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!
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.
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.
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.
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. A tautology? A deepity? I must be missing something.
It's a play on the "unreasonable effectiveness of math" essay right?
I took it to mean the "unreasonable" ingredient which makes math so effective is context independence - since that's something which is not so easily attainable in other fields.
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."
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].
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)
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.
>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?
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.
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.
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 number theory work that eventually ended up appearing in cryptography, but if something in the real world happens ahead of it, math will expand to swallow it.
Think of a scientific theory that isn't described with some kind of math. I'm not sure it can be done. My sense is that whatever you think of, even if it's completely new, will be called math. For instance general relativity relied on some quite new concepts at the time, but nobody would point at it and say it wasn't math.
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 ;)
> 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.
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 wrong at some point. If math was just "anything that works", then we'd be regularly purging stuff from the "math" label, but I can't think of anything that was called "math" in history and not called "math" now.
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.
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.
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.
[+] [-] red_admiral|4 years ago|reply
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!
[+] [-] Aerroon|4 years ago|reply
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.
[+] [-] nicoburns|4 years ago|reply
[+] [-] xtiansimon|4 years ago|reply
_"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...
[+] [-] drewcoo|4 years ago|reply
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. A tautology? A deepity? I must be missing something.
https://rationalwiki.org/wiki/Deepity
[+] [-] pvg|4 years ago|reply
https://hn.algolia.com/?dateRange=all&page=0&prefix=true&que...
it's also a meme-ish title thing, sort of like 'considered harmful'
https://hn.algolia.com/?dateRange=all&page=0&prefix=true&que...
[+] [-] skohan|4 years ago|reply
I took it to mean the "unreasonable" ingredient which makes math so effective is context independence - since that's something which is not so easily attainable in other fields.
[+] [-] pmichaud|4 years ago|reply
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."
[+] [-] bannedbybros|4 years ago|reply
[deleted]
[+] [-] ukj|4 years ago|reply
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...
[+] [-] giomasce|4 years ago|reply
(yes, I know I am oversimplifying something, take this at the "philosophical" level)
[+] [-] andrewflnr|4 years ago|reply
[+] [-] danielmorozoff|4 years ago|reply
I wonder how you square this idea of generalization with Godel's incompleteness theorems?
https://plato.stanford.edu/entries/goedel-incompleteness/
[+] [-] guimplen|4 years ago|reply
[+] [-] szundi|4 years ago|reply
[+] [-] bannedbybros|4 years ago|reply
[deleted]
[+] [-] lordnacho|4 years ago|reply
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 number theory work that eventually ended up appearing in cryptography, but if something in the real world happens ahead of it, math will expand to swallow it.
Think of a scientific theory that isn't described with some kind of math. I'm not sure it can be done. My sense is that whatever you think of, even if it's completely new, will be called math. For instance general relativity relied on some quite new concepts at the time, but nobody would point at it and say it wasn't math.
[+] [-] agarsev|4 years ago|reply
[+] [-] d_tr|4 years ago|reply
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.
[+] [-] _xnmw|4 years ago|reply
[+] [-] unknown|4 years ago|reply
[deleted]
[+] [-] ggm|4 years ago|reply
The context is sometimes everything.
[+] [-] giomasce|4 years ago|reply
[+] [-] _xnmw|4 years ago|reply
[+] [-] anomosmarxist|4 years ago|reply
[deleted]