Can Fish Count?

CHAPTER 1

THE LANGUAGE OF THE UNIVERSE

Galileo (1564–1642) said that the universe is written in mathematics and we cannot read it unless we become familiar with the characters in which it is written. Eight hundred years earlier, the great Persian mathematician Al-Khwarizmi (in Latin, Algorithmi) (c.780–c.850) wrote ‘God set all things in numbers.’ In 1960, physics Nobel Laureate Eugene Wigner (1902–1995) wrote a famous article entitled ‘The Unreasonable Effectiveness of Mathematics in the Natural Sciences’. Mathematics, he said, had an uncanny ability to describe and predict phenomena in the physical world. The suggestion is that maths isn’t just a tool to describe the world, but rather there is something about it that is profoundly mathematical. This is an idea that goes back even further than Al-Khwarizmi to Pythagoras (c.570–495 BCE), who allegedly said that all things are made of numbers.

In one sense, this is obvious nonsense, but perhaps there is a deeper truth here. Pythagoras may have been the first to observe the numerical structure of musical pitch, and we still use terms such as harmonic mean and harmonic progression. He also documented the relationship between numbers as shapes, and again we still use his terms: squares, cubes, triangular numbers and pyramidal numbers. Once we get into the mind of the Pythagoreans, it is possible to think of the world as being built, in a rather atomic or molecular way, out of these numerically defined objects.

Finding mathematical structures in the world is the work of people we would now call scientists, and it is them that Galileo, Al-Khwarizmi and Pythagoras were addressing. It has also been assumed that scientists in the rest of the universe, provided they were intelligent enough, would be able to read the language of the universe. If they wanted to show us that they were indeed intelligent, they would broadcast something numerical. The challenge of communicating with aliens by radio was taken up enthusiastically by Nikola Tesla (1856–1943), who claimed to have intercepted a signal from ‘another world, unknown and remote’. It began with counting: ‘One… two… three…’.1 The American scientist Carl Sagan (1934–1996), in his science fiction novel Contact, had his extra-terrestrials sending a sequence of prime numbers.

In 1960, the Dutch mathematician Hans Freudenthal (1905–1990) published his Lincos code (Lingua Cosmica) that encoded not just numbers, by the number of pulses, but relations between them, such as equals to, greater than, and so on, to prove to the recipient intelligences that we were an equally advanced civilization.1 In the movie version of Contact (1997), SETI (search for extra-terrestrial life) astronomers receive a radio transmission from space that has a Lincos-like dictionary embedded in the message.

But do you really have to be an advanced civilization, or even very smart, to have some understanding of the language of the universe? Can the relatively dumb fellow species of our own world also read the language of the universe, at least the type of characters proposed in Lincos, the whole or natural numbers 0, 1, 2, 3, and the relations among them?

In the physical world, whole numbers are fundamental: the water molecule has three atoms, two hydrogen and one oxygen; nitric oxide NO (an important cardiovascular signalling molecule) has two atoms, nitrogen and oxygen; nitrous oxide N2O is an anaesthetic and has three atoms, two nitrogen and one oxygen; nitrogen dioxide NO2 is a nasty pollutant, and has one nitrogen and two oxygen. We have four limbs, insects have six, spiders and octopi have eight. We have two eyes, but some spiders have eight.

These are real properties of the real world. Things would be very different if these numbers changed – for example, if we had three arms and three eyes. The mathematical structure of the world is important to us as scientists, but it could be important to other creatures too. Consider the following numbers in the real world.

I can see three ripe fruit on that tree and five on this tree. I can hear five invaders to my territory, but there are only three of us. There are three little fish like me over there, and five here. I can hear five croaks to a phrase over there and six near the lilypad. I passed three big trees between home and the food source.

All these numbers have an evolutionary significance, and if a creature can recognize them, this could afford an adaptive advantage. Foragers benefit from selecting the tree with five ripe fruit over the one with three, and the female frog benefits by mating with the male able to produce six croaks in a breath over the one that could only manage five (see Chapter 8). Lions are more likely to survive and reproduce if they only attack invaders when they outnumber them (see Chapter 5).

These ideas are the starting point for this book. Can our unique mathematical abilities have an evolutionary basis? How is it possible to tell if creatures without language can respond to the numerical structures of the universe?

In fact, there has been a hundred years of research into animal abilities to read the mathematical structure of their worlds. Now that doesn’t mean that their abilities, if they exist, are evolutionary antecedents of our own. That would require a genetic link between us and them. There is an example that could serve as a clue. We know that there are genes for timing that have been conserved for more than 600 million years, from before the invertebrate line (insects, spiders and so on) separated from the vertebrate line (fish, reptiles and mammals). They are helpfully labeled CLOCK, PER (for period) and TIM (for regulating the time of the daily circadian rhythm). We can find these genes in fruit flies (Drosophila melanogaster) and the descendants of these genes from a common ancestor in humans. Timing is a mathematical property of the world because duration can be represented by a number. Thus we may find genes for numerical abilities to go with the timing genes.

What is number? What is counting?

Before delving any deeper I had better come clean about what I mean by numbers and by counting. All readers of this book think they know what a ‘number’ is. They may think of the words one, two, three, or of symbols – 1, 2, 3 – or both. Because we have been brought up in a numerate culture and have learned to count with counting words, we may habitually think of counting as necessarily reciting ‘One, two, three…’. Scientists have to be more specific.

Of course, there are many kinds of number: positive whole numbers, sometimes called natural numbers; integers which include negative numbers; fractions, reals (decimals), imaginary (i, the square root of –1) and even the late, great John Conway’s surreal numbers. Among the whole numbers are ordinals for ordered sequences, like pages in a book or house numbers in a street, which do not directly reference magnitude. So my house is number 44, but it is exactly the same size as my neighbours’ in 42 and 46. This page is the same size as the next. There are also numerical labels for TV channels and phone numbers. These refer neither to the magnitude nor order. And it doesn’t make sense to ask whether my phone number is larger or smaller than yours, nor to ask if it comes before or after yours. The type of number that does indicate magnitude is a cardinal number. These denote the size of a set.2

The idea of a set that underlies the idea of cardinal number needs some further explaining. Think of the set of three things – for example, the set of three coins in the fountain. Sets and their sizes do not depend on what the objects are: they can be three coins (physical objects), three knocks on the door (sounds) or three wishes (thoughts). The important thing is that these sets have nothing in common apart from their threeness. This leads to an ancient philosophical difficulty I will return to in the final chapter.

For the rest of the book when I talk about numbers I will mean cardinal numbers unless I specify otherwise. However, I want to introduce a new term, numerosity, to refer to set size rather than the logical and mathematical term cardinality. This is because we are talking about what goes on inside the brain of an animal, not about logic or mathematics.

I follow the eminent scholar Randy Gallistel in his proposal for assessing whether an animal, or a human, is actually capable of representing numerosity in their brains.3 He sets out two criteria:

(a) Do they represent numerosity as a distinct property of a set, separable from the properties of the items that compose the set?

This is exactly in line with what I have described.

It is not enough to represent numerosity; you also have to be able to do something with it. You have to be able to calculate, to do what Gallistel calls ‘combinatorial operations’ of a certain type.

(b) Do they perform with the representatives of numerosity combinatorial operations isomorphic [equivalent] to the arithmetic operations that define the number system (=, <, >, +, –, ×, ÷)?

Thus, we can ask whether the animal can in some way recognize that two sets have the same numerosity (=), and that set A is larger than set B (A > B), and that the sum of sets A and B is equal to set C (A + B = C). Division and multiplication you might think to be much more difficult, but for an animal to calculate how frequently food or a predator occurs this is a matter of division (e.g. 3 appearances per day = 24 hours/8-hour intervals). When it comes to navigation – ‘dead reckoning’ in sailing vocabulary – this involves quite complicated calculations.

Now these are fairly tough criteria. But I would elaborate (a) to ask to what extent can an animal abstract from a particular set to a novel set? In other words, can they assess whether sets with different types of object have, for example, the same or different sizes? For instance, can the animal notice that a set of sounds has the same numerosity as a set of food items? Other species, indeed some humans, may be able to apply numerosities only to some sorts of objects, perhaps those vital to one’s life but not to other things. They may not be able to tell whether a set of one type of object, for example petals on a flower, has the same numerosity as other types of objects, such as landmarks, as I will try to explain a bit later.

Gallistel’s two criteria reflect current philosophical thinking about the foundations of mathematics.2 They also reflect how arithmetic is typically taught around the world: 1–1 enumeration of sets of objects, ensuring that enumeration does not depend on the nature of the objects (abstraction), and then working with the arithmetical consequences of operations on sets – comparing, combining and adding, splitting sets, subtraction, and so on.

WHAT IS COUNTING?

Most readers of this book, if they think about counting at all, will think of it as an activity that is intentional, purposeful, conscious and usually accompanied by the use of counting words. And an intention and purpose of this activity is establishing the numerosity of a set. This characterization rules out all non-human counting, and as we will see in the next chapter, some human counting. No other creatures have counting words – apart from the parrot Alex in Chapter 6 on birds – and ascribing intentions, purposes or consciousness to non-humans is, to say the least, controversial. We may be willing to ascribe them to the great apes or to pet dogs, but to fish or insects? No way.

Suppose you have to count the number of dolls on the table, and do this by counting out loud one, two, three dolls. This enables you to establish the size of the set of dolls on the table: three. Rochel Gelman and Randy Gallistel in their ground-breaking book, The Child’s Understanding of Number (1978), listed the ‘counting principles’ that characterize human counting.4 The ‘cardinal principle’ states that the counting process yields the cardinality of the set with the last word of the count, provided of course that all the objects are counted and each object is counted once and only once (1–1 enumeration). That is, there is a strict one-to-one correspondence between the counting words and the objects in the set. They also note that to have a sense of set size, it doesn’t matter which object you start counting with; the size will always be the same. So for the set of three objects A, B, C, it doesn’t matter whether you start with A, B or C, you will still end up with the numerosity of three. They call this the ‘order-irrelevance principle’. Finally, they note that sets can be sets of anything, three dolls, three chimes, or three wishes. This they call the ‘abstraction principle’. These principles are required for humans to be competent counters using counting words. They bring together the cultural tools – the counting words – with the concept of set size. I’ll say more about how children learn to count with counting words in the next chapter.

Now consider one way in which we count the members of a set without necessarily using counting words, by using a very simple and cheap device called the tally-counter (see Figure 1). The button on the top is pressed once for each object counted – the one-to-one correspondence principle. The total of the count is given in the readout from the last button press – the cardinal principle. Anything that is countable can be counted (the abstraction principle) and the set can be counted starting with any member (the order-irrelevance principle).

To use the tally-counter a person also has to count. That’s the hard part. Suppose you are a shepherd counting sheep but not goats – you have to be able to decide which is which. Then suppose you have to decide whether you have more sheep than goats – you will have to count the goats with a separate counter and then inspect the readout to see which number is larger. The tally-counter has a memory – a readout – that indicates the number of objects counted. Of course, you could use the same tally-counter twice, once for sheep and once for goats, but then you would need a memory for the sheep, set the counter to zero and start again for goats. In both cases, you also need a mechanism for carrying out the comparison. I’ll show how this might all work in a moment.

The English philosopher John Locke (1632–1704) made an early attempt to characterize number and counting in a way that anticipates our tally-counter in An essay concerning human understanding (1690). He said the simplest idea is one; one can be repeated, ‘and by adding repetitions together’ – like repeating the button pushes – we get the complex idea of larger numbers. ‘Thus, by adding one to one, we have the complex idea of a couple,’ etc.5

This is an example of recursion, a procedure or function, that calls itself to do it again. Locke is proposing a particular form of recursion called ‘tail recursion’, where a procedure generates the last item (the tail) by adding one and then calls itself to carry out the procedure again, adding another one.

As to counting words, Locke says we should give each complex idea ‘a name or sign, whereby to know it from those before and after’. The tally-counter can be thought of as an implementation of Locke’s proposal. Each button press is a repetition of one, and the sum of button presses is given by a ‘sign’, in this case a readout in digital form.

The other issue is the degree of abstraction involved in numbers and counting. Big Ben chimes five times at five o’clock and we have five fingers on each hand, but chimes and fingers have no other properties in common. So how can brains, even tiny ones, deploy these abstract ideas, and how can we tell that they do so? We can ask humans to count out loud or say whether the number of fingers and chimes is the same, but we can’t ask fish. To what extent can other creatures generalize the numerosity in one modality – sounds, for example – to numerosities in other modalities – visible objects, actions and so on? This will be a matter for a selector rather than the counter. Although I have given this component a special name, it is really based on widespread ideas in cognitive science and neuroscience. It is a way of focusing on or attending to an object or an event. The selecting process does not have to be conscious or even intentional, which are controversial concepts when applied to non-human animals. It simply picks out one or more objects from the environment for further action.6 We can use a single tally-counter to count the chimes and the fingers, as long as it can access two memory locations.

A NEURAL COUNTING MECHANISM

Of course, we don’t literally have tally-counters in our heads, but do we have some kind of neural equivalent? The tally-counter is a