Human beings possess several distinct skills that have to do with numbers. First, we can attach labels to different numbers of things: we know that three oranges and three skyscrapers both share the label ‘three’. Another way of saying this is that we recognize that there is a property, ‘three-ness’, that very different sets of objects can take on. Second, we have what is called an ordinal concept of number. This means that we think of numbers as having an order: one comes before two, which comes before three, and so on. Finally, we can do arithmetic with our numbers (well, some of us can). There are more possible sub-divisions of numerical skills, but let’s stick with those three.
Quite early on in the study of animal cognition, people wondered whether animals could also use numbers. George Romanes, a student of Darwin, wrote in 1888 that he had
“…succeeded in teaching [a] Chimpanzee… to count correctly as far as five. The method
adopted is to ask her for one straw, two straws… or five straws… Thus, there can be no
doubt that the animal is able to distinguish receptually [meaning conceptually] between
the numbers 1, 2, 3, 4, 5, and understands the name for each… But the ape is capricious,
and, unless she happens to be in a favourable mood at the time, visitors must not be
disappointed if they fail to be entertained by an exhibition of her learning” .
As you can see, Romanes demonstrated only that his chimp had the first of the three different numerical skills listed above: putting labels to numbers of objects. Later work has, however, shown that chimps and a few other animals will also arrange numbers ordinally and can even do simple arithmetic .
What about fish? We’ve actually known for a while that fish also possess some numerical skills. About 20 different species of fish have now demonstrated the ability to tell apart two sets that differ only on number . The usual method of doing this is to present the test fish with two groups of other fish, on opposite sides of its tank. Fish prefer to be part of a larger group, because it is safer, and so will swim towards the more numerous of the two groups, if they can tell the difference. You can also train fish to choose, between two cards with dots on them, the more (or less) numerous one (see the image at the top of this post, of a goldfish doing just that; ), which suggests an ability to order numbers. Nobody, as far as I know, has yet shown that fish can (or can’t) do arithmetic.
One of the interesting things about how humans (and probably most other mammals) estimate numbers is that we actually use two separate systems. For small numbers, up to about 4, we do something called ‘subitizing’, which is a little magical and involves simply seeing the number of items, all at once. For numbers larger than that, assuming we don’t have time to count them off one by one, we use an approximate system. The accuracy of this second system decreases as the number of items gets bigger, following something called Weber’s Law. Basically, your ability to tell apart two groups of objects (each of more than 4) depends on the ratio of their numbers, not the numbers themselves. So, telling 10 from 20 is as easy as telling 20 from 40, and both are far easier than telling 10 from 15.
As I mentioned, there is quite a bit of evidence that other mammals also have two similar systems for estimating number. However, there is currently a lively debate about what fish have. Some experiments seem to show that fish accurately represent small numbers and use ratio for large numbers  – just like humans – but other experiments show no dependence on ratio for any number . There’s even a suggestion that this depends on the age and experience of the fish: one day old guppies (one day!) can tell 2 from 3 but only develop the ability to tell apart larger numbers as they age, and do so more quickly if they are raised in a group (where they can practice counting how many friends they have; ).
So, we’ve arrived at one of those places where science gets really fun: we know that fish can use a concept of number, that they have at least the first of the three skills I listed above, but we really don’t know how they do it. Are they using the same two systems as mammals, which would suggest that these systems both evolved a very long time ago, or do they just have one system, which might mean that our other system (whichever one they don’t have) evolved after we parted ways about 400 million years ago? We don’t know yet, but we’ll keep looking. Count on it.