What Makes Mathematics Mathematics?
IAN HACKING
We have seldom paused to ask what counts as mathematics. Certainly we have thought a good deal about the nature of mathematics—in my case, for example, about the logicist claim that mathematics is logic.1 But we took “mathematics” for granted, and seldom reflected on why we so readily recognize a conjecture, a fact, a proof idea, a piece of reasoning, or a sub-discipline, as mathematical. We asked sophisticated questions about which parts of mathematics are constructive, or about set theory. But we shied away from the naive question of why so many diverse topics addressed by real-life mathematicians are immediately recognized as “mathematics.”
Mathematics. Originally, the collective name for geometry, arithmetic, and certain physical sciences (as astronomy and optics) involving geometrical reasoning. In modern use applied, (a) in a strict sense, to the abstract science which investigates deductively the conclusions implicit in the elementary conceptions of spatial and numerical relations, and which includes as its main divisions geometry, arithmetic, and algebra; and (b) in a wider sense, so as to include those branches of physical or other research which consist in the application of this abstract science to concrete data. When the word is used in its wider sense, the abstract science is distinguished as pure mathematics, and its concrete applications (e.g. in astronomy, various branches of physics, the theory of probabilities) as applied or mixed mathematics.
The Oxford English Dictionary (OED) definition is an excellent one. Why not stop right here, and answer our question by quoting the dictionary? Because the kinds of things we call mathematics are, in a word, so curiously miscellaneous. The dictionary already hints at that, in part by its implication that the concept of mathematics itself has a history, with the name applying in different ways to different categories over the course of time.
A Mathematician's Miscellany2
The arithmetic that all of us learned when we were children is very different from the proof of Pythagoras' theorem that many of us learned as adolescents. When we began to read Plato, we saw in the Meno how to construct a square double the size of a given square, and realized that the argument is connected to “Pythagoras.” But that is totally unlike the rote skill of doubling a small integer at sight, or a large one by pencil.
Both types of examples are unlike the idea that Fermat had when he wrote down what came to be called his last theorem. We nevertheless seem immediately to understand his question about the integers. The situation is very different from the proof ideas that lie behind Andrew Wiles' discovery of a way to prove the theorem. Few of us have mastered even a sketch of that argument. Is it “the same sort of thing” as the proof that there is no greatest prime? I am not at all sure.
The mathematics of theoretical physics will seem a different type of thing again, but we should not restrict ourselves to theory. Papers in experimental physics are rich in mathematical reasoning. Take a recent very successful field, very cold atoms, virtually at absolute zero (f. Hacking 2006). There are two fundamentally different types of entity, bosons and fermions. All elements but one have isotopes that are bosons. Ions of any chosen species of boson go into the same ground state when they lose enough energy. Then they all have the same wave function, which leads some experimenters to speak of macroscopic wave functions, which would once have seemed to be a contradiction in terms. This all started in 1924 when Einstein read a letter about photons from S. N. Bose, and saw from the equations that something very strange should happen near 0° K: there would be a new kind of matter. Only in 1995 was it possible to get a few thousand trapped ions of rubidium-47 cold enough to create what we now call Bose-Einstein condensate.
Although the experimental results in this field are brand new, the mathematics that Einstein took for granted was mostly old-fashioned, taken from what has been compared to a physicist's toolkit (Krieger 1987, 1992). Much of it had been around for more than a century when Einstein made use of it. The mathematics in the toolbox—and the way it is used—is very different from that of the geometer or number theorist.
The mathematical part of the physicist's toolbox is mostly old, but something entirely new has been added. We have powerful computational techniques to make approximate solutions to complex equations that cannot be solved exactly. They enable practitioners to construct simulations that establish intimate relations between theory and experiment. Today, most experimental work in physics is run alongside simulations. Is the simulation of nature by powerful computers (applied) mathematics, in the same way that modeling nature using Lagrangians or Hamiltonians is called applied mathematics?
Modern condensed matter physics, of which the theory and practice of cold atoms is a part, employs sophisticated mathematical models of physical situations. Economists also construct complicated models. They run computer simulations of gigantic structures they call “the economy” to try to guess what will happen next.The economists are as incapable of understanding the reasoning of the physicist as most physicists are of making sense of modern econometrics. Are they both using mathematics?
We are not really sure whether to say that programmers writing hundreds of meters of code are doing mathematics or not. We need the programmers to design the programs on which we solve, by simulation and approximation, the problems in physics or economics. What part is mathematics and what part not? We do not dignify as mathematics the solving of chess problems, white to mate in three. Few people will call programming a computer to play chess an instance of mathematics. Arithmetic for carpentry or commerce seems very different from the theory of numbers. What, then, makes mathematics mathematics?
Only Wittgenstein Seems to Have Been Troubled
It is curious that Wittgenstein seems to have been the first notable philosopher ever to emphasize the differences between the miscellaneous activities that we file away as mathematics. “I should like to say,” he wrote in his Remarks on the Foundations of Mathematics, that “mathematics is a MOTLEY of techniques of proof” (Wittgenstein 1978: III § 46, 176). When he repeated this idea, saying that he wanted “to give an account of the motley of mathematics,” he went out of his way to emphasize it. Where the translators use a single capitalized word, Wittgenstein's German has a capitalized adjective followed by an italicized noun: “ein BUNTES Gemisch von Beweistechniken.”
In Luther's Bible, bunt is the word for Jacob's coat of many colors, and the word in general means parti-colored, and, by metaphor, miscellaneous. “Motley” is an apt translation of Wittgenstein's double-barreled phrase, “buntes Gemisch,” for it implies a disorderly variety within a group. The German noun “Treiben” denotes bustling activity; “ein buntes Treiben” is emphatic, meaning a real hustle and bustle with all sorts of different things going on. Likewise, “ein buntes Gemisch” is not just a mixture, but rather a mixture of all sorts of different kinds of things. When the adjective is in capital letters, and the noun in italics, BUNTES Gemisch, Wow!
Thus the metaphor captures an aspect of grouping different from Wittgenstein's well-known family resemblances. This is not to deny that there are family resemblances between the motley examples of mathematics that I have just given; it is instead to suggest that mere family resemblance is not enough to collect them together.
There are of course disagreements about how to use Wittgenstein's notion of family resemblance, and about what he himself intended. If, as writers as diverse as Renford Bambrough (1961) and Eleanor Rosch (1973) have suggested, virtually all general terms are family resemblance terms, then the notion gives no help with the identity of mathematics in particular. I favor the “minimalist” interpretation firmly advanced by Baker and Hacker (1980: 320–43). They argue that Wittgenstein's use of the notion is critical rather than constructive, and that he used it chiefly for what they call psychological and formal concepts. The latter includes language, number, Satz. He also used the idea implicitly in his discussion of names. Baker and Hacker also mentioned passages where the notion of a family resemblance occurs in accounts of “the concepts of proof, mathematics, and applications of a calculus, and of the ways in which mathematics forms concepts” (Baker and Hacker 1980: 340; Hacker deleted this observation in his revised edition, 2005). None of the citations bears on the motley of mathematics, or the question of what makes mathematics mathematics. That phrase, “BUNTES Gemisch,” was making a positive point different from Wittgenstein's remarks about family resemblance.
In an essay that Timothy Smiley invited for a British Academy symposium on Mathematics and Philosophy, I connected Wittgenstein's “motley of mathematics” with some of his other thoughts about mathematics (Hacking 2000). I shall not pursue those ramifications here. I wanted only to notice that one philosopher had taken my question seriously. I shall not mention him again.
We do take for granted that the answer to the title question of what makes mathematics mathematics, will not be a set of necessary and sufficient conditions for being mathematics.
The Philosophy of Mathematics
An innocent abroad, who consulted the online Stanford Encyclopedia of Philosophy, and then the Routledge Encyclopedia of Philosophy, would conclude that there is no question about what counts as mathematics. She might be left wondering about what counts as the philosophy of mathematics itself. Stanford has a heading, “Philosophy of Mathematics” (Hor- sten 2007). Routledge does not. Instead it has “Mathematics, Foundations of,” which covers something of the same waterfront (Detlefsen 1998). This is not about what cognoscenti would call Foundations of Mathematics, as represented say by the excellent and very active online FOM: “a closed, moderated, e-mail list for discussing Foundations of Mathematics.” It is about the philosophy of mathematics.
Many of the topics discussed in the one article are discussed in the other. Yet although between them they list some 170 articles and books in their bibliographies of classic contributions, only 13 of these items occur in both lists. Only eight items cited by Stanford were published after Routledge appeared, so that does not explain the discrepancy.
A complete accounting of all related entries in the two encyclopedias generates a little more overlap, but the initial contrast is striking. The prudent innocent will judge that the philosophy of mathematics covers a lot of topics, and that different philosophers have different opinions of what is most central or interesting. Perhaps the discrepancy between the two encyclopedias is due in part to different ideas about what mathematics “is.” Only in part, for sure, but that is food for thought.
A Different Type of Demarcation Problem:
What Counts as a Proof?
The title question, “What makes mathematics mathematics?” looks like a demarcation problem. Our explorer will not find the boundaries of mathematics discussed in the two encyclopedias, but she will encounter some other boundary questions. This is because there are different opinions about what mathematical arguments count as sound proofs.
Only some mathematics is constructive; for brevity I group intuition- ist criticism as constructivist, although the motivations are not identical. Constructivists of various stripes can tell a classically sound proof when they see one. They seem not to deny that classical arguments are the product of mathematical insight, or that they are produced by mathematicians. Perhaps there have been iconoclasts, somewhere, sometime, arrogant enough to say, “That is not mathematics at all,” but that does not seem to be a common reaction. We may call the constructivist critiques of classical proofs retroactive, because they apply to proofs, inter alia, that had been on the books long before the criticisms were made. It does not seem that these retroactive critiques address the larger question of what makes mathematics mathematics.
There are also what I shall call conservative criticisms of new mathematical techniques. One is described in the final paragraphs of the Stanford essay, which in turn refers to its first citation, to Appel, Haken, and Koch (1977). That was a first publication of the proof of the conjecture that any map can be colored using only four colors. The proof relies on a computer to check that some 1,936 graphs have a certain property, and relies on dozens of pages of written argument to show that any map falls into one of these 1,936 categories. “The proof of the four-color theorem gave rise to a debate about the question to what extent computer- assisted proofs count as proofs in the true sense of the word” (Horsten 2007: end of third-to-last paragraph). The need for computers focused a general debate. Topologists, struggling with the map problem itself, seem to have been more vexed by the length and imperspicuity of the hand-checking that was also needed. The problem is not about using a computer. Many topologists now use computers to generate counterexamples and eliminate red herrings. Conservatives insist only that in the end, no lemma in a proof should call for a computer confirmation.
Perhaps some conservatives say that computer-assisted proofs that never mature into perspicuous proofs are not mathematics. The issue is, nevertheless, not about the boundaries of mathematics, but about the boundaries of admissible proofs. Indeed we count mistaken proofs as mathematics. Thus Sir Alfred Kempe's 1879 and P. G. Tait's 1880 non- proofs of the four-color conjecture stood for a decade until flaws were found. Even today they are on any common understanding, “mathematics.” There are live current questions, about what makes a proof a proof, but they are prima facie distinct from my question, of what makes mathematics mathematics.
Three Kinds of Answer
There are three inviting answers to my question. They represent different attitudes, perhaps three different casts of mind:
(1) Mathematics has a peculiar subject matter, which people versed in the discipline simply recognize.
(2) Mathematics is a cognitive field ultimately determined by a domain-specific faculty or faculties of the human mind. It is a task of cognitive science and of neurology to investigate the faculty(ies) or “module(s)” in question.
(3) Mathematics is constituted less by its content than by disciplinary boundaries that have emerged in the course of contingent historical practices.
These three answers are compatible. It will occur to any unsophisticated person that mathematics obviously has a peculiar subject matter, which is investigated by means of one or more mental faculties. Perhaps, as Kant thought, there is a distinct faculty for arithmetical reasoning, and another for geometrical reasoning. The disciplinary boundaries in our teaching and our professions mark our present grasp of that subject matter.
There seems to remain, in this terminus of temporary good-will, a chicken-and-egg question about (1) and (2) that tacitly ignores (3). Is the peculiar subject matter of mathematics a consequence of our mental faculties, so that in some curious sense there is no mathematics without the human brain to process it? Or are the mental faculties simply honed to accord with a human-independent body of fact? Here we get two fundamentally opposed attitudes to mathematics, both of which take present disciplinary boundaries as irrelevant.
The Two Attitudes or Viewpoints
These attitudes are better represented not by the philosophers' distinctions between “realism” and “antirealism” but by a classic debate between a mathematician and a neurologist. One author, Alain Connes, won a Fields Medal for mathematics.The other, Jean-Pierre Changeux, directed the research laboratory in Molecular Neurobiology at the Institut Pasteur. Both were colleagues at the College de France, and so were able to be frank and direct without the rancor that sometimes attends such discussions. Connes mentions one phenomenon that impresses him deeply:
Here we come upon a characteristic peculiar to mathematics that is very difficult to explain. Often it's possible, though only after considerable effort, to compile a list of mathematical objects defined by very simple conditions. Intuitively one believes that the list is complete, and searches for the general proof of its exhaus- tiveness. New objects are frequently discovered in just this way, as a result of trying to show that the list is exhausted. Take the example of finite groups. (Connes and Changeux 1995: 19)
Mathematicians first thought that there are just six types of finite groups, a list complete by the end of the nineteenth century. During a period quite late in the twentieth century, exactly 20 more types were discovered—the sporadic groups. And that is all there are: end of the story. The last finite group to be found is called the monster, for such it is, with more than (8) X (10!) elements. Assuming that there is no deep underlying mistake in the proof, it feels as if this last idiotic group was just there all the time, laying in wait for us, with a monstrous grin on its face. And if human beings had not been smart enough to figure this out, the monster would still have been there, happy as a clam, indifferent to our stupidity.
The neurobiologist agrees that phenomena like this are astonishing. But they are the consequence of cognitive procedures that are formed within the human genetic envelope of possibilities. It is a contingent fact that human beings devised group theory, within which certain structures would form, ultimately based on combinatorial practices to which our minds are given.3
Realism and Antirealism
The Routledge Encyclopedia addresses closely related issues in its entries for “Antirealism in the philosophy of mathematics” (Moore 1998), and for “Realism in the philosophy of mathematics” (Blanchette 1998). The difference between these two philosophies is represented in terms of a question: Are there mathematical truths that we could never know? Realists are defined as those who answer “Yes,” and antirealists as those who answer “No.”
Among philosophers, the classic realist stance is platonism. I use a lowercase “p,” because the name, which was invented by Bernays in 1936, derives more from folk-knowledge of Plato than from historical texts. Given these two limited options, Plato would prefer realist pla- tonism to antirealism. Connes is a platonist, but not a Platonist.
Among British philosophers, the antirealist analysis due to Michael Dummett has become classic, although he is not even mentioned in the Stanford article. The “antirealist” neurologist finds Dummett's version of antirealism uninteresting and perhaps unintelligible. Even to most cognitive scientists it is just so much conversation about language. It does not, they think, get to the heart of the matter, namely the fact that mathematics is a by-product of our brain and its “genetic envelope,” namely the field of possible structures that it can construct. To which the platonist makes the obvious retort: “Exactly so, 'structures'! Structures that we reconstruct, perhaps, but the form of the structure was there for us to discover.”
A third quiet voice might be heard here, that of the under-represented “applied” mathematician.These structures were mostly discovered when investigating nature. They are not merely useful representations of nature, says an heir to Galileo: they are found in nature.
At any rate, even though (1) and (2) appear to be compatible, they stand for very different ways of thinking about mathematics. Neither has much truck with (3), the idea that what counts as mathematics is the product of a contingent history of human endeavors and the emergence of disciplinary boundaries. The three answers betoken different interests and, in the case of (3), a research program that befits the new discipline of Science Studies. I shall connect this third perspective with the first two, not because I profess Science Studies, but because it widens our understanding of the title question. I shall do this by a route that is anathema to most sociologists of science, for I proceed through Im- manuel Kant.
Kant
An unexpected paragraph comes right at the start of Detlefsen's survey in the Routledge Encyclopedia. It follows his assertion that Greek and medieval thinkers “continue to influence foundational thinking to the present day”:
During the nineteenth and twentieth centuries, however, the most influential ideas [in the philosophy of mathematics] have been those of Kant. In one way or another and to a greater or lesser extent, the main currents of foundational thinking during this period—the most active and fertile period in the entire history of the subject—are nearly all attempts to reconcile Kant's foundational ideas with various later developments in mathematics and logic. (Detlefsen 1998: 181)
Kant does not loom so large in most other introductions to the subject. He is not even mentioned in Horsten's “Philosophy of Mathematics” in the Stanford Encyclopedia.
I accept that there is something absolutely right in Detlefsen's stage- setting. For among Kant's innumerable legacies was the conviction that there is a specific body of knowledge, mathematics, of striking importance to any metaphysics and epistemology. In Bertrand Russell's words of 1912: “The question which Kant put at the beginning of his philosophy, namely 'How is pure mathematics possible?' is an interesting and difficult one, to which every philosophy which is not purely sceptical must find some answer” (Russell 1912: 85). We know what troubled him. “This apparent power of anticipating facts about things of which we have no experience is certainly surprising.”
Detlefsen singles out two other problems, namely mathematics' “richness of content and its necessity.” These are among the mathematical phenomena that have made mathematics loom so large in the work of some, but only some, figures in the canon of Western philosophy. In Hacking (2000) I emphasized the alleged a priori character of mathematical knowledge, and the alleged necessity of mathematical truths. Richness of content should, of course, be added. Alain Connes's reaction to the finite group called “the monster” is a fine example. In an axiomatic account of mathematics, we start with what seem to be rather trivial assertions, of the sort that John Stuart Mill called “merely verbal,” and proceed by leaps and bounds to all sorts of astounding discoveries. Richness of content beyond belief. Kant seldom mentions this, but he surely had it in mind (see e.g. Kant 1787: B 16f).
On this occasion I shall hardly touch on these three topics of richness, necessity, and the a priori character of mathematics. They are not why I pick up on Kant. Instead I focus on the clause to which we never pay attention, the conviction that there is a specific body of knowledge, mathematics, of striking importance. Perhaps Kant helped to lodge that proposition in our heads, so that mathematics is just a given, a domain that makes some philosophers curious. Of course mathematics has mattered to philosophers all the way back at least to Plato, but, as we shall see, Plato's own demarcation of mathematics is different from our own.
Kant's Vision of the Ur-History of Mathematics
Kant had already published the Critique of Pure Reason when he sat back, and reflected, that even reason has a history. That pivotal moment, between the first and second editions of the Critique, took place when Europe turned from the timeless reason of the Enlightenment to the historicist world that we still to some extent inhabit. In his new Introduction for the second edition, Kant betrayed a wonderful enthusiasm for a defining moment in the history of human reason (as he saw it). Kant, of all people, has become a historicist. He used such purple prose that I quote it in full:
In the earliest times to which the history of human reason extends, mathematics, among that wonderful people, the Greeks, had already entered upon the sure path of science. But it must not be supposed that it was as easy for mathematics as it was for logic—in which reason has to deal with itself alone—to light upon, or rather to construct for itself, that royal road. On the contrary, I believe that it long remained, especially among the Egyptians, in the groping stage, and that the transformation must have been due to a revolution brought about by the happy thought of a single man, the experiment which he devised marking out the path upon which the science must enter, and by following which, secure progress throughout all time and in endless expansion is infallibly secured. The history of this intellectual revolution—far more important than the discovery of the passage round the celebrated Cape of Good Hope—and of its fortunate author, has not been preserved. But the fact that Diogenes Laertius, in handing down an account of these matters, names the reputed author of even the least important among the geometrical demonstrations, even of those which, for ordinary consciousness, stand in need of no such proof, does at least show that the memory of the revolution, brought about by the first glimpse of this new path, must have seemed to mathematicians of such outstanding importance as to cause it to survive the tide of oblivion. A new light flashed upon the mind of the first man (be he Thales or some other) who demonstrated the properties of the isosceles triangle. The true method, so he found, was not to inspect what he discerned either in the figure, or in the bare concept of it, and from this, as it were, to read off its properties; but to bring out what was necessarily implied in the concepts that he had himself formed a priori, and had put into the figure in the construction by which he presented it to himself. If he is to know anything with a priori certainty he must not ascribe to the figure anything save what necessarily follows from what he has himself set into it in accordance with his concept. (Kant 1787: B x-xii, 19)
We no longer countenance the hero in history, “be he Thales or some other.” Even if there was a historical Thales in whose head the legendary penny dropped, there had to be uptake, there had to be people to talk to, to correspond with, to turn a transient thought into knowledge that endured. And thanks to the labors of generations of scholars, we can no longer dismiss the Egyptians and the peoples of Mesopotamia as in “the groping stage.”
We can now turn Kant's prose into something closer to the historical facts, thanks to Reviel Netz (1999). He would prefer Eudoxus to Kant's Thales, but the important point is that there was a moment of radical change in the human mastery of mathematics. Kant got that right, by present lights. Using the metaphor recently favored in paleontology, Netz suggests “that the early history of Greek mathematics was catastrophic”— a sudden change in the very “feel” of mathematical thinking. In a lower key: “A relatively large number of interesting results would have been discovered practically simultaneously” (Netz 1999: 273). Netz suggests a period of at most eighty years. We have no need to dismiss the Babylonians, Egyptians, and others who taught mathematics to the Greeks, in order to see that at the time of “Thales or some other” a revolution in reason was wrought.
What Was So Revolutionary?
In the modern spirit of iconoclasm called post-modern, post-colonial, and so forth, the Greeks are no longer “that wonderful people.” Their elevation to a central role in Western history, as the only begetters of all things wise and beautiful, was (it is now said) an act of European imperialism. It was all the more important in German thought in the epoch before the German-speaking lands had become sufficiently united to exert their power outside of central Europe. Let us not argue the point.
Greeks may have been glorified in the era of European triumphalism, but there really was something revolutionary, “catastrophic,” that happened in those eighty years of Mediterranean history. Some Greeks, including Ionians, discovered new mathematical facts and structures, but that, from Kant's point of view, was not what counted most. The revolutionary discovery that made it all possible was proof. Greeks uncovered what may well be an innate capacity of all human beings, the ability to make demonstrative proofs.
A New Metaphor: Crystallization
Kant was absolutely right. The discovery of proof was revolutionary. But because the idea of scientific revolution has been so over-worked since the days of Thomas Kuhn, it is wise to choose another metaphor, saying that a crystallization of mathematics occurred at the time of Eudoxus (“or some other”).4
This metaphor is intended to capture several aspects of what happened. First, the new method of demonstrative proof did not lack precursors or anticipations. As in the case of literal crystallization, there was a great deal going on before the new structure appeared. Second, this event inaugurated a whole new way of doing things, stabilized within its own local and historically contingent practices, and yet capable of transfer to new civilizations, where it would be stabilized in very different social environments. Like many a crystal of purest ray serene, it could be thrust into darkness before being uncovered again in a period of reopening and rebirth.
A Style of Scientific Thinking
Demonstrative proof is a distinctive style of scientific thinking. Because the very word “style” is so evocative, the expression, “style of scientific thinking” can be used in many ways. I use it in an artificially narrow sense that I acquired from the historian of science Alistair Crombie (1994). He proposed some six distinct fundamental “styles of scientific thinking in the European tradition” that emerged and matured at distinct times and places. Brief tags for his six styles are: mathematical, hypothetical modeling, experimental exploration, taxonomic, statistical, and genetic. I believe he taught an important insight, but I shall not argue the case here. I mention it because my argument below is part of a larger analysis of the history of scientific reason. It can be seen as a continuation of Kant's original “historicist” thought stated in the long paragraph quoted above.
Indeed in the very next paragraph after that, Kant turns to natural science (Naturwissenschaft) which, he tells us, “was very much longer in entering upon the highway of science (Wissenschaft).” He takes us to the world of Galileo and of Torricelli, and speaks of a “discovery [that] can be explained as being the sudden outcome of an intellectual revolution,” another crystallization (Kant 1787: B xii.). Following Crombie, I file Galileo under the style of hypothetical modeling and Torricelli under the style of experimental exploration. Then I proceed to the synthesis that Kant probably intended, what I call the laboratory style of scientific thinking. That is all a matter of footnotes to Kant's recognition of these “intellectual revolutions.” They were not only intellectual; they were also a matter of new things we could do with hand and eye. They briefly changed the role of Europe in world history, and permanently changed the role of our species on our planet.
The metaphor of “crystallization” may suggest something too rigid, too mineralogical, and too fixed. Styles of scientific thinking evolve, do they not? Perhaps we should extend the metaphor to the science of near-life. It is often observed that viruses are equivocal: some of the time they are inanimate, but when they find a host they are alive. They become alive by exploiting molecules in the host cell to create a kind of metabolism that serves them well. They are not parasites, which are autonomous organisms, but individuals that thrive by incorporating themselves into the lives of their hosts. When alive, they evolve far more rapidly than their hosts do. When inanimate, they are said to be like crystals. I seize upon the analogy. A style of scientific thinking is like a virus, a crystallization that can evolve in a host, a community, a network of human beings.
Enough of metaphor. To return to real people, I emphasize that what Eudoxus and company did, was not only to establish some new mathematical facts, techniques, and proof ideas. They also discovered a new way to find things out, namely by reasoning and proof. This was not a mathematical discovery, but the discovery of a human capacity of which our species had, in earlier times, only glimmerings here and there. It was the discovery and then exploitation of a mental faculty or faculties, of precisely the sort that cognitive science and neurobiology is now investigating.
Netz's book is widely admired for its reconstruction of the diagrams that are notoriously missing from surviving ancient texts. Few readers attend to his subtitle, A Study in Cognitive History, yet that aspect is exactly what is fundamental and wholly original. It is the first detailed analysis of the cultural history of the discovery of a cognitive capacity.
Some cognitive scientists conjecture that mental modules—one or more—enable us to engage in mathematical reasoning. Netz himself argues against excessive modularity, and sides with Jerry Fodor in favor of more general processing devices (Netz 1999: 4—7). He urges never to forget Fodor's “First Law of the Nonexistence of Cognitive Science.” Fodor's first law preaches humility. “The more global…a cognitive process is, the less anybody understands it” (Fodor 1983: 107). The cognitive processes needed for mathematical demonstration are pretty global.
Mention of cognition returns us to the second answer to our title question: (2) Mathematics is a cognitive domain ultimately determined by a domain-specific faculty or faculties of the human mind. The above discussion enriches answer (2), but is wholly compatible with the “pla- tonist” answer (1). Now we pass to reflections that depend on a wholly fortuitous aspect of cultural history, thus directing us towards answer (3).
Why Should Greeks Have Cared?
Why did some Greek thinkers think that the newly discovered capacity for demonstrative proof was so important? (This is different from the question, of why future Europeans such as Kant and Bertrand Russell thought that it was important.) Netz (1999: 209ff), following Geoffrey Lloyd (1990), suggests an answer. City-states were organized in many ways, but Athens is of central importance. It was a democracy of citizens, all of whom were male and none of whom were slaves. It was a democracy for the few; but within those few, there was no ruler. Argument ruled. If you could make the weaker argument appear the stronger, you won.
Athenians were the most consistently argumentative bunch of self- governors of whom we have any knowledge. We read Aristotle for his logic and not for his rhetoric. Greeks read him for his rhetoric; his logic was strictly for the Academy. The trouble with arguments about how to administer the city and fight its battles is that no arguments are decisive. Or they are decisive only thanks to the skill of the orator, or the cupidity of the audience. But there was one kind of argument to which oratory seemed irrelevant. Any citizen, and indeed any young slave who was encouraged to take the time, and to think under critical guidance, could follow an argument in geometry. He could come to see for himself, perhaps with a little instruction, that an argument was sound. He could even create the argument, find it out for himself. In geometry, arguments speak for themselves to the inquisitive mind.
Cynics will say that this is a lie from the start. A “little” instruction? The instruction is just a kind of rhetoric, mere oratory. Look at the classic, the demonstration to be found in the Meno, of how to double a square. The slave boy is said to discover the technique by himself, unaided. He is coached by leading questions from Socrates.
But in addition to prompting there is something else: the extraordinary phenomenon, accessible to almost any thoughtful reader of the Meno, of seeing that the square on the diagonal is twice that of the given square. In company with the diagram, and talk about the diagram, there is a new kind of experience, of conviction based solely on the perception of a new truth. Geoffrey Lloyd remarked that this phenomenon is truly impressive to members of an argumentative society that has no recourse to a ruler, and whose final criterion is nothing more than talk and persuasion.
Plato, the Kidnapper
“So what?” asks the politician in the public arena. You can prove only recondite or useless facts. Quite aside from the uselessness of proof in political debate, it is not even useful to the architect. It is no good saying that geometrical theorems could be useful to a surveyor. (a) The surveyors already knew most of the practical facts required, for they had been acquired from empirical Egyptian mathematicians. (b) Netz reports that not a single surviving text suggests a connection between the problems and solutions of the geometers, and the practical interests of architects. Only later did mathematics become “useful.” Maybe Archimedes used it in the famous problem of burning mirrors. Military mathematics came into its own only in the age of Napoleon. The great mathematicians such as Laplace solved problems connected with artillery, but they were interested not in proofs but in solutions to problems of motion.
Nobody debating military strategy or the tax on corn in the Agora was able to use geometrical proof. So why should Greeks have cared about proof? An answer may be that hardly anyone did. Netz's book is about an epistolary tradition involving a small band of mathematicians exchanging letters around the Mediterranean Sea. They cared about new discoveries and new proofs, but not about the very idea of proof. Enter Plato, kidnapper.
I take the label from Bruno Latour's brilliant critical exposition of Netz's book. His opening sentence reads:
This is, without contest, the most important book of science studies to appear since Shapin and Schaffer's Leviathan and the Air-Pump. (Latour 2008: 441)
Alongside Netz, Latour is referring to Shapin and Schaffer (1985): the book subtitled Hobbes, Boyle, and the Experimental Life. I completely agree with Latour's judgment, but for reasons quite opposite to his. Latour sees both books as magnificent illustrations of his network theory of knowledge—which they certainly are. But I also see Shapin and Schaffer as having presented a decisive crystallization of the laboratory style of scientific thinking, and Netz as presenting a decisive crystallization of the mathematical style of scientific thinking which we call demonstrative proof. In both cases a new style of scientific thinking became established in the practices of discovery, of creating knowledge, or, to be more colloquial but more exact, in the human repertoire of finding out.
Latour rightly takes Netz's analysis as a compelling example of knowledge sustained by a network of creators and distributors of that knowledge. Nowhere is that better illustrated than by Archimedes, who, working out of Syracuse in Sicily, created and maintained an unparalleled body of new understanding, and yet had only a handful of disciples and correspondents around the Mediterranean. But what specially fascinates Latour is the isolation of this network from the rest of the ancient world, be it learned, political, or vernacular.
To the great surprise of those who believe in the Greek Miracle, the striking feature of Greek mathematics, according to Netz, is that it was completely peripheral to the culture, even to the highly literate one. Medicine, law, rhetoric, political sciences, ethics, history, yes; mathematics, no. (Latour 2008: 445)
The Greek and Hellenistic mathematicians were a handful of specialists talking with and writing to each other around the Mediterranean basin, and no one else cared:
—with one exception: the Plato-Aristotelian tradition. But what did this tradition (itself very small at the time) take from mathematicians?…Only one crucial feature: that there might exist one way to convince which is apodictic and not rhetoric or sophistic. The philosophy extracted from mathematicians was not a fully fledged practice. It was only a way to radically differentiate itself through the right manner of achieving persuasion. (Latour 2008:445)
Latour overstates his case. The philosophical tradition took a good deal from mathematics: what about the golden mean, for example, or the profound role of proportion in ethical theory? That is irrelevant to La- tour's case. He proposes that the philosophers focused on proof in order to differentiate themselves from the common herd. Thus their use of proof as above rhetoric was nothing more than a rhetorical trick.
Latour pays little heed to the ways in which the philosophers were profoundly impressed by the human capacity to prove. They were, as I like to put it, bowled over by demonstrative proofs. In consequence they vastly exaggerated the potential of proof. It is easy to argue that the ensuing theory of knowledge impeded the growth of scientific knowledge from the time of Archimedes to the time of Galileo. To defeat the lust for demonstrative proof, we needed another crystallization. Or in Kant's phrase, we needed the “intellectual revolution” that he associated with Galileo and Torricelli. That was the discovery of other human talents—not purely intellectual ones—and it led to the laboratory style of scientific thinking. The definitive history of that crystallization was Leviathan and the Air-Pump, the very book that Latour rightly pairs with Netz's Shaping of Deduction.
Let us here agree with Latour: Plato kidnapped a certain idea of proof and made it a dominant theme in Western philosophical thought. But let us not grant to Latour the idea that proof is unimportant. Let us not allow Latour to kidnap Netz, that is, to allow us to forget Netz's own fundamental concern, cognitive history (which Latour barely mentions).
On the other hand, let us extend Latour's insight. Kant codified, for the modern world, Plato's kidnapping of mathematics. He made the a priori, the apodictic, and the necessary the hallmarks of mathematics, even though they are noticeable only here and there in the motley of mathematical activity. This leads us, for a final observation about ancient times, back to Plato's own demarcation of mathematics.
Plato on the Difference between Philosophical
and Practical Mathematics
An important tradition in reading Plato on mathematics derives from Jacob Klein (1968). He argued that Plato made a fundamental distinction between the theory of numbers and calculating procedures. Here is a brief summary of the idea, due to one of Klein's students:
Plato is important in the history of mathematics largely for his role as inspirer and director of others, and perhaps to him is due the sharp distinction in ancient Greece between arithmetic (in the sense of the theory of numbers) and logistic (the technique of computation). Plato regarded logistic as appropriate for the businessman and for the man of war, who 'must learn the art of numbers or he will not know how to array his troops.' The philosopher, on the other hand, must be an arithmetician 'because he has to arise out of the sea of change and lay hold of true being'. (Boyer 1991:86)
We need not subscribe either to the terminology or the details of the interpretation to propose that (“real”) mathematics, for Plato, did not include the arithmetic we learned in school, and later applied in business transactions, or ordering supplies for the troops. A redescription, owing more to Netz than to Klein, would be that Euclid's Elements made a decision, to emphasize diagrammatic proofs rather than numerical examples. Hence despite the depth of some work on numbers to be found in Apollonius and Archimedes, there was no tradition of “advanced arithmetic” in antiquity, in the way in which there was “advanced geometry.”5
Plato, then, put to one side the daily uses of arithmetic in technologically and commercially advanced societies such as those of Greece or Persia. Those uses are what Klein for his own reasons called “logistic.” In my opinion we should avoid the notion that computation is for practical affairs in “the sea of change.” That is the philosophical gloss of appearance and reality all over again. A primary point, closer to the experience of doing or using mathematics, is that computation is algorithmic. It proceeds by set rules. One does not understand a calculation: one checks that one has not made a slip. There is no experience of proof as in the theory of numbers or geometry.
Quite possibly there were manuals that taught how to calculate, complete with shortcuts. They would be comparable to what we sophisticates dismiss as “cookbooks,” mere “how to do it” instructions that do not convey insight or understanding of “how it works.” We may conjecture that that sort of text has not been preserved, partly because on Platonic and then Aristotelian authority, it did not present “science,” scientia. Just possibly, if there had been a classical text of advanced arithmetic in antiquity, the questions of a priori knowledge, apodictic certainty, and necessity would have been posed in that context. I like to imagine that those questions might, instead, have been exposed as pseudo- questions, at least in the context of the theory of numbers.
These cursory remarks suggest that Plato (or his heirs) created a disciplinary boundary between mathematics, the science that every philosopher must master, and computation, the technique of commerce and the military. This bears some relationship to the recent distinction between pure and applied mathematics, but the fundamental difference is that the one involves perspicuous proof, insight, and understanding, while the other involves routine.
It is important to add that, from the perspective of twentieth-century British analytic philosophy, with its talk of “puzzlement” in philosophy, the results of calculation are not “puzzling,” in the way that proof can be experienced as puzzling (f Hacking 2000: §4.3). Notice also that in Plato's vision (version Klein), there is far less of a motley of mathematics than in ours—because the routine computational side of mathematics is not (real) mathematics at all.
Pure and Mixed Mathematics
Francis Bacon was his usual prescient self when he devised that now abandoned term, “mixed mathematics,” which appears at the end of the OED definition.6 He was captured by the powerful image of the Tree of Knowledge. Every “branch” (as we still say) of knowledge had to have its place on a tree. He needed a branch of the main limb of mathematics on which optics and mathematical astronomy could flourish, for they were mathematics in the older sense of the term, as noticed by the OED. These he called mixed, which also included music, architecture, and engineering.
They were not mixed, I think, because they mixed deduction and observation. It was rather a matter of the sphere to which they applied. Mixed mathematics was not pure mathematics “applied” to nature, but an investigation of the sphere in which the ideal and the mundane were intermingled. Both the mixed and the pure were that part of natural philosophy that fell under metaphysics, viz. the study of fixed and unchanging relations.
The Enlightenment was an era of classification, where Natural History had a proud place as the science of nature observed. Hierarchical classifications, which we now conceive as branching trees, were the model for all knowledge, be it of minerals or diseases; the tree was also the model for the presentation of knowledge itself. Bacon's Tree of Knowledge was hardly new; Raymond Lull's was more graphic. We now think of tree- diagrams as one of the most efficient ways to represent certain kinds of information, but preserved tree-diagrams begin to appear surprisingly late in human history (Hacking 2007). By Bacon's time, however, they flourished in genealogy, logic, and many other fields, and were cast in wonderful glass on cathedral windows. That was the past: Bacon's own Tree of Knowledge was a benchmark for the future.
The image of a branching Tree of Knowledge was to persist for centuries after Lull and Bacon. It is most notably incarnate in D'Alembert's preface to La grande encyclopedic. It is a prominent pull-out page of Au- guste Comte's Cours, the massive 12-year production that continued the encyclopedic project of systematizing all knowledge and representing its growth in both historical and conceptual terms. The Tree of Knowledge that was planted so firmly in the early modern world by Francis Bacon has long been institutionalized in the structure of our universities with their departments and faculties.
Probability—Swinging from Branch to Branch
Many a new inquiry had to be forced on to the tree. Where would the Doctrine of Chances, a.k.a. the Art of Conjecturing, fit? It was by definition not about the actual world, nor about an ideal world. It was about action and conjecture; it was the successor to a non-theory of luck. There was no branch on a Tree of Knowledge on which to hang it. Probability was uneasily declared a branch of mixed mathematics, less because of its content than because of its practitioners, such as the Ber- noullis, who were mathematicians par excellence. The mixed, as we shall see, morphed into the applied. Hence the residual place for the “theory of probabilities” as “mixed or applied” mathematics alongside astronomy and physics in the OED entry.7
The Tree of Knowledge became the tree of disciplines. This may have a somewhat rational underlying structure, of the sort at which Bacon or D'Alembert aimed, but it is largely the product of a series of contingent decisions. This can be nicely illustrated by the location of probability theory in various sorts of institutions around the world. It was once a paradigm of the mixed, so you would expect it to continue as applied mathematics. That is certainly not what happened in Cambridge, where the Faculty of Mathematics is divided into two primary departments. One is Applied Mathematics and Theoretical Physics, the home of Newton's Lucasian chair. The other is the Department of Pure Mathematics and Mathematical Statistics. Probability appears to have jumped from branch to branch of the Tree of Knowledge. In truth, to continue the ancient arboreal metaphor, it is an epiphyte. It can lodge and prosper anywhere in a tree of knowledge, but is not part of its organic structure at all.
Pure and Applied
There is no space, here, to adumbrate the transition of nomenclature from “mixed” mathematics to “applied” mathematics. Perhaps the switch was from an idea of mixing mathematics and the study of nature, to one of applying mathematics to nature. That picture may well be too anachronistic or at least too simple a vision.
Galileo's own famous image is a compelling alternative. The Book of Nature is written in the Language of Mathematics. Galileo did not apply abstract structures to nature. He found the structures in nature, and articulated their properties, thereby reading the Book of Nature itself. Husserl (1936) rightly seized upon what Galileo was doing as radically new, and said that Galileo mathematized nature. Galileo might have retorted that Husserl had things upside down: “I did not mathematize Nature, for she is already mathematical, and waiting to be read.”
Galileo's contribution may have been, as Netz puts it, a footnote to Archimedes (Netz and Noel 2007: 26). It was certainly not a footnote to Plato. Galileo had no truck with Plato's conception of mathematics as outside this world. I realize that this statement flies in the face of a received tradition established by Alexandre Koyre, according to which Galileo was permeated by Platonism. Let us compromise, and say that in the world of Galileo, mathematics had an entirely new role.
The situation looks more straightforward half a century later. Newton distinguished practical from rational mechanics. He took geometry to be a limiting case of practical mechanics, important to builders and architects. Geometry, the very possibility of which so astonished Plato, was placed alongside the practical arts, which Plato did not count as mathematics at all.
There is little reason to think that Newton, the greatest mathematician of his age, cared much about the phenomenon or experience of proof which Plato had made central to his fetishism of mathematics. To continue Latour's metaphor, slightly tongue-in-cheek, we may venture that Galileo and Newton liberated mathematics from the philosophical bonds in which kidnapper Plato had enslaved it.
Newton's rational mechanics was among other things the general theory of motion, and hence of what is constant underneath ever-changing Nature. That can be presented as conforming to Plato's imperative, to discover the reality behind appearance. Whatever it was, it was mathematics, set out in a book with an unambiguous title: Philosophic Naturalis Principia Mathematica, the mathematical principles of natural philosophy.
Pure Kant
“Pure”—rein—evidently plays an immense role in Kant's first Critique, starting with its title. The primary contrast for both the English and the German adjectives is “mixed.”8 Hence Bacon's branching of mathematics into pure and mixed. The next, moralistic sense of being free from corruption or defilement, especially of a sexual sort, comes a close second. At the start of his rewritten Introduction for the second edition of the first Critique, Kant emphasizes what, for him, was the primary contrast: “The Distinction Between Pure and Empirical Knowledge” (Kant 1787: B 1).
Kant's question, which Russell repeated with such enthusiasm, was “How is pure mathematics possible?” What contrasts with “pure” on this occasion? We hear “applied.” Galileo and Newton did not speak of applied mathematics. Kant's opposite of pure mathematics was empirical.9 In fact Kant asked a pair of questions, one after the other:
How is pure mathematics possible?
How is pure science of nature (Naturwissenschaft) possible? (Kant 1787:B 20)
Today we are puzzled, and some are baffled, by the idea of a pure science of nature. In his footnote Kant clearly contrasts it to “(empirical) physics.”10 Kant cites, as an example of pure science of nature, primary propositions “relating to the permanence in the quantity of matter, to inertia, to the equality of action and reaction, etc.” On one reading, Kant is talking about Newtonian mechanics.
Kant embedded pure mathematics in the Transcendental Aesthetic, the launching pad for his entire theory of knowledge. Plato had made mathematics a matter of Ideas in a realm other than that of appearance. Kant made it part of transcendental idealism, and arithmetic and geometry conditions of all possible experience. This was a radical innovation, and yet a continuation of Plato's leitmotif. Kant was restoring a Platonic vision of pure mathematics as something utterly separate and absolutely fundamental to the nature of knowledge. Kant was a kidnapper too. He kidnapped mathematics from the mathematicians by insisting that some of what they did was pure.
The great mathematicians of the generation that flourished in the era of the first Critique, men such as Lagrange, Legendre, and Laplace, did not see things that way. They were mathematicians. In general the scientists of that era made no difference between pure and applied. It was the tidy Kant who put a category of pure mathematics up front.
That is the core truth behind Detlefsen's starting point in the Rout- ledge Encyclopedia, the statement that during the nineteenth and twentieth centuries, the most influential ideas in the philosophy of mathemat ics have been those of Kant. The philosophy of mathematics, as many of us understand it, starts from an unquestioned assumption that there is pure mathematics. We then proceed with Plato's and Kant's vision as something to accept, to modify, to explain, or to reject. The philosophy of mathematics is implicitly about the philosophy of pure mathematics, with a coda, asking how some of it is so applicable to nature.
The separation of the pure from the applied could not happen on Kant's say-so alone. It also called for some highly contingent events in disciplinary organization.
Applied Mathematics
Our idea of applying pure mathematics to nature should not be read back into Kant or Newton. We do have a convenient benchmark for the distinction between pure and applied. In 1810 Joseph Gergonne (1771 — 1859) founded what is usually regarded as the first mathematics journal, Annales de mathematiques pures et appliquees. Most of the articles were contributions to geometry, Gergonne's own field of expertise. Germany followed suit in 1826, when A. L. Crelle (1780–1855) founded the Journal JUT die reine und angewandte Mathematik. The focus was quite different. Crelle published most of Niels Henrik Abel's (1802–1829) papers that transformed analysis.
The nominal distinction between pure and applied did not take hold for some time. I very much doubt that readers of Gergonne's and Crelle's journals knew which articles were pure and which were applied, if, indeed, they asked the question at all. Lagrange inventing Lagrangians, and even Hamilton inventing Hamiltonians, did not think they were applying mathematics to nature. They were investigating nature mathematically. Only after the fact do we abstract the mathematics from nature and “purify” it, and then, retroactively, speak of applying the pure mathematics to scientific problems.
Pure Mathematics
Here I should like to be not just insular but local. The analytic tradition in the philosophy of mathematics is properly traced back to Frege, but a lot of the stage-setting is grace of Whitehead and Russell. Their opus is a third benchmark. Compare their book with Newton's. Two great works are titled Principia Mathematica. They are entirely different in content and project.
Perhaps nobody really believed in Whitehead and Russell's great book. Possibly only the authors read all three volumes. But even the great German set-theorists set themselves up with that work as a monument, even if it turned out, to everyone's surprise, to be essentially incomplete. So it is worth the time to consider the mathematical milieu that Whitehead and Russell took for granted—and their conception of pure mathematics for which they hoped to lay the foundations.
At least in British curricula we can locate the point at which “Pure Mathematics” became a specific institutionalized discipline. In 1701, Lady Sadleir had founded several college lectureships for the teaching of Algebra at Cambridge University. In 1863, the endowment was transformed into the Sadleirian Chair of Pure Mathematics, whose first tenant was Arthur Cayley. This coincided with an important shift in the teaching of mathematics. The old Smith's Prize, founded in 1768, was the way in which a young Cambridge mathematician could establish his genius. In the old days, up until 1885, it was awarded after a stiff examination in what would now be called applied mathematics. One Wrangler who went on to tie for the Smith's Prize became the greatest British mathematician of the nineteenth century. But what we call mathematics has changed. We do not call him a mathematician but a physicist. I mean James Clerk Maxwell. Many names hallowed in the annals of physics, such as Stokes, Kelvin, Tait, Rayleigh, Larmor, J. J. Thomson, and Ed- dington won the Smith's Prize for mathematics.11
Yet after 1863, what was called mathematics at Cambridge was increasingly pure mathematics rather than Natural Philosophy. It was within this conception of mathematics that Russell came of age. Likewise it was in this milieu that G. H. Hardy became the preeminent local mathematician, whose text, Pure Mathematics, became a sort of official handbook of what mathematics is, or how it should be studied, taught, examined, and professed at Cambridge. Russell's vision of mathematics was not determined by Hardy's, or vice versa, but the two visions are coeval, a product of a disciplinary accident in the conception of mathematics.
Contingency, Necessity, and Neurology
I have sketched only the beginning of an argument, that what is counted as mathematics depends in part on a complex and very contingent history. I do not mean to imply that the history could have gone any way whatsoever. It was constrained by its content, and by human capacities. They no longer constrain in the same way. The advance of fast computation is changing the entire landscape of human knowledge, including that of mathematics. That is a topic for the future. Here I have been concerned with the past.
Our picture of the philosophy of mathematics is of philosophical reflection on a definite and predetermined subject matter. I suggest that the subject matter itself is much less determinate than we have imagined. This does not undercut the debate between the two attitudes I have mentioned, (1) platonic and (2) neurobiological, or the traditional philosophical debate between “realists” and “antirealists” of mathematics. It will surely go on as before. I urge only that the more difficult but perhaps more answerable question should now become: how have the platonic and neurobiological constraints jointly interacted with the contingent history of mathematics from “Thales” to now?
Notes
1.This essay appeared originally in a festschrift forT. J. Smiley of Cambridge University. The “we” of this first paragraph refers to we, his students. My own exploration of logicism is Hacking (1979), a long-delayed by-product of a research paper (Hacking 1963) that profited from Smi- ley's constant advice.
2. This phrase is the title of J. E. Littlewood's (1953) charming potpourri of mathematical anecdote and examples.
3. I got the phrase “genetic envelope” from a conversation with Changeux; I do not think he has used it in print.
4. The idea of a crystallization of a style of scientific thinking was introduced in Hacking (2009), and will be developed in a continuation of that book. My Crombian theme of styles of scientific thinking was launched long ago (Hacking 1982, 1992).
5. These remarks are my version of a short discussion with Netz in March 2009.
6. The assertion that the term “mixed mathematics” is original to Bacon is due to Brown (1991).
7. Two late quotations from the OED speak of applying, but differ in their meaning. In 1706: “Mixt Mathematicks, are those Arts and Sciences which treat of the Properties of Quantity, apply 'd to material Beings, or sensible Objects; as Astronomy, Geography, Navigation, Dialling [sundials], Surveying, Gauging &.” In 1834, in Coleridge: “We call those [sciences] mixed in which certain ideas of the mind are applied to the general properties of bodies.”
8. OED: Without foreign or extraneous admixture: free from anything not properly pertaining to it; simple, homogeneous, unmixed, unalloyed. Grimm's Deutsches Worterbuch: frei von fremdartigem, das entweder auf der Oberflache haftet oder dem Stoffe beigemischt ist, die eigenart trubend.
9. I know of only one occasion where he spoke of applied mathematics, namely in his Lectures on Metaphysics (delivered from the 1760s to the 1790s). “Philosophy, like mathematics as well, can be divided into two parts, namely into the pure and into the applied” (Kant 1790–1: 307).
10. Kant wrote “eigentilichen (empirischen) Physik,” which Kemp Smith renders “(empirical) physics, properly so called.” Like the English noun “physics,” Physik in Kant's time still meant natural science in general. Kant might have meant something more like “real (empirical) physics.”
11. Partly thanks to T. J. Smiley, by 1961 an essay in modal logic could win a Smith's Prize. Robert Smith, Plumian Professor in Astronomy, doubtless turned over in his grave. He intended the encouragement of Mathematics and Natural Philosophy. It may amuse students of today's economy to know that Smith endowed the prize with profits from the South Sea Bubble. The structure of the competition for the Smith's Prize and related prizes was reorganized in 1998.
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