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Let No One Ignorant of Geometry Enter

August 26, 2012

Those are the exact words of the gate of ancient Plato Academy.  More exactly,the complete version reads « mèdeis ageômetrètos eisitô mou tèn stegèn ».  Following that another line that said “Let non one unjust sneak in here”.  Wait, what has geometry has anything to do with “justice”?  To quote part of an article in platoDialogue [dot] org

“…geometry, as well as all other mathematical sciences, is not an end in itself, but only a prerequisite meant to test and develop the power of abstraction in the student, that is, his ability to go beyond the level of sensible experience which keeps us within the “visible” realm, that of the material world, all the way to the pure intelligible. And geometry, as can be seen through the experiment with the slave boy in the Meno (Meno, 80d1-86d2), can also make us discover the existence of truths (that of a theorem of geometry such as, in the case of the Meno, the one about doubling a square) that may be said to be “transcendant” in that they don’t depend upon what we may think about them, but have to be accepted by any reasonable being, which should lead us into wondering whether such transcendant truths might not exist as well in other areas, such as ethics and matters relating to men’s ultimate happiness, whether we may be able to “demonstrate” them or not.

So I went in search of the actual geometry that Plato was referring to.  To tell you the truth, right after my Math degree in NUS, I was still unsure of the 5 axioms of Euclid.  I was not able to quote any of the all important geometry axioms.  And shamefully, I usually knocked my head in agreement when I attended graduate school in Maths and some philosophers and mathematicians alike spoke of the great work of the parallel postulate, also called Euclids fifth postulate.  In fact, all through Primary school to JC, I did not distinguish between algebra from geometry.  I simply took my ‘A’ level Math exams and I did not even know Mechanics from Statistics.

“Isn’t Math just problems you solve in 10-year series and get an answer?”

Yes, you can say it’s pathetic.  And until I have a job, I did not spend time to get in this very exciting and intellectual cross between philosophy and elements of Plato’s or Euclid’s geometry.  Even at graduate school level, I just assumed that if some kind of GRAND postulate was removed then there is no need to panic, all the other 4 axioms still remain consistent. Yes, it’s true the world will not be Euclidean anymore, it simply becomes hyperbolic.   All the sin(x), cos(x) and tan(x) have an extra edge (no puns intended) to it, they simply become sinh(x), cosh(x) and tanh(x).

Seriously again, up until graduate school, I did not see the philosophical shift in all these argument, and simply accept the world with an extra “h”.  “What’s the big deal?”  I took it as if we just discovered the irrational numbers and the world should not collapse or come to an end just because of that.  And since I studied Einstein’s Special Theory of Relativity in details mathematically, I thought “Hey, this wasn ‘t so bad, Einstein already said we are in this hyperbolic Minkowski geometry, so Dude, just chill, no need to panic!  \cosh^2(x) - \sinh^2(x) is still going to give us the number ONE!

So? If you want, try digesting these 3 axioms, where :

  1. Every two distinct points must describe a line.
  2. Every two distinct line must intersect at a unique point.
  3. There exists 4 points that are not lying on the same line (or non-collinear).

Only when I think very deep and started coming up with examples and non-examples then I realize what all these is about.  That’s what the education I received was lacking.  It did not challenge me to think and come up with system by myself.  It throw “knowledge” at me, make me very familiar with it and then went on to test me.  Appreciation comes rarely if not never.  (If you want to know what is the answer, it gave rise to the axioms for projective planes.)

“Hitherto, I still dare not enter this Gate of the Plato’s Academy.”

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