In the preface to Number: The Language of Science, its author, Tobias Dantzig, writes that
our school curricula, by stripping mathematics of its cultural content and leaving a bare skeleton of technicalities, have repelled man a fine mind. It is the aim of this book to restore the cultural content and present the evolution of number as the profoundly human story which it is.
Anything math-related that begins by placing math firmly among the rest of human culture—something I wasn’t expecting from a book first published in 1930—will catch my attention. On the other hand, Dantzig ends the book by concluding that though most of math is not logically necessary or environmentally dictated, it works and therefore we must accept its primacy as a method of argument. In other words, he undermines math as an arbiter of absolute truth and simultaneously affirms its power in the world—a slick trick. I’m sympathetic to his argument and don’t completely agree. His nod to math’s culturally-bound nature had me thinking his critique would be deeper, but ultimately he is an apologist for math’s success in the world despite any philosophical or logical problems it may have. Hence, Number is both more than I expected and less than I hoped for.
As Dantzig points out, the fundamental power of math is in “correspondence and succession.” Humans needed to count things—say sheep in your herd or stones in your hand. One way of doing this is to match the stones in your hand with the sheep in your herd. If they both run out at the same time, then both collections have the same amount in them. However, this basic correspondence method gets old fast: it’s quite a pain to carry around more than a few stones and much easier to have a word for the number of stones or sheep that corresponds to both collections. This idea may seem trivial to you now, but think for a moment about the mental leap required. Once made, we have a word that links five sheep with five stones with five trees with five people with five ideas—and ideas aren’t really “things” at all.
Where is this “five” that applies to every collection of “five”? It is, of course, an abstraction, and a very useful one. Furthermore, we can then add another “object” to this collection of five and get “six”—succession is born. And it doesn’t matter how big a collection of objects we have, i.e., it doesn’t matter how large a number we have, we can always add another. We take it for granted, but succession, the assumption that there is always a bigger number, is the basis of arithmetic.
Dantzig goes on to describe the origin of zero as “a gift from blind chance,” a happy accident resulting from the need to remember that there was once something there. It is a testament to his narrative skill that other kinds of numbers arise out of similar, relatively organic necessity—negative integers, fractions, algebraic numbers, transcendentals, and complex numbers all fall into his scheme. But, as smooth as his history of the origins of numbers is, Dantzig really shines when he discusses questions of philosophy: infinity, continuity, and the relationship between mathematics and reality.
The discussion of Zeno’s paradoxes gives the clearest explanation of the issue that I have seen: it comes down to a conflict between the fundamentally discreet nature of reality and the idealized continuum that is purely a product of our imaginations. For example, Zeno’s “arrow” paradox notes that an arrow in flight is not moving at any particular instant; the “motion” is therefore made up of an infinite number of “motionless” instances—how is motion possible if it is actually a series of motionless moments? Dantzig notes that the underlying assumption is that time is infinitely divisible into arbitrarily small instants. In fact, time is not infinitely divisible, so the idea of the arrow’s “motionless instant” is in our minds only; no such moment actually exists, so there is no paradox. Assumptions of infinite divisibility—of time or of space—are often useful, but remembering their imaginary status is the key to resolving all of Zeno’s questions.
Dantzig has a surprisingly post-modern view of the relationship between math and reality:
The mathematician is only too willing to admit that he [sic] is dealing exclusively with acts of the mind. To be sure, he is aware that the ingenious artifices which form his stock in trade had their genesis in the sense impressions which he identifies with crude reality, and he is not surprised to find that at times these artifices fit quite neatly the reality in which they were born. But this neatness the mathematician refuses to recognize as a criterion of his achievement. . . .
(I apologize for the gendered text; I chose to leave it so you would know what you were getting, should you choose to read the book.) He goes on to warn that when anyone “discovers a law of striking simplicity or one of sweeping universality which points to a perfect harmony in the cosmos, he will be wise to wonder what role his mind has played in the discovery.” Is this discovery truly telling us something about the universe, or is it “but a reflection of [our] own minds?”
With this caution in mind, Dantzig critiques the concept of infinity: first it is possible to construct a consistent mathematics without infinity; and, second, there is no evidence of any actual infinities; thus, infinity is “not a logical necessity and . . . all experience protests its falsity”—the conclusion must be that “the application of infinity to mathematics must be condemned in the name of reality” [emphasis in the text]. Dantzig continues concerning our ability to understand the rules that govern mathematics and the world around us: “We cannot change the rules of the game, we cannot ascertain whether the game is fair. We can only study the player at his game; not however, with the detached attitude of a bystander, for we are watching our own minds at play.”
If at this point you are shaking in your epistemological boots, never fear—Dantzig does not leave it there. In fact, he retreats to a almost radically humanist stance: “It is not the immediate evidence, nor the laws of logic, that can determine the validity of a mathematical concept. The issue is: how far does the concept preserve and further the intellectual life of the race?” That is, in the end, he doesn’t care if the foundations of math are shaky, as long as we as a “race” continue to grow and develop.
I like this turn to the real politik, but I would much rather see the focus on human rights and social justice than on some liberal ideal of intellectual legacy, despite the fact that a connection can be drawn between the two. To suggest that the arbiter of an idea’s usefulness is the degree to which it perpetuates itself and produces other ideas is solipsistic and ignores a world in which injustice and suffering are all too real. It is essential to consider the practical consequences of our ideas and actions, not in terms of some academic standard, but in terms of their ability to improve conditions for people at every level of society. Perhaps it was too much to hope that Dantzig, an academic, would see beyond his own concerns to the larger world.