Some Musings on Mathematics

Recently in a forum I read regularly and contribute to occasionally, a discussion arose about Cantor's theorem of the uncountability of the reals. From there we talked about the real numbers themselves, and someone, quite reasonably, said:

    > Isn't the whole point of math that you expect at
    > some point to analyze the result of the symbol
    > manipulation and "read off" an interpretable
    > meaning from the answer?

Not always, no.

If you're doing applied math then yes. If you're doing theoretical physics then yes. But if you're doing pure math, then no.

When Pure Math is explained to non-mathematicians, the audience always asks "Why?" and "Of what use is it?" The result is that mathematicians always have to motivate their explanations and give applications for the results. Pure Number Theory is motivated by applications in cryptography, pure Calculus is motivated by applications in ballistics and weather forecasting, pure Combinatorics is motivated by analysis of computer networks and data processing, pure Statistics is motivated by life assurance, insurance and gambling, pure Linear Algebra is motivated by optimization problems and Google's Page Rank algorithm.

The truth is far simpler. Mathematicians are solving puzzles, and some don't come from the real world at all, and can't be motivated in that way.

Why do we care that there are only five Platonic Solids? The true answer is because there is an answer, and it would be intolerable not to know it.

Why do we care if every even number from 4 onwards can be written as the sum of two primes? Answer: We don't, really. But not knowing is an itch to scratch, and who knows what might turn up in our efforts to solve the problem.

It was said by E.C.Titchmarsh:

  • It can be of no practical use to know that Pi is irrational, but if we can know, it surely would be intolerable not to know.
The whole point of pure math is that there are problems to solve, and you're working to solve them. On the way you might generate all sorts of stuff that has no real relevance to the real world at all. The interesting thing is that stuff from pure math ends up being useful anyway, even when they were studied just because they were interesting at the time.

Part of the issue here is that in math, when you have a tough problem and you're working away at it you follow your instincts and create structures and objects and processes and relationships, never quite knowing where it will take you. Some of these arise very naturally, very easily, and then seem to have application beyond the specific problem you're working on. You get a sense that you haven't created them, but they exist independently, and you've uncovered them.

Back to the original question ...

The reals are a bit like that. When you study equations you want solutions to things like x+5=8 and that's easy if you only have the natural numbers. But something very similar like x+8=5 doesn't, so you're forced to invent the negative numbers. You won't think that's a big deal, but 0 and the negative numbers turned up very late in our history, evidence that they were an "unnatural" idea at the time.

Interestingly, the quadratic has a formula to solve it, and so does the cubic, so you start to wonder about quartics. Can they be solved by equation? Yes, they can! Fantastic! Maybe all polynomials can be solved with an equation and process.

Then it turns out no, they can't. From the fifth power onwards, not every polynomial can be solved exactly. How can you prove something like that? How can you show that no matter how clever, talented, knowledgeable or just plain lucky someone is in the future, that it's impossible to solve exactly every polynomial of power 5 or more?

Well, it was done, and on the way an area known as Group Theory was invented/discovered/used for the first time. [Galois]

Who cares? Surely that's all completely useless. Well, no. Group Theory is fundamental to understanding modern cryptography.

Then you want to solve things like 3x=5 and so suddenly you need fractions. Then you want to solve quadratics and cubics (which arise naturally in astronomy) so you need square roots and cube roots.

We've now got the numbers needed to express the solutions to all polynomials. They're called the algebraic numbers, and even though we might not have formulas to find the answers to polynomials, at least we have the numbers to express the answers.

And so we turn to sequences of numbers. Here's one that turns up in nature:

1/1, 2/1, 3/2, 5/3, 8/5, 13/8, ...

If you plot these on a number line it looks like they get closer and closer to some number, and they do. They approach the Golden Ratio, which is known to be the solution to the equation x^2-x-1=0 so that's a number we already have. It's an algebraic number. There's a success.

But what about this sequence:

1, 2, 5/2, 8/3, 65/24, 326/120, 1957/720, 13700/5040, ...

They are the convergents of the continued fraction for e.
The pattern isn't as obvious here, but it derives from a single, simple formula. Writing the numbers as decimals to six places we get ...

1.0, 2.0, 2.5, 2.666667, 2.708333, 2.716667, 2.718056, 2.718254, ...

Again, the numbers seem to be getting closer and closer together, they even seem to be piling up as if against a barrier beyond which they won't, or can't, pass.

Suppose we plot them all on a number line, and you give me a really, really small disk. I can find a place where your disk covers everything from then on. Then if you give me an even smaller disk, I can find a place where your new disk covers everything from then on. I'd like to think that the numbers are approaching a limit.

But they don't.

Why not? Because there is no algebraic number that fits the bill. For every algebraic number you think of, the numbers in that sequence either never get close to it, or they end up moving away from it and never come back.

It's almost as if there's a gap in the algebraic numbers. It's almost as if the sequence does approach a limit, but the limit is a kind of number we don't have yet.

So why don't we do what we did before? When we couldn't solve equations like x+8=5 we invented the negative numbers. When we couldn't solve equations like 3x=5 we invented fractions. When we couldn't solve equations like x^2-x-1=0 we invented the algebraic numbers. So let's invent the numbers that are the limits of sequences of rationals.

The process doesn't stop there. We still can't solve equations like x^2+1=0, for example, which leads to Complex Numbers.

And there's more.

And that gives you what we call the Real Numbers.

And as an invention, the Real Numbers stand on their own. They complete the number line. We now have no gaps (yes, we can prove that) and while it's true that they turn out to be useful in their own right, they don't have to. In a sense they have an independent right to existence.

The fact that they are used in calculus, astronomy, fluid dynamics, predicting the weather, modelling the economy (for better or worse) and a myriad of other applications doesn't make them mere tools.


Further reading:

Niccol˛ Fontana Tartaglia came up with a general solution to the cubic equation and, after some severe pressure, told it to Gerolamo Cardano. Cardano later saw an unpulished work by Ferro who had found the same solution, and so even though Tartaglia had sworn him to secrecy, Cardano felt justified in published it, prompting a life-long feud with Tartaglia. Cardano's student, Lodovico Ferrari, solved the quartic equation. In Cardano's book Ars Magna he also acknowledged the existence of what are now called imaginary numbers, although he did not understand their properties.

╔variste Galois while still in his teens found necessary and sufficient conditions for being able to solve polynomials using just the algebraic numbers. He then died in a duel aged just 20 and a half. Postumous examination of his work showed major contibutions to several areas of mathematics.

The number e (approximately 2.718281828459...) is sometimes known as Euler's number. It turns up unexpectedly in many areas of mathematics, but is most often associated with compound interest, probability and combinatorics (specifically, the "hat check problem")

This entire article is related to and affected by an article published in 1960 by the physicist Eugene Wigner entitled "The Unreasonable Effectiveness of Mathematics in the Natural Sciences"


The item that started this is here: Are The Reals Really Uncountable?

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