Continuing my flogging of the issue: is it the essence of a mathematical proof to be persuasive, such that someone who fails to persuade has failed to engage in “mathing?”
I ran across this fun little Numberphile video which raises in passing an interesting and important point. Fermat came up with an idea (not his super-famous one) about some primes being the sum of two squares (like 17=16+1). What the video goes on to mention is that many mathematicians after Fermat–the super-heavyweights like Euler and Gauss and Dedekind and Co.–all came up with proofs of this idea.
Each of those proofs is different. Very different. If the goal of mathematical proof were simply to persuade, the proofs would be valued for getting different “mathematical demographics” to agree to the truth of the conclusion. Or perhaps, even more simply, one could insist that everyone should agree to the conclusion of the first, rational proof and then get on with life.
But this is not the role of mathematical proofs, any more than it is the role of the scientific method or logical argumentation. The various proofs are valued because each of them illuminates different aspects of the problem as well as different areas of the wide world of mathematics. Continue reading Is Math Persuasive?
Irrational numbers can never be real magnitudes.
There, I said it. Irrational numbers cannot exist as magnitudes of length or weight or whatever in the world of mobile substance. My stock example: you can never forge a sword with a blade whose length is √2 feet.
(Aside: that’s a gimmick in a to-be-written short story of mine–a magical Sword of Impossibility whose blade really is √2 feet in length and instantly annihilates whatever it cuts.)
Back to my math heresy. Why do I think this?
Well, here’s one intuitive appeal: Continue reading Math Heresy
Brandon roused me from my mathematical slumber with a post on one of my hobby-horses, prime numbers. Just enough impetus to put down a thought I was mulling over last week.
Primes are the delightful irreducibles of the number world. As a kid I thought of them as weird exceptions to good, common-sense mathematics–the kind of things you memorized and played goofy games with. But that has things almost backwards.
The Fundamental Theorem of Arithmetic says that any integer can be expressed as a unique prime factorization (this is what Brandon was posting about, so I shan’t repeat him). Said in reverse, primes are the building blocks of the number world. Everything traces back to them. That got me thinking about primes as an example of first movers like one would use to explain the First Way or STA’s explanation of the act of the will. Continue reading Prime Movers
If you want to be a forensic accountant (or learn to foil forensic accountants), check this out. Ok, or if you just like numbers! I saw a few videos on Benford’s Law and picked what I thought was the simplest to follow. A very cool, very interesting property of numbers.
I wondered, less mathematically and more philosophically, if the strong “preference” for 1 has to do with the way we construct or think about units. But this is just a quick post, not a think piece, and the video speaks for itself nicely.
More on my love affair with primes. This one is a terrific, infallible test for prime numbers:
For any prime number P
(P-1)! + 1 = x
X is divisible by P (the original prime)
I’d never heard this one before. Made me sit up in my chair when the video started rolling. Primes always pass this test, composites always fail it. The video goes on to show a small group of primes where the result is divisible by P twice. Enjoy the vid and math on.
I’m moving up about fifteen weight classes when I dabble in the Riemann Hypothesis (link is to the Riemann zeta function). It’s really more a matter of what the problem represents, and my (perhaps) odd views on infinity, than it is about the zeta function itself or my ability to hack the math. I won’t be the one solving it, that’s for sure!
This video gives you a neat look at the problem although it’s a bit brief on the setup:
I thought I’d comment on why I find RH interesting.
It tickles me to no end that at the foundation of some really profound mathematics–mathematics, that realm of certainty beyond anything the physical universe allows–we have an unsolved and possibly unsolvable problem. That mystery and beauty could–and for now, in some ways, do–undergird the realm of the rational appeals to me in a highly iconoclastic way. Anselm was right and Boso was wrong! Conveniens is the more fundamental proof! (That’s a Cur Deus Homo joke.
There’s also some low-brow humor mixed in, Pratchett-style. “Whoops, I guess everything we know about primes is wrong!” gives me a giggle. I don’t think it will turn out that way, but the idea of it delights me. Just like dying and finding out that the solar system really is geocentric would.
It’s also a really neat illustration of the difference between induction and deduction. We’ve tested trillions of candidates and they all conform to the Riemann Hypothesis…but that’s not a proof! Not in math anyway. Maybe in one of those filthy natural sciences, but Never. In. Math!
Enjoy vid, and math on.
I’ve been thinking about how to explain the principle of non-contradiction to annoying teenagers. Of course, Aristotle has some amazing suggestions for people who deny this first principle of thought. If there’s no difference between throwing yourself off a cliff and not throwing yourself off a cliff, why not go do so and spare us your inane jibber-jabber?
And I could probably get some mileage out of shaming adolescents into accepting the PNC. But that’s not very a) philosophical, b) practical, or c) Catholic. Besides, where’s the fun in solving a problem by chopping a sword through it? (Never overlook the Fun Principle in teaching)
It’s a tough world when they say “No” to everything involving truth. I could get them to at least consider the idea that Hitler raping babies might be ok, but watch them recoil at the idea of something being true. So there’s a long list of tricks we use to try to get kids to see this for the first time. Maybe I’ll kick them around in another post some time.
But lately I’ve been having a simple thought. It’s not a demonstration of the PNC, but maybe its most elemental appearance in logic. Or the simplest way to point it out, maybe?
1. A line segment is either rectilinear or curved.
2. The line segment AB is not rectilinear.
3. Therefore the line segment AB is rectilinear.
PNC: you can’t affirm in line three what you just denied in line two. Why is 3. an incorrect conclusion? Because it violates the PNC.
Perhaps this is the way to go. Put that syllogism on the board and ask the boys why the conclusion is false.
Pi is one of our oldest and most recognizable math accomplishments. Fancy-pants mathematicians think we should move away from using it because there are numerous counter-intuitive features of Pi as the math gets more advanced. Tau v. Pi is a battle to reconceptualize circle math in terms of motion rather than in terms of solids. Continue reading Cool Math: Is Pi Outdated?
I was torn between posting a video explanation of how they found the recent super-massive Mersenne Prime and this shorter bit on Lucas Numbers, which is how they found it. Since this video has more explanation–and some very cool commentary on the nature of mathematics toward the end–I went with it. I’ll save the super-Mersenne for another day.
Lucas Numbers are related to the Fibonacci Sequence. Both are related to the Golden Ratio, which I definitely need to post on at some point. Like many Numberphile videos, this one does a good job conveying infectious love of numbers. Enjoy! Continue reading Cool Math: Lucas Numbers
I am a huge fan of Greek geometry. That’s the real stuff. Real geometry involves an unmarked straightedge and a compass. If it has numbers, you are doing it wrong. One of my colleagues is a geometrical genius–he’s actually one of the most famous origamists in the world (that’s right, I name-dropped an origami master!). I like to tease him when they get into “analytic” stuff late in the year after the proofs are all done. “Oh look, Algebra! I thought you were teaching Geometry this year!”
Doubling the Cube? Squaring the Circle? Neusis Constructions? Now we’re talking. The Greeks thought geometrically in a way that borders on the mystical. Come to think of it, they actually did make a religion out of it. Continue reading Cool Math: (IR)Regular Polygons