Beautiful Math

Back in high school I discovered this by accident (yes, I was really bored!). I suppose it's nothing new, but it turns out that this works for more than simple squares and cubes:

Given any sequence of numbers, keep finding differences of differences until you hit a constant; the number of iterations needed is the maximum exponent in the formula that produced the numbers. That is, this works even if there are other terms, regardless of whether any or all terms have coefficients other than 1.

None of this was really new content to me, and yet I enjoyed reading it immensely. It must be what going to church is sometimes like for a theist, the reassurance hearing a sermon on beautiful things you already know. :-)

But what was the bias to be overcome? Mathematical platonism?

Any sequence of numbers Ak = f(k), where f(k) is a polynomial of degree n, will have its nth differences a constant. This is the method of "finite differences"; in fact, taking the differences of a sequence of numbers is roughly analogous to differentiation, and taking partial sums is analogous to integration.

It's an interesting fact about the way mathematics has historically developed that the analogous statement about polynomials viewed as functions of real numbers seems much more obvious to most people that have some mathematical training.

I love math. It's the only reason I sometimes wish I'd stayed in school. When I get rich, I want to hire a mathematician to live in my basement and tutor me. I bet they can be had for cheap.

Pure math is potentially a perfect idea. Applied math; not so much. When you see that line of 2's, how do you know it continues forever? You don't. You're making an induction; a beautiful guess. It's only because you peeked at the real answer-- an answer you yourself created-- that you can confidently say that you "predicted" the sequence with your method.

I'm much more interested in sequences produced in a simple deterministic way that are extremely difficult to crack. The move from "it makes no sense" to "it's obvious" is a critical dynamic in human thought. I'd like to see you write about that.

As Polya would say, solving these problems is a heuristic process. The reason you think you find order when you dig down far enough is that you systematically ignore any situation where you don't find order. Your categories have order built into them. You are drawn to order. There are probably a host of biases influencing that: availability, ontology, instrumentalism, and hindsight among them.

There's lots of order to be found. There is also infinite amounts of disorder, unprovable order, and alternate plausible order. Occam's razor helps sort it out-- that's also a heuristic.

Thanks for the beauty, it feels good. Some thinking out loud. I can't help but feel that the key is in the successive layers of maps and territories : maths is (or contains) the map of which physics is the territory, physics is the map of which 'the real world' is the territory, 'the real world' is the map our brains create from the sensory input concerning the territory which is the 'play of energies' out there, while that in itself is another map. Antony Garrett Lisi's proposal, as an example, would be the most elegant meta-map yet. What these maps have in commmon is : being created by the human brain, a wet lump of nervous tissue comprising ad-hoc purpose specific modules. It has specific ways of making maps, so small wonder all these layers of maps are coherent. Now if the 'mathematics' layer of maps has unforeseen and self-consistent properties, it could be just a manifestation of the nature of our map-making modules : they are rules driven. So, is the Universe a geometric figure corresponding to a Lie E8 group, or does that just happen to be the way the human brain is built to interpret things ?

So... Should it be obvious that the nth difference is n factorial? I'm afraid I don't have intuitive knowledge of all the properties of the factorial function.

To go even further, if you take the sequence A[k] = k^n and, for each n, take differences until you reach a constant, then the list A[n] of those constants is n factorial (n!).

This reminds me of when I first started learning about topological spaces, and then we added a metric, suddenly all the theorems and lemmas we'd had to prove in fiddly ways with lots of epsilons and deltas in first year analysis were blindingly and beautifully obvious. The sheer glorious interconnectedness was so overwhelming that I very nearly orgasmed in the lecture theatre!

Josh, if you think about a picture like the one Eliezer drew (but in however many dimensions you like) it's kinda obvious that the leading term in the difference between two n-cubes consists of n (n-1)-cubes, one per dimension. So the leading term in the next difference is n(n-1) (n-2)-cubes, and so on. But that doesn't really give the n! thing at a glance. I'm not convinced that anything to do with nth differences can really be seen at a glance without some more symbolic reasoning intervening.

James Bach, I suspect that the really good mathematicians can't be had for cheap because doing mathematics is so important to them, and the quite good mathematicians can't be had for cheap because they've taken high-paying jobs in finance or software or other domains where a mathematical mind is useful. But maybe it depends on what you count as "cheap" and what fraction of the mathematician's time you want to take up with tutoring...

Isabel, I think perhaps differentiation really is easier in some sense than differencing, not least because the formulae are simpler. Maybe that stops being true if you take as your basic objects not n^k but n(n-1)...(n-k+1) or something, but it's hard to see the feeling that n^k is simpler than that as mere historical accident.

Martin Gardner has a chapter on these "look-see" proofs in Knotted Donuts.

The beauty of it all never fails to shock me, and I'm very much a novice. 'The Road To Reality' is sitting on my shelves looking darkly at me, waiting to be attacked for the first time. I can only imagine what being at the cutting edge must be like.

But maybe it depends on what you count as "cheap" and what fraction of the mathematician's time you want to take up with tutoring...

Such a mathematician's answer.... ;)

Carl Linderholm notes in Mathematics Made Difficult that the next number in 1 2 4 8 16 ? has to be 31, based on just those differences.

Lautreamont on mathematical objects.

Quote from a friend: "Mathematicians are Platonists pretending to be formalists."

Words that ring true in every math department I've ever been to...

"To say that human beings "invented numbers" - or invented the structure implicit in numbers - seems like claiming that Neil Armstrong hand-crafted the Moon. The universe existed before there were any sentient beings to observe it, which implies that physics preceded physicists."

No, there's a conflation of two things here.

Have you ever really looked at a penny? I'm looking at a 1990 penny now. I know that if you look at the front and you see the bas-relief of Lincoln, and the date 1990, and it's a penny, then you can be sure that the back side will have a picture of the Lincoln memorial. It works! And you can find all sorts of connections. Like, there's a single "O" on the front, in the name GOD in the phrase IN GOD WE TRUST. And there's a single "O" on the back, in the phrase ONE CENT. One O on the front, one O on the back. A connection! You could make lots and lots of these interconnections between the front and the back of the penny, and draw conclusions about what it means. You could invent a discipline of Pennyology if only somebody would fund it.

Is it true that Pennyology is implicit in pennies? In a way. Certainly the pennies should exist before the Pennyology. But the pennies are only whatever they are. The existence of pennies doesn't tell us much about what the practitioners of the discipline of Pennyology will actually notice. They might never pay attention to the pair of O's. There could be a fold in Lincoln's coat that after the proper analysis provides a solution to the whole world crisis, and they may never pick up on it. While it's predictable that different independent attempts at Pennyology would have a whole lot in common since after all they all need to be compatible with the same pennies, still they might be very different in some respects. You can't necessarily predict the Pennyology from looking at the penny. And you can't predict what mathematics people will invent from observing reality.

You can predict some things. A mathematics that invents the same 2D plane we use and that proves a 3-color theorem has something wrong with it. But you can't predict which things will be found first or, to some extent, which things will be found at all.

If there's a reality that mathematics must conform to, still each individual version of human mathematics is invented by humans.

Similarly with physics. Our physics is invented. The reality the physics describes is real. We can imagine a platonic-ideal physics that fit the reality completely, but we don't have an example of that to point at. So for example before Townsend invented the laser, a number of great physicists claimed it was impossible. Townsend got the idea because lasers could be described using Maxwell's equations. But people thought that quantum mechanics provided no way to get that result. it turned out they were wrong.

Actual physics is invented. Certainly incorrect physics must be invented. There's nothing in reality that shows you how to do physics wrong.

J Thomas - thought experiment: how different could alien mathematics possibly be?

I don't see how Pennyology proves your point: a pennyologist discovers that there's an O on the front and an O on the back. Perhaps he invents meaning to attach to this fact, but the fact was first discovered.

Or is that what you were saying: that we discover mathematical facts, but in fact what we call "mathematics" is the invention of meaning that we attach to certain favored facts?

"So... Should it be obvious that the nth difference is n factorial? I'm afraid I don't have intuitive knowledge of all the properties of the factorial function."

I don't remember writing this. Did somebody else write this using the same handle as me or did I write this and forget it?

"I don't remember writing this. Did somebody else write this using the same handle as me or did I write this and forget it?"

My son's name is also Bort.

Chris and Ben, we create axiom systems and we discover parts of "mathematics". There are probably only a finite number of theorems that can be stated with only 10 characters, or 20 characters, or 30 characters, provided we don't add new definitions. But the number of possible theorems quickly gets very very large. Will each independent group of mathematicians come up with the same theorems? Probably not. So we get different mathematics.

How different could alien mathematics be? I don't know. We could look at a variety of alien mathematics and see. Except, we don't have much of that. We presumably had different mathematical traditions in china, india, and europe, and we got some minor differences. But they were solving similar real-life problems and they could have been in communication. If you want trade in silk then you need a lot of it for it to make much difference. A very few mathematicians traveling could spread ideas easily.

It's easy to see alternate physics, and alternate technology is kind of arbitrary. I think alien math might be pretty different depending on which theorems they proved first. But I don't have good examples to demonstrate it.

The Pennyologist who notices the O's is not that different from the Pennyologist who notices there's one L on each side. A particular Pennyology might notice one of those, or the other one, or both. Out of the many relationships you could pick out from the penny, which ones will people pay attention to?

Beautiful post, thanks :-)

The differences method is of course strongly related to differential calculus. The first difference corresponds to the first derivative, etc. And just like in differences, the nth derivative of a x^n term is sure enough a constant.

Eliezer sometime ask something that I would now like to ask him: how would the world looks like if mathematics didn't precede mathematicians? And if it did?