Where do numbers come from?

When I was addressing this lunacy about how God exists because minds and mathematics are supernatural, I was also thinking about a related set of questions: biologically, how are numbers represented in the brain? How did this ability evolve? I knew there was some interesting work by Ramachandran on the representation of digits and numerical processing, coupled to his work on synesthesia (which is also about how we map abstract ideas on a biological substrate), but I was wondering how I can have a concept of something as abstract as a number — as I sit in my office, I can count the vertical slats in my window blinds, and see that there are 27 of them. How did I do that? Is there a register in my head that’s storing a tally as I counted them? Do I have a mental abacus that’s summing everything up?

And then I realized all the automatic associations with the number 27. It’s an odd number — where is that concept in my cortex? It’s 33. It’s the atomic weight of cobalt, the sum of the digits 2 and 7 is 9, the number of bones in the human hand, 2 times 7 is 14, 27 is 128, my daughter’s age, 1927 was the year Philo Farnsworth first experimentally transmitted television pictures. It’s freakin’ weird if you think about. 27 isn’t even a thing, even though we have a label and a symbol for it, and yet it’s all wrapped up in ideas and connections and causes sensations in my mind.

And why do I have a representation of “27” in my head? It’s not as if this was ever useful to my distant ancestors — they didn’t need to understand that there were precisely 27 antelope over on that hillside, they just needed an awareness that there were many antelope, let’s go kill one and eat it. Or here are 27 mangoes; we don’t need to count them, we need to sort them by ripeness, or throw out the ones that are riddled with parasites. I don’t need a map of “27” to be able to survive. How did this ability evolve?

Really, I don’t take drugs, and I wasn’t sitting there stoned out of my head and contemplating 27. It’s a serious question. So I started searching the scientific literature, because that’s what one does. There has been a great deal of work done tangentially to the questions. Human babies can tell that 3 things is more than 2 things. An African Grey parrot has been taught to count. Neurons in the cortex have been speared with electrodes and found to respond to numbers of objects with differing degrees of activity. The problem with all that is that it doesn’t actually address the problem: I know we can count, I know there is brain activity involved, I can appreciate that being able to tell more from less is a useful ability, but none of it addresses the specific properties of this capacity called number. Worse, most of the literature seems muddled on the concept, and confuses a qualitative understanding of relative quantity for a precursor to advanced mathematics.

But then, after stumbling through papers that were rich on the details but vague on the concept, I found an excellent review by Rafael Núñez that brought a lot of clarity to the problem and summarized the ongoing debates. It also lays out explicitly what had been nagging me about all those other papers: they often leap from “here is a cool numerical capability of the brain” to “this capability is innate and evolved” without adequate justification.

Humans and other species have biologically endowed abilities for discriminating quantities. A widely accepted view sees such abilities as an evolved capacity specific for number and arithmetic. This view, however, is based on an implicit teleological rationale, builds on inaccurate conceptions of biological evolution, downplays human data from non-industrialized cultures, overinterprets results from trained animals, and is enabled by loose terminology that facilitates teleological argumentation. A distinction between quantical (e.g., quantity discrimination) and numerical (exact, symbolic) cognition is needed: quantical cognition provides biologically evolved preconditions for numerical cognition but it does not scale up to number and arithmetic, which require cultural mediation. The argument has implications for debates about the origins of other special capacities – geometry, music, art, and language.

The author also demonstrates that he actually understands some fundamental evolutionary principles, unlike the rather naive versions of evolution that I was recoiling from elsewhere (I’ll include an example later). He also recognizes the clear differences between estimating quantity and having a specific representation of number. He even coins a new word (sorta; it’s been used in other ways) to describe the prior ability, “quantical”.

Quantical: pertaining to quantity related cognition (e.g., subitizing) that is shared by many species and which provides BEPs for numerical cognition and arithmetic, but is itself not about number or arithmetic. Quantical processing seems to be about many sensorial dimensions other than number, and does not, by itself, scale up to produce number and arithmetic.

Oops. I have to unpack a few things there. Subitizing is the ability to immediately recognize a number without having to sequentially count the items; we can do this with a small number, typically around 4. Drop 3 coins on the floor, we can instantly subitize them and say “3!”. Drop 27, you’re going to have to scan through them and tally them up.

BEPs are biologically evolved preconditions.

Biologically evolved preconditions (BEPs): necessary conditions for the manifestation of a behavioral or cognitive ability which, although having evolved via natural selection, do not constitute precursors of such abilities (e.g., human balance mechanisms are BEPs for learning how to snowboard, but they are not precursors or proto-forms of it)

I think this is subtley different from an exaptation. Generally, but not necessarily, exaptations are novel properties that have a functional purpose that can be modified by evolution to have additional abilities; feathers for flight in birds are an exaptation of feathers for insulation in dinosaurs. Núñez is arguing that we have not evolved a native biological ability to do math, but that these BEPs are a kind of toolkit that can be extended cognitively and culturally to create math.

He mentions snowboarding as an example in that definition. No one is going to argue that snowboarding is an evolved ability because some people are really good at it, but for some reason we’re more willing to argue that the existence of good mathematicians means math has to be intrinsic. He carries this analogy forward; I found it useful to get a bigger picture of what he’s saying.

Other interesting data: numbers aren’t universal! If you look at non-industrialized cultures, some have limited numeral systems, sometimes only naming quantities in the subitizing range, and then modifying those with quantifiers equivalent to many. Comparing fMRIs of native English speakers carrying out a numerical task with native Chinese speakers (both groups having a thorough grasp of numbers) produces different results: “the neural circuits and brain regions that are recruited to sustain even the most fundamental aspects of exact symbolic number processing are crucially mediated by cultural factors, such as writing systems, educational organization, and enculturation.”

Núñez argues that many animal studies are over-interpreted. They’re difficult to do; it may require months of training in a testable task to get an experimental animal to respond in a measurable and specific way to a numerical task, so we’re actually looking at a plastic response to an environmental stimulus, which may be limited by the basic properties of the brain being tested, but aren’t actually there in an unconditioned animal. It says this ability is within the range of what it can do if it is specifically shaped by training, not that it is a built-in adaptation.

What we need is a more rigorous definition of what we mean by “number” and “numerical”, and he provides one.

Strangely, because this is one case where I agree with human exceptionalism, he argues that the last point is a signature of Homo sapiens, but…that it is not hard-coded into us, and that it may also be possible to teach non-humans how to do it. I have to add that all of those properties are hard-coded into computers, although they currently lack conscious awareness or intent, so being able to process numbers is not sufficient for intelligence, and an absence of the cultural substrate to enable numerical processing also does not imply a lack of intelligence.

The paper doesn’t exactly answer all of my questions, but at least it provides a clearer framework for thinking about them.


Up above, I said I’d give an example of bad evolutionary thinking from elsewhere in the literature. Conveniently, the Trends in Cognitive Science journal provides one — they link to a rebuttal by Andreas Nieder. It’s terrible and rather embarrassing. It’s not often that I get flashed by a naked Panglossian like this:

Our brain has been shaped by selection pressures during evolution. Therefore, its key faculties – in no way as trivial as snowboarding – are also products of evolution; by applying numbers in science and technology, we change the face of the earth and influence the course of evolution itself. The faculty for symbolic number cannot be conceived to simply ‘lie outside of natural selection’. The functional manifestations of the brain need to be adaptive because they determine whether its carrier survives to pass on its genes. Over generations, this modifies the genetic makeup of a population, and this also changes the basic building plan of the brains and in turn cognitive capabilities of the individuals of a population. The driving forces of evolution are variation and natural selection of genetically heritable information. This means that existing traits are replaced by new, derived traits. Traits may also shift their function when the original function becomes less important, a concept termed ‘exaptation’. In the number domain, existing brain components – originally developed to serve nonverbal quantity representations – may be used for the new purpose of number processing.

I don’t think snowboarding is trivial at all — there are a lot of cognitive and sensory and motor activities involved — but just focus on the part I put in bold. It’s absurd. It’s claiming that if you find any functional property of the brain at all, it had to have had an adaptive benefit and have led to enhanced reproduction. So, apparently, the ability to play Dungeons & Dragons was shaped by natural selection. Migraines are adaptive. The ability to watch Fox News is adaptive. This really is blatant ultra-adaptationism.

He also claims that “it has been recognized that numerical cognition, both nonsymbolically and symbolically, is rooted in our biological heritage as a product of evolution”. OK, I’ll take a look at that. He cites one of his own papers, “The neuronal code for number”, so I read that, too.

It’s a good paper, much better than the garbage he wrote in the rebuttal. It’s largely about “number neurons”, individual cells in the cortex that respond in a roughly quantitative way to visual presentations of number. You show a monkey a field with three dots in it, no matter what the size of the dots or their pattern, and you can find a neuron that responds maximally to that number. I can believe it, and I also think it’s an important early step in working out the underlying network behind number perception.

What it’s not is an evolutionary study, except in the sense that he has a strong preconception that if something exists in the brain, it had to have been produced by selection. All he’s doing in that sentence is affirming the consequent. It also does not address the explanation brought up by Núñez, that these are learned responses. With sufficiently detailed probing, you might be able to find a small network of neurons in my head that encode my wife’s phone number. That does not imply that I have a hard-wired faculty for remembering phone numbers, or even that one specific number, that was honed by generations of my ancestors foraging for 10-digit codes on the African savannah.

Nieder has done some valuable work, but Núñez is right — he’s overinterpreting it when he claims this is evidence that we have a native, evolved ability to comprehend numbers.


Núñez RE (2017) Is There Really an Evolved Capacity for Number? Trends in Cognitive Sciences 21(6):409–424.

Nieder A (2017) Number Faculty Is Rooted in Our Biological Heritage
Trends in Cognitive Sciences 21(6):403–404.

Nieder A (2016) The neuronal code for number. Nat Rev Neurosci 17(6):366-82.

Sapolsky’s explanation for the benefits of religiosity

I agree with most of what he says.

One exception, but I think it’s just a minor wording problem.

And believing that there is something, someone responsible for it at least gives some stress reducing attributes built around understanding causality. If on top of that you believe there is not only something out there responsible for all of this but that there is a larger purpose to it, that’s another level of stress reducing explanation.

If then on top of it you believe that individual out there is benevolent—even more so control and predictability. Benevolent and listens to human entreaties? More elements of control.

Benevolent, listens to human entreaties, and prefers to listen to people like you who look like you, pray like you, request like you? Even more so. They’re just all these levels of control, predictability; they’re stress reducing.

Where he says “control”, I’d say “illusion of control”, and I also think that’s key to answering his final question: what’s up with those atheists who don’t embrace the stress-reducing benefits of religion? I think the answer is simple. If the purpose of this belief is to help us feel in control and reduce stress, it fails on both counts if you see through the illusion and realize that prayer and worship to an invisible being do nothing.

It’s like driving along in the passenger side of a car when a deer darts out in front of you; you may slam your foot down on the floorboards as if you’ve got a brake on your side, but you don’t, and it doesn’t matter. Atheists choose not to believe in an invisible parallel brake pedal on the passenger side of the car that you can push to assist the driver in stopping faster, and we don’t see what genuine virtue would be attached to pretending there is one.

But still, there are lots of people who clutch armrests and stomp on the floor to give themselves that feeling of control.

Never when I’m driving, of course.

Who’s the zombie here?

Jonathan Wells has a “new” book — he’s rehashing that dreadful crime against honesty and accuracy, Icons of Evolution. It’s titled, ironically, Zombie Science: More Icons of Evolution, and apparently, from Larry Moran’s discussion, it sounds like pretty much the same old book, flavored this time with an obstinate refusal to honestly consider any new evidence, or previous rebuttals of his previous distortions.

I like the cover, though, but only for its stunningly oblivious lack of self-awareness and self-referential nature. Wells is kind of a zombie, his arguments were destroyed 20 years ago, and he just keeps shambling back.

A nicely done critique of Murray and Harris

It tries hard to be generous to both Murray and Harris, but this analysis of their racist claims also doesn’t cut any corners in tearing apart their claims.

Harris is not a neutral presence in the interview. “For better or worse, these are all facts,” he tells his listeners. “In fact, there is almost nothing in psychological science for which there is more evidence than for these claims.” Harris belies his self-presentation as a tough-minded skeptic by failing to ask Murray a single challenging question. Instead, during their lengthy conversation, he passively follows Murray to the dangerous and unwarranted conclusion that black and Hispanic people in the US are almost certainly genetically disposed to have lower IQ scores on average than whites or Asians — and that the IQ difference also explains differences in life outcomes between different ethnic and racial groups.

In Harris’s view, all of this is simply beyond dispute. Murray’s claims about race and intelligence, however, do not stand up to serious critical or empirical examination.

I’m far less charitable in my opinion of the two. It also doesn’t help that after I posted my criticisms of their bad science, I got flooded with racist email — the people who love Harris and Murray most are not dispassionate, objective scientists, but rather a motley assortment of unhinged bigots. Among other things they did, they subscribed me to goddamned awful newsletters and mailing lists, such as the one from American Renaissance. I got an invitation to attend their 2017 conference, featuring such illustrious racists as Peter Brimelow, Jared Taylor, and John Derbyshire.

That is the company Harris and Murray keep. Let’s not pretend they’re serious scholars anymore, ‘k?

David Clarke and his pretty, pretty flair

Sheriff David Clarke is fond of wearing a flashy uniform, with lots of badges and crap splattered all over it. A man with military experience analyzes the stuff — it’s all meaningless garbage. I don’t think you can exactly call it “stolen valor”, though, since it’s not actually copies of anything militarily meaningful.

He never served in the military, either.

And guess what? I don’t think he’s a cowboy.

Fake all the way through. No wonder Trump likes him — they’re kindred spirits.

Happy belated Syttende Mai!

Yesterday was Norwegian Constitution Day, which, weirdly, communities all over the upper midwest of the United States celebrate every year. It’s all these damn immigrants who refuse to assimilate, don’t you know.

Unbelatedly, today is also the 37th anniversary of the Mt St Helens eruption. Seems like just yesterday I was sweeping ash off my car in Eugene and we were getting daily reports from wife’s family, who all lived in the Vancouver, Washington area.

Has any conspiracy theorist thought to link the two? Mad Norwegians expats in Washington getting carried away in their celebration, maybe?