The Unreasonableness of Math Is Context Independence
bellmar.medium.com
bellmar.medium.com
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!
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.
_"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...
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.
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.
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...
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].
1. https://en.wikipedia.org/wiki/Universe_(mathematics)
2. http://math.andrej.com/2012/10/03/am-i-a-constructive-mathem...
(yes, I know I am oversimplifying something, take this at the "philosophical" level)
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...
>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?
Multiple algorithms (proofs) of theorems. Multiple different theories explaining one and the same phenomenon.
The principle of equifinality is everywhere in open systems.
Only exemplified further that Logic, Programming and Mathematics are essentially three different tools for the same job.
I wonder how you square this idea of generalization with Godel's incompleteness theorems?
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.
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...
Nonetheless many of the examples in the above links still fit your criteria.
In any case, there is still the foundational crisis in the late 19th and early 20th century that‘s worth a mention.
> 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.
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.
One thing experts can do is tell you when a theory is applicable.
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.
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.
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?
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!
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.
The context is sometimes everything.