Mathematicians Like to Optimize

To get cafe au lait one should carry coffee to milk and not milk to coffee.

The idea is of the kind that mathematicians are prone to conceiving. Indeed, we are talking numbers here. If you go to a fridge to pick up a carton of milk, then back to the table where your cup of coffee is getting colder with every step, pour your milk into the cup, hurry towards the fridge and finally rush back to the table - this would take 4 trips between the table and the fridge. On the other hand, if the first time you carry your cup along, pour milk by the refrigerator and return back to the comfort of the table, sipping the coffee on the way and enjoying its warmth too - all in two trips - would not you think the second way is by far more preferable.

Two aspects of the opening sentence deserve our attention.

  1. Are mathematicians really liable to think up something of this sort, do they see numbers everywhere? A testimony to this effect comes from a chapter in the J. A. Paulos' book Beyond Numeracy.

  2. Regardless, what I just did was solving an optimization problem: one is sitting by the table with a cup of coffee. Somewhat away from the table a refrigerator is situated which is known to contain a half full carton of milk. What is the least number of trips between the table and the refrigerator would it take to prepare cafe au lait?

Much of Mathematics is devoted to optimization problems. Furthermore, optimization permeates the whole philosophy of math development. Now, metamathematics is the science that studies mathematics. Rota makes the following metamathematical observation:

It is an article of faith among mathematicians that after a new theorem is discovered, other simpler proofs of it will be given until a definitive one is found.

A discussion of what is exactly meant by a definitive proof is too philosophical for my purposes. However, we may be sure (and the history seems to confirm the assertion) that mathematicians will not rest until a sufficiently simple proof is discovered. I wish to suggest that optimization is germane to simplification. Somehow simplified proofs deepen our knowledge of whatever Mathematics is about.

References

  1. J. A. Paulos, Beyond Numeracy, Vintage Books, 1992
  2. G.-C. Rota, Indiscrete Thoughts, Birkhäuser, 1997

Related material
Read more...

A Sample of Optimization Problems

  • 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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