Every field of human endeavour exists for a reason, but often its origins get lost in the mist of time. Mathematics is no exception. The most ancient mathematical texts that made it to us (from ca. 2000 BC) do not mention how mathematics appeared, although this is not difficult to guess. Keeping track of domestic animals and performing commercial transactions are some of the activities likely responsible for the birth of arithmetic. Geometry probably sprouted from needs related to the division of agricultural land and the building of large constructions. The curious mind of the prehistoric thinker then got detached from practical purposes and started investigating the properties of numbers, triangles, circles and other abstract mathematical objects, a process that continues today at a much higher level and with incredible benefit for our civilization. Without mathematics, we would still live in the Bronze Age.
So much for “why mathematics exists at all.” Ian Hacking, a former professor of philosophy at the University of Toronto and the Collège de France, chooses for a title of his latest book an even more intriguing question: Why Is There Philosophy of Mathematics at All? The answer is not difficult to give. Every field of human endeavour is divided into schools of thought grouped around a set of answers to some basic questions. In mathematics, for instance, it is natural to ask: Do we discover or invent mathematical objects? Or, more specifically, do the numbers 1, 2, 3, etc., exist independently of us, like the laws of physics, or are they the product of our brain? Those who accept the former answer are the Platonists; the others are the materialists (or naturalists). No matter whose side we take, by pursuing questions like this we have entered the realm of mathematical philosophy.
A more difficult question to answer is: why would anyone care about the philosophy of mathematics? Unlike physics, which can be made or unmade by philosophy (Einstein came to his theories thanks to his novel philosophical approach), mathematicians seem to be immune to philosophical thought when engaging in research. But this is just an appearance. They unconsciously follow the philosophical approach embedded in their minds during the training years. Nevertheless, while the huge benefits of mathematics to our society are not in doubt, what is the contribution of mathematical philosophy? Without ever asking this question (except obliquely in his title), Hacking is on a pursuit to show why this activity matters.
The beginnings of the link between philosophy and mathematics can be traced to the ancient Greeks. The Pythagoreans were both philosophers and mathematicians, in the modern sense of the words. They attached physical meaning to the five Platonic solids (earth to the tetrahedron, water to the cube, etc.), and apparently went through an existential crisis when learning that there are numbers, such as √2, that cannot be written as fractions. Until then, they strongly believed in the existence of a unit of measure that can be used to precisely evaluate every length, a belief that collapsed after this discovery.
The Greeks also made a crucial philosophical step when requiring proof for every mathematical statement. The birth of democracy in the city-states of the Peloponnese put argument above the rule of autocracy, a social change that also influenced mathematical thought, which in turn kept the taste for justification alive in society. Earlier cultures, such as Babylon and Egypt, had stated theorems as facts, with no attempt to defend them. But an even more important development flourished from the idea of proof: Euclid’s imposition of the axioms (or postulates) to which every theorem of geometry had to be reduced by logical reasoning. In Euclid’s view, an axiom was a self-evident statement, such as “through any two distinct points we can draw only one straight line.”
The philosophical idea of reducing every proof to some axioms imposed not only an unprecedented level of rigour, but also allowed the lifting of mathematics to levels that would be impossible to reach otherwise. An illuminating example is that of the attempts to eliminate Euclid’s Postulate 5, which is equivalent to saying that “through a point lying outside a given straight line, we can draw a single parallel to that line.” To Euclid this statement was obvious, but his followers tried for more than two millennia to show that it follows from his other axioms. All the attempts to find a proof failed until the 1830s when two young mathematicians, János Bolyai and Nikolai Lobachevsky, independently showed that the claim was not true. If Postulate 5 is replaced by the axiom “through a point lying outside a given straight line, we can draw at least two parallels to that line,” a new geometry emerges. That this new axiom transcends imagination is irrelevant. It matters only that the new geometry is free of contradictions. Bernhard Riemann showed some three decades later that yet another geometry can be built with the axiom “through a point lying outside a given straight line, we can draw no parallel to that line.” These developments opened the door to more geometries and to a novel way of thinking.
Without these mathematical achievements, Einstein could have never come up with his general relativity, which led to our current understanding of the large-scale universe. These developments also influenced literature, the visual arts, in fact our entire culture. And this mathematics was catalyzed by the philosophical idea of imposing axioms, a fact that underlines the importance of mathematical philosophy. Although obscure and with few actual practitioners (mathematicians are not inclined to philosophize and few philosophers know enough mathematics to analyze it), the philosophy of mathematics has offered several times in history the spark that kindled new research directions or was the fuel that kept alive the interest in certain mathematical questions.
Ian Hacking does a good job in treating many of the problems that concern those few thinkers who pursued them, whether they had been trained as philosophers or as mathematicians. While I do not always agree with his views, and sometimes find his mathematical background shaky (as when he mixes up algebra with arithmetic while discussing Descartes’ making of analytic geometry), the deeper my reading progressed, the more I enjoyed his book. Unlike most philosophers who venture into this field, Hacking does not restrict himself to the foundations of mathematics, but dares to cover both the breadth and the depth of mathematical philosophy. From the various visions on the concept of proof to the apparent contradiction between determinism and chaos, he takes the reader to exotic places surrounded by new questions, which look like rugged mountains impossible to climb.
In spite of Hacking’s good writing, this volume is not for everyone: the nature of the subject is forbidding. Without a solid background in mathematics and a keen interest in philosophical reasoning, a curious reader would abandon this book early. But those who have the necessary skills will enjoy it. They should not miss the opportunity.