Surfaces and Essences

  Lagrange’s first ideas about substitutions were extended by Paolo Ruffini in Italy and Niels Henrik Abel in Norway, among others. In 1799, Ruffini published a proof that the fifth-degree equation was in fact unsolvable in terms of radicals — that is to say, using square roots, cube roots, fourth roots, and so on. To prove that a goal in mathematics was unattainable was almost unprecedented at that time. Unfortunately, in a couple of passages in Ruffini’s proof there were some gaps, and for that reason, few of his colleagues took his result seriously. It took another thirty years until Abel published a more complete (though still not flawless) proof of the same result. Finally, in 1830, the very young French mathematician Évariste Galois put the crowning touch on all this work by writing an article that spelled out the precise conditions under which a polynomial equation would or would not be solvable using radicals.

  At the heart of Galois’ work was the notion of a symmetric group, which is to say, the set of all permutations of a finite set of entities (in this case, the N solutions of a given polynomial equation). Among these permutations are swaps, for instance, in which just two entities are interchanged (A ↔ B), and which are thus analogous to reflections of a human face in a mirror. Then there are cycles, such as A → B → C → A. This cycle, which carries us back to our starting point after three steps, is analogous to a rotation of an equilateral triangle through 120 degrees. Any physical object whose appearance doesn’t change under a reflection or a rotation has some kind of symmetry, and Galois understood the crucial importance of the symmetry that was possessed by the abstract “object” constituted by the set of all N solutions of an Nth-degree polynomial. In making this connection, Galois was consciously generalizing the idea of the symmetries of a physical object, extending it to collections of abstract algebraic entities. The “reflections” and “rotations” of the set of all N solutions to a given polynomial were transformations that could be applied one after another, and they reminded Galois of a series of arithmetical calculations carried out one after another.

  This reminding soon gave rise to the central analogy that guided Galois in his quest. Having written down a “multiplication table” of all the different permutations of the solutions of a particular polynomial, he proceeded to study the patterns in his table. For example, in the case of the most general quartic equation, Galois knew that the group of symmetries of its four solutions consisted of all the different ways of permuting four indistinguishable objects (ABCD, ABDC, ACBD, …… DBCA, DCAB, DCBA). There are 24 such permutations, so the multiplication table of this group has 24 rows and 24 columns. Galois suspected and then showed that the secrets of polynomial equations resided in hidden patterns in the multiplication tables of such groups of permutations.

  The history of groups is far too long to tell here, but for us what is important is the idea that the structure of groups themselves became a new domain for research in mathematics. Specifically, Galois discovered that inside a group there are often smaller groups — “subgroups” — and inside subgroups there can be subsubgroups, and so on. He saw that there could be jewels nested inside jewels going many levels down: an incredible feast for his inquisitive young mind!

  Once mathematicians had understood and absorbed Galois’s highly novel ideas, they collectively passed a point of no return. They made the leap of moving away from the study of concrete, visualizable objects, like the regular polyhedra, and towards the study of more abstract entities such as rotation groups (which reflected the hidden symmetries of concrete objects) and then substitution groups (which reflected the hidden symmetries of abstract entities). And it was the teen-aged Galois who, around 1827, first understood the tight connection between the nesting pattern of subgroups of the group of permutations of the solutions of a polynomial equation and the possibility of solving the given equation with radicals. As the twentieth century unfolded, groups started to pop up in every field of physics, like wildflowers in spring meadows, thus showing the profound prescience of the intuitions of this genius who died at just twenty.

  Fields, Rings, N-dimensional Knots…

  The very same teen-aged Galois also came up with the theory of fields, which are groups that have two different operations, one being analogous to addition in its familiar sense and the other to multiplication. The key connection linking these two operations is the distributive law: a x (b + c) = a x b + a x c. (Note: The symbols “x” and “+” do not stand for the usual operations of multiplication and addition, but for analogous operations that act on the entities belonging to the field.) Once again we see that the jump to a new kind of entity (sets of things having two operations linked by a distributive law) came thanks to an analogy with an already-familiar entity (ordinary numbers). The most familiar of all fields is the infinite set of real numbers, and another example of a field — smaller though still infinite — is given by the rational numbers (a subset of the reals). Certain other familiar sets of numbers, however, such as the whole numbers, are not fields, because if one tries to carry out division, one is led outside of the set (for instance, 2/3 is not a whole number).

  Out of the blue, drawn forward solely by his sense of beauty, Galois made a new analogical leap. He observed that fields, like the groups of symmetries that he had just invented, did not have to have an infinite number of members. He saw that there could be finite fields as well, such as the set of integers from 0 through 6, which is a seven-element field, provided that the “sum” of two elements m and n is defined to be the remainder after the ordinary sum m + n is divided by 7. Thus 4 + 5 would not be 9 but just 2 — and similarly, 4 x 5 would not be 20 but just 6 (the standard term for these operations is “addition and multiplication modulo 7”).

  In any field, whether infinite or finite, one can write down polynomials and ask about the solutions of these polynomials. All the natural questions discussed above in connection with “real” polynomials arise once again in these new worlds. Just as in the field of real numbers, there can exist polynomials that have no solution inside the field in question. Évariste Galois, at age 17, took a marvelous flight of fancy, deciding that a field lacking such a solution could be enriched by adding to it one or more “imaginary” elements modeled on the classic “imaginary” number i, which solved the quadratic equation x2 + 1 = 0, which had no solution in the classic field of real numbers.

  What the young Galois dreamt up opened the door to a new style of mathematics characterized by a constant drive towards greater abstraction. One takes a familiar phenomenon — say, the set of prime numbers 2, 3, 5, 7, 11, 13, 17, 19, … — and one tries to pinpoint its most essential essence, its deepest crux. Then one tries to locate this exact essence in other structures that are less familiar. A good example is given by the “Gaussian primes”, which are complex numbers of the form a + bi, with a and b being ordinary integers, and which share with the classical prime numbers the quintessential property of having no divisors except themselves and unity. The study of such generalized prime numbers is a rich vein of research in contemporary mathematics.

  Not long after the tragic death of Galois, mathematicians came up with a new kind of algebraic structure that was similar to a field but in which division was not always possible. The idea lying behind these new structures, called “rings”, came from the attractive — indeed, well-nigh irresistible — analogy that said that every field had to contain a special subset that played the role played by the integers in the classic field of the rational numbers. Whereas you can always divide any rational number by any other to get a third one, this does not hold for integers — dividing 5 by 3 does not yield another integer. Similarly, in a ring, division of two arbitrary elements is not always possible. Thus, given any specific ring, one could imagine its set of “prime numbers” — those entities that are not the product of any elements of the ring except themselves and unity. The study of the prime elements of rings attracted many researchers in the second half of the nineteenth century, inspiring them to define structures and phenomena that were
ever more abstract and that seemed ever more distant from “real life”. And yet many of these ultra-abstract phenomena turned out, decades or centuries later, to be profoundly implicated in the laws governing the physical universe, and so in the end they turned out to be every bit as real as the more familiar mathematical notions that everyone learns in school.

  The study of groups furnishes a good example. Galois noticed that under certain special conditions, a group can be “divided” by one of its subgroups (of course we are not speaking of the familiar arithmetical notion of division, but of a far more abstract analogue — an unmarked sense of division, to use the term we used in Chapter 4). This process gives rise to what is called a “quotient group”.

  In order to get some sense of what Galois’s new kind of abstract division is like, imagine a 100-floor skyscraper all of whose floors are identical. We now wish to “divide” the building by one of its floors. What we will get is a skeletal skyscraper, a tall vertical structure no longer having 100 floors but simply 100 dots. Et voilà! This abstract structure is the “quotient skyscraper”, the result of “dividing” the original skyscraper by its generic floor. If you are able to carry this rarefied notion of division from the world of skyscrapers back to the much more abstract world of groups, then you will have some feel for what “dividing” a group by one of its subgroups is all about. This process of division by a subgroup can be repeated until one finally obtains a chain of “prime factors” of the original group. Thus one is led back to the familiar notions of factorization and prime number, or at least their very distilled essences, now instantiated in this abstract and austere context.

  The theory of knots, launched by Leonhard Euler in 1736 and ever since then profoundly enriched by thousands of researchers, furnishes another good example of this kind of push towards abstraction via analogy. At a certain point, the concept of factorization was extended into the world of knots (and here, we really do mean knots on a string — a very concrete idea, for once!), which meant that a complicated knot could be seen as the “product” of two simpler knots, and so forth. This process of reducing complex knots to simpler and simpler knots then led, naturally, to the idea of prime knots. However, no sooner had this idea been suggested than it was inevitably generalized to intangible knots inhabiting N-dimensional spaces, and all at once things became completely unimaginable to anyone except to those few hardy souls who are able to survive in the forbidding world of such high abstractions.

  As we’ve tried to show, modern mathematics is pervaded by an unrelenting pressure to move towards ever more abstract ideas, but unfortunately the ubiquity and intensity of this pressure are almost impossible to convey to non-mathematicians. The modus operandi of mathematical abstraction is pretty much as we have described it above: you begin with a “familiar” idea (that is, familiar to a sophisticated mathematician but most likely totally alien to an outsider), you try to distill its essence, and then you try to find, in some other area of mathematics, something that shares this same distilled essence. An alternative pathway towards abstraction involves recognizing an analogy between two structures in different domains, which then focuses one’s attention on the abstract structure that they share. This new abstraction then becomes a “concrete” concept that one can study, and this goes on until someone realizes that this is far from the end of the line, and that one can further generalize the new concept in one of the two ways just described. And thus it goes…

  Mechanical Mathematical Manipulations: Also the Fruit of Analogy

  Before we take our leave of mathematics, we would like to comment on the use of standard techniques for manipulating mathematical expressions. Take, for example, Cardano’s shifting of a term from one side of an equation to the other, by which he got the term to change sign. This is a general strategy or technique that children today are taught in beginning algebra classes. For instance, to solve the equation 3x – 7 = x + 3, one carries the variable “x” across from the equation’s right side to its left side while changing its sign, and (analogously) one carries the number –7 across from the left side to the right side while changing its sign. The upshot is the equation 2x = 10, and here one divides both sides by 2, thus winding up with the solution x = 5. But what, if anything, do such routine algebraic operations have to do with analogy-making?

  A great deal! Such operations come from analogies that reside at the lower end of the creativity spectrum, very much like perceptual acts such as looking at an object sitting on a table and mentally calling it a “a paperweight”. For someone to be able to recognize objects and to assign them labels, their memory has to be efficiently organized, allowing new experiences to evoke, by analogy, linguistic labels that long ago were attached to old experiences. Such labels seem so self-evident that one can easily fall prey to the seductive illusion that one is doing nothing other than mechanically allowing the intrinsic labels of the various items in the situation to emerge explicitly.

  The very same illusion arises in routine mathematical situations as well. The recognition, when one is in a brand-new mathematical situation, that it is crying out for a certain standard memorized technique is much like the everyday recognition that one is in a new situation that is crying out for a standard linguistic label, or for a standard canned kind of behavior. Recognizing that the equation 3x – 7 = x + 3 is crying out for two of its terms to be transposed to the other side (and for their signs to be changed), and then recognizing that the resulting equation, 2x = 10, is in turn crying out for division of both its sides by 2, is no different, in principle, from recognizing that a suitcase “cries out” to be lifted (an “affordance”, as defined in Chapter 6), or that a certain situation calls for hitting a nail into a wall with a hammer.

  There are all sorts of standardized techniques for manipulating algebraic equations, such as the ones just mentioned; indeed, mathematics abounds with routine techniques at all levels of abstraction. Among these are various ready-made, off-the-shelf types of logical argument, such as the famous schema called reductio ad absurdum, which basically says that if you wish to prove that an idea X is true, see what happens if you presume that the opposite of X is the case, and if this assumption leads to an absurdity (more specifically, a self-contradiction), then X must be correct. This mode of reasoning is a standard, often-used ingredient in every mathematician’s toolkit. Likewise, one of many standard techniques for doing integrals in calculus is that of making “trigonometric substitutions”. One recognizes that a certain situation “smells” as if it needs such a substitution, and so one goes for it. The fact that it eventually takes on a recipe-like quality, however, does not disqualify it from being a case of analogy-making, since if one has a good deal of mathematical experience, the formulaic manipulations that one uses all the time were committed to memory a long time earlier. Now, thanks to one’s ability to categorize situations efficiently, these mathematical techniques are evoked “automatically” — that is to say, so effortlessly that it feels as if analogy plays no role. But this is simply the same illusion that says that the paperweight on the table evokes the label “paperweight” independently of one’s analogy-making capacity.

  How many times have you played Monopoly or a similar game? Probably very often. And of course your style of play today is consistent with, and indeed comes from, all those times when you played it before. You know when you feel comfortable taking risks buying pieces of property or railroads or houses or hotels, and you know when it feels too dangerous. Although such decisions occur in a much more concrete world than that of a mathematician, they have much in common with a mathematician’s decisions about whether to try using a certain technique or not in a given situation. Exploring the technique might waste a lot of valuable time and effort, but it might pay off in the end. Is it worth it? It all depends on how the unfamiliar and nebulous new situation “smells” to the seasoned mathematician. These are subtle qualities that come only from long experience and from having slowly built up a refined repertoire of categories
. And as in Monopoly, so in mathematics.

  One day, two math graduate students were arguing about a famous unsolved problem in number theory. One of them was insisting that the highly elusive and mysterious distribution of prime numbers must play a crucial role in the matter; the other argued that that couldn’t possibly have any relevance at all. As it happens, neither student had any intention to devote years of their life to this thorny problem, and so their clashing opinions on what was of the essence versus what was irrelevant were merely idle words without import; but if someday they both decided to tackle this challenge independently, then their diametrically opposite intuitions about its nature would have profound impacts on the directions that their research would follow. Thus whichever of them had “sniffed” the true crux of the problem would wind up following far more promising pathways than the other one would. And what would have given them these powerful, make-or-break intuitions (whether they were correct or incorrect)?

  Analogies in mathematics range widely in sophistication. At the lower end of the spectrum is the evocation of standard canned recipes such as we have been discussing. At the upper end lie the strokes of genius by great mathematicians, such as the importation of the idea of imaginary number into the realm of finite fields, or the search for prime numbers (or prime knots!) in highly abstract structures. Perhaps mathematics seems at times to be merely a symbol-manipulating game, but it is important to realize that even the most rote-seeming manipulation of symbols relies on analogy.