Short Reflections on Fable's Math Capabilities

Had some time to reflect on my chats with Fable and it doesn’t move my already longer timelines, which tbf i only had a short time frame to play around and evaluate.

It’s very good at breadth as reiterated many times but it clearly lacks depth in niche subjects as it still requires a lot of babysitting and supervision by human experts to squeeze the models full capabilities, which also seems to resemble the experience of some of my friends who are at the very top in their respective niche.

For example what I’ve seen happening quite often is that they treat or confuse numerical equivalence with linear equivalence, which on paper often looks very similar but the model underestimates which mathematical consequences this can carry.

One example of such a local failure mode is that they silently replace one equivalence relation by a stronger one. for example they may treat

D \equiv E

as if it meant

D \sim E.

but numerical equivalence only preserves intersection data, while linear equivalence controls the actual line bundle $\mathcal O_X(D)$ rational functions and global sections.

Let’s say on an elliptic curve two points $p$ and $q$ are numerically equivalent as divisors because they have the same degree. This doesn’t make them generally linearly equivalent. treating

p \equiv q

p \sim q

would mean pretending there is a rational function with divisor $p - q$ which is false unless $p = q$.

D \equiv E \not\Rightarrow \mathcal O_X(D) \cong \mathcal O_X(E).

and if you wrongly assume it does then you may wrongly identify the sections or linear systems

H^0(X,\mathcal O_X(D)) \qquad\text{and}\qquad H^0(X,\mathcal O_X(E)).

you can kinda teach the model to a certin degree to not make an mistake in your domain your trying to teach it twice and slowly work your way up from there before its loosing context again. i’ve read some threads here on X talking exactly abt this.

I think this to be only an issue for now and not the future but i’m still not sure wether the transformer is the right foundational structure in order to get rid of some fundamental issues i see with it and i still heavily advocate researchers to try different out of the box novel routes to eventually reach an intelligent enough system to call AGI even if it’s not seen as cool in the current research paradigm.