The Distance to Look Your Best

For every object there is a distance at which it looks its best.

In his wonderful book A Mathematician's Apology G. H. Hardy, one of the foremost analysts of the first half of this century, puts forward a thesis that ... a mathematical achievement is the most enduring of all. ... Archimedes will be remembered when Aeschylus is forgotten, because languages die and mathematical ideas do not."

At the end of the book, Hardy cites a remark by Dr. C. P. Snow with regard to a thesis to the effect that mathematical fame is a little too anonymous to be wholly satisfying. "We could form a fairly coherent picture of the personality of Aeschylus (still more, of course, of Shakespeare and Tolstoi) from their works alone, while Archimedes and Eudoxus would remain mere names."

Following is the final paragraph of the book.

"Mr. J. M. Lomas put this point more picturesquely when we were passing the Nelson column in Trafalgar Square. If I had a statue on a column in London, would I prefer the column to be so high that the statue was invisible, or low enough for the features to be recognizable? I would choose the first alternative, Dr. Snow, presumably, the second."

This is how it often happens in real life. The one who could ask the right question did not care to. While the one who apparently would care to find out did not know to inquire. I think if it comes to having a statue, the important question is What size of the column the statue looks its best?

While I can't give a definite answer to this question I can assert with complete certainty (disregarding, of course, other people tastes) that, given a statue, there indeed exists a certain distance (perhaps not defined uniquely) from which the statue would look most attractive. This is what is known in mathematics as a proof of pure existence. Its application underlies another Hardy's thesis that Mathematics, a few minor applications aside, is completely useless.

I base my assertion on the idea that any object's attractiveness is a continuous function f(x) of the distance x from which the object is viewed. I would call this Statement A. Statement B claims that f(x) attains its maximum. I'll show that the implication A=>B is a consequence of a certain property of continuous functions. We already had an example how a simple property of continuous functions helps prove a nontrivial result. Here I plan to use another one. But this is to be understood at the outset. Assume A=>B has been established. One obvious way to derive B is to apply modus ponens. Which means first proving A, then proving A=>B. B will then follow.

Now we come to the reason why Hardy considered Mathematics useless. In so far as A is a statement about "real world" phenomena, Mathematics is helpless to prove it. Mathematics can supply reasons to believe this statement (e.g., small change in the distance results only in a small change in attractiveness) but there is absolutely no way to prove A rigorously. In the words of J. E. Littlewood, "... A by the nature of the case is incapable of deductive proof, for the sufficient reason that it is about the real world..." Thus it's a question of accepting or not accepting A in the first place. However, note again that (in Hardy's view) once A has been accepted as a valid mathematical statement it actually ceased to be directly related to the "real world" phenomena. Thus Hardy would insist that what follows still has nothing to do with any real statue or any real column.

First I make two further assertions:

  1. Placed at infinity, the statue is invisible and, therefore, has attractiveness of 0.
  2. At the distance 0 too, one can sense the statue, perhaps smell it but definitely not see it. So f(0)=0.

The function f is defined for all x≥0. From 1 we can also say that f()=0. (There is no cheating here believe me. I just cut corners a little to avoid getting carried away.) If we bend now the semiline x≥0 into a circle (topologically this is OK) we get a continuous function defined on what's is called a compact set. The following theorem then applies.

Theorem

A continuous function defined on a compact set attains on this set its maximum and minimum values.

Remark

As one would expect, there are various problems related to viewing a statue placed on a pedestal. Regiomontanus (1436-76) sought the distance from which the statue on the plinth subtended the largest angle.

Compactness

Compact sets are ubiquitous in Mathematics with a variety of different properties many of which are often equivalent. Thus there is a good deal of various definitions as well. The one which is important here is that a set is compact iff it's both bounded and closed.

Aeschylus (525-456 B.C.)

A Greek tragic dramatist. Wrote 90 plays of which only 7 have survived: The Suppliant Women, The Persians, the Seven Against Thebes, Prometheus Bound, a trilogy Orestia (Agamemmon, Libation Bearers, Eumenides.)

Eudoxus (400-350 B.C.)

A student at Plato's Academy. Known as a mathematician and astronomer who substantially advanced number theory, and gave the first systematic explanation of the motion of the Sun, Moon, and planets.

References

  1. G. H. Hardy, A Mathematician's Apology, Cambridge University Press, 1994.
  2. Encyclopaedia Britannica.
  3. Littlewood's Miscellany, B. Bollobas (ed), Cambridge University Press, 1990.
  4. P. J. Nahin, When Least Is Best, Princeton University Press, 2007 (Fifth printing).

Related material
Read more...

A Sample of Optimization Problems

  • Mathematicians Like to Optimize
  • Reshuffling knights - castle defenders
  • Isoperimetric Theorem and Inequality
  • Viewing a Statue: the Problem of Regiomontanus
  • Fagnano's Problem
  • Minimax Principle Demonstration
  • Maximum Perimeter Property of the Incircle
  • Extremal Problem in a Circular Segment
  • Optimization in Four Variables with Two Constraints
  • Daniel Dan's Optimization in Three Variables
  • Problem in a Special Trapezoid
  • Cubic Optimization with Linear Constraints
  • Cubic Optimization with Partly Linear Constraints
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