I really liked this, and it's also why it's so hard to teach mathematics, which is part of my current job.<p>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!
I chased 2-3 linked articles deep am still wondering what is meant by reasonable here. Or "reasonably effective."<p>Is that just an example of the ineffective reasonability of essays?<p>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.<p><a href="https://rationalwiki.org/wiki/Deepity" rel="nofollow">https://rationalwiki.org/wiki/Deepity</a>
Math is not context-independent.<p>The meta-mathematical assumptions (axioms) are the context.
Different axioms produce different truths; or if you want - they produce different Mathematical universes [1].<p>Maths is relative like Physics is relative - it depends on your frame of reference [2].<p>1. <a href="https://en.wikipedia.org/wiki/Universe_(mathematics)" rel="nofollow">https://en.wikipedia.org/wiki/Universe_(mathematics)</a><p>2. <a href="http://math.andrej.com/2012/10/03/am-i-a-constructive-mathematician/" rel="nofollow">http://math.andrej.com/2012/10/03/am-i-a-constructive-mathem...</a>
>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.<p>I wonder how you square this idea of generalization with Godel's incompleteness theorems?<p><a href="https://plato.stanford.edu/entries/goedel-incompleteness/" rel="nofollow">https://plato.stanford.edu/entries/goedel-incompleteness/</a>
IMO math seems effective because everything that works is called math. So yeah, that quote from the beginning is right, it's selection.<p>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.<p>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.
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.<p>The context is sometimes everything.