Why are math tests stupid? Because computers could do math tests too.
If the problems are given in a computer-readable format, which problem that you can do a computer cannot do. And if you are not fast, I believe a properly programmed computer should do them faster.
If the problem is a calculation problem, I wouldn't think we will out-perform the computer albeit indefinite integral or whatever.
If it's a proof where we would think we are better. I still wonder if we are really at an advantage. In a proof, we basically have a few assumptions, and we reach a conclusion by the logic, if all A is B, if all B is C, then all A is C. To support our argument, we use a finite number of theorems to show all A is B, all B is C, etc. And the computer can do finite depth or width searches given the proper input.
One question is that whether we can have more complicated logic structures such as mathematical induction and proving the contrapositive is true. If so, each of these will make the search take longer time but still finish within finite number of steps.
Let's examine induction, epsilon related kind of questions and an algebra question.
A typical induction kind of question involves if n is an integer, and A is the object concerned, then B is true. The input of the program similar to the A is B, B is C then A is C kind of logic should be: if n is an integer, A is C, then D is E (because the problem is basically: if n is an integer and we have A satisfy some conditions, ie, A is C, then p holds, ie, some other object D is E).
Given the input A, in our knowledge base, so equally, in the computer's "knowledge", A is a finite number of things. And because we have finite knowledge (there's really a philosophical question involved, do we really understand infinite? According to modern science, we are finite objects. I feel that the way we observe infinite is that given a number, it is larger than that. But that is a very finite thing. You are given a FINITE number, and you just imagine your "infinite" to be larger.) so for any object H, it is only a finite number of other things I. For any object J, there can only be a finite number of objects K so that K is J.
So it appears to me we could do a depth search and reach at the conclusion.
With the induction problem, we just add one extra test at each search step. We examine the possibility that when hypothesis 1 is given, we can reach the conclusion (or whatever the conclusion is). (A typical math theorem consists of several hypotheses. It is abstracted like if A1, A2...An, then L is M. ) And we do another test if an integer is given, integer plus one make the conclusion hold.
There's a little bit of a kink here is that in this approach, we cannot let integer be anything. A computer tests problems in the way that it has to be given an numerical value. Here we abstract an integer just as we say circle. It is an object. The computer doesn't know what it is (and you think we do but we really don't know the way the thing actually is. For example, you don't understand integers the way integers are mathematically constructed which I take the way the integers actually are.)
So we could do the hypothesis induction when we know if something holds for n, then it holds for n+1 simply because we know something is true for n then it is true for n+1. Suppose you are to prove 1+2+...n = Sn = n(n+1)/2. then you say suppose this is true, then 1+2+...+n+1 = n(n+1)/2 + n+1 = .... Then the computer could do the same calculation and evaluated x(x+1)/2 for n+1.
A real problem here I think if a computer is able to understand 1,2,3... is an integer. My way to go around this is just to tell the computer a "word" consists of 0-9 without a dot is an integer. Then I think we will have a perfect pseudo code to work out any induction kind of problem.
The sum of first n integers is an easy example, someone propose a harder one and see if we can work out an induction pseudo code? Or are there any objections to this?
A bigger problem seems to be with seperating cases. A seperation of cases is essentially this. Suppose we have hypotheses A and we want to say B is C. Then a seperation of cases works like if D and A, then B is C and if complement D and A, then B and C. It looks peculiar how we find D and it is also an amazing choice most of the time. However, considering we only have a finite number of theorems, if we do a search on theorems that contain contain condition A, there are finitely many. Then there are only finite number of choices of D corresponding to the theorems.
It seems that computers are not to make abstractions however. And I saw the inductive proof of a closed connected combinatorial surface with one boundary is "equivalent" to a disc where the seperation of cases seems not so computer capable.
The reason I had this wonder in the first place is that we've been working on a kind of typical algebra qual problem where you are asked to prove that a certain group is solvable. I mean there's almost an algorithm to solve it and it is still an unavoidable question in quals all the time. That's...
If you want to see the algorithm see here.
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