Does the education of mathematics exclude too many of its malcontents and political thinkers?
Most math classes begin with something like the following. “We’re here to study mathematics. Mathematical truth can be proved to anyone who accepts the universal logic that 2 + 2 = 4.
Maybe that’s only true in that simple sense. When it really counts — pun intended — mathematical proofs only rise to importance when they are accepted by cultural norms.
I’m thinking about all this because of these two videos on the subject of number series. The difference of opinions begin with Numberphile’s Astounting 1+2+3+… = -1/12 video. It has 6.1 million views. It was challenged by Mathologers: Numberphile v. Math: the truth about 1+2+3+…=-1/12.
My takeaway is that the Numberphile video looks at an “infinite series” in a different way than Mathologer. I can’t determine who is right, or wrong. I suspect they are both right in the context of what they’re trying to prove.
I enjoy watching Mathologer’s criticism because, I too, have felt, in math classes, that crucial issues were glossed over. Why did I have to accept that one could prove something by comparing two infinite series by adding 1 to the end of each. I felt I understood the +1 part, but not the infinite series part which was, to me, a fantastical idea which I could not match to anything in the real world.
Whenever a math teacher said to me “imagine an infinite series of primes” I always wanted to burst-out “is that like an infinite series of fairies eating pixie dust?” Or, no, I can’t imagine!
I had been taught, or assumed, that mathematics was a subject where mathematicians added to the findings of those before them. Mathematical truth grew over time. In that view, Newton knew more than Euclid and Einstein knew more than Newton.
Oswald Spengler challenged that. From his 1922 “The Decline of the West”.
It remains still to mention the corresponding difference, which is very deep and has never yet been properly appreciated, between Classical and modern mathematics. The former conceived of things as they are, as magnitudes, timeless and purely present, and so it proceeded to Euclidean geometry and mathematical statics, rounding off its intellectual system with the theory of conic sections.
We conceive things as they become and behave, as function, and this brought us to dynamics, analytical geometry and thence to the Differential Calculus. The modern theory of functions is the imposing marshalling of this whole mass of thought. It is a bizarre, but nevertheless psychologically exact, fact that the physics of the Greeks — being statics and not dynamics — neither knew the use nor felt the absence of the time-element, whereas we on the other hand work in thousandths of a second.
As someone who has read deeply into modern investment analysis, with its assumptions based on growth rates (calculus), Spengler articulated my subconscious doubts. That is, the Greeks or Romans didn’t use calculus in their financial world, not because they couldn’t understand it, they just didn’t see what practical use it was.
I would argue that calculus in modern finance has more to do with the cultural comfort of believing in the idea of “growth”, where every person is gaining more wealth over time, than in any real financial benefit of calculus over algebra/geometry.
In other words, the mathematical concepts that are used on Wall Street are not picked by the qualities of mathematics, but by cultural influences. We’re a culture of growth. The Greeks were a culture of stability. Culture leads, math follows.
Let me continue by making the analogy that calculus is to geometry what fiat-currency is to gold. Indeed, the latter is an incompleteness theorem all investors and economists appreciate. The two contradictory systems of money were immortalized by William Jennings Bryan, in his 1896 speech, You shall not crucify mankind upon a cross of gold. In other words, real commodities, like gold, are the basis of value. You cannot create another system (and its various calculus-type mathematics) based on money created by the government. They are contradictory.
Take Godel’s Incompleteness Theorem. It is another one of those complex subjects, if flipping through the pages of that proof are an indication, that is obvious to any sentient human being.
Your rules will contradict my rules. Your evidence will contradict my evidence. Although we will be able to connect our systems together, there will always be a point where they don’t meet.
Maybe to mathematicians, there is more objectivity in Godel’s proof, but again, what significance to everyone else? That’s the rub. Is mathematical education biased towards defending the profession of mathematics, both in research and education, as if it has no responsibility to culture influences?
Another similar over-complication I experienced was Noam Chomsky’s Syntactic Structures. Is language hard-wired in humans? He delivers page after page of symbolic proofs. I thought he was a genius until I had children. Then I had the “duh” realization.
I don’t know what education is like in mathematics these days. My guess is that much depends on the individual teacher. I do know that mathematicians are no less secure than anyone else. No matter what they prove, cultural influences will always have their say, as they do about everything else.
I would have found it easier to get into mathematics if I there was more discussed, about its cultural aspects and biases. Both numberphile and mathologer, in their differing approaches, give room to these cultural questions.