- Pages: 5
- Word count: 1014
- Category: Mathematics
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For those who are unaware, at least 1 and 2 are thanks to Georg Cantor and his invention of set theory. He had to endure a lot of resistance from the mathematical luminaries of his day – Kronecker and apparently, even Poincaré! Like Nietzsche, he died in a mental institution, the vicious attacks on his ideas (mostly by Kronecker) having supposedly played a part in Cantor’s breakdown.
His mathematical ideas also ran afoul of some religious “philosophers” who, as is usually the case, are never ever open minded enough to consider truly groundbreaking ideas (virtually by definition, seeing as they are religionists). because “many mathematicians agreed with Kronecker that the completed infinite may be part of philosophy or theology, but that it has no proper place in mathematics”.
Even the great Gauss, according to the wikipedia article was saying that “Infinity is nothing more than a figure of speech which helps us talk about limits. The notion of a completed infinity doesn’t belong in mathematics”, e.g. “the only access we have to the infinite is through the notion of limits, and hence, we must not treat infinite sets as if they have an existence exactly comparable to the existence of finite sets.”
The irony, of course, is that when the notion of infinitesimals/limits were first introduced, they were ALSO subject to a very similar kind of criticism! (And as usual, religionists figured in such critcisms, most notably Bishop Berkeley).
Our perceptions can tell us “instantly” that all the chairs are taken, but beyond a certain single-digit number, we cannot conclude instantaneously how many chairs there are or people sitting on them.
Such a perceptual limitation seems to be what spurs humans to come up with the whole machinery of formal computation just as our physical limitations spurs us to invent the universe of devices we make use of in modern civilization. There are at least two “computers” in our brain. The visual one and the one that does symbolic manipulation.
The visual one seems to be far more natural and powerful, and interestingly it seems to reside on the right brain. We seem to need nothing but our “eyes” (visual cortex is probably the more accurate term here) to confirm to us “instantaneously” that a square’s sides are all equal in length, or that opposite sides of a rectangle are equal in length, whereas if we were to depend on symbolic manipulations to prove these, it would be a plodding (and boring) affair.
Prior to the advent of computers, we had a very hard time exploiting the visual cortex to do mathematics. I think that even now, we are still mostly ignoring the potentials in trying to do so. For most mathematicians, symbolic manipulation is the only “true” mathematics – a bit akin to relying on serial computers when parallel ones are available.
It is funny and ironic how mathematicians prefer to trust their scribblings more than they do their eyes (visual cortex) when, as the Lakoff-Nunez book reveals, our mathematics IS completely rooted on and derived from our perceptual mechanisms. e.g. the assertion being that creatures with a different sensory apparatus will not necessarily develop a similar mathematics. The way they interplay is really interesting IMHO. For example, some mathematical concepts (like the aforementioned Riemann-Zeta function and the mathematics surrounding the irrational real number e), were mostly derived using forays into imaginary realm, and yet e is a huge part of modern mathematics and science and the Riemann-Zeta function may help us understand primes more.
The problem is that the world just doesn’t really appreciate math that it cannot see, and that’s what turns off aspiring mathematicians.
All of those set theory results are indeed from Cantor. It’s so fundamental in math that in your first proving class, after proving that square root of two is irrational, you will proceed on proving Cantor’s results in the next lessons. But I think the philosophical objections about infinity are as valid as they were in Cantor’s time. I also think that the successful application of set theory in math is no more a refutation than the success of science is of the critiques on induction. Even the success of Newton and Leibniz’s calculus failed to dispel suspicions about the infinitesimals. Actually, the limits interpretation, due to Cauchy, is a clever way of doing calculus without infinitesimals. The only reason why we bother with epsilon-delta proofs is just to avoid the controversial infinitesimal. Perhaps, we would also one day (although I think it unlikely) find ourselves doing away with Cantor’s set theory.
As to the visual-symbolic divide. I think for the most part of human history, mathematics was done visually—that is geometrically. Even something as characteristically symbolic today like multiplying or dividing numbers, you can actually achieve by geometric construction. Then Descartes discovered analytic geometry, the eponymous Cartesian plane, and that changed everything. His symbolic/analytic approach served to greatly advance mathematics. Such was the power of the method that in less than a century, Newton would discover the calculus. Newton used the analytic approach in deriving the results in his Principia, but because geometry was still very much in fashion at that time, he was obliged to translate analytic proofs to synthetic geometric proofs.
Although the divide is worth noting, the two methods just serve to complement each other. Symbolic manipulation does make life easier but at the same time geometry does seem the more intuitive. Actually, despite what’s been said, geometry is as relevant as ever. We shouldn’t underestimate the influence of geometric arguments in math. The primitive concepts of geometry like ‘distance’, ‘perpendicular’, ‘convex’, ‘dimension’ are the very essence of linear algebra. I think the only fun way of thinking of linear algebra is to think of vectors and spaces as points, lines, planes or hyperplanes.
Anyway, a well-reviewed book that is worth checking out is Tristan Needham’s Visual Complex Analysis. He went out of his way to make the most beautiful subject in mathematics extra “visual.”