# Can Neural Net Solve Math Problems?

By Xah Lee. Date: .

### Does Automated Theorem Prover Exist? Can Neural Net Solve Math Problems?

can neural net solve the Collatz conjecture problem?

The Collatz conjecture is a conjecture in mathematics that concerns a sequence defined as follows: start with any positive integer n. Then each term is obtained from the previous term as follows: if the previous term is even, the next term is one half the previous term. Otherwise, the next term is 3 times the previous term plus 1. The conjecture is that no matter what value of n, the sequence will always reach 1.

[2017-12-14 the Collatz conjecture Collatz conjecture]

this is a interesting question, because it is a simplified version of whether neural networks based AI can answer math problems. “Math problems” here means, for example, any of

Millennium Prize Problems Millennium Prize Problems

The above math problems are complex to computerize. Computerizing math is not a solved problem.

So, we might ask instead, whether neural network can solve math problems that are simple to computerize. For example,

are there odd perfect number? (a positive integer that's equal to its divisors excluding itself)

or group theory List of unsolved problems in mathematics#Group theory

my understanding is, no. Neutral networks cannot answer any such question. Unless, an AI reaches what's called Strong AI (which basically means human level intelligence). But, at that level, the AI is simply like a human, that it isn't using neural net to solve a specific given problem, rather, it has become something else, even possibly sentient.

Also, i think it's understood that we cannot learn much from neural net other than observing the results. Other AI approaches (such as say brute force, or genetic algorithm), we can actually learn something from it. For example, by brute force, we have solved many chess end game problems, and know some theory about it.

Chess endgame

we have also learned, that some stalemate rules are bad. That our rule says after x moves the game is a draw, but computer by brute force have found forced checkmate that require more than x moves. (i think the x is 50)

AlphaGo (the precursor of AlphaZero), even though it beat world champion Ke Jie and many other international champions of go early in 2017, and have given us many games it played with itself. However, we did not learn anything about theory of go. What we did learn, is only by observing its play, and theorize ourselves.

other “simple” math problems are, for example,

is there an odd perfect number Perfect number#Odd perfect numbers

“a perfect number is a positive integer that is equal to the sum of its proper positive divisors, that is, the sum of its positive divisors excluding the number itself”.

or Boolean satisfiability problem Boolean satisfiability problem

traveling salesman problem Travelling salesman problem

am wondering, with the seemingly powerful AlphaZero, how's such neural network based AI can tackle these kind of concrete, absolute, seemingly simple, math problems. Or, was it agreed that neural network simply cannot work on these kinda problems?

if neural net can deal with these problems, what's the approach? are there examples?

i don't know much about netural net, but it seems, it is never used on these type of problems.

as for working on disease and other general human problems, am just wondering, what exactly are the problems in concrete terms? Since “disease” is quite general, nothing like chess or go.

suppose you write a chess program. And by brute force, you completely solved chess. That is, you've determined, the optimal move for every position. That is, automated theorem proving.

That, is the idea, and beginning, of automated theorem proving.

of course, we cannot brute force all the way, since there are more ways than we can fathom. Therefore, we try to cut corners, and be smarter, in our ways of enumeration, such as the neural networks of AlphaZero.

aside from that, of mathematics, we cannot even begin to brute force or neural net, since math is not codified as chess or go is. The problem, of turning a human math question into logic and into computer, is itself, not a solved problem. Before we can automate prove theorems, we need codification of math, and that's in the realm of foundation of math.

and in this realm, even though we made a lot progress, or none, relative to the cosmos, there are still mysteries and unbelievers and glory holes. We make do what we can. Thus, we have “conjecture” searchers, “assisted” provers, alterantive foundations such as homotopy type theory and such. Their meaning and context, evolves. Few, knew what they are talking about, reality speaking.

### can neural net solve math problems?

Dear Lu, here's a problem you might find illuminating.

suppose you went to RadioShack and built a tiny neural networks Artificial Intelligence software. In just 1 hour of playing with itself, it plays so good at tac-tac-toe that it never loses.

Now, that's some accomplishment. But, now, how to solve, say, x + 1 = 2, for arbitrary 1 2, with your neutral net?

Can your neural net solve such math problem?