The Limits of Mathematical Knowledge

My good friend Jason Rosenhouse had an interesting discussion a few days ago on whether or not modern mathematics is reliable. The gist is that the deepest of maths are increasingly built on a foundation of theorems and proofs that are so complex that it effectively becomes an act of faith to take them as true and correct. And when further theorems are built on these theorems, how confident can you be in the structure of mathematical knowledge?

It’s an interesting point, and it gets to something I think about from time to time - the limits of mathematical knowledge.

The idea is that humans, when doing math, have an understandable bias towards proofs that are readable and (easily) checkable. The question then is what portion of mathematical ‘truth’ can be expressed in such a way.

Linear Recurrences And … Stuff

So, I know I’ve been a little quiet lately. This is mainly do to the confluence of two factors - one, I’ve actually been doing some work lately. I’ve been doing some programming (and avoiding doing some programming), and mulling over a series of papers on the four color theorem, written between 1892 and 1947. Interesting stuff - the author basically transforms the map coloring problem into solving a system of simple linear equations. I may talk about that some later.

Secondly, I’ve been wrestling with my own frustrations about a much simpler problem. I’m sort of chasing my tail on this one.

To begin with, imagine a function f(x) that satisfies the following equation.

The problem is to prove that it is periodic. Further, what other values besides give rise to periodic functions.

Science in the Times

The New York Times has a good piece today, tonight? tomorrow?, with Dennis Overbye answering questions on physics. Many good topics are covered, like time travel, the big bang, dark energy, the LHC, and the nature of the universe. Interesting stuff and a fairly good overview of a lot of areas of interest, I think.

I think the most amazing thing I have covered lately is dark energy. The galaxies are falling up. How amazing is that? But I have already gone on about that above. In general, I am continually amazed at the passion, resourcefulness and stubbornness of humans, both individually and in groups, and how they manage to pick away at finding out things you would have thought were impossible to find out. There is a robot baking dirt on Mars as I write this. Cosmologists are finding traces of physics from when the universe was less than a trillionth of a second old and hotter than any particle accelerator will ever recreate in a fuzz of microwaves in the sky left over from the Big Bang. They fixed the Hubble Space Telescope, again and again and again.

But then, I’m a mathematician, what do I know ^^

Divisibility Rules: Extended and Sideways

First, a few words on modular arithmetic. I really wanted to avoid talking about modular arithmetic, but I tried writing this post without it, and it became overly complicated and I used the word remainder, and remainders of remainders, about eleventy trillion times. So here it is. Modular arithmetic is basically shuffling around integer multiples of things. We say that two numbers A and B are ‘congruent mod k’, if A = n*k + B for some integer n. So, for example, 17 and 7 are congruent mod 10 since 17 = 10*1 + 7. 107 is also congruent to 7. -3 is also congruent to 7, since 7 = 10*1 + -3. Notice the link to division and remainders. 107 gives a remainder of 7 when divided by 10 and is congruent to 7 mod 10, because in each case you’re subtracting the largest possible multiple of 10 from 107. When two things differ by an integer multiple of k, they are congruent mod k. We can think of divisibility then as when a number N is congruent to 0 mod k. 100 is congruent to 0, since 100 = 10*10 + 0, 100 gives a remainder of 0, 100 is divisible by 0, it’s all saying the same thing.

There are a small set of rules people can use to determine whether an integer is divisible by numbers like 2, 3, 4, 5, 6, 8, 9, 10, etc. Rarely does anyone go much beyond this, or even address the lonely 7. For the most part, these rules are simply shortcuts for calculating congruences. Congruencies? For example, we know that any number N can be written in the form N = 10*A + B, where B is the last digit of the number, and A is the rest of the digits. We can also say then, N = 5*(2*A) + B. Notice that B Thus, N is congruent to B mod 5. If N is divisible by 5, it is congruent to 0 mod 5, so we want B to be congruent to 0 mod 5. The only values from 0 to 9 (since B is a single digit) that are congruent to 0 mod 5 are 0 and 5. Thus, N is only congruent to 0 (and thus divisible by 5) if it’s last digit is a 0 or a 5.

All the other divisibility rules accomplish the same thing, finding a shorter way to calculate these congruences with the divisor. Dividing by 3, it turns out that N is congruent to, and thus leaves the same remainder as, the sum of the digits of N.

But there’s a way at getting into the problem indirectly, sort of sideways. Calculating whether or not a number is congruent to 0 mod the divisor without actually calculating the remainder.

In brief, start with your number N, (7931, for example). Take the last digit, double it, and subtract it from the remaining digits. 793 - 2*1 = 791. Repeat. 79 - 2*1 = 77. Eventually you will get to a number that is obviously divisible by 7, or obviously not. If you reach a number divisible by 7, the original number is as well. We reached 77, 7*11, so 7931 is divisible by 7. And if you check it is. This is very fast, as you effectively lose a digit with each step ^^

Below, I explain why this works, and extend the same trick to derive simple divisibility tests for 11, 13, 17, and 19.

Irrationals Do … What? (Warning: Graphic!)

As I see it, this is a very graphic demonstration of how irrational numbers relate to rational numbers. It definitely concerns some inherent property of rational numbers. The problem is, I really don’t know what that property is or how to describe it.

We start with simple periodic functions like sine and cosine. Being periodic, they repeat with a certain period : ) But that period can be controlled. Below, Cos(x), Cos(3/2*x), and Cos(2*x) are all plotted on top of each othe, the first in blue, second in red, third in yellow-ish. As x increases, the functions oscillates and eventually repeat.

The Clearly, Cos(2*x) oscillates twice as fast as Cos(x) and has a period half as long.

Basic stuff, but then we start combining things.

Math as Language

Via slashdot, I found this article on an Amazonian tribe that does not seem words to describe numbers and counting. Rather, they have relative quantifiers that indicate a quantity from 1 to 4, 5 or 6, or ‘many’. In effect, small, medium, or large.

Of course, it’s probably a bit of a stretch, but as counting forms much of the basis of math (though I wouldn’t mind hearing discussions on that), this relates, as I see it, to whether mathematics is something inherent to the universe itself, or rather something more akin to an evolutionarily advantageous artifact of our brains. Some of our brains, anyway.

Evolution almost certainly plays a role here, as the language in question is spoken by about 300 people, and they likely haven’t spent a lot of time doing commerce with other cultures.

There were some interesting thoughts raised in the comments. For instance, one person noted that most people can easily recall about five separate objects at a glance, and the perception of quantity is likely linked to how easy it is to remember the objects. Another noted that this may be a different kind of language artifact - the researchers were having them count foreign objects, but should’ve asked them to count how many children they had, or some other familiar kind of object, the idea being that perhaps different words were used for different types of things. This is interesting, because linking quantity and type descriptors would greatly diversify what could be expressed.

There are far more questions I’d like to ask. For example, the researches only went up to 10 objects. What if they went straight from 10 to 1000? While both are ‘many’, there’s a pretty significant visual difference from the sheer amounts.

Also, I do wonder whether or not, even with the limited range of quantifiers, if the speakers here make use of any kind of computational-esque rules. For example, two ‘medium’ groups combined would always produce a ‘large’ group. Very fuzzy sort of addition. Of course, two ’small’ groups combined don’t necessarily produce anything definite.

It’s all very interesting.

Zeroless Squares

This is really a return to a problem I addressed over a year ago. It came up again, quite by chance, and I came up with another proof that makes my original post … almost embarrasing.

The idea is that for n = 0, 1, 2, 3, …, for numbers of the form

their squares contain no zeroes.

For example, for n = 1, (10 + 2)/3 = 4, and 4*4 = 16, 16 contains no 0’s. But the same holds for -all- n, none of the squares contain 0’s.

Calculus and Geometry: A Family of Ellipses

The answer to the question never directly stated in the previous post is the following equation.

Where a varies from 0 to 1.

It’s an interesting equation, so I’m just going to consider it by itself, independent of the problem not really discussed.

Plotting that equation on the xy plane for various values of a (namely, 1/10 to 9/10 at increments of 1/10) produces the following picture.

Some nice symmetries going on there. But notice the way there seems to be a very definite boundary to the shape formed by all the ellipses overlayed. Is there a nice mathematical description of that shape?

Of course, there likely is, or I wouldn’t be talking about it.

Not a Couch in a Stairwell.

No, it certainly isn’t.

Coming up short on ideas lately, so here’s something I worked out last night while I was trying to fall asleep.

The paths certain points on a ladder trace out as it slides from vertical to horizontal.

Shorter: Twisted Kinematics

This is a relatively simple problem, though it pestered me for a good while. I’m not even sure my answer is correct, but it certainly seems reasonable.

The idea is that acceleration is usually given in terms of velocity per time. Falling in gravity, you accelerate at a rate of 32 feet per second every second. And if you’re falling at a constant rate ‘a ft/s/s’, it’s easy enough to work out what your speed is at any time, and where you are at any time.

But suppose instead of feet/second/second, your acceleration was given in terms of feet/second/foot? That is, you’re slowing down based on how far you’ve gone instead of how long you’ve gone. So of course, the further you go, the slower you go, which means the longer it takes you to travel more distance, which means the slower you decelerate. But what does all this add up to?

So, we’ll say that you’re initially going at 100 m/s, decelerating at a rate of 10 m/s/m. Describe how position changes, velocity changes, and how are you actually decelerating in the traditional sense?

Here’s my solution, with no guarantee it is correct.