Cool Math: Riemann’s Zeta Function

One of my long-standing love affairs in math is with Riemann’s Zeta Function.  I use it as an illustration of a few points in my high school classes.  Since I name-dropped it in my recent series on Cur Deus Homo, I figured I’d cover a bit more about it.  Here’s a fantastic video explanation to a very difficult and fun bit of math.

You should give this guy a follow; he does fantastic work.

Math on!

Cool Math: The Basel Problem

William Whewell has been on my mind.  Not Brandon level, since he lives and breathes the guy, but as part of my perennial dissatisfaction with modern educational theory and practice.  Anyway, as part of Whewell’s insistence on the pedagogical priority of geometry over algebra, I present this: a beautiful summary of a famous mathematical problem solved by Euler in the 18th century.

Why Whewell?  Well, Euler and Gauss and company all followed the Primacy of Geometry education and it served them reasonably well.  But enough of edu-theory.  Enjoy the cool video!

Why a sum involving nothing but natural numbers involves pi: Continue reading Cool Math: The Basel Problem

Is Math Persuasive?

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?

Math Heresy

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

Prime Movers

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

Cool Math: Benford’s Law

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.

Math on!

Cool Math: Wilson’s Theorem

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.

Cool Math: Riemann Hypothesis

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.  Sorry)

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.

Non-contradiction: Geometry

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.