In our previous article, we proposed the difficult challenge of thinking about a world without numbers. Although the task is indeed difficult, it is not without its merits. One way to tackle it would be to ask, since we clearly live in a world that is dependent on numbers, what was the process that brought us here, starting from the very beginning? In other words, since there clearly was a time without numbers in the history of mankind, what did it look like and what were the actual first steps taken towards numbers?
This article also comes with a book recommendation, which is not (only) about the history of mathematics, but the history of numbers. In fact, even this characterization is not entirely accurate: the author proposes a history of the number concept, being convinced that it means much more than the history of mathematics: it is the history of mankind itself. Here’s a revealing excerpt from the first chapter:
Numbers have become so integrated into our way of thinking that they often seem to be a basic, innate characteristic of human beings, like walking or speaking. But this is not so. Numbers belong to human culture, not nature, and therefore have their own long history. For Plato, numbers were “the highest degree of knowledge” and constituted the essence of outer and inner harmony. The same idea was taken up in the Middle Ages by Nicolaus Cusanus, for whom “numbers are the best means of approaching divine truths”. These views all go back to Pythagoras, for whom “numbers alone allow us to grasp the true nature of the universe”.
In truth, though, it is not numbers that govern the universe. Rather, there are physical properties in the world which can be expressed in abstract terms through numbers. Numbers do not come from things themselves, but from the mind that studies things. Which is why the history of numbers is a profoundly human part of human history.
— Georges Ifrah, The Universal History of Numbers, first edition, 1985
Before deciding whether we agree with the author or not, it is worth thinking again about the possibility of a world without numbers. The difficulty of this question tips the scale towards the necessity of numbers, or more precisely, of the number concept.
However… The evidence of objects was kept since the dawn of mankind using uniform cuts and grooves (called tallies or scores, whence the modern term). One of the earliest discoveries of this method is the Ishango bone, found in Congo and dated around 20,000 years ago.
The scores were grouped and when they became too many, symbols were invented as placeholders for various multitudes, i.e., multiples. These multiples became bases of numbering. Since digit counting (that is, using fingers) was and still is a fundamental method of learning numbers, most of the bases were multiples of 10 (10, 20, and 60 being the most well-known examples). However, the subject we are steering towards now is that of the one-to-one correspondence.
Before there even existed any signs or words, how could one communicate or keep track of numbers, that is, of discrete quantities? Through a method that is both simple and essential in mathematics: the one-to-one correspondence, also called bijective. The concept is straightforward: think about sign language. When one wants to buy three objects, they show the seller three fingers. Let us ignore the millennia-long evolution which automatically makes us associate three objects with the numbers and the word “three” and imagine that we are at the dawn of mankind. Such an interaction is based on the premise that for each finger, the buyer wants one object[1]. Therefore, we have an example of a one-to-one correspondence between the buyer’s fingers and the objects they want.
Similarly, let’s make another mental exercise and imagine that the prehistoric man wanted to provide for his family of, say, ten members, so he went gathering and hunting. Assume each member wanted to eat one rabbit and one fruit. The hunter-gatherer could have gone, well, hunting and gathering with a bone, a piece of tree bark or some other material on which he had made ten grooves or signs. He hunted rabbits and gathered fruits until he could associate them one-to-one with the signs on his “delivery list”, so to speak. He didn’t have to know what “ten” means, nor whether there is any word, sound, or sign to characterize it. In fact, he could have used the same procedure when making the list itself: he could line up his family members and draw a groove on his primitive list while passing in front of every one of them.
It should be clear that this explicit method makes the one-to-one correspondence seem laborious and hard to use. Furthermore, in reference to our previous article, it assumes the existence of “one”, which is repeated as many times as needed. But the essence of saying “for each X exactly one Y” is what matters here. This was further developed mathematically in an abstract form throughout the centuries and millennia, and nowadays, it is one of the most important and powerful methods used in all branches of mathematics. The correspondence is no longer made explicit by listing the pairs of objects that correspond to each other, but instead, rules are used. For example, one correspondence between the sets {0, 1, 2, 3, 4, 5} and {0, 1, 4, 9, 16, 25} could be made explicitly, associating 0 to 0, 1 to 1, 2 to 4 etc., but the method limits us to a small number of objects. If we draw the correspondence through a rule, such as for each number, one associates its square, one could extend it to sets with an arbitrary number of elements. In particular, it could be used for infinite sets! The rule above could be used to show that there exist “as many” natural numbers as perfect squares. The quotes must be added, since both sets are infinite. However, there exists a one-to-one correspondence between the two.
In an abstract formulation, mathematical objects that could be put in a one-to-one correspondence are “the same” — the specific term is isomorphic, whose etymology means the same shape. The existence of such a correspondence between two objects reduces them to one from the point of view of their “type”, each being like a copy of the other or both being a copy of the same isomorphism type, as it is called.
What if one cannot make a one-to-one correspondence between two sets though? There could be two possible reasons. Assume we have two sets, A and B, and we try to make their elements correspond one-to-one. If we can’t, it could be that:
Either we are left with elements of A which can no longer be associated with elements of B, since the latter has no more elements. This means that A has more elements than B.
Or we are left with elements of B which have no elements of A associated, since the latter has no more elements. This means that B has more elements than A.
Such connections were fundamental in understanding infinite sets in the eighteenth and nineteenth centuries, through Georg Cantor, Richard Dedekind, David Hilbert, and their peers. Moreover, mathematicians proved that infinity itself has a hierarchy: there exist infinite sets which could be put in a one-to-one correspondence, but there are also some for which it is not possible. What’s striking is that this discussion holds true unmodified for infinite sets alike. In other words, and to put it popularly, it means that some infinities could be larger or smaller than others. This is the topic we explore in our next article.
[1] To be mathematically precise, we have to add that for each finger, the buyer wants exactly one object, and the objects have to be different, i.e., they cannot take the same object for different fingers, nor two or more objects for the same finger. Usually, this is understood in common language, but it is worth making it mathematically precise as well.