Surfaces and Essences: Analogy as the Fuel and Fire of Thinking (104 page)

Specialists of any ilk, be they experts in psychology, mathematics, physics, or any other field, do not belong to a different species from ordinary people; they make naïve analogies not only in their daily lives, but also in their professional lives, even ones that involve the concepts with which they are the most proficient.

In short, we sympathize with those readers who still have some doubts about the identity of categorization and analogy-making. It is, after all, a counterintuitive view, and a little voice inside, prompted by a beguiling naïve analogy, continually whispers, “It’s wrong! It’s wrong!” Nonetheless, we harbor fond hopes that in the remaining pages of our book, we might still get some doubters to swing around to our view.

In any case, speaking of the remaining pages, it is high time that we moved from the often troubling world of naïve analogies to the ever-admirable role of analogies in scientific discovery. And yet in so doing, will we really leave naïve analogies far behind?

In
Chapter 7
, we saw that students who encounter new mathematical ideas for the first time lean heavily on analogies at every step, doing their best to keep from taking tumbles in the abstract world into which they are tremulously treading. We also saw that naïve analogies have a way of sticking in people’s minds for a long time. Indeed, naïve mathematical analogies often last a lifetime in the minds of non-mathematicians, tending to lead their makers into dead ends, confusions, and errors. To avoid this fate, one has to gradually refine one’s category system as one is exposed to mathematical notions having ever higher levels of sophistication and abstraction. But what about professional mathematicians? Do they, too, rely on naïve analogies in order to keep from stumbling left and right, or would such a vision of their professional lives itself be a result of making too naïve an analogy between beginners and experts?

To some people, it might seem far-fetched to imagine that any role at all might be played by analogical thinking in the professional activity of a mathematician. After all, of all intellectual domains, mathematics is generally thought of as the one where rigor and logic reach their apogee. A mathematical paper can seem like an invincible fortress with ramparts built from sheer logic, and if it gives that impression it is no accident, because that is how most mathematicians wish to present themselves. The standard idea is that in mathematics, there is less place for intuitions, presentiments, vague resemblances, and imprecise instincts than in any other discipline. And yet this is just a prejudice, no more valid for mathematics than for any other human activity.

The great French mathematician Henri Poincaré devoted much thought to the nature of scientific creativity. In a commentary on mathematicians, he wrote:

This observation about mathematical thinking strikes your authors as being spot-on — but we don’t expect readers to take Poincaré’s word for it. In support of his thesis, and as antidotes to any possible skepticism about the role of analogies in mathematics, we now offer a bouquet of anecdotes.

For starters, let’s go back to the early part of the sixteenth century, to a time when European mathematicians were struggling with the challenge of solving third-degree polynomial equations (of the form “
ax
³
+
bx
²
+
cx
+
d
= 0”), otherwise known as “cubics”. For more than fifteen centuries, the formula that gave the general solution of
second
-degree polynomial equations (that is, quadratic equations, which have the form “
ax
²
+
bx
+
c
= 0”) had been known, and at the heart of this now world-famous formula there was a square root — that of a certain quantity (specifically,
b
²
– 4
ac
) calculable from the three coefficients (
a, b
, and
c)
of the specific quadratic in question. For the few people who cared about such arcane matters, it was therefore natural to wonder if
something of the same general sort
might not also be the case for cubic equations.

But why would a mathematician entertain such a blurry, imprecise thought? The answer is simple: any serious mathematician would suspect that these two equations, so similar in form, must be linked by a hidden connection — an
analogy.
More specifically, one would expect that there would be a general formula for the cubic and that it would contain, at its core, the
cube root
of some quantity analogous to
b
²
– 4
ac
, and that this quantity would involve the
four
coefficients (
a
,
b, c
, and
d
) of the given cubic equation.

Such an analogy is irresistible here, just as were the
me-too
analogies described in
Chapter 3
. Mathematicians, even the most exceptional ones, are also ordinary human beings, and without any conscious thought, they automatically anticipate that there will be an analogy between, on the one hand, the
two
solutions of any
quadratic
equation, which one can calculate by extracting the
square root
of a special number determined by the equation’s
three
coefficients, and, on the other hand, the
three
solutions of any
cubic
equation, which one can calculate by extracting the
cube root
of some special number determined by the equation’s
four
coefficients. This little guess, sliding a couple of times from
two
-ness to
three
-ness, and also once from
three
-ness to
four
-ness (which in itself comes from a mini-analogy: “4 is to 3 as 3 is to 2”) seems like an utter triviality, but without very simple-seeming conceptual slippages of this sort, which crop up absolutely everywhere in mathematics, it would be impossible to make any kind of progress at all.

Let’s return to the story of the solution of “the” cubic equation (the reason for the quote marks will emerge shortly). It all took place in Italy — first in Bologna (Scipione del Ferro), and a bit later in Brescia (Niccolò Tartaglia) and Milan (Gerolamo Cardano). Del Ferro found a partial solution first but didn’t publish it; some twenty years later, Tartaglia found essentially the same partial solution; finally, Cardano generalized their findings and published them in a famous book called
Ars Magna
(“The
Great Art”). The odd thing is that, as things were coming into focus, in order to list all the “different” solutions of the cubic equation, Cardano had to use thirteen chapters! Nowadays, by contrast, the whole solution is covered by just one formula that can be written out in a single line, and which could easily be taught in high schools. What lay behind such diversity, of which we no longer see any trace today?

The problem was that no one in those days accepted the existence of negative numbers. For us today, it’s self-evident that the coefficient in the third term of the equation
x
³
+ 3
x
²
– 7x
= 6 is the
negative number
–7. It jumps right out at us, since we are completely used to the idea that a subtraction is equivalent to the addition of a negative quantity. We could rewrite the equation as follows:
x
³
+ 3
x
²
+ (–7)
x
= 6. For us who live five centuries after Cardano, these two equations are trivially interchangeable. The conceptual slippage on which their equivalence is based is so minute that we don’t even perceive it at all. But for the author of the vast tome on the third-degree equation, the concept of
negative seven
simply didn’t exist. For him, the only legitimate way to get rid of a subtraction in a polynomial (that is, a term with a negative coefficient) was to move the misfit term to the other side of the equation, thus yielding a different but related equation — namely,
x
³
+ 3
x
²
= 7
x
+ 6 — all of whose coefficients are
positive
.

The upshot of all this is that before he could handle all the different cases of the general problem of cubic equations, Cardano had to move terms around so that there were no more subtractions anywhere, thus obtaining new equations that had only plus signs (and therefore positive coefficients). As it turns out, this procedure gave rise to thirteen different types of cubic equation, each one being — in the eye of specialists of the time —
essentially different
from the twelve others. And thus, in order to publish the general solution to “the” cubic equation (now the
raison d’être
for the quote marks should be apparent!), Cardano had to write thirteen chapters, each of which contained a complicated recipe covering one of the thirteen types of cubic. All in all, Cardano’s book on cubic equations,
Ars Magna
, was a long, heavy, and formidable tome, but its reception was, nonetheless, positive, shall we say.

From a contemporary viewpoint, what Cardano did is comparable to someone who invents thirteen kinds of can-openers, each one working for just one type of can. It was a great feat, but what was lacking was an umbrella formulation, laying bare the hidden unity lying behind all this apparent diversity. That is, what was missing for “the” cubic equation was its universal can-opener. But this goal was unthinkable until someone recognized that all these different equations
were
, in fact, just one equation.

Indeed, although all thirteen of Cardano’s recipes were somewhat different, there were nonetheless striking similarities between them — analogies, that is, that inspired his successors to try to combine them all into just one formula. However, in order for such a unification to come about, some concept was going to have to stretch, expand, or bend. In this case, the concept in question was the most basic of mathematical concepts — that of
number.
The unification of Cardano’s recipes for solving cubic equations depended on a conceptual extension, and a quite significant one, which would allow thirteen different types of algebraic recipes, as seen by as highly skilled a mathematician as Cardano, to melt down into just a single one.

Obviously, what was needed was a new conceptual leap, this time extending the category
number
to include negative numbers. This was by no means an easy step to take. Ever since the ancient Greeks, it had been known that there were very simple equations that lacked solutions, such as
2x
+ 6 = 0. The idea of giving such equations solutions had been considered but was always rejected (at least in Europe). Cardano himself understood that “fictitious” numbers (as he referred to them) could satisfy such an equation, but he rejected the idea with disdain. To him, the concept of
negative three
, being in no way visualizable, was an absurdity, somewhat like the concept of an object that violated the laws of physics. Such an idea might be stimulating to the mind, but it had to be recognized as absurd, because there was no way of actually realizing it in the world. Since Cardano was unable to associate negative numbers with any kind of entity in the real world, he labeled them “fictitious” and discarded them.