AlphaGeometry and the threat of AI’s takeover of mathematics | Explained Premium

AlphaGeometry, an AI developed by DeepMind, solves complex geometry problems at the level of the International Mathematical Olympiad.

A few weeks ago, an animated discussion unfolded in a WhatsApp group whose members are mathematicians interested in the Indian Mathematical Olympiad. The spark was a Nature paper that announced a Google DeepMind artificial intelligence (AI) named AlphaGeometry had achieved a milestone: it could solve geometry problems at the level of the International Mathematical Olympiad, nearly matching the prowess of gold medallists.

The news evoked a mix of awe, fear, and wonder among us, especially in light of how AI tools like ChatGPT have started to reshape education. Some mathematicians wondered if the advent of AlphaGeometry signals the start of AI’s ascendancy in mathematics.

Is this truly the beginning of an AI takeover in mathematics? To answer this question, let’s take a look at the inner workings of AlphaGeometry.

The Nature paper was coauthored by two computer scientists at New York University and two DeepMind researchers. AlphaGeometry is one of DeepMind’s array of AI systems – perhaps the most popular of which is AlphaZero, a deep-learning algorithm that excels at playing chess. Programs like these are part of researchers’ efforts to work up a ladder of complexity, building tools that can perform more complex tasks more reliably.

The AlphaGeometry team has published supplementary information describing the proofs generated by AlphaGeometry for some geometry problems, showcasing its ability to create hundreds of logical steps in proof construction.

Let’s start with a simple example from school mathematics. Suppose we only know that for any number a, a + 0 = a. From this, we will be able to prove that for any number a × 0 = 0. How? If a + 0 = 0 for any number a, then we should have 0 + 0 = 0. Thus a × 0 can be written as a × (0 + 0), which is the same as a × 0 + a × 0. So we have the equality a × 0 = (a × 0) + (a × 0). Cancelling a × 0 on both sides of the equation, we can conclude that a × 0 = 0.

Here, the entire proof is simply derived from the hypothesis using the rules of logic. Many computer programs can execute such a process but AlphaGeometry stands apart because of its ‘Deductive Database’ – a method that significantly reduces the number of steps in a proof.