Maths is built on concepts, all the way down

Some of us think numbers are just concepts, but even then we have differences

A while ago I offered my own (fairly short) solution to the foundational question in the philosophy of mathematics: what sort of thing are numbers and other mathematical objects?

The traditional approaches to this question turn on whether numbers (and similar objects) actually exist. Platonists hold that they do (e.g. there is a real thing that is the number 2). Constructivists, by contrast, hold that numbers are simply human creations and are therefore largely arbitrary. A third school, the formalists, typically treat mathematics as a subset of logic, so a number is a just a type of formal logical construct.

My view, as I wrote previously, is that “mathematics is the systematic exploration and elucidation of the inevitable (or logical) consequences of different foundational concepts.” The good news is that I’m not the only person on Substack arguing for a ‘conceptualist’ account. David Bessis, a trained theoretical mathematician, entrepreneur and author, has been arguing for a position like this and has also explored the implications with respect to current AI systems.

Conceptualism

Bessis argues that the objects of mathematics are akin to ‘universals’ in language, which are “abstractions that are detached from any particular individual: beauty, roundness, youth…” The conceptualist position on universals, on his reading, is that “universals are products of human cognition” and he argues that the same holds for mathematics. That is, mathematical objects, map to things in our brains or cognition. They don’t have an external existence (Platonism) but aren’t entirely arbitrary conventions (constructivist) as they reflect something that genuinely exists.

There are some potentially uncomfortable consequences of this approach for mathematicians, that Bessis himself accepts. He argues that mathematics “is an imaginary activity supported by symbolic writing systems”. This means that mathematics isn’t true in the sense it describes anything in the world, but it works as a field because we treat it as if it were true. For Bessis, one huge benefit of mathematics for humans is that it is a large driver of neuroplasticity - it makes our minds and thinking better.

A stronger conceptualism

I agree broadly with his conceptualist approach, that mathematics is about understanding what follows from various concepts. However, I’d argue that Bessis underplays the importance and role of concepts in all human cognition. Concepts are not simply things that exist in our brains, but are the foundational structures of all human knowledge and, if we take Kant seriously, all human experience. Causation, for example, is a concept that we cannot make sense of the world without but also something we cannot define or see in the world.

Even if you don’t accept the Kantian point that concepts structure human experience, it still holds that concepts are the building blocks of all of our knowledge and theories about the world, especially scientific theories. As I’ve noted before, if you examine any scientific theory you’ll find it is built on key concepts. Without concepts, there are no theories.

This foundational importance of concepts explains one astonishing discovery in the history of science - the critical importance of mathematics. There is no a priori reason why the world should turn out to be mathematical. For example, many Greek philosophers saw the world as flawed while mathematics was perfect. For them, there couldn’t be any use for mathematics in science.

Yet not only is mathematics foundational for science, but weird mathematical fields have scientific applications that we would never expect. A good example is the way that Complex Numbers, which are based on what are known as Imaginary Numbers, are essential for electrical engineering.

Let me explain a little for those who don’t know what any of this means. A square number is any number that is the result of one number multiplied by itself. One basic property of square numbers is that they are all positive (as two negatives multiplied together produce a positive number). The concept of an Imaginary Number was discovered when mathematicians asked: “What if we assume that there is a number that you multiply by itself and the result is -1?” This number is called i - for imaginary - and it creates a whole new field of mathematics. The unexpected thing is that this field, formed by creating an imaginary number that can’t really exist, is indispensable for understanding real world phenomena.

Yet the existence of situations like this isn’t so bizarre if we take a conceptualist position seriously. The building blocks of all scientific theories are concepts, which can interact in weird and interesting ways. Mathematics is the study of the consequences of different concepts, so is the rigorous study of how concepts interact and what flows from them. It therefore provides a series of worked out concepts and conceptual structures for scientists to explore and test for fit with the real world.

As we know but sometimes forget, the world often surprises us and scientific theories often go beyond our human concepts. The exploratory nature of mathematics has supplied us with a range of different conceptual structures that we have used in our scientific theories. It is hard to predict in advance which will be useful and which won’t. There is no a priori reason why Complex Numbers are useful, but also no a priori reason why not.

In one sense, mathematics is, to follow Bessis, imaginary as it is investigating and exploring objects and structures that we don’t know whether they exist in the physical world or not. On the other hand, it rigorously explores the core building blocks which we use to build and test scientific theories, so isn’t fanciful or a wild flight of fantasy. To ask whether mathematical objects are real is therefore a bit like asking whether architectural drawings are real. They both describe something that could turn out to exist, but not always. Some architectural drawings are impossible to actually build, just as some mathematical fields cannot be applied. But that may change in the future, and we cannot know without exploring the those fields.

What about AI?

Whether you agree with my stronger account or not, a conceptualist account of mathematics has an important consequence for the practice of theoretical mathematics and research that Bessis also writes about at length. Historically, mathematicians have celebrated proofs as the pinnacle of mathematical achievement: the goal of the mathematicians is to prove difficult and interesting theorems.

However, Bessis argues that the real work was the conceptual work involved in new ideas, new definitions and new insights. As he wrote “Mathematicians created value by introducing new concepts, but the rule was that only theorems could put bread on the table.” If we take a conceptualist account of mathematics seriously, it naturally follows that exploring and elucidating foundational concepts is the work of mathematics. Proofs and theorems are the demonstration of coherence, not the bulk of the productive work. New proofs are often most valuable when they create new concepts or elucidate old ones. The proof itself is less important.

This has significant consequences for the future success of AI systems in mathematics. There has been a lot of buzz over the past couple of years about generative AI systems solving various theoretical mathematical problems, including research level questions. AI systems, or more precisely teams of researchers using AI systems, have solved various research level questions. However, the strengths and weaknesses of these systems is significant and it lines up nicely with the conceptualist account I am exploring here.

Looking at one competition, the First Proof project, various AI systems were put to work and, as a useful commentary noted, it “seems likely that somewhere between 6 and 8 [of the 10] problems were solved correctly if one combines all attempts.” However, “Even the best models/scaffolds seem not to be able to reliably detect when they are producing nonsense.” That is, the AI systems could generate correct proofs but would also generate nonsense - and couldn’t tell the difference between the two.

This strongly suggests that there is a foundational limit to the ability of generative AI systems. They can manipulate words, logic and work with formal systems, but do not have the ability to grasp or comprehend the concepts that sit behind them. This explains whey they cannot tell whether they are providing valid proofs or nonsense. They are in the same boat as all of us who are not research level mathematicians: we cannot grasp the concepts at play and so have no idea whether any proof is correct or not.

So AI systems may be able to generate new and interesting mathematical proofs, even if humans will have to decide if they are correct or not.¹ However, they will not be able to do the most valuable work human mathematicians do: elucidate new concepts and definitions. It comes back to the problem I have been writing about this year: AI systems cannot grasp concepts like humans can.

The act of grasping a concept, the “sudden flash of meaning”, is something we are all personally familiar with but unable to explain in any real sense. We don’t know how it works and what goes into it, and we definitely cannot teach it to a digital system. It is something foreign to any current AI system, both in their experience and in the logical structure of their training or code base. AI systems do statistical analysis, logical deduction, correlations and various other sophisticated operations, but they don’t grasp concepts.

1. There is a caveat to note here: some proofs can be expressed in fully formal languages, and so a computer could provide a proof that is demonstrably correct but we cannot check.