What is this:
a^2 + b^2 = c^2
The Pythagorean Theorem.
Right. What else is it?
An equation in three unknowns.
Do you know some solutions to this equation?
Sure. (3,4,5) and (5,12,13).
Right. Those are common pythagorean triplets. Do you know others?
Well, Google is my friend, let’s see. (typing) It looks like (7,24,25) and (9,40,41) all satisfy the equation.
Have you noticed that you’ve only supplied integer solutions?
Oh, right. I suppose that there are a whole load of non-integer solutions.
Have you heard of Diophantus?
Is that a Greek name?
Yes. Diophantus was interested in equations that had integral solutions. We call such equations: Diophantine equations.
a^2 + b^2 = c^2 is a Diophantine equation?
Yes. And there are many others. For example:
a^3 + b^3 = c^3
Oh, sure. And what are some solutions?
There aren’t any.
Yes. None. That has been proven. In fact it has been proven that
a^n + b^n = c^nhas no integral solutions for
n>2. This is known as Fermat’s conjecture.
Huh. OK, well this is kinda interesting I guess, but why should I care?
What is a digital computer?
What do you mean? This thing that you and I are conversing on is a digital computer.
Yes, but what does a digital computer do?
Uh. It computes digitally?
Precisely! And the word digitally means…?
Um. With digits?
Exactly! And are the number of digits finite?
Of course, though very very large nowadays.
…and a finite number of digits is…?
Oh, I think I see where you are going. A finite number of digits is an integer.
Right. A digital computer is a computer that computes with integers. Nothing but integers.
Well, wait. What about floating point numbers and rational numbers?
They are represented by integers in the computer. The computer deals with integers, only integers.
OK. sure. Integers. But what does this have to do with Diophantine equations?
What are the inputs to a computer program?
There are lots of kinds. Keyboard characters, mouse movements, mouse clicks, network packets. You name it.
They are all made up of integers aren’t they?
Um. Yeah, I guess they are. OK, so every input to a computer program is integral.
And what about the outputs?
Well, yes, pixels, characters, network packets. They are all composed of integers too.
So a digital computer program takes in integers and returns integers.
Right. That’s right. It’s all integers.
A digital computer program, therefore, represents a Diophantine equation.
Integers in. Integers out.
OK. sure. But it’s one big complicated Diophantine equation.
Actually, the specification of the program is the equation. The program finds the solutions to that equation.
Yeah, yeah, ok. That’s right. The specification of a program is a great big Diophantine equation in a bazillion unknowns, and the program that meets that specification finds solutions to that ginormous equation. Is this useful to know?
Who is David Hilbert?
You mean that guy who designed that funny recursive curve that looks like mosquito netting?
(Ahem.) That was one of his accomplishments yes. He also helped Einstein with the General Theory of Relativity. He was a very great mathematician.
And he did something with Diophantine equations I’m guessing.
Indeed he did many, many things. Among them was a very famous question. The question of “Entscheidung” – decidability.
What did he want to decide?
Remember Fermat’s conjecture?
You mean that equation that has no solutions.
a^n + b^n = c^n where
Yes, that’s the one. For a long time there was no proof that
n=2was the only solution. How could you disprove that conjecture if you thought it was untrue?
I could write a program to find counter examples. Like, maybe
n=999,999,999 might work.
Right. And if you found such a solution, you’d have disproven Fermat’s conjecture. But how long would it take to PROVE the conjecture using that method?
The program would run forever. I couldn’t prove it that way.
Correct. What Hilbert wanted was a finite algorithm to determine whether or not a solution exists. He wanted a way to “decide” whether or not a search, such as the one you suggested, was practical.
Wait, wait. What? He wasn’t asking for the solutions, he was asking for a way to know if there were any solutions?
Right. He wanted a finite algorithm that could tell him whether a given diophantine equation had solutions or not. That algorithm would not supply the solutions; it would just supply the decision.
That’s why he called it “decidability”?
Harumph! Now. Who do you think solved the problem of decidability?
I think you’re about to tell me.
Two people whom you’ve heard of. The two great founders of modern computer science. Alonzo Church, and Alan Turing.
Church! That’s the guy who invented functional programming, right?
In a manner of speaking, yes.
And Turing! He won world war 2 right?
He certainly contributed. The two of them proved, using very different techniques, that there was no general and finite solution to decidability.
That must have disappointed Hilbert.
Perhaps. But that’s not the issue.
Yeah, just what is the issue here?
When you are given a program specification, i.e. a Diophantine equation, what is the first thing you are asked to do?
Estimate it of course. Folks want to know how long it will take to write the program.
And the program is what again, in terms of a that Diophantine equation?
The program is the … solution … to the … OH!
(smile) I perceive you’ve gotten the point.
Yeah, like, they are asking me to DECIDE. An estimate is a decision.
And is there a finite method for finding that decision in every case?
No! OH, that’s hilarious.
Right. The founding documents of computer science are documents that prove that there is no finite mechanism for deciding if a program can even be written. The founding of computer science was based on the proof that estimates were not guaranteed.
Yeah, but we CAN estimate.
Yes, we can. That’s because most specifications are estimable.
So this has just been a cute little mathematical diversion with no pragmatic result.
I suppose you could say that. But I enjoyed it. And, after all, I think it’s deliciously ironic that it was the proof of NOESTIMATES that founded computer science.