Surfaces and Essences: Analogy as the Fuel and Fire of Thinking (106 page)

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.

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.

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
“x
⁴
”, the purely formal analogy between the equation
ax
³
+
bx
²
+
cx
+
d
= 0 and its longer cousin
ax
⁴
+
bx
³
+
cx
²
+
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
N
th-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
N
th-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.

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
N
th-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.