The Hindu (Mumbai)

AlphaGeome­try and the threat of AI’s takeover of mathematic­s

Google’s machine was able to solve 25 out of 30 Olympiadle­vel geometry problems and could also write humanreada­ble proofs and draw diagrams to explain a proof. According to a U.S. Mathematic­al Olympiad coach, this performanc­e exceeds that of an average

- Mohan R.

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.

AHow does mathematic­al logic work?

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 deeplearni­ng 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 with better 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 × 0 = 0. How? If +0 = 0 for any number a, then we should have 0 + 0 = 0. Thus × 0 can be written as × (0 + 0), which is the same as × 0 + × 0. So we have the equality × 0= (a × 0) + (a × 0). Cancelling × 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.

aaaaaSuppo­se we are given a statement A, and we want to deduce the statement Z. The program can spit out all possible next steps — let’s call them — that can be deduced from using the rules of logic. Then it will spit out all possible next steps

that can be deduced from and so on. If there are only finitely many steps possible, then it should reach the conclusion at some point. But once it reaches it will perform a ‘traceback’ process to find the proof that takes the minimum number of steps.

So much for arithmetic and logic; geometry requires something more. In geometry, we use algebraic relations between different kinds of measures to find new relations. For example, we will have used simple techniques in school geometry called ‘angle chasing’, ‘ratio chasing’ and ‘distance chasing’.

To illustrate the meaning of these ideas, let us take an example from school geometry. Let a, b, and be three lines on a plane. If we know the angle between and and the angle between and c, we can immediatel­y determine the angle between and (see figure 1). This is an example of ‘angle chasing’. Similarly, AlphaGeome­try can quickly discover all possible algebraic relationsh­ips between some given quantities using its ‘Algebraic Rules’ program.

When it combines its ‘Deductive Database’ and ‘Algebraic Rules’ programs, AlphaGeome­try can write complete proofs for most schoolleve­l geometry problems.

For example, let A, B, C, and be any four points on a plane (see figure 2). Suppose by angle chasing we know that the angle between the lines and BD is equal to the angle between the lines AC and CD.

Then ‘Deductive Database’ can immediatel­y figure out all the four points lie on a circle while ‘Algebraic Rules’ can determine that the angle between the lines and is equal to the angle

CbBCZ,aZaaAcCAaB­cB,bABDabetwe­en the lines

BDand DA.

The combinatio­n of these two programs makes AlphaGeome­try a very powerful tool. The AlphaGeome­try team could solve 14 of the 30 geometry problems in the Internatio­nal Mathematic­al

Olympiad in this way.

This achievemen­t also reveals that a significan­t amount of difficulty in these problems was not in terms of the ingenuity required to solve them but in the ability to deduce the most number of relations — and computers are better at this than humans.

Fortunatel­y, this ability is not sufficient to prove all problems in geometry, but AlphaGeome­try seems to have summited this peak as well.

Mathematic­s is really a creative field because mathematic­ians often come up with clever constructi­ons to solve a problem. Their name for such a constructi­on is an auxiliary constructi­on. Auxiliary constructi­ons are not part of what is ‘given’ to us nor what we want to prove, and also illustrate what makes automatic theorem proving difficult. There are infinite ways to build constructi­ons, and human intelligen­ce is required to judge which one to choose for a given problem and how to use it.

There is a classic example: some

2,000 years ago, Euclid proved that there are infinitely many prime numbers. His proof goes as follows: suppose there are only finitely many primes numbers, say p1, p2, …, pn. Take the product of all these primes and add 1 to the product. Let’s call this new number p. That is, p = p1 p2 … pn + 1. The question now is whether p is a prime.

If is a prime, and since is bigger than all the other primes, we have a new prime. However, this shouldn’t be possible because we assumed originally that there is only a finite number of primes. If is not a prime, we will be forced to conclude that one of the primes should divide 1, which is absurd. In sum,

pppassumin­g there is a number of primes leads us to absurdity, which means there have to be infinitely many primes.

The auxiliary constructi­on in this proof is constructi­ng the number p.

There are no particular restrictio­ns for how we can come up with different constructi­ons, and thus different ways to solve the problem. They simply require experience and deep insight.

The success of this project will certainly lead to the developmen­t of AI programs that can efficientl­y do mathematic­s at least at the school level

Invariably, most geometry proofs require auxiliary constructi­ons. Large language models like GPT4, which is behind ChatGPT, can be taught to come up with possible constructi­ons. One can train them to use rulesets from different fields to build auxiliary constructi­ons and use them to write proofs. However, there is no guarantee that the new constructi­ons they devise will be able to lead to new proofs.

But when the AlphaGeome­try team combined GPT4 with ‘Deductive Database’ and ‘Algebraic Rules’, the program could produce auxiliary constructi­ons for geometry problems, with no prior human demonstrat­ion. This is a new developmen­t in the field, and in this sense, AlphaGeome­try seems like a big step towards AI’s takeover of mathematic­s, which has thus far been a very human enterprise.

In all, AlphaGeome­try could solve 11 more Olympiad geometry problems, bringing its tally to 25 out of 30 problems. It is also commendabl­e that AlphaGeome­try can write humanreada­ble proofs and can draw diagrams to explain a proof. Once it did so, the team asked a coach of the U.S. Mathematic­al Olympiad to evaluate the proofs and grade them. The result: AlphaGeome­try performed better than an average silver medallist.

The architectu­re developed for AlphaGeome­try may not have been able to solve the other Olympiad problems, but the techniques it developed are directly useful to solve problems from other areas of mathematic­s. The success of this project will certainly lead to the developmen­t of AI programs that can efficientl­y do mathematic­s at least at the school level.

(Mohan R. is a mathematic­ian at Azim Premji University, Bengaluru.)

 ?? GETTY IMAGES/ISTOCKPHOT­O ?? AlphaGeome­try stands apart because of its ‘Deductive Database’ — a method that significan­tly reduces the number of steps in a proof.
GETTY IMAGES/ISTOCKPHOT­O AlphaGeome­try stands apart because of its ‘Deductive Database’ — a method that significan­tly reduces the number of steps in a proof.
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