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Would it be fair and/or relevant to say that all areas of math can be represented geometrically
by cee_el2 10y ago
Would it be fair and/or relevant to say that all areas of math can be represented geometrically
- lawpoop 10y agoAs a layperson, I don't know how you could represent infinite sets, unless you just pick a random, finite symbol.
- jonsen 10y agoAny line segment, for example, is an infinite set of points.
- lawpoop 10y agoYes, as a mathematical object, that is true. But any symbol we render in the physical world is going to be finite. Drawing a straight mark and calling it an infinite set of points isn't really different from drawing a sideways 8 and calling it an infinite set. It's a symbol that represents something else.
- nkassis 10y agoCheck out the Riemann sphere https://en.wikipedia.org/wiki/Riemann_sphere https://en.wikipedia.org/wiki/Riemann_sphere I thought it was cool when I first saw it. Infinity in the complex plane is one of the poles.
- thanatropism 10y agoI dunno. There's a lot of conceptual metaphors and sometimes even structure from geometry in higher maths, but this is sometimes very abstract and merely makes reference to non-geometric ideas about geometry previously developed. For example, in the kind of stochastic calculus quant finance people learn, there's an isometry (like a transformation that preserves size in some sense) between two very different kinds of continuous, non-enumerable spaces. Geometric intuition is of no help there -- you did learn what an isometry was in high school so you know the word, but you can't see function spaces in any way, shape or fashion. OTOH what I'm working on for my dissertation is "geometric integrators" for certain kinds of differential equations where the isometries and references to geometry are more direct; basically, the most common numerical solvers for initial value problems in ODEs preserve certain invariants and are useful for many many problems, but sometimes you want computation to preserve some sense of volume -- clasically, in mechanics. So even though many problems are too abstract to be seen, the notion of volume conservation is the best way to acquire the basic notions of the field.
- cee_el2 10y agoThanks for your insightful reply, quant isometry sounds like a great counterexample, will read up on it
- cee_el2 10y agohttps://www.quantamagazine.org/20130917-a-jewel-at-the-heart-of-quantum-physics/ https://www.quantamagazine.org/20130917-a-jewel-at-the-heart... To even hint that the above breakthrough could possibly be relevant to this thread's subject would qualify as a form of quantum mysticism, wouldn't it ?
- threepipeproblm 10y agoEvan Miller has a fascinating analysis of this question via a critical analysis of a Bret Victor essay. http://www.evanmiller.org/dont-kill-math.html http://www.evanmiller.org/dont-kill-math.html