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Paul Halmos states:

It is frequent in mathematics that every instance of a concept of seemingly great generality is in essence the same as a small and concrete special case.

What are examples of such: Small, concrete, individual special cases that turn out to be unexpected archetypes covering wide ranges of unexpected generality?

One example is the Cantor Set. This seems like a very unique, artificial, one-off construction. Until you learn the topology showing that huge ranges of sets are equivalent to or built from it (those who know more topology than me can fill in the details.)

What other examples exist of "small and concrete special cases" that turn out to capture "every instance of a concept of... great generality"?

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    $\begingroup$ I remember back in middle school, whenever we had a theorem about triangles, the teacher would draw a particular triangle on the blackboard and illustrate the theorem on this particular example. And we'd all pretend that instead of a particular triangle, the triangle on the blackboard was somehow "general" and representative of all triangles. $\endgroup$
    – Stef
    Commented Mar 19, 2023 at 21:33

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Another example is given by the Dirac delta function. This is the simplest example of a Schwartz distribution and displays a few paradoxical properties that can be satisfied by these generalized functions: zero at any non-zero points, infinite at the origin, with integral equal to one.

Laurent Schwartz said that one of the early surprises of his theory was the fact that smooth functions are dense in the space of distributions. The Dirac function is approximated by mollifiers.

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To say that it captures every instance would be wrong of course, but I think understanding the structure and representation theory of $\mathfrak{sl}_2$ goes a long way towards understanding the structure and representation theory of all semisimple Lie algebras over $\mathbb C$, and even further than that. I bet that most people have maybe looked at $\mathfrak{sl}_3$ for a more general case, but that is enough to build intuition.

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    $\begingroup$ While it is undeniable that it is important to thoroughly understand the $\mathfrak{sl}_2$ case, to say that every general instance is in essence like it seems like a stretch. A bit like saying that cyclic groups (or dihedral groups to be a bit more credible) well represent the general notion of a finite group. For instance, $\mathfrak{sl}_2$ gives a rather simplistic example of the nature and role of the Weyl group. $\endgroup$ Commented Mar 20, 2023 at 14:21
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    $\begingroup$ I was told by a great Lie theorist that to understand things you have to do sl_4. $\endgroup$ Commented Mar 22, 2023 at 23:49
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The Smale's horseshoe is a famous example in the theory of hyperbolic dynamical systems.

The work of Smale on the Poincaré conjecture was using gradient flows and led him to conjecture that "most" flows on varieties have only finitely many periodic points. Norman Levinson was fast to point out an example of a dynamical system studied by Cartwright and Littlewood with infinitely many periodic points, and such that any small perturbation of the system also has infinitely many periodic points. Littlewood said that this work was the most difficult one of his whole career.

Smale devised his horseshoe during his investigation of the Cartwright Littlewood paper. The story is recalled in an issue of the Mathematical Intelligencer, in an article entitled Finding a horseshoe on the beaches of Rio. Here is an excerpt:

I worked day and night to try to resolve the challenge to my beliefs that the letter posed. It was necessary to translate Levinson's analytic arguments into my own geometric way of thinking. At least in my own case, understanding mathematics doesn't come from reading or even listening. It comes from rethinking what I see or hear [...] I eventually convinced myself that indeed Levinson was correct, and that my conjecture was wrong. Chaos was already implicit in the analyses of Cartwright and Littlewood. The paradox was resolved, I had guessed wrongly. But while learning that, I discovered the horseshoe!

Smale's horseshoe is now a landmark in the theory of chaotic systems.

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  • $\begingroup$ When talking about dynamical systems, the Mandelbrot set comes to mind, but I would not say that it is small. It embodies the unexpected complexity that can result from the iteration of a simple function though. $\endgroup$
    – coudy
    Commented Mar 19, 2023 at 11:59
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Two examples come from elementary geometry: Euclid's Fifth Postulate and the "ambiguous case" of triangles.

Euclid's Fifth Postulate sounds on the surface like a limited special case, especially the way Euclid phrased it:

If a line segment intersects two straight lines forming two interior angles on the same side that are less than two right angles, then the two lines, if extended indefinitely, meet on that side on which the angles sum to less than two right angles.

Yet it's equivalent to wide ranges of geometry, such as scale invariance, the existence of squares, the existence of similar but not congruent figures, the Pythagorean theorem, and, shockingly, the seemingly undeniable version of Tarski:

Given any angle and any point v in its interior, there exists a line segment including v, with an endpoint on each side of the angle.

Next: The "ambiguous case", consisting of two triangles with side lengths $a, b, c$ and $a, b, c'$ respectively, $c \neq c'$, and angles $A, B, (\pi - B) - A$, and $A, \pi - B, B - A$. The fact that two (but no more) triangles can have such similar, but distinct, properties comes up in in many unexpected ways in geometry, both Euclidean and neutral. Often promising lines of proof fail, and it turns out because they somehow run a foul of this fact; fixing the proof requires limiting it to avoid this case.

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    $\begingroup$ On the topic of Euclid, because the ancient Greeks lacked a notation to describe the general term of a sequence or an arbitrary finite product, many proofs in the number theory sections of Euclid's Elements that are written today with induction only treat a specific instance of the inductive step, usually going from 2 terms to 3 terms rather than from $n$ terms to $n+1$ terms. The great generality of the modern inductive step is presented only in a concrete special case, but it's enough to see how the general case would proceed. $\endgroup$
    – KCd
    Commented Mar 19, 2023 at 20:52
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The Von Koch snowflake is a classical example of a curve of infinite length and no tangent. It is also a well-known fractal.

Von Koch snowflake

CC by SA 4.0 Juarez Bochi.

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