Fetishizing Formulas

In much recent popular writing on mathematics and physics there is a tendency to worship extravagantly those mathematical results that can easily be expressed as an equation or, much less frequently, as an inequality. Euler's equation e πi + 1 = 0, in particular, comes in for a lot of this.

Thus there are books like

Similarly, a couple of years there was a much publicized neuroscience experiment by neuroscientists Semir Zelki and John Paul Romaya, physicist Dionigi Benincasa, and mathematical Michael Aatiyah, which demonstrated that the contemplation of a beautiful equation by a mathematician stimulates the same areas of the brain as a beautiful work of art. Mathematicians (and a control group of non-mathematicians) were shown a collection of 60 mathematical formulas and were asked to evaluate their beauty, while their brain state was measured by fMRI. Of the 60, Euler's equation ended up ranked most beautiful and a rather horrible infinite series for 1/π due to Ramanujan was ranked ugliest.

No question, these equations are mostly wonderful things. But an obsessive focus on equations ends up giving a somewhat one-sided view of mathematics and in particular of beauty in mathematics. There are many theorems in mathematics that are not naturally expressed as equations which, to my mind at least, are no less beautiful. Here, off the top of my head, are seventeen.

  1. The square root of 2 is irrational.
  2. There are infinitely many prime numbers.
  3. If the 8 triples of points A,B,C; D,E,F; A,X,E; D,X,B; A,Y,C; D,Y,C; B,Z,F; E,Z,C are all collinear then X,Y,Z are collinear.)
  4. Every polynomial has a root in the complex plane.
  5. Every finite group is a subgroup of a permutation group.
  6. Every finite field has order pn for prime p.
  7. The quintic equation is not solvable by radicals.
  8. Every closed curve separates the inside from the outside (Jordan curve theorem).
  9. The figure that encloses the maximal area for a given perimeter is the circle.
  10. If function f is once differentiable in the circle B(x,r) in the complex plane, then the power series to f computed at x converges to f throughout B(x,r). series.
  11. π is irrational.
  12. π is transcendental.
  13. The real numbers are uncountable.
  14. If a is algebraic and not 0 or 1 and b is algebraic and irrational then ab is transcendental. (Gelfond-Schneider).
  15. If every chain in a partially ordered set has an upper bound, then the set has a maximal element (Zorn's lemma).
  16. Any compact surface is homeomorphic, either to the sphere, or to the connected sum of tori, or to the connected sum of projective planes.
  17. There are unprovable true statements in the first-order theory of the natural numbers (Gödel).
  18. Inference in first-order logic is complete. (Gödel).
  19. Every finite game has a Nash equilibrium.
  20. There is no algorithm to solve Diophantine equations.
  21. The first-order theory of real arithmetic is decidable (Tarski).

    I certainly don't insist on these; it's of course a matter of personal taste. Personally, I find these, overall, more fascinating, more compelling, more thought-provoking, more inspiring, more mysterious in the positive sense, than almost all of the 60 equations in the Zeki et al. experiment.

    The focus on equations also leads to an emphasis on some areas of math at the expense of others. By my count, of the 60 equations in the Zeki experiment, 27 were from real analysis, 8 from complex analysis, 9 from geometry, 4 from set theory (and 3 of those were trivial), 5 from number theory, 4 from algebra, 2 from linear algebra, and 1 from probability theory.

    Which are more beautiful, the equations or the theorems? It might be worthwhile trying the neuroscience experiments on these statements to see what those would show. One might say, perhaps, that they have a different kind of beauty. The equations are more like music; the theorems are more like language. Alternatively, perhaps the beauties of the equations are like the beauties of Scarlatti, Mondrian, and Emily Dickinson; whereas the beauties of the theorems are like the beauties of Wagner, Rembrandt, and Kafka. But I have no confidence, actually, that those kinds of comparison are in any way valid or useful. (I certainly doubt that they can be validated with brain imaging.)

    I don't at all suppose that this overemphasis on equations is having any impact whatever on actual mathematical practice. I do worry, though, that it may be distorting the view of mathematics in the larger world of math buffs, and, separately, distorting the study of beauty in mathematics — a niche subject area, but not a wholly negligible one.