With OpenAI’s recent firing and immediate rehiring of Sam Altman, the debate over the development and use of artificial intelligence (AI) is once again in the spotlight. What’s even more unusual is that a major topic in media reporting has been the ability of AI systems to do mathematics. Apparently, some of the drama at OpenAI was related to the company’s development of a new AI algorithm called Q*. This system has been talked about as an important advancement and one of its main features was the ability to reason mathematically. But isn’t mathematics the foundation of AI? Given that computers and calculators can perform mathematical operations, how can AI systems have trouble with mathematical reasoning? AI is not a single entity. It is a patchwork of strategies to perform calculations without direct instruction from humans. As we will see, some AI systems are capable of mathematics.
Several nuances to consider
However, one of the most important current technologies, large language models (LLMs) behind AI chatbots like ChatGPT, has struggled so far to simulate mathematical reasoning. This is because they are designed to focus on language. If the company’s new Q* algorithm can solve unseen mathematical problems, it could be a significant breakthrough. Mathematics is an ancient form of human reasoning that large language models (LLMs) have thus far struggled to emulate. LLM is the technology that underlies systems like OpenAI’s ChatGPT. At the time of writing, details of the Q* algorithm and its capabilities are limited, but highly interesting. So there are many nuances to consider before considering Q* a success.
Greater reasoning abilities than humans
These AI systems can be said to be capable of mathematics. However, it is likely that Q* is not being used to help academics in their work, but for some other purpose. Nevertheless, even if Q* is unable to push the boundaries of state-of-the-art research, there is still great potential to be found in the way it is designed that could raise attractive opportunities for future development. Increasingly Comfortable As a society, we are becoming comfortable using expert AI to solve predefined types of problems. For example, most people will be familiar with digital assistants, facial recognition and online recommendation systems. What remains elusive is so-called “artificial general intelligence” (AGI), which has broader reasoning capabilities than humans.
math a basic skill
Mathematics is a fundamental skill that we wish to teach to every schoolchild, and it will certainly prove to be a fundamental milestone in the pursuit of AGI. So how else will mathematically capable AI systems be helpful to society? The mathematical mindset is relevant to many applications, for example coding and engineering, and so mathematical reasoning is an important transferable skill to both humans and artificial intelligence. One irony is that AI is based on mathematics at a fundamental level. For example, many of the techniques implemented by AI algorithms ultimately boil down to a mathematical field known as matrix algebra. A matrix is simply a grid of numbers, of which a digital image is a familiar example. Each pixel is nothing more than numerical data.
most likely to follow
Large language models are also inherently mathematical. Based on a huge sample of text, a machine can learn the probabilities of the words that are most likely to follow a prompt (or question) from the user to the chatbot. If you want a pre-trained LLM to specialize in a particular subject, this could be fine-tuned in mathematical literature, or another area of learning. An LLM can produce text that reads as if it understands mathematics. Unfortunately, doing so produces an LLM who is good at bluffing, but bad at details. The point is that a mathematical statement, by definition, is one that can be assigned an unambiguous Boolean value (that is, true or false). Mathematical logic is equivalent to the logical deduction of new mathematical statements from previously established statements.
Formal verification in one way architecture
Naturally, any approach to mathematical logic that relies on linguistic possibilities will run out of its way. One way to do this might be to incorporate some system of formal verification into the architecture (exactly how LLMs are built), which continuously checks the logic behind the progress made by larger language models. A clue that this is what may have come from the name Q-Star, which may refer to an algorithm developed in the 1970s to aid in deductive reasoning. Alternatively, Q* may refer to Q-learning, in which a model can improve over time by testing and rewarding correct conclusions. But many challenges exist in building mathematically capable AI. For example, some of the most interesting mathematics involves highly unlikely events. There are many situations in which one might think that a pattern exists based on small numbers, but when one examines enough cases it unexpectedly breaks down. It is difficult to incorporate this capability into any machine.
play devil’s advocate
Another challenge may be surprising: mathematical research can be highly creative. This must be so, because practitioners need to invent new concepts and still remain within the formal rules of an ancient discipline. Any AI method trained only to find patterns in pre-existing mathematics can never possibly create truly new mathematics. Given the pipeline between mathematics and technology, this appears to prevent the conception of new technological revolutions. But let’s play devil’s advocate for a moment, and imagine if AI could actually create new mathematics. The previous argument against this has a flaw in that even the best human mathematicians were trained specifically on pre-existing mathematics. Large language models have surprised us before, and will continue to do so.