Sunday, October 30, 2022

281: Pascal Vs Mathematics

 Audio Link

Welcome to Math Mutation, the podcast where we discuss fun, interesting, or weird corners of mathematics that you would not have heard in school.   Recording from our headquarters in the suburbs of Wichita, Kansas, this is Erik Seligman, your host.  And now, on to the math.

If you have enough interest in math to listen to this podcast, I’m pretty sure you’ll recognize the name of Blaise Pascal, the 17th-century French mathematician and physicist.   Among other achievements, he created Pascal’s Triangle, helped found probability theory, invented and manufactured the first major mechanical calculator, and made essential contributions to the development of fluid mechanics.   His name was eventually immortalized in the form of a computer language, a unit of pressure, a university in France, and an otter in the Animal Crossing videogame, among other things..   But did you know that in the final decade of his life, he essentially renounced the study of mathematics to concentrate on philosophy and theology?   

According to notes found after his death in 1662, Pascal had some kind of sudden religious experience in 1654, when he went into a trance for two hours, during which time he claims that God gave him some new insights on life.   At that point he dropped most of his friends and sold most of his possessions to give money to the poor, leaving himself barely able to afford food.   He also decided that comfort and happiness were immoral distractions, so started wearing an iron belt with interior spikes.   And, most disturbingly, he decided that math and science were no longer worthy of study, so he would devote all his time to religious philosophy.   He was still a very original and productive thinker, however, as in this period he wrote his great philosophical work known as the Pensees.   

There were a couple of reasons why he may have decided to give up on math and physics at this point.    Part of it was certainly just a change in emphasis:  he was concentrating on something else now, which he considered more important.   He also made comments about worldly studies being used to feed human egos, which means nothing in the eyes of God.   At one point he stated that he could barely remember what geometry was.

He never completely suppressed his earlier love of mathematics though.     Ironically, at several points in the Pensees he uses clearly mathematical ideas.    Most famously, you may have heard of “Pascal’s Wager”, where he discusses the expected returns of believing in God vs not believing, based on his ideas of probability theory.   You have two choices, to believe or not believe.   If you choose to believe, you may suffer a finite net loss from time spent on religion, if God doesn’t exist— but if he does, you have an infinite payoff.   Choosing not to believe offers at best the savings from that lifetime loss.   Thus, the rational choice to maximize your expected gain is to believe in and worship God.

As many have pointed out since then, there is at least one huge hole in Pascal’s argument:   what if you choose to believe in the wrong God, or worship him in the wrong way?   Many world religions consider heresy significantly worse than nonbelief.   He has an implicit assumption of Christianity, in the form he knows, being the only option other than agnosticism or atheism.   I think Homer Simpson once refuted Pascal’s Wager effectively, when trying to get out of going to church with his wife:  “But Marge, what if we chose the wrong religion? Each week we just make God madder and madder.”

Another surprising use of math in the Pensees is Pascal’s comments on why the study of math and science may be pointless in general.   He compares the finite knowledge that man may gain by these studies against the infinite knowledge of God:   “… what matters it that man should have a little more knowledge of the universe?  If he has it, he but gets a little higher.  Is he not always infinitely removed from the end…?   In comparison with these Infinites all finites are equal, and I see no reason for fixing our imagination on one more than on another.”   While he doesn’t write any equations here, the ideas clearly have a basis in his previous studies related to finite and infinite values.   We could even consider this a self-contradiction:  wouldn’t the fact that his math just gave him some theological insight mean that it was, in fact, worthy of study to get closer to God?

Pascal did also still engage a few times during this period in direct mathematical studies.   Most notably, in 1658, he started speculating on some properties of the cycloid, the curve traced by a point on a moving circle, to distract himself while suffering form a toothache.   When his ache got better, he took this as a sign from God to continue his work on this topic.   That excuse seems a bit thin to me:  clearly he never lost his inbuilt love for mathematics, even when he felt his theological speculations were pulling him in another direction.   

And this has been your math mutation for today.


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