The Unreasonableness of Math Is Context Independence • Hacker News Board

Re: The Unreasonableness of Math Is Context Independence

#21

Post by guimplen » Sat, Feb 12, 2022, 11:21 AM UTC

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/

Why it is even relevant? Godel's incompletness theorem applies to almost any strong formal system with self-referential abilities, so if you want to do a good formalization, you bound to have one that satisfy Godel's theorem requirement.

Re: The Unreasonableness of Math Is Context Independence

#22

Post by d_tr » Sat, Feb 12, 2022, 11:34 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…

> they don't necessarily have to correspond with anything real, ...

I would just say they correspond to encoded thought processes, encoded reasoning. If you can take a thought process and describe it in terms of sets and relations (i.e. subsets with certain properties), you have a mathematical structure and you can start trying to prove theorems.

You spend time thinking about a problem, then hopefully you start recognizing patterns, then you take the reasoning, clean it up, abstract it and generalize it to increase its ultimate utility, and package it for others to reuse and build upon.

> everything that works is called math

It is just fortunate that people have been able to "package" a lot of stuff this way. Like Riemann did with his geometry for example. It is not that mathematicians just decide to "take over" everything.

Re: The Unreasonableness of Math Is Context Independence

#23

Post by red_admiral » Sat, Feb 12, 2022, 11:38 AM UTC

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 confused stare followed by "what's a McGuffin?". Except of course for the one kid who has no problem with the math because they misheard it as McMuffin and could visualise that!

Re: The Unreasonableness of Math Is Context Independence

#24

Post by nicoburns » Sat, Feb 12, 2022, 11:43 AM UTC

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…

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.

Re: The Unreasonableness of Math Is Context Independence

#25

Post by morelisp » Sat, Feb 12, 2022, 11:59 AM UTC

nicoburns wrote: post #24

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…

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.

Re: The Unreasonableness of Math Is Context Independence

#26

Post by morelisp » Sat, Feb 12, 2022, 12:01 PM UTC

_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.

You can't open the conversation offering "the entire history of math since ancient times" and then demand thoroughly modern things like "an actual published result that was cited by other results" as counter-evidence.

Nonetheless many of the examples in the above links still fit your criteria.

Re: The Unreasonableness of Math Is Context Independence

#27

Post by szundi » Sat, Feb 12, 2022, 12:11 PM UTC

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/

Easy: have you ever had a an epiphany how idiot you were and did not understand what all people told you for a long time? Then you realized something and everything just snapped into place. You now understand all. Paff! An axiom or condition was just changed in you by an experience. Suddenly you get now what others told you. Welcome to an other system of understanding.

Re: The Unreasonableness of Math Is Context Independence

#28

Post by agarsev » Sat, Feb 12, 2022, 12:34 PM 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…

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 those sciences there are a lot of things which are not well-defined, but rather rely on human intuition and language. If you go "upwards" in the stack, you find things like biology where there is a lot of very formal scientific knowledge which is not maths. And I work in linguistics, so just imagine what it's like at this level ;)

Re: The Unreasonableness of Math Is Context Independence

#29

Post by _xnmw » Sat, Feb 12, 2022, 1:18 PM UTC

TheOtherHobbes wrote: post #20

_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. 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 t…

Sure there are incomplete proofs and dead ends. But I have yet to see an example of an "established mistake" which disproved an entire line of research that depended on the mistake. Sure, mistakes have been published. But it never led to an entire branch of "knowledge" based on a false belief -- something that happens regularly in other fields.

Re: The Unreasonableness of Math Is Context Independence

#30

Post by _xnmw » Sat, Feb 12, 2022, 1:20 PM UTC

morelisp wrote: post #26

_xnmw wrote: post #14

Earlier quoted context omitted.

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.

You can't open the conversation offering "the entire history of math since ancient times" and then demand thoroughly modern things like "an actual published result that was cited by other results" as counter-evidence. Nonetheless many of the examples in the above links still fit your criteria.

A published result that cites another result is not "a thoroughly modern thing". Mathematicians have been citing each other since Pythagoras and Avicenna.