." The point of the whole essay was that math was was spookily good at modelling reality, empirically speaking, but that fact is super weird considering we have no rational basis to expect that to be the case--ie. we have no first principles based in physical reality that we could use to deduce that math/twiddling with the relationship between symbols using rules we basically just made up should be able to model reality the way it does, and isn't that a strange mystery to contemplate.

Most essays that use the phrase really just mean "surprisingly effective," which is a pet peeve of mine, but I think this essay gets a pass because it's trying to actually address that "strange mystery."

Re: The Unreasonableness of Math Is Context Independence

#40

Post by leonry »

_xnmw wrote: post #14
jacobolus wrote: post #13

Earlier quoted context omitted.

> 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.

How about Hilbert‘s 16th problem, would that satisfy your conditions for a counterclaim to your assertion? For a short summary, see https://mathoverflow.net/a/116530/60775 or https://valentermz.github.io/documents/slides/olivetti-2013-... for instance.

In any case, there is still the foundational crisis in the late 19th and early 20th century that‘s worth a mention.

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