ON MATHEMATICS AND THE NATURE OF THINGS

George Anastaplo

[Adapted from George Anastaplo’s The Constitutionalist:  Notes on the First Amendment

(Dallas:   Southern Methodist University Press, 1971), pages 806-808
(Chapter 9, Note 39, the concluding note of the book).]

            We have been moving, as we have prepared to close (if not throughout this book [The Constitutionalist]), from the mundane concerns of the politically-minded Publius to the more enduring concerns and diversions of the human being, even to a preliminary examination of that very activity of the reason which may be the most remarkable product as well as the ultimate justification of the community established and preserved by political men. That activity itself, dedicated at its best to understanding for its own sake, can be decisively affected by chance—and not only because it seems to depend upon the existence of human beings and community (which are themselves to some extent subject to chance).Thus, chance may be seen even in the emergence of the problems one happens upon, problems which may in turn point to enduring questions, if not to eternal things. (Chance may be seen as well both in the teachers one discovers or is discovered by and in the capacities one is provided by nature.)

Consider, for example, the following challenge (as reported by a mathematician) which one may chance to encounter:

Collective models of the nucleus [of the atom] have not lacked successes. But many details have eluded them. Take, for example, the seemingly haphazard numbers 2, 8, 20, 28, 50, 82, and 126. Physicists call them magic numbers, a name harking back to the days when the numbers were less well understood than they are now. The magic numbers have special nuclear significance; among the many hundreds of known nuclei, those containing just these numbers of neutrons or protons stand out from the rest because of their greater stability and other tell-tale signs. Clearly they reflect fundamental properties of the possible configurations of nuclear matter. They present a prime challenge to any nuclear theory. [Banesh Hoffmann, The Strange Story of the Quantum, 2d ed. (New York: Dover Publications, 1959), p. 239]

The author goes on to explain that “the shell model theory,” which “goes back to the earliest days of nuclear quantum theory,” “brilliantly met the challenge.” However that may be, one can continue to be intrigued by this series of seven numbers and to speculate about it in a manner appropriate (in an age which boasts of its “scientific” political science) for the conclusion of this study [The Constitutionalist], a conclusion which should have the merit and hence run the risk of pointing beyond both this study and the merely political. We can try to move, that is, from that which is occasional to that which is permanent, including the decisive permanent question of what it is to know (whether the knowledge be of Adam or atoms). See the very end of [Hobbes’s] Leviathan for the return of one of my predecessors, at the conclusion of his study of political things, to an “interrupted [[page 807]] Speculation of Bodies Naturall.” (See, also, the opening exchange of [Hobbes’s] Dialogue between a Philosopher and a Student of the Common Laws of England.) Cf., for the perils of such speculation, Isaac Disraeli, “Hobbes’s Quarrels with Dr. Wallis, the Mathematician,” Quarrels of Authors (New York:  Eastburn, Kirk & Co., 1814), 2: 145-46 (“[Hobbes’s] Amata Mathemata was a war of idle ambition; it became his pride, his pleasure, and his shame”), 146-47, 148 (“he had always much to say, from not understanding the subject of his inquiries”), 162 (“[Hobbes,] though a most energetic reasoner, [was] so little skilful in these new studies [geometry], that he could never know when he was confuted and refuted”). Cf., also, Plato, Phaedo 96A ff. But see Plato, Republic 546B-D, 549C ff., Laws 737C ff., 771A ff. See, also, Plato, Phaedo 96E-97C, Timaeus 53A ff., Epinomis 976C ff. (Is it certain, by the way, that Hobbes was altogether mistaken?)

One obvious feature of the challenging series of numbers which happens to be offered us by the mathematician (2, 8, 20, 28, 50, 82, and 126) is that the seven are all even numbers. Is evenness, one must wonder, a necessary but, of course, not a sufficient cause of the stability reported? Perhaps it would be instructive to consider these nuclei as doubled versions of simpler arrangements. That is, why should we not consider evenness the result of a doubling (or “folding over”) effect, thereby “permitting” the object to be less vulnerable (or more cohesive) in that it is “able” to present two similar “sides” “to the world”? (It is not unusual in nature to find unity or strength in pairs, to see a doubling or coupling as a completion. One is reminded of the intuition of Plato’s Aristophanes in the Symposium. See chap. 5, n. 126, above. See, also, Genesis 2: 18. Cf. ibid., 3: 12. Nor is it unusual to find in both nature and literature that number is a key to understanding. See, e.g., “The Catalogue of Ships” in Book 2 of the Iliad, which has the contingent of the prudent Odysseus central to the array of Achaean ships at Troy, with the contingents of the two great contenders, Agamemnon and Achilles [each with his special claim to preeminence], at an equal distance [in number of contingents] on either side of the unifying Odysseus. The contingents among the Trojans, we also notice, are one-half those of the much more complicated [and hence much more interesting] Achaeans.)

By halving our seven numbers, then, we can be said to elicit the “simpler arrangements” I have posited. Thus, we are left with this series to conjure with: 1, 4, 10, 14, 25, 41, and 63. Now we have both odds and evens—and perhaps an even more challenging series of numbers, if only in that there has been eliminated one obvious feature (evenness) which had contributed to the unity of the original series.  How are these seven numbers distinctive? That is, in what are they distinctively alike? Each one is the sum of different squares of whole numbers:  that is, each is the sum of squares of whole positive numbers in which sum no square is used more than once. This seemed to me when I first figured it out (it happened to be in Athens in the summer of 1966) rather remarkable, considering the fact that within the range that these numbers fall (1-63) there are 40 numbers which are sums of disparate squares, 23 which are not. This still seems to me remarkable. (Even more remarkable may be the explanation which the original seven numbers may merely reflect.)

I note in passing—and what happens to be “in passing” in such matters can turn out to be more interesting than the main line or original purpose of one’s inquiry—that I have discovered, upon calculating further, that there are only 31 whole numbers which are not sums of squares (in my sense of this designation, since obviously every whole number is the sum of squares if particular squares of whole positive numbers, and especially 1, may be used more than once to make up the number). The 31 numbers which I believe to be thus are 2, 3, 6, 7, 8, 11, 12, 15, 18, 19, 22, 23, 24, 27, 28, 31, 32, 33, 43, 44, 47, 48, 60, 67, 72, 76, 92, 96, 108, 112, and 128. That is, I believe it can be demonstrated that all whole numbers greater than 128 are sums of disparate squares. I have been intrigued and even reassured to discover, in the course of my calculations, that central to this series of 31 special numbers is the number 31 itself. Why all this should be so—and, indeed, what “why” means here—I do not yet know:  it is almost as if “number” reasserts itself (here as elsewhere?) by “ruling” in this manner even the “unruly” numbers. See Plato, Epinomis 990C; chap. 6, n. 43, chap. 9, n. 12, above. (Nor do I yet know whether others before me have noticed these things:  but I do know that the instruction and [[page 808]] entertainment one may derive from such inquiry do not depend, ultimately, on that form of “self-expression” which is so self-centered as to emphasize “originality.” The existence of such relations among numbers, it should be noticed, does not depend on will but rather reflects the nature of things:  will and chance may affect their discovery, not their being. [My calculations with respect to the distribution of sums of squares have recently been confirmed for me on an IBM 1130 computer through the courtesy of Hans Neumann and Michael J. Carone of C. F. Murphy Associates. It should also be noticed, in passing, that there are thousands of numbers which are not sums of disparate cubes:  at the level of 10,000, I find that one-fourth of the numbers still are not sums of cubes.]) Let us return to the point from which we digressed at the end of the preceding paragraph.

Thus, although more than one third of the numbers (between 1 and 63) within which the second series of seven (halved) numbers falls are not sums of disparate squares, all seven of these numbers are sums of disparate squares. Why should this be so? It is difficult (as well as uninteresting and unproductive) to dismiss this remarkable uniformity as a mere coincidence. (This uniformity becomes even more striking when one notices that more than one-half of the numbers [between 1 and 25] within which the first five numbers [1, 4, 10, 14, 25] fall are not sums of disparate squares.) Perhaps this uniformity has something to do—and on this the “shell model theory” may be, for all I know, useful—with the stability reported in the nuclei from which these numbers are said to be drawn. (The original series of seven numbers, it should be noticed, does not have this uniformity but rather approximates what would be a random distribution:  that is, only four of those seven numbers are sums of disparate squares. But consider the remarkable uniformity of evenness in that original series.)

A series of questions comes to mind recapitulating and extending what I have said here. These questions, which point to further possible inquiry by both the physicist and the political scientist, suggest perhaps even in their very formulation and arrangement still another “likely story” which can be instructive, salutary, and entertaining (see p. 11 and chap. 2, n. 1, above):

May it be said that the reported material stability of each of the seven nuclei (and perhaps, ultimately, of the universe?) is dependent upon a “simple” (or halved) arrangement which is restricted to a combination of disparate squares—which simple arrangement is itself, because of its reliance on squares, a configuration of elements which “look” the same on all “edges” (as distinguished from the two “sides”)?

Thus, does evenness (on the two “sides”) insure sameness “front and back” (or, if one prefers to designate the “sides” otherwise, “top and bottom”), while “squareness” (on the “edges”) insures sameness “all around”?

May it not even be said, therefore, that “evenness” promotes (or reflects) stability in one dimension and that “squareness” promotes (or reflects) stability in another dimension (however those “dimensions,” as well as “squareness” and “evenness,” are understood or defined)?

Is stability, then, dependent upon the doubling, or “folding over,” of elements which are themselves (despite their differences) essentially alike in one critical respect, in that they in turn are made up of disparate “squares”?

Should we not go further and say (skipping as we do, at least for the moment, the steps which would take us explicitly from the previous questions to this one and to the one following) that there must be an orderly limitation upon variation, upon “liberty,” if a complex body is to be relatively stable in the world we know?

Should we not go even further and say that a free community, if it is to be stable, must rest both upon the “evenness” of such associations as the family (which distinguishes us from members of other families and of other associations) and upon the “squareness” which makes all citizens the same in some essential respect (despite their divergent families, classes, or other associations), with the attractive differences (the liberty we cherish?) being “no more” than surface manifestations but manifestations which are (considering human nature) nevertheless vital to a good life if not to life itself?

Or is it, in the words of the prudent Horatio, “to consider too curiously, to consider so”?

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