Are the Natural Numbers Supernatural?

Steven Landsburg writes:

…It is not true that all complex things emerge by gradual degrees from simpler beginnings. In fact, the most complex thing I’m aware of is the system of natural numbers (0,1,2,3, and all the rest of them) together with the laws of arithmetic. That system did not emerge, by gradual degrees, from simpler beginnings….

…God is unnecessary not because complex things require simple antecedents but because they don’t. That allows the natural numbers to exist with no antecedents at all—and once they exist, all hell (or more precisely all existence) breaks loose: In The Big Questions I’ve explained why I believe the entire Universe is, in a sense, made of mathematics. (“There He Goes Again,” The Big Questions Blog, October 29, 2009)

*   *   *

The existence of the natural numbers explains the existence of everything else. Once you’ve got that degree of complexity, you’ve got structures within structures within structures, and one of those structures is our physical Universe. (If that sounds like gibberish, I hope it’s only because you’re not yet read The Big Questions, that you will rush out and buy a copy, and that all will then be clear.) (“Rock On,” The Big Questions Blog, February 8, 2012)

With regard to the first quotation, I said (on October 29, 2009) that

Landsburg’s assertion about natural numbers (and the laws of arithmetic) is true only if numbers exist independently of human thought, that is, if they are ideal Platonic forms. But where do ideal Platonic forms come from? And if some complex things don’t require antecedents, how does that rule out the existence of God … ?

I admit to having said that without the benefit of reading The Big Questions. I do not plan to buy or borrow the book because I doubt its soundness, given Landsburg’s penchant for wrongheadedness. But (as of today) a relevant portion of the book is available for viewing at (Click here and scroll to chapter 1, “On What There Is.”) I quote from pages 4 and 6:

…I assume — at the risk of grave error — that the Universe is no mere accident. There must be some reason for it. And if it’s a compelling reason, it should explain not only why the Universe does exist, but why it must.

A good starting point, then, is to ask whether we know of anything — let alone the entire Universe — that not only does exist, but must exist. I think I know one clear answer: Numbers must exist. The laws of arithmetic must exist. Two plus two equals four in any possible universe, and two plus two would equal four even if there were no universe at all….

…Numbers exist, and they exist because they must. Admittedly, I’m being a little vague about what I mean by existence. Clearly numbers don’t exist in exactly the same sense that, say, my dining-room table exists; for one thing, my dining-room table is made of atoms, and numbers are surely not. But not everything that exists is made of atoms. I am quite sure that my hopes and dreams exist, but they’re not made of atoms. The color blue, the theory of relativity, and the idea of a unicorn exist, but none of them is made of atoms.

I am confident that mathematics exists for the same reason that I am confident my hopes and dreams exist: I experience it directly. I believe my dining-room table exists because I can feel it with my hands. I believe numbers, the laws of arithmetic, and (for that matter) the ideal triangles of Euclidean geometry exits because I can “feel” them with my thoughts.

Here is the essence of Landsburg’s case for the existence of numbers and mathematics as ideal forms:

  • Number are not made of atoms.
  • But numbers are real because Landsburg “feels” them with his thoughts.
  • Therefore, numbers are (supernatural) essences which transcend and precede the existence of the physical universe; they exist without God or in lieu of God.

It is unclear to me why Landsburg assumes that numbers do not exist because of God. Nor is it clear to my why his “feeling” about numbers is superior to other persons’ “feelings” about God.

In any event, Landsburg’s “logic,” though superficially plausible, is based on false premises. It is true, but irrelevant, that numbers are not made of atoms. Landsburg’s thoughts, however, are made of atoms. His thoughts are not disembodied essences but chemical excitations of certain neurons in his brain.

It is well known that thoughts do not have to represent external reality. Landsburg mentions unicorns, for example, though he inappropriately lumps them with things that do represent external reality: blue (a manifestation of light waves of a certain frequency range) and the theory of relativity (a construct based on observation of certain aspects of the physical universe). What Landsburg has shown, if he has shown anything, is that numbers and mathematics are — like unicorns — concoctions of the human mind, the workings of which are explicable physical processes.

Why have humans, widely separated in time and space, agreed about numbers and the manipulation of numbers (mathematics)? Specifically, with respect to the natural numbers, why is there agreement that something called “one” or “un” or “ein” (and so on) is followed by something called “two” or “deux” or “zwei,” and so on? And why is there agreement that those numbers, when added, equal something called “three” or “trois” or “drei,” and so on? Is that evidence for the transcendent timelessness of numbers and mathematics, or is it nothing more than descriptive necessity?

By descriptive necessity, I mean that numbering things is just another way of describing them. If there are some oranges on a table, I can say many things about them; for example, they are spheroids, they are orange-colored, they contain juice and (usually) seeds, and their skins are bitter-tasting.

Another thing that I can say about the oranges is that there are a certain number of them — let us say three, in this case. But I can say that only because, by convention, I can count them: one, two, three. And if someone adds an orange to the aggregation, I can count again: one, two, three, four. And, by convention, I can avoid counting a second time by simply adding one (the additional orange) to three (the number originally on the table). Arithmetic is simply a kind of counting, and other mathematical manipulations are, in one way or another, extensions of arithmetic. And they all have their roots in numbering and the manipulation of numbers, which are descriptive processes.

But my ability to count oranges and perform mathematical operations based on counting does not mean that numbers and mathematics are timeless and transcendent. It simply means that I have used some conventions — devised and perfected by other humans over the eons — which enable me to describe certain facets of physical reality. Numbers and mathematics are no more mysterious than other ways of describing things and manipulating information about them. But the information — color, hardness, temperature, number, etc. — simply arises from the nature of the things being described.

Numbers and mathematics — in the hands of persons who are skilled at working with them — can be used to “describe” things that have no known physical counterparts. But  that does not privilege numbers and mathematics any more than it does unicorns or God.

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