Teach AI to Think in Math

In our previous discussion on Why LLMs Aren't Good at Math, we explored some limitations of large language models (LLMs) in mathematical reasoning. But what exactly makes math different from natural language skills like reading, writing, listening, and speaking?

One insight is that “the essence of math is about symmetry.” Here, symmetry goes beyond just geometric patterns; it refers to a broader concept in which, after certain transformations, the result remains equivalent to the original. This idea underlies why math is so focused on equations, which express that one thing is equal to another—essentially a statement of symmetry.

Consider geometric symmetry as an example: if we flip a shape and it remains identical to the original, we call it mirror symmetry. If rotating a shape results in the same form, that’s rotational symmetry.

But there are many other types of symmetry too. For instance, if we have a force of 10N, after some transformation, we can express it as 10,000 mN—equivalent to the original, of course. Another profound example is in networks: two networks may be equivalent, or specific parts of one network may mirror parts of another.

Douglas Hofstadter claims that “Analogy [is] the Fuel and Fire of Thinking.” Analogy, then, can be seen as occupying middle ground—it is useful in natural language, yet it also represents equality after transformation by focusing on essential similarities while ignoring superficial differences. (By the way, his book is an interesting read: Surfaces and Essences: Analogy as the Fuel and Fire of Thinking.)

So, for AI to excel in math, it first needs to understand the language of math. Current LLMs have done a good job of this. Going futher in this direction requires specifically trained LLMs that can grasp expressions in forms beyond language as a stream of tokens, including geometry, networks, and other abstract structures.

Additionally, training AI on *equalities *is crucial so it can learn to recognize equivalences. By immersing an AI in the concept of equality within the language of math, it could start to “sense” what equals what.

I look forward to a future where AI becomes truly adept at math.