31 January 2016


A well-designed quote conveys interesting and often powerful ideas and makes one think for a time disproportionately longer than the length of the quote.

I have decided to add a quotes collection on my page. I try to restrict to quotes pertaining to Academia, but I can't promise I won't divert.

Anyway, here is a quote that impressed me today:
  • Most [people] discover that they have often be working in the affine plane without realizing that it could be so designated. (H.S.M. Coxeter)
Coxeter is the Grand Man of classical geometry who lived in the 20th century. Some of his books should be compulsory in middle/high school education. Alas, exactly the opposite is happening: to my total dismay I learned that Swedish schoolchildren never learn why the angle bisectors of a triangle pass through the same point. Instead, I was told, a Stockholm school made computer games a mandatory course. Not designing computer games, mind you, but playing them. No wonder that university students have no clue that there is a proof of the Pythagorean theorem. Yes, they can state it (and so can the greengrocer) but not only do they not know a proof, but--what's worse--they're not even aware that a proof is needed.

Back to the affine plane, however, even under ideal schooling circumstances, what makes the affine plane so elusive is the quick passage from Euclidean geometry to coordinate geometry (thanks Descartes!). Typically (I guess not any more), a schoolchild would learn a lot of Euclidean geometry in school but then pass on quickly to linear algebra in his/her first-year university course. The affine (and so goes for the projective) plane, responsible for a lot of elementary mathematics, would go by very quickly, if at all.

Pondering the Coxeter quote carefully is, perhaps, all is needed in order that the affine plane be re-surfaced from the stack of one's toolset.

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What measure theory is about

It's about counting, but when things get too large.
Put otherwise, it's about addition of positive numbers, but when these numbers are far too many.

The principle of dynamic programming

max_{x,y} [f(x) + g(x,y)] = max_x [f(x) + max_y g(x,y)]

The bottom line

Nuestras horas son minutos cuando esperamos saber y siglos cuando sabemos lo que se puede aprender.
(Our hours are minutes when we wait to learn and centuries when we know what is to be learnt.) --António Machado

Αγεωμέτρητος μηδείς εισίτω.
(Those who do not know geometry may not enter.) --Plato

Sapere Aude! Habe Muth, dich deines eigenen Verstandes zu bedienen!
(Dare to know! Have courage to use your own reason!) --Kant