Surfaces and Essences

by Douglas Hofstadter

  Nonetheless, his successors — most especially Raffaello Bombelli in Bologna — were powerfully driven to find the elusive unity in Cardano’s troublingly diverse set of thirteen chapters, and in the end they wound up accepting the notion of negative numbers on the same level of reality (or nearly so) as positive numbers. This move yielded an enormous simplification in the solution for the cubic equation, as the thirteen families were gracefully fused into a single family, and the thirteen recipes associated with them were also fused into a single, compact recipe.

  This is an excellent illustration of the recurrent theme in earlier chapters that the human mind is forever driven to transform its categories, not just to use them as givens, and that intellectual advances are dependent on conceptual extensions. In this particular case, welcoming negative numbers into the fold was a decision that grew out of a desire for unification, and it led to such a gratifying simplification that no one would have wished to go back to the previous stage, with its long list of special cases.

  Nonetheless, the welcoming of negative numbers into the category number was not immediate or universal. Even 250 years later, the English mathematician Augustus De Morgan, a central figure in the development of symbolic logic, was still resisting, as this passage from his 1831 book On the Study and Difficulties of Mathematics shows:

  “8 – 3” is easily understood; 3 can be taken from 8 and the remainder is 5; but “3 – 8” is an impossibility; it requires you to take from 3 more than there is in 3, which is absurd. If such an expression as “3 – 8” should be the answer to a problem, it would denote either that there was some absurdity inherent in the problem itself, or in the manner of putting it into an equation…

  DeMorgan’s comment is reminiscent of what a seven-year-old girl once said to one of us when she was a participant in experiments on subtraction errors. To explain why she’d written “0” at the bottom of a column containing the numerals “3” and “8”, she said, “If I had three pieces of candy in my hand and I wanted to eat eight, I’d eat the three I had and there wouldn’t be any more left.” Despite the passage of several centuries, the sizable age gap, and the immense amount of mathematical sophistication separating our two commentators, their reactions still share a common essence.

  Further on in the same chapter of his book, De Morgan gives a slightly more concrete example, and comments as follows on it:

  A father is 56 and his son 29 years old. When will the father be twice his son’s age?

  De Morgan translates this word problem into the equation 2 (29 + x) = 56 + x, where x is the number of years that will have to transpire before the described moment arrives. Then he solves the equation, easily obtaining the value of –2 for x. But for him, this result is absurd. What could it mean to say “–2 years will pass”? What could “–2 years from now” mean? Absolutely nothing — or perhaps even less! He then explains that we shouldn’t have fallen into the trap lurking in the words “the father will be”. Instead, we should have thought of x as the number of years that have passed since the doubling, which would have led us to write down the equation 2 (29 – x) = 56 – x, whose solution is x = 2, which corresponds to the fact that two years ago, the father’s age was twice his son’s age.

  Only at this point is De Morgan happy, admitting that the idea “–2 years will pass” is equivalent to the idea “2 years have passed”. He thus does accept the idea that a numerical value can be negative, but not that a length of time can be negative. All this would lead one to think that De Morgan had no qualms about negative numbers within pure mathematics, even if he didn’t think they applied to the real world — and yet, a little later in his book, when he deals with the quadratic equation, instead of considering it as one single, unified problem, he breaks it up into six different families of equations, insisting (in perfect Cardano style) that all three of its coefficients must be positive! De Morgan thus finds that there are six different quadratic formulas, rather than one universal one. And all of this nearly 300 years after Cardano!

  De Morgan’s qualms reveal that the extension of any concept, driven by the mental forces of analogy and by a quest for esthetic harmony based on unification, is a gradual and subjective process, and that even the most insightful of minds within a domain can balk at certain extensions that, some time later, may strike other minds as being as innocent as baby lambs.

  In the preceding chapter, we showed that naïve analogies made in early years of school, such as multiplication is repeated addition or division is sharing, continue to influence the reasoning of adults, including students in college. In this chapter, we have seen that some highly regarded mathematicians, such as De Morgan, could vacillate on the nature of the concept number, swinging back and forth between welcoming and barring negative numbers. For him, the legitimacy of the entities being manipulated depended on what they represented in the situation described in words.

  Today, the notion of negative numbers seems commonsensical, even bland, which shows how tightly linked the concrete everyday world is to the abstract mathematical world. We are all familiar and comfortable with negative temperatures in the winter, with basements and subbasements whose floor numbers are negative, and with credits on bills, which are indicated with minus signs, meaning that you owe the company a negative amount of money, which is to say, the company owes you a positive amount of money. Children who encounter negative numbers in natural contexts of this sort have no trouble absorbing the general idea. And thus, over time, what was once a most daring intellectual insight turns into a commonplace, unreflective habit.

  Enter Complexities

  Raffaello Bombelli, after he had fully accepted the existence and reality of negative numbers (around 1570), found himself forced to confront an even greater mystery, which flowed directly out of his acceptance of such strange numbers. The source of the problem was that the formula for the solution of the cubic equation sometimes required taking square roots of negative numbers. Bombelli understood the multiplication of signed numbers better than anyone else, having been the first person ever to state the rules for it. He knew that multiplying two negative numbers always gives a positive result, as does multiplying two positive numbers, and thus that there is no number (today we would say “there is no real number”) whose square is a negative number. In short, all squares are positive (or zero), and so negative numbers do not have square roots.

  All this would be fine, but the problem was that square roots of negative numbers cropped up in the formula for the solution of the most innocent cubic equations. For example, the formula giving the solutions to the equation “x3 – 15x = 4”, one of which is x = 4 (as can easily be checked), was a long algebraic expression inside of which was found (not just once but twice) the square root of–121, which is to say, . What to do, when faced with such seeming nonsense? And yet Bombelli knew beyond a shadow of a doubt that this long, curious, and troubling algebraic expression somehow had to stand for the extremely familiar and very real number 4.

  Seeing this paradox as a subtle hint, he took the courageous step of accepting the mysterious square root point blank, manipulating it just as he would any other number, using the standard rules of arithmetic (such as the commutativity of addition and multiplication, and so forth). Having adopted this new attitude, he discovered that the solution formula, using the mysterious quantity twice and thus seeming to embody a piece of mathematical nonsense, did indeed act formally like the real number 4. To be more concrete, when he plugged the strange expression into the polynomial x3 – 15x — that is, when he formally cubed it and then subtracted 15 times it from the result — he discovered, to his amazement, that all the frightening square roots of –121 either were squared, simply giving –121 and thus losing their fearful fangs, or else canceled each other out in pairs, leaving him in the end with just the number 4, exactly as the cubic equation’s right-hand side said it should. Bombelli thus realized that he could manipulate just as he would manipulate any “genuine” number, and this analogy with
more familiar numbers made the new quasi-number quasi-acceptable for him. From that moment on, Bombelli began to accept the square roots of negative numbers, although he didn’t have the slightest idea what they actually were.

  As mathematicians gradually got used to the fact that these mysterious expressions acted in many ways just as ordinary numbers do, that they were not going to lead people into paradoxical waters, and moreover that they enriched our collective understanding of the world of mathematics, the objecting voices slowly faded away and the mathematical community opened up to them, although not unanimously. Here, for example, is how Gottfried Wilhelm von Leibniz, co-inventor with Isaac Newton of the infinitesimal calculus, described, in 1702, the numbers that René Descartes had dubbed “imaginary”: “an elegant and marvelous trick found in the miracle of Analysis: a monster of the ideal world, almost an amphibian located somewhere between Being and non-Being.” And even Leonhard Euler, the Swiss genius who deserves enormous credit for putting the theory of complex numbers on a solid footing, declared, on the subject of the square roots of negative numbers, that they are “not nothing, nor less than nothing, which makes them imaginary, indeed impossible.”

  In any case, despite the protestations of lesser and greater minds, imaginary numbers slowly took hold over the course of the next couple of centuries after Bombelli’s early explorations, thanks largely to the discovery of a way of visualizing them as points on a plane, which led to an elegant geometrical interpretation of their addition and multiplication. (This crucial development is vividly described in Fauconnier and Turner’s The Way We Think.) It would be hard to overstate the importance of geometrical visualization in mathematics in general, which is to say of attaching geometrical interpretations to entities whose existence would otherwise seem counterintuitive, if not self-contradictory. The acceptance of abstract mathematical entities is always facilitated if a geometrical way of envisioning them is discovered; any such mapping confers on these entities a concreteness that makes them seem much more plausible.

  N-dimensional Spaces

  For example, the notion of square number owes its name to the geometrical image of a square on a plane, all of whose sides have the same length, and whose area is therefore the product of that length by itself. In like manner, a cubic number was originally understood as the volume of a physical cube: the product of three equal lengths. But no one dared to go beyond the case of the cube — at least not if such a quantity was going to take its name from a visualizable object. Thus the arithmetical operation written as “5 x 5 x 5 x 5” was perfectly fine, so long as one didn’t try to attach any geometrical interpretation to it. (This is reminiscent of division by a number between 0 and 1, discussed in the previous chapter; most people can carry out the operation formally, but only a small minority of adults, even university students, ever come to understand division clearly enough that they are able to dream up a real-world situation that such an operation could represent.) When the audacious analogical suggestion was made that the fourfold product 5 x 5 x 5 x 5 might stand for something that was somewhat like an area or a volume, but associated with a four-dimensional space, people objected strenuously, feeling that this violated the essence of what space was. Even at the start of the nineteenth century, that is how many mathematicians protested. As this shows, a reliance on concrete analogies to make sense of abstract mathematical operations is not limited to children or non-mathematicians. The lack of spatial analogues to such arithmetical expressions thus kept the concept of dimension from being extended beyond three, even for mathematicians, until well into the nineteenth century.

  Once the ice had been broken, however, it didn’t take long for the general idea of N-dimensional spaces — 4, 5, 6, 7,… — to be accepted, thanks to tight analogies between the theorems that hold in higher-dimensional spaces and those that hold in familiar low-dimensional spaces. Indeed, one of the best ways to imagine four-dimensional space is to try to put oneself in the place of a poor two-dimensional being striving mightily to visualize a three-dimensional space. One easily recognizes oneself in this creature valiantly struggling against its own limitations; and just like that more limited being, one uses one’s analogy-making skills to extend one’s mental world. One thinks, “My limitations are just like its limitations, only slightly more elaborate!”, and one attempts to transcend one’s own limitations by imagining how that two-dimensional being would transcend its limitations. Once this barrier has been broken, the analogy is so strong that there is no going back. Pandora’s box is open and one jumps readily from four to five dimensions, then six, and so on, all the way to infinity.

  “What?! A space with an infinite number of dimensions? Balderdash!” Thus reacted many mathematicians at the end of the nineteenth century, yet such objections would just bring smiles to the lips of their counterparts today, for whom the idea seems self-evident. In fact, this is just the tip of the iceberg, for after the work of the German mathematician Georg Cantor, it became a commonplace that there is not just one infinity, but many infinities (of course, there are an infinite number of different infinities).

  Spaces with a countably infinite number of dimensions (this is the smallest version of infinity, and mathematicians would say that it is the cardinality of the set of natural numbers — that is, of the set of all whole numbers) are called “Hilbert spaces”, and for theoretical physicists, quantum mechanics “lives” in such a space; that is to say, according to modern physics, our universe is based on the mathematics of Hilbert spaces. This connection to the physical world lends plausibility to the notion of infinite-dimensional spaces.

  Let’s not forget that between the integers there are plenty of other numbers (for example, 1/2 and 5/17 and 3.14159265358979…, etc.), and mathematicians in the early twentieth century who were interested in abstract spaces — especially the German mathematician Felix Haussdorff — came up with ways to generalize the concept of dimensionality, thus leading to the idea of spaces having, say, 0.73 dimensions or even π dimensions. These discoveries later turned out to be ideally suited for characterizing the dimensionality of “fractal objects”, as they were dubbed by the Franco–Polish mathematician Benoît Mandelbrot.

  After such richness, one might easily presume that there must be spaces having a negative or imaginary number of dimensions — but oddly enough, despite the appeal of the idea, this notion has not yet been explored, or at any rate, if it has, we are ignorant of the fact. But the mindset of today’s mathematicians is so generalization-prone that even the hint of such an idea might just launch an eager quest for all the beautiful new abstract worlds that are implicit in the terms.

  How Analogies Gave Rise to Groups

  The enrichment of the real numbers by the act of incorporating i, the square root of –1, was a great step forward, because the two-dimensional world that was thus engendered — the complex plane — turned out to be “complete” in the sense that any polynomial whose coefficients are numbers lying in the complex plane always has a complete set of solutions within the plane itself. To state it more precisely, any complex polynomial of degree N has exactly N solutions in the complex plane; one never needs to reach beyond the plane to find missing solutions. One can imagine that at this stage mathematicians might well have joyfully concluded that they had at last arrived at the end of the number trail: that there were no more numbers left to be discovered. But the predilection of the human mind to make analogies left and right was far too strong for that to be the case.

  The discovery of the solution of the cubic by the Italians in the sixteenth century inspired European mathematicians to seek analogous solutions to equations having higher degrees than 3. In fact, Gerolamo Cardano himself, aided by Lodovico Ferrari, solved the quartic — the fourth-degree equation. Even though there was no geometric interpretation for an expression like “x4 ”, the purely formal analogy between the equation ax3 +bx2 + cx + d = 0 and its longer cousin ax4 + bx3 + cx2 + dx + e = 0 was so alluring to Cardano that he could not resist tackling
the challenge. And in short order Cardano and Ferrari, using methods analogous to those that had turned the trick for the cubic, came up with the solution. However, there were some curious surprises lurking in the formula they discovered.

  Most strikingly, there was no fourth root anywhere to be seen, although the natural (but naïve) analogy would lead anyone to expect one. Instead, there was a square root, and underneath the square-root sign there was a complicated expression in which was found another square root, and then underneath the inner square-root sign there was another complicated expression containing a cube root, and finally, underneath the cube-root sign there was another long expression that contained another square root — all in all, a fourfold nesting of radicals! Who could have predicted such a curious, complicated nesting pattern? Why did only square roots and cube roots appear? Why no fourth root? Why were four radicals involved, and not two or three or five — or twenty-six? And why in the order “2–2–3–2” rather than, say, “2–2–2–3”? For that matter, why not “3–3–3–2”? The unexpected and unexplained pattern of these nested radicals in the solution formula for the quartic equation suggested that the general solution for the Nth-degree polynomial must contain some deep secret. And thus was launched the quest for the general solution to polynomial equations of any degree.

  Despite the intense efforts of many mathematicians for more than 200 years, nothing worked even for the fifth-degree equation, let alone for its higher-degree cousins. But finally, toward the end of the eighteenth century, the Franco–Italian mathematician Joseph Louis de Lagrange began to intuit the nature of the subtle reason behind this striking lack of success, although he was unable to pin it down precisely. Lagrange saw that there were tight relationships between the N different solutions of the Nth-degree polynomial — so much so that these numbers, though not identical, acted indistinguishably in certain respects. Like Tweedledee and Tweedledum, if they were interchanged, there was a sense in which no effect was observable. This meant that there was a new kind of formal symmetry involving such numbers. Lagrange looked into what happens if one permutation of an equation’s N solutions was followed by another, then yet another, and so forth. He thereby opened up the theory of “substitutions”, planting the seed of what turned out later to be the theory of groups, one of the linchpins of modern mathematics.