By Keith Devlin @KeithDevlin@fediscience.org, @profkeithdevlin.bsky.social

How (and when) did human beings acquire the ability to do mathematics? Someone raised that question at the end of a workshop I participated in just before Christmas. It’s a question that has nagged me ever since Richard Dawkins’ book The Selfish Gene (1976) made me aware of just how evolution works (at least, to a first approximation). Dawkins’ bestseller came out just five years after I got my Ph.D., so by then I had sufficient experience of doing mathematics to really appreciate physicist Eugene Wigner’s famous observation (actually, the title of an essay) “The Unreasonable Effectiveness of Mathematics in the Natural Sciences”. How did we acquire that powerful tool?

On the scale of human evolution, mathematics is very recent. Based on the archeological evidence uncovered to date, our species, Homo sapiens, has been around for some 300,000 years. Mathematics, as we are familiar with today, is much younger. The European tradition traces back around 2,500 years, exemplified in the works of Thales (ca. 540BCE), Euclid (ca. 300BCE) and Archimedes (ca. 250BCE). (Earlier than that, there is archeological evidence unearthed by Denise Schmandt-Besserat, showing that around 8,000BCE the ancient Sumerians had a physical-tokens monetary system for trading and an associated symbolic notation to keep records.)

Even ten-thousand years is a very short time by evolutionary standards. How then do we explain what on an evolutionary scale is essentially an instantaneous appearance of mathematics?

Specifically, the questions splits into three parts:

• How did the human brain acquire the ability to do mathematics?

• When did it acquire that ability?

• What evolutionary advantage did the ability confer?

I thought about the question on and off until, in the late 1980s, I started collaborating with cognitive scientists, linguists, psychologists, and sociologists, and looking at the way people use mathematics in various human activities. Based on what I learned from those experts, I was able to develop an approach I thought might provide an answer. (My involvement in the PBS television series Life By The Numbers stemmed from, and became part of that collaborative work.)

There was, of course, no way to be sure my theory was correct. Though it was/is possible to generate hypotheses based on the theory that could be tested experimentally on mathematics learners (such did occur), my project was no more than a process of rational reconstruction. But to my mind, a plausible theory was better than a mystery. (And like any scientific theory, it is open to refutation. To date, that has not occurred, but then it took several centuries for classical physics to fall, and even so it still remains useful and much-used.)

My approach was to split mathematical ability into simpler mental capacities for each of which I asked two questions:

• What led to the human brain acquiring that capacity?

• When was that capacity acquired?

By digging around in the research literature on evolutionary brain development in the animal kingdom, I was able to identify several mathematics-enabling capacities for which those two questions could be answered, leaving open one big question:

• How and when did those capacities come together to give mathematical thinking?

Working with that final goal in mind, I ended up with a list of nine mental capacities that, taken together, can yield a mathematical mind:

1. Number sense

2. Numerical ability

3. Spatial reasoning ability

4. Relational reasoning ability.

5. A sense of cause and effect

6. The ability to construct and follow a causal chain of facts or events

7. The ability to handle abstraction

8. Algorithmic ability

9. Logical reasoning ability

 I convinced myself (and others I talked with) that those capacities suffice to give mathematical thinking. The key questions then were:

• What survival value did each capacity offer?

• What brought them together to give the ability for mathematical thinking?

In my book The Math Gene: How Mathematical Thinking Evolved And Why Numbers Are Like Gossip, published in 2001, I looked at each of those capacities in some detail, and presented evidence in support of my argument. Here, briefly, is a “PowerPoint-style summary” of the answers I came up with.  (For a full discussion, see the book.)

1. Number sense. This involves having a sense of the size of a collection. It does not require numbers. (Though mathematics educators use the term in a context were numbers are involved.) Studies have shown that it is possessed by many creatures, and is has obvious benefits in terms of natural selection. (Are we outnumbered? Which tree has the most fruit  to warrant climbing it? Etc.)

2. Numerical ability. This requires numbers. Only humans have it (apart from limited forms). It depends on language (in addition to number sense).

3. Spatial reasoning ability. Any creature that moves needs this ability.

4. Relational reasoning ability. We use this to understand human relationships. It is the way evolution has equipped us to live by cooperation and planning (Homo sapiens’ major survival trick).

5. Causality. Understanding causality (or more generally, acting in accordance with an acquired sense thereof) offers clear survival benefits. Even very primitive creatures respond to environmental factors in a way that benefits them.

6. Chaining. Homo sapiens inherited the ability to chain cognitive capacities from their much earlier ancestor Homo erectus, who acquired it as part of the (unique to the species) capacity to throw a projectile (such as a spear) with accuracy at high speed. Much has been written about this capacity, including the connection to language (which is how I came to know of it and use it to complete my theory of the development of mathematical capacity).

A good source that was not available to me at the time is Working‐Memory Capacity and the Evolution of Modern Cognitive Potential Implications from Animal and Early Human Tool Use, by Miriam Noël Haidle, Current Anthropology, Volume 51, Number S1, June 2010, “Working Memory: Beyond Language and Symbolism”.

The natural election advantage of accurate throwing is obvious. But to perform such a feat, the thrower’s brain has to construct in memory a chain of detailed muscle commands and then run them like a computer program; the speed at which the brain operates is simply far too slow for accurate throwing of fast projectiles to be achieved by a real-time feedback mechanism, such as the way we maintain our balance when walking along a beam. The chain of instructions has to be specified and queued up prior to execution.

We can, it seems, give thanks to the spear-throwing ability of our early ancestors not just because it provided our species with a source of food when plants were out of season, but it also prepared our brains for the modern capacities of language and mathematics.

7. Abstraction. This is the key capacity to do mathematics. It is equivalent to having the capacity for language. (It was only when I came across the work of the linguist Derek Bickerton that I realized this connection. It was reading his 1995 book Language and Human Behavior that brought all my ideas together and led me to write The Math Gene.)

Once you have abstraction, you have algorithmic and logical reasoning ability (the last two items on my list) as refinements and/or extensions of previous capacities in the list.

You acquire abstraction when the brain—which originally developed in order to mediate input stimuli prior to initiating response outputs—is able to run processes without input stimuli and without initiating output response actions; what I referred to as “offline thinking”. As Bickerton observed, this is effectively equivalent to having a (linguistic) grammar, namely a cognitive structure for performing and following actions not by acting on things in the world but by cognitively manipulating mental tokens thereof. Mathematics is just a special case of that.

The crucial step in the development of mathematical ability was then, handling increased abstraction—not a greater complexity of thought processes.

To understand mathematics, you should view it as a fictional analogue of parts of the real world, both the physical world and the social world. We take mental capacities developed to negotiate the physical and the social world and apply them to reason about a fictional, abstract world our mind creates.

A natural question to raise at this point was, by what specific steps do the above capacities come together to give mathematical thinking? Back in 2000, my plan to deal with that was to explore the issue in the Q&A discussions that I would become embroiled in on the author’s tour to promote The Math Gene (author tours was still a thing back then, now it’s all TikTok videos and podcasts), and then write a follow-up book. But as things turned out, at the same time I was finishing up my book, George Lakoff and Rafael Nunez were completing their book, Where Mathematics Comes From.

Their book was in fact being published by the same publisher as mine, though neither they nor I were aware of the other project. Yet they began their account pretty well exactly where I left off, with language and how it relates to the world providing the common thread that connects the two books. As a result, my book tour inevitably functioned not as a research seminar but more as a promotion for their book. (I actually disagreed with their account when they got to very abstract, modern mathematics involving the infinite, but for the most part their book serves as a continuation of mine.) The Basic Books editor invited Lakoff and me to dinner in Berkeley just before both books were published. I had not previously met or even interacted with Lakoff, though I had been a fan of his work ever since the mid-80s when I read his 1980 book Metaphors We Live By, written jointly with Mark Johnson.

At the time I wrote The Math Gene, I was very much enamored by John Brockman’s notion of the “Third Culture”, whereby scientists and thought leaders conduct their work by way of books written for a so-called “general audience”, rather than (or as well as) the more traditional way of scholarly papers and books. (Dawkins’ Selfish Gene was an early exemplar.) With that playground in mind, I explained my theory of the acquisition of mathematical ability with the analogy: a mathematician is someone who views mathematics as a soap opera. (Hence my book’s subtitle.)

The “characters” in the mathematical soap opera are not people but mathematical objects — numbers, geometric figures, topological spaces, vectors, analytic functions, etc.

The facts and relationships of interest are not births, deaths, marriages, love affairs, and business relationships, but mathematical facts and relationships about mathematical objects.

As any mathematician can attest, mathematical facts and relationships all come in pretty well one of four forms, answering the question (actually, one is an instruction):

• Are objects A and B equal?

• What is the relationship between objects X and Y?

• Find an object X having property P.

• Do all objects of type T have property P?

View the letters in those templates as denoting people and social types/properties and you have the basic stuff of television soap operas. (The point of this analogy is that there is a lot of research on the critical role played by gossip in cementing the human relationships that are necessary for a species that survived by cooperation. It’s why soap operas are so popular.)

To sum up, mathematicians don’t have different brains. Our species has found a way to use a standard-issue brain in a slightly different way. Mathematicians think about mathematical objects and the mathematical relationships between them using the same mental faculties that everyone uses to think about physical space and about other people.

Dr. Keith Devlin is an emeritus mathematician at Stanford University, a co-founder and Executive Director Emeritus of the Stanford H-STAR institute, a co-founder of the Stanford mediaX research network, and a Senior Researcher Emeritus at CSLI.