I have. I understand that it's partly a theorem prover. From what I understand, theorem provers can, in principle, prove (or try to prove, given limited time resources) any theorem in mathematics, no matter how abstract, since all of mathematics is apparently reducible to FOL. In practice, however, this would require thousands of axioms, and impossible amounts of time for any non-trivial theorem.

That's why it may be beneficial for a theorem prover to integrate/communicate with Mathematica. However, no such link exists with Coq. The only theorem prover I know of which does integrate with modern versions of Mathematica is Theorema 2, which is in too early a stage of development to be useful for non-trivial proofs. (There's also Analytica, which is old, and doesn't work on Mathematica versions after 4 or so.)

Although I'm not sure if anything here applies to my case, it does seem to be a case of abstract mathematics represented in Mathematica. I will look into it further.

Thank you very much for taking the time to answer!

I do not know how it is possible, from my question, to conclude that I expect Mathematica to know all of mathematics, and prove the necessary theorems for every question that might be asked. It seems to me that I have presented 4 axioms, only 3 of which I have presented completely, and am asking if Mathematica can represent those 3, or even just one, of the axioms, in the Wolfram Language.

The obvious way to answer in the affirmative would be to provide a translation of one of those axioms. This would be very helpful to me, and I'd really appreciate it if someone could spare a few mins to do it (it probably wouldn't take more than a few mins for an expert).

Yes, I am asking if the Wolfram Language can represent the axioms of specific field (which is different from saying that I expect the axioms to be built into Mathematica - I am merely asking for the tools to program them in myself). The axioms I provided are 3 from about 20 axioms of the Newton-Poisson theory of gravitation, as formulated by Mario Bunge in his book "Philosophy of Physics" (1973).

I find it very interesting that the axiom system of Grassmann algebra can be represented in Mathematica, although that does necessarily mean that my axioms can. You also seem to suggest that you can deduce Euclidean theorems from the axioms. I like this idea a lot - could you please point me a publication where I can learn about it? Also, could you show me an axiom or two represented in the Wolfram Language, as illustrative examples?

Thanks!

Yes, it certainly seems to be. I was aware of it, but just wanted to confirm (by asking this question) that the kind of things I'm trying to do are indeed the ones Wolfram states (in the blog entry) that Mathematica cannot presently do.

In that case, my question is: is there any add-on to Mathematica, or an entirely different system (Magma, maybe?) which CAN do things like this? Theorema (mentioned above) might aspire to this end, but is in a very early stage of development, and not helpful to me as of now.

Here is the direct quote:

"In the calculational case, one’s typically dealing with an operation to be performed. In the Wolfram|Alpha-like pure mathematical case, one’s typically just giving a description of something. In some cases that description will be explicit. A specific number. A particular equation. A specific graph. But more often it will be implicit. It will be a set of constraints. One will say (to use the example from above), “Let F be a field,” and then one will give constraints that the field must satisfy.

But to really be useful in pure mathematics, there’s something else that’s needed. In addition to being able to deal with concrete mathematical objects, one also has to be able to deal with abstract mathematical structures. Countless pure mathematical papers start with things like, “Let F be a field with such-and-such properties.” We need to be able to enter something like this—then have our system automatically give us interesting facts and theorems about F, in effect creating a whole automatically generated paper that tells the story of F.

So what would be involved in creating a system to do this? Is it even possible? There are several different components, all quite difficult and time consuming to build. But based on my experiences with Mathematica, Wolfram|Alpha, and A New Kind of Science, I am quite certain that with the right leadership and enough effort, all of them can in fact be built."

Consider the exact example he gives of something Mathematica cannot represent - "Let F be a field with such-and-such properties". Consider the first axiom I have in my original post - "M3 is a metric space with a metric tensor g". From the point of view of level of abstraction, they seem to be similar.

Your example shows that theorem proving in possible using simple propositional calculus and first-order logic constructs. This is well known to me. However, my question is whether Mathematica can provide a True/False answer (which is a strictly weaker requirement than actually proving that answer. I would love to have a proof, but at this point I don't expect it) to statements involving abstract mathematical structures like fields. A first order logic statement does not involve abstract mathematical structures. From the above quotation, Wolfram's answer appears to be that this is not possible at present.

It seems that True/False answers involving arithmetic are possible. I was able to ask Mathematica if the square root of 2 is rational, and it replied with False, although I don't believe a proof can be generated at present. Theorema (an add-on package for Mathematica) should eventually be able to provide proofs of things like this, but cannot at present.

I would LOVE to be proved wrong. In fact, I posted this question hoping that I'll be proved wrong. But that would involve someone providing a Wolfram Language translation of say, the first axiom I provide. If someone knows of an add-on or different algebra system (Magma?) that makes this possible, I'd be delighted as well.

Thank you for your reply. However, I have already experimented with Theorema, and while its ultimate goal might be to handle cases like mine, it is nowhere to close to that right now. At this moment, it does not even have axioms for set theory built in, let alone metric spaces, tensors and integrals.