Surfaces and Essences: Analogy as the Fuel and Fire of Thinking (105 page)

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:

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.

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
“x
³
– 15
x
= 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
x
³
– 15
x
— 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.”