Early 80's I learned Logo and Basic at a local store that sold Apple computers. It was called "The Logical Choice" and had a classroom tucked away behind a false wall and an English Sheep dog "Sir Isaac Newton" that would hang around the store!
Nice, I just found a story on the store, page 5 of [1].
Money isn't the issue here, I'm trying to determine if they are able to offer a better value/speed than I can currently extract from renting a 3+ month block of cards myself. Time is a bit more important to me
Maybe in the future mathematicians could be the ones proposing different axiomatic foundations (e.g. ZF vs ZF + C vs ZF + C + CH vs ...) and then using computers to examine the consequences of these differing foundations?
I'm sure AI could contribute to this, but this is already a well-developed field of mathematics, and most of the consequences of additional axioms have been worked out. (The most productive hypothesis has been what's called "projective determinacy", if you're curious.)
Mathematicians have also gone in the opposite direction, and tried to work out what are the weakest foundations where different results hold. This is called "reverse mathematics".
What name does this "well-developed field of mathematics" go by? (I just want to get a taste of what the field is like.)
I also thought that there are an infinite set of possible extra axioms, e.g. axiomize any statement that's true but not provably so via Gödel's First Incompleteness Theorem, though maybe the vast majority of such axioms are "uninteresting".
"Descriptive set theory" is a good starting point, though it's the bulk of what set theorists in general do.
It's true that there's an infinite possible set of axioms. It does seem that the types of axioms that have consequences that humans are interested in fall into simple families. For example, many seemingly unrelated questions are settled by assume the existence of very large sets (larger than can normally constructed in set theory).
Looking a bit more into this, it doesn't seem your claim "most of the consequences of additional axioms have been worked out" holds up.
Yes, metamathematics is well-developed, but I don't think that most of the consequences of any particular additional set of axioms have been worked out. Each such new set requires re-deriving all of this alternate mathematics from scratch. This is a lot of work!
So I think my original claim---mathematicians select interesting axioms and AI figures out their implications---still seems a possible way forward.
PS: I'd guess descriptive set theory under determinacy is the one place where projective determinacy, as you stated, pays off.
I don't see how you came to that conclusion, since I'm telling you the actual state of play. There's a big literature on what results require the Axiom of Choice, for example. (The book Handbook of Analysis and Its Foundations covers this thoroughly.) There are many results on what follows from the Continuum Hypothesis or other cardinal arithmetic axioms. There is a big literature on what follows from assuming the existence of large cardinals. There's a separate literature on adding "forcing axioms", like Martin's maximum. There are hundreds of papers on open questions that are settled by adding additional axioms to ZFC, and to identifying the weakest axioms to add to settle various open questions.
In another direction, there's even a literature on what happens when you allow sets to contain themselves as members, like Aczel's Anti-Foundation Axiom. There's literatures on purely constructive versions of set theory, where everything has to be computable. Like I mentioned before (reverse mathematics), there's work on what happens when you adopt much weaker axiom sets, like second-order arithmetic but weak choice principles such as taking Kruskal's tree theorem as an axiom.
So while AI would accelerate this work, the existing body of work on alternate axioms is tremendous. A surprisingly large amount of it translates between systems, and there are precise tools to measure how weak or strong a system is, relative to its competitors.
While the existing body of work on alternate axioms is tremendous, it's finite. The set of possible axiom systems is infinite. Does current the current body of work cover all consequences of all possible axiom systems?
> I am currently being funded by the EPSRC to formalize a proof of Fermat’s Last Theorem, and a naive reaction to the news above is that I no longer have any work to do. This is not the case. The work certainly achieves some of the aims of the EPSRC project, and indeed it goes much further in terms of what is formalized (I only promised the EPSRC that I would reduce FLT to the 1980s; this repo proves the whole thing). But I also promised several other things to EPSRC: firstly, that I would be making pull requests to Lean’s mathematics library, adding fundamental objects from modern number theory; this is ongoing. And secondly, and perhaps most importantly, that I would be creating a dynamic document enabling humans to explore the modern proof. My guess is that it is unlikely that Anthropic are going to do this; they will feel that their job is done with the formalization (and they did not formalize the modern proof anyway).
> Note that mathematically this work of anthropic tells us essentially nothing: I am on record as saying that I am 99.9% sure that the proof of FLT is OK, and most people in the number theory community are 100% sure (formalization has made me more paranoid about the mathematical literature than most). From my understanding of the argument, the formalization just faithfully follows the early literature on the proof and adds nothing.
> We shared the resulting proof with Kevin Buzzard, who said:
> > This extraordinary autoformalization achievement, which Anthropic researchers say only took 11 days, proves Fermat’s Last Theorem with no assumptions other than the axioms of mathematics. Along the way we see autoformalization of algebra, harmonic analysis, geometry and number theory, and we learn that AI autoformalization artefacts are now robust enough to be built upon; the proof is multi-layered.
The article seems like an interesting Gedankenexperiment. However, I think it overrotates on the GR analogy.
For example "..ARC captures the logical leap, it misses the manipulative component—the physical sensation and embodied simulation..." makes lots of assumptions on how such a discovery must occur, e.g. through "physical sensation and embodied simulation". Results matter, not the path there.
For example, quantization of energy, at the core of QM, wasn't discovered through "physical sensation and embodied simulation" at all. Planck simply found that if energy is quantized, then one obtained the observed black-body radiation spectrum. There was no "physical sensation and embodied simulation".
