“Mathematics and the Metaphysicians,” pages 74-96
[In the Preface (pages v-vi), Russell tells us that this chapter originally appeared in 1901, and that necessary updates are indicated in footnotes.]
Despite what you may have heard, pure mathematics is a recent discovery, made by Boole in 1854. Though unbeknownst even to Boole, mathematics and formal logic are equivalent. Pure mathematics is about general statements, along the lines of “If (some proposition) A is true, then (some proposition) B is true,” but precisely what A is and whether in fact it is true or not are issues for applied, not pure, mathematics. “Thus mathematics may be defined as the subject in which we never know what we are talking about, nor whether what we are saying is true [p. 75].”
The method familiar from geometry – start with some primitive definitions and axioms, and then commence deducing – is not descriptive of pure mathematics, except as that method applies to unalloyed logic. The branches of mathematics (geometry, arithmetic, and so on) develop from the building blocks of logic. It used to be an item of contention in philosophy as to whether mathematics could be built solely upon logic; the mathematicians have ended the contention by actually doing the building.
Aristotle and his syllogism founded formal logic, which for centuries advanced no further. Since 1850, we advance more every decade than the cumulative progress “from Aristotle to Leibniz [p.76];” Charles Sanders Peirce’s Logic of Relatives has been instrumental in expanding the scope of logic.
Symbolic logic allows us to get at the foundations of mathematics, through the paradoxical method of making the initial part of the path more difficult. The symbolism eliminates any obviousness from even the simplest propositions (such as 2+2=4), so we must rely on mechanical operations. We can thereby uncover the minimal set of definitions and axioms to generate algebra, for instance. At first, it might seem frivolous to rigorously prove that 2+2=4, but by connecting obvious statements through the non-obvious applications of rigorous methods, we are learning. One thing that is learned is that obvious truths sometimes are false. For numbers in general, for instance, it is not the case that the addition of one item leads to a greater number of items (thanks to transfinite numbers).
Giuseppe Peano is at the forefront of mathematical logic. [Russell inserts a footnote indicating that in the original version of this chapter, he was unfamiliar with the work of Gottlob Frege, but that Frege should be included as a contemporary leader in logic.] Peano dispenses with words (including “therefore,” and “let us assume”) in developing most (and soon all) of mathematics to symbols. Excepting geometry, most mathematics needs only three primitives: zero, number, and “next after.” And even these three primitives can be replaced by two ideas, relation and class.
Leibniz glimpsed the method that Peano has developed, but Leibniz’s progress was constrained by his unwillingness to accept that Aristotle made logical errors. Though lampooned, Leibniz’s vision of philosophical disagreements resolved by calculations has, to a significant extent, been realized, at least in mathematical philosophy. What used to be mysteries (such as the nature of infinity) are now certainties.
For centuries it was believed that Aristotle had effectively answered the paradoxes of Zeno of Elea, but with the work of Karl Weierstrass, we learn that Zeno largely was right. Zeno’s sole mistake was to believe (if he did believe it) that the non-existence of a state of change implies an unchanging reality. Weierstrass’s use of mathematics avoids any mistaken inferences, with the result that Zeno’s paradoxes appear as straightforward statements, though perhaps at the cost of removing the delight that can accompany Zeno’s enigmas.
Zeno’s paradoxes fundamentally implicate “the problems of the infinitesimal, the infinite, and continuity [p. 81].” For centuries no serious progress was made on these problems, until Weierstrass, Dedekind, and Cantor solved them: “[t]his achievement is probably the greatest of which our age has to boast [p. 81].” Weierstrass, in particular, showed that the infinitesimal which had bedeviled thinkers for millennia could safely be dispensed with. We can always divide a length more finely without ever reaching a single point. We cannot say where a body in motion will be in the next instant, because there is no such thing as the next instant (p. 84).
Recent advances on infinity have taken rather the opposite path than new thinking on infinitesimals: in the case of infinity, a concept that once was thought to hold inherent logical contradictions has been rendered perfectly understandable by Dedekind and Cantor. Their first step was to offer a precise definition of infinity; Cantor then demonstrated that the standard contradictions disappear if a faulty line of reasoning employed in the proofs of the contradictions is rejected. That commonsensical but incorrect notion was that a proper subset of a collection has fewer items than the original collection. The notion is correct for a finite collection, but provides the definitional distinction for infinity: “A collection of terms is infinite when it contains as parts other collections which have just as many terms as it has [p. 86].” Since every positive integer has a unique double (which is even), the positive integers and the even integers have the same number of terms – and hence there is an infinite amount of positive integers.
In dealing with the infinite, we can’t determine the size of a set just by counting the terms – that process would never end. But of two infinite collections, we can ask (and determine) if one has more terms than the other. The method is like the even-number example: we look for a one-to-one mapping between the collections. If such a mapping exists, then the two infinite sets are of the same size. Some infinite collections are larger than others, however. Is there a greatest infinite number? Cantor says no, but his proof is mistaken. [Russell adds (page 89) a footnote in 1917 indicating that Cantor’s proof actually is correct, and that it was Russell who was mistaken.]
The paradoxcial Zenovian notion that fleet Achilles cannot catch a slow tortoise that possesses a head start goes away when we see that proper subsets do not have to be smaller than their parent collection. Why are people led to think that there is a serious argument that Achilles cannot catch the tortoise? Because people recognize that at each instant after the start of the race, Achilles must be in precisely one spot and the tortoise must be in precisely one spot. But with the head start, the tortoise necessarily has been in more spots along the race course than has Achilles. As each passing instant adds one new spot for Achilles and one more spot for the tortoise, Achilles can never have been in a greater number of spots than the tortoise: he cannot catch the tortoise. Once we recognize that Achilles’s portion of the course does not possess fewer spots than the tortoise’s longer portion of the course, however, the argument crumbles.
Russell offers “the paradox of Tristam Shandy [p.90],” building on Tristam’s recognition that as it takes him longer to write about a period of his life than the period itself, his autobiography will become increasingly further from completion even as he makes progress on it. But as we can match each day of Tristam’s life with the year that it takes to write about it — that one-to-one mapping again — then over infinite time, the autobiography is complete. For a similar reason, with enough time, the slow tortoise will go as far as fleet Achilles. With Cantor as a guide, the paradoxes that once seemed inherent to infinity no longer look so paradoxical -- just as scientific advances have rendered it uninteresting that people can live on the other side of the earth despite their ”necessity” to live upside down.
The puzzles of continuity likewise have given way to Cantor’s exactitude: continuity is one type of order. It is order, and not quantity, that now seems fundamental in mathematics; much can be accomplished without introducing numbers. Limits, formerly expressed as quantities, are now based upon order. “Thus, for instance, the smallest of the infinite integers is the limit of the finite integers, though all finite integers are at an infinite distance from it [p. 92].”
Geometry, too, appears in a different light as its non-Euclidean variants have multiplied. Geometry is not about the space in which we live: it is about the valid conclusions that can be drawn from some set of starting principles, where those principles need not accord with what we see in the real world. The leading books in geometry now do not even rely upon figures to demonstrate their arguments. Indeed, figures mislead, as they suggest conclusions that seem obvious from the figures, but do not follow with necessity from the first principles.
Modern geometry does not start with an assumption of some large space; rather, a point is assumed, and then a second distinct point, with lines and other points building upon these beginnings, so nothing is assumed to exist unless it is necessary for the next step of reasoning. The mathematicians responsible for much of this improvement are Peano and Fano. Euclid’s work itself now seems error-filled, in the sense that, strictly speaking, many of his theorems do not follow from his axioms alone. The difficulty of Euclid’s book, along with its errors, renders it unfit for any consideration beyond the historical; it should not be thrust upon English schoolboys. “A book should have either intelligibility or correctness; to combine the two is impossible, but to lack both is to be unworthy of such a place as Euclid has occupied in education [p. 95].” [Russell’s own youthful introduction to Euclid was both intelligible and revelatory. Recall that this chapter of Mysticism and Logic originally appeared in 1901; in 1902, Russell published a short essay with a more detailed critique of Euclid. Even earlier, Russell had dealt with the empirical validity of Euclid’s postulates, an issue he regards in Mysticism and Logic as “a comparatively trivial matter [p. 94].”]
Modern formalism and symbolic logic have brought rigor to mathematics, a rigor that was absent since the Ancient Greeks. Mathematical advances in the interim were so alluring that the foundations of the subject were unexamined. Weierstrass and his ilk were to mathematicians what Hume was to Kant, the prod that awoke a slumbering intellect. Formalism can seem pedantic, but its record in uncovering errors provides its justification.
Kant’s metaphysics cannot survive the fact that mathematics (including geometry) are but elements of symbolic logic: his theory of knowledge was meant to complete Euclid. Now that we know that Euclid is wrong, not simply incomplete, the Kantian theory is not viable. What is needed is for mathematical logic to come to full flower, and then for philosophy to repurpose the same rigorous tools. If this process is successful, the future might see a golden age in philosophy that parallels the recent era of advance in mathematics, and matches the glory that was philosophy in Ancient Greece.