It had long been sought in vain, he said, to demonstrate the axiom known as Euclid’s fifth postulate and this search was the start of the crisis. Euclid’s postulate of parallels, which states that through a given point there’s not more than one parallel line to a given straight line, we usually learn in tenth-grade geometry. It is one of the basic building blocks out of which the entire mathematics of geometry is constructed.Pirsig, Robert M.. Zen and the Art of Motorcycle Maintenance (pp. 234-235). HarperCollins. Kindle Edition.
All the other axioms seemed so obvious as to be unquestionable, but this one did not. Yet you couldn’t get rid of it without destroying huge portions of the mathematics, and no one seemed able to reduce it to anything more elementary. What vast effort had been wasted in that chimeric hope was truly unimaginable, Poincaré said.
Finally, in the first quarter of the nineteenth century, and almost at the same time, a Hungarian and a Russian—Bolyai and Lobachevski—established irrefutably that a proof of Euclid’s fifth postulate is impossible. They did this by reasoning that if there were any way to reduce Euclid’s postulate to other, surer axioms, another effect would also be noticeable: a reversal of Euclid’s postulate would create logical contradictions in the geometry. So they reversed Euclid’s postulate.
Lobachevski assumes at the start that through a given point can be drawn two parallels to a given straight. And he retains besides all Euclid’s other axioms. From these hypotheses he deduces a series of theorems among which it’s impossible to find any contradiction, and he constructs a geometry whose faultless logic is inferior in nothing to that of the Euclidian geometry.
Apologies for the long quote, but I remain fascinated by this excerpt. Must be more than twenty years ago that I first read it, but it still boggles my mind that we don’t teach students in school that the sum of all angles in a triangle can be equal to, less than or more than one hundred and eighty degrees.
Now, these alternate geometries aren’t “wrong”. They simply become possible once you get it into your head that Euclid’s fifth postulate need not be true. That “through a given point there’s not more than one parallel line to a given straight line” isn’t written in stone.
And once you imagine a world in which the fifth postulate doesn’t hold, you can build up a perfectly reasonable, entirely consistent, completely non-contradictory alternate geometry. A completely different geometry from the one that we did battle with in school, but an entirely valid one.
A different point of view, in other words, and a whole other reality that emerges once you broaden your mind enough to accept it.
As Mark Twain put it, ““It ain’t what you don’t know that gets you into trouble. It’s what you know for sure that just ain’t so.”
Euclid, you could argue, knew for sure that parallel lines don’t meet. And that got him into trouble, in the sense that he couldn’t then imagine the worlds that Bolyai and Reimann constructed.
Which brings me to my question of the day: What are your blind spots? Or put another way, which are your fifth postulates?
What do you know for sure that just ain’t so?
Are markets always and everywhere efficient?
Is your favorite political leader superman (or superwoman)?
Are governments necessarily better than markets?
Are your political opponents always wrong simply because they are your opponents?
I can go on and on, but the point is to ask yourself some hard, uncomfortable questions. What do you think you know for sure that just ain’t so? The more you choose to not ask yourself this question, the more you’re setting yourself up for trouble.
Here are some of mine:
- Second helpings are a good idea, third helpings are even better.
- Caste is a stupid concept
- The Pune Municipal Corporation will never build a footpath in my neighboorhood.
- All politicians (past, present and future) are driven by votes and the desire to acquire and retain power.
I have more, of course, this is a very incomplete list. I’m always willing to listen to arguments, and especially so when it comes to my axioms, but know that you will have to work extra hard to convince me about turning my back to any one of these. You probably won’t succeed, and this means that I, like Euclid, have trouble imagining a world in which my axioms might not hold.
And if I can’t imagine a world in which they don’t hold, I find it impossible to analyze it, empathize with it, or help better it.
Difficult as it may seem, all of us owe it to ourselves to imagine worlds different from the ones that we have constructed in our own heads. Learn to try and let go of your fifth postulates, and a good place to begin is by asking ourselves what these might be.
All the best!