This is kind of a big picture question. I just counted up all the symbols used in normal mathematics and, give or take, there are probably around 150 of them, tops. And that's really stretching things. I am including:

  • the transpose symbol in linear algebra,
  • the gamma function symbol,
  • the direct sum symbol for two vector spaces,
  • the tensor product symbol,
  • ...

I am including a lot of seemingly esoteric stuff! Even so, no matter what formula spits out of Wolfram Alpha, or no matter where you look in the Handbook of Mathematical Functions, for practical applied math purposes there really aren't that many symbols. That got me to thinking: we have these famous numbers $e$, $i$, and $\pi$. They are related by the famous Euler formula which blows everybody's mind when they first see it, except for, reportedly, Gauss.

$\pi$ relates to the circle. $e$ relates to rate of change -- relates to integration and differentiation somehow. $i$ gives us an extra number dimension to solve problems.

Question: Why aren't there more of these numbers? Is it the case that $99$% of all important mathematics in practice is covered by the rationals, operations like taking rational roots, the vast swath of anonymous, non-name-worthy irrational & transcendental reals, and $e$, $i$ and $\pi$?

What is it about these three numbers that makes them, in effect, practically the only important numbers in mathematics other than those that can be expressed in terms of regular numbers? Is it because this circle relationship, this extra dimension thing with $i$, and this rate of change thing with $e$, covers all the really important relationships between the dimensions? I really want someone to break this down for me and tell me why this is the case.

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    $\begingroup$ First, mathematics is hardly about numbers anymore. Second, if you are not a professional mathematician how can you tell which symbols are esoteric? $\endgroup$ – Asaf Karagila Jan 19 '13 at 8:11
  • $\begingroup$ Asaf, re math not being about numbers anymore, I am not sure I understand that statement completely. Don't all the letters ultimately represent sets of numbers, in the end? On the second point, I am sure I am wrong about calling anything "esoteric", maybe a bad choice of words. So I withdraw that adjective. It is just really interesting to me that so few numbers are so important and I wonder if anyone has a perspective on why that is the case. $\endgroup$ – user58450 Jan 19 '13 at 8:27
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    $\begingroup$ Mathematics is about numbers as much as linguistics and literature are about letters. $\endgroup$ – Asaf Karagila Jan 19 '13 at 8:33
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    $\begingroup$ If you're just counting symbols, there are not that many because it's hard to invent a new symbol when it's not available in standard fonts/keyboards/etc. Even if you're inventing something new, it's easier to just re-use an existing symbol and explain the meaning. $\endgroup$ – Ted Jan 19 '13 at 8:48

Have a look at http://en.wikipedia.org/wiki/Mathematical_constants where you will find any number of important mathematical constants which have no known expression in terms of $e$, $i$, and $\pi$. The Euler-Mascheroni constant, $\gamma$, is a biggie in analytic number theory. $\zeta(3)$ was immortalized by Apery. Feigenbaum's constant $\delta$ is super-important in dynamical systems and transition to chaos. Khinchin's constant $K$ is big in continued fractions. And so on.

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    $\begingroup$ Many thanks! I should have checked before I stated there were only three important constants. What I was really hoping for with this question was to see if some kind of deep geometric intuition exists -- e.g. perhaps the circle, the relationship via e between differentiation and integration, and i might (I was thinking) essentially encapsulate a huge amount of geometric information about the relationships between the dimensions, and therefore maybe that's why they show up all the time in formulas. But perhaps it's not so, as you have pointed to many examples of other important constants. $\endgroup$ – user58450 Jan 23 '13 at 9:11

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