VII.12 Analysis, Mathematical and Philosophical

John P. Burgess

1 The Analytic Tradition in Philosophy

Philosophical problems are never solved for the same reason that treasonous conspiracies never succeed: as successful conspiracies are never called “treason,” so solved problems are no longer called “philosophy.” Philosophy, which once included almost every subject in the university (every subject in which the highest degree is Ph.D.), has thus been shrunk by success. The greatest shrinkage occurred during the seventeenth and eighteenth centuries, when natural philosophy became natural science. Philosophers of the period, all intensely interested in the emergence of the new science, differed over issues of scientific method. Philosophy had always been understood to differ from, for instance, theology, by restricting itself to the methods of reasoned argument and the evidence of experience, without appeal to authority, tradition, revelation, or faith. But philosophers of the era of the scientific revolution disagreed about the comparative importance of reason and experience.

In introductory histories, philosophers are accordingly divided into the rationalists, or the party of reason, and the empiricists, or the party of experience. The former, mainly from Continental Europe, were dominant in the seventeenth century, while the latter, mainly from the British Isles, predominated in the eighteenth. The rationalists, who included the mathematicians DESCARTES [VI.11] and LEIBNIZ [VI.15], were impressed by the apparent ability of pure thought—logical deduction from self-evident postulates—to achieve, as it seemed to do in geometry, substantive results with worldly applications; and they were tempted to adopt similar methods in other areas. Spinoza even wrote his Ethics in the style of EUCLID’S [VI.2] Elements, a world-historic peak of the influence of mathematics on philosophy. The empiricists, who included that acute critic of the calculus, Berkeley, recognized that in physics one cannot proceed as the rationalists wished to. The principles of physics are not self-evident, but must be conjectured from and tested against systematic observation and controlled experiment. What puzzled leading empiricists such as Locke and Hume was how pure thought was able to succeed in any area, as it seemed to in geometry. Thus, while for rationalists mathematics was a source of methods, for empiricists it was the source of a problem.

An influential formulation of that problem was offered by Kant, whose system attempted a synthesis of rationalism with empiricism. On the one hand, Kant claimed, geometry and arithmetic are a priori rather than a posteriori, by which he meant that they are knowable in advance of experience rather than dependent on it. On the other hand, they are synthetic rather than analytic, which is to say that they are more than mere logical consequences of the definitions of concepts, statements whose denials would amount to contradictions in terms. Philosophy of mathematics, today a smallish specialty within philosophy of science, itself a smallish specialty within epistemology or the theory of knowledge, played a much more important role for Kant, who in his own summary of his system gave pride of place to the question, “How is pure mathematics possible?” as the first case of the question, “How is synthetic a priori knowledge possible?” Kant’s proposed solution was based on the insight that our knowledge must be shaped as much by the nature of ourselves, the knowers, as by that of what is known. Kant concluded that space, the subject matter of geometry, and time, according to him the ultimate subject matter of arithmetic, were not features of things as they are in themselves, but rather of things as we must perceive and experience them, given the nature of our sensibility. Synthetic a priori knowledge is ultimately self-knowledge, knowledge of the forms that we supply, and into which reality independent of us pours content. This distinction between phenomena, or things as we experience them, and noumena, things beyond our experience, about which we can wonder but never know, was central to Kant’s entire system, his ethics as much as his metaphysics.

Such is the history, painted in quick strokes and with a broad brush, of early modern philosophy. After Kant, the story no longer has as clear a plotline. System building continued for another generation, down to Hegel. But eventually, and inevitably, his system collapsed under its own weight, and in the ensuing reaction philosophers went off in all directions. Outside academia, striking figures sporadically appeared on the borders of philosophy and literature, notably Nietzsche. Meanwhile, academic philosophy, rather like Victorian architecture, experienced a number of revivals, of which the Kantian was the most prominent. But even as neo-Kantianism prevailed in the schools, the Kantian conception of mathematics was under attack. First, though the development of consistent non-Euclidean geometries in itself only confirms Kant’s claim that geometry is synthetic, those who developed alternatives to Euclid were quickly led to question whether Euclidean geometry is really a priori, as Kant had claimed. GAUSS [VI.26] had already concluded that geometry is a posteriori, or, as he put it, of the same status as mechanics, and RIEMANN [VI.49] argued at greater length that an examination of the hypotheses that lie at the foundation of geometry must lead us into the domain of the neighboring science of physics. Second, while few doubted Kant’s claim that arithmetic is a priori, a challenge arose to the claim that it is synthetic in the work of GOTTLOB FREGE [VI.56] and (slightly later, but largely independently) BERTRAND RUSSELL [VI.71], who both attempted a derivation of arithmetic from logic along with an appropriate definition of number.

Frege’s work long remained less well-known than it deserved to be, despite the publicity given it by Russell once he became aware of it himself. As a result, Frege, though very influential at present, is more a precursor of the tradition in philosophy within which he stands than a founder, the founders being rather Russell and his contemporary and colleague G. E. Moore. That pair began by rebelling against the philosophy of their teachers, a late nineteenth-century aberration called absolute idealism, a kind of Hegel revival; but it soon became apparent that the rebels were aiming at more than just a return to the traditional empiricism of British philosophy from Bacon to Mill. Meanwhile, Edmund Husserl was developing the first form of what was to become the great rival to the Russell–Moore tradition in twentieth-century philosophy. Like Frege, Husserl had begun his career with work in the philosophy of arithmetic, work of which Frege himself had taken notice, and no one in the early twentieth century expected that Husserl’s and Frege’s heirs would, within a generation, split into two noncommunicating lines of descent.

The two lines of development or traditions are oddly named, with a stylistic label, “analytic,” for one, and a geographical label, “Continental,” for the other. This odd labeling reflects the historical fact that the principal representatives of the analytic style in Continental Europe (Ludwig Wittgenstein, Rudolf Carnap, and others) were forced to go into exile in the English-speaking world in the 1930s, as a result of the process generally known as the Nazification (but celebrated by Husserl’s estranged student Martin Heidegger as the “self-affirmation”) of the German university. This physical separation—more than Heidegger’s break with his teacher, hostility toward science, rebarbative prose style, or loathsome politics—created a split that no one could have anticipated twenty years earlier.

With the years the gap has widened, as later writers on each side tend to read and cite only predecessors on that side. Indeed, the divide has extended backwards in time. For while Borges has said that in literature great writers create their own predecessors, in philosophy even not-so-great writers can do so, and the two twentieth-century traditions came to see different nineteenth-century figures as leading up to themselves, thus extending the division between them right back to the death of Kant (with Hegel rather than Heidegger being identified as the first distinctively Continental philosopher). The gap between the reading lists of students in the two traditions has become so large that nowadays for a student trained in one to take up the other is virtually to switch disciplines.

The word “tradition,” rather than “school” or “movement,” is used advisedly, for each tradition has contained several movements, as well as individuals who defy classification by school. It would be a serious mistake to suppose that there is any doctrine or method on either side of the analytic/Continental divide that all philosophers on that side uphold. In particular, analytic philosophy should not be confused with logical positivism, a Viennese–American school defunct for more than half a century, nor should Continental philosophy be confused with existentialism, a literary-philosophical movement out of fashion in Paris for nearly as long. Logical positivism and existentialism were indeed varieties of analytic and Continental philosophy respectively, and perhaps the most prominent varieties half a century or so ago; but each was even then far from being the only variety. In assessing the influence of mathematics on philosophy in the twentieth century, one must take into account divisions within each tradition as much as the division between the two traditions.

It may be true that since the early work of Husserl there has been comparatively little contact between mathematics and philosophy on the Continental side, though the label “structuralist” is broad enough to take in both the mathematics of BOURBAKI [VI.96] and the various anthropological and linguistic doctrines that became influential in France after the eclipse of existentialism; but it is also true that the direct influence of mathematical ways of thinking on many individuals and groups within the analytic tradition has been negligible. Thus, just as there are distinguishable German and French subtraditions within the Continental tradition, so within the analytic tradition one may distinguish a more technically oriented subtradition, including Frege (who was himself a professor of mathematics), Russell (who as an undergraduate had concentrated on mathematics before turning to philosophy), and the logical positivists (who had mostly been trained as theoretical physicists), from a nontechnical or antitechnical subtradition, including Moore, Wittgenstein, the so-called ordinary-language school of mid-century Oxford, and others. (Wittgenstein even went so far as to claim that mathematicians always make bad philosophers, a sweeping judgment condemning many right back to Thales and PYTHAGORAS [VI.1], though the immediate target was Russell.) However, there has been very much more communication and influence back and forth between the two subtraditions within each tradition than between the two traditions.

Even among the more technical analytic philosophers the influence of mathematics after the period of the founders has been occasional and sporadic, and has come mostly from areas such as mathematical logic, computability theory, probability and statistics, game theory, and mathematical economics (as in the work of the philosopher–economist Amartya Sen), which are rather far from the core of pure mathematics as mathematicians see it. Thus it is hard to imagine the solution to any of the Millennium Prize Problems (except perhaps the P vs NP problem, the one question coming from theoretical computer science rather than core mathematics) having measurable impact even among the most susceptible analytic philosophers. In contrast to this limited direct influence, the indirect influence of mathematics, resulting from its effect on the thought of the early figures Frege and Russell, has been over-whelming even among the less technically oriented analytic philosophers. The branches of mathematics that influenced Frege and Russell were geometry and algebra and, above all, the third great branch of core mathematics, “analysis,” in the mathematical rather than the philosophical sense, the branch beginning with differential and integral calculus. (Frege and Russell were not influenced by mathematical logic: rather, they created it, and mathematical analysis was a key influence on its creation.)

2 Mathematical Analysis and Frege’s New Logic

Let us turn, then, to consider the state of mathematical analysis in the days of Frege and Russell, beginning our account with a quick look back at the situation ca. 1800. As rich as its results were, and as powerful its applications, mathematics at the beginning of the nineteenth century was concerned with but a few structures: the natural, rational, real, and complex number systems; and the Euclidean and projective spaces of dimensions one, two, and three. All that changed quickly when the work of Gauss, HAMILTON [VI.37], and others introduced the first non-Euclidean spaces and first noncommutative algebras, after which a proliferation of new mathematical structures rapidly ensued. This generalizing tendency went hand in hand with a rigorizing tendency, since the proliferation of novelties persuaded mathematicians that they needed to adhere more strictly than had become customary to the ancient ideal of rigor, according to which all new results in mathematics are to be logically deduced from previous results, and ultimately from a list of explicit axioms. For without rigor, intuitions derived from familiarity with more traditional structures might easily be unconsciously transferred to new situations where they are no longer appropriate.

Generalization and rigorization went hand in hand not only in geometry and algebra, but also in mathematical analysis. Generalization in mathematical analysis took place in two directions. The eighteenth-century notion of “function” had been that of an operation applying to one or more real numbers as inputs or “arguments” and yielding a real number as output or “value,” according to a certain formula, such as f(x) = sinx + cosx or f(x, y) = x² + y². On the one hand, nineteenth-century mathematicians generalized by dropping the requirement of an explicit formula. On the other hand, Cauchy, Riemann, and others extended the notion to allow as arguments not only real numbers but also complex numbers, that is, numbers of the form a + bi, where a and b are real numbers and i is the “imaginary” square root of −1.

Rigorization in mathematical analysis also took place on two levels. First, for each theorem it had to be clearly stated just what special properties were being assumed for the functions to which the result was supposed to apply, since special properties such as definability by a formula (or continuity or differentiability) were no longer being built into the highly general notion of function itself; moreover, the relevant properties themselves had to be clearly defined (leading to the so-called WEIERSTRASS [VI.44] epsilon–delta definitions of such concepts as “continuity” and “differentiability” in freshman calculus), since, as POINCARÉ [VI.61] remarked, until one has rigor in one’s definitions one cannot have rigor in one’s theorems. Second, the properties assumed for the numbers to which the functions apply had also to be clarified and stated explicitly as axioms, with the properties of complex numbers being derived by logical definition and deduction from properties of real numbers (by Hamilton), which themselves in turn were derived from properties of rational numbers (by DEDEKIND [VI.50] and CANTOR [VI.54]), which themselves in turn were derived from properties of the system of natural numbers 0, 1, 2, and so on.

Here Frege wished to press still further, and to do what Kant had said could not be done, and derive the properties of the natural numbers themselves from pure logic. For this purpose he needed to become more self-conscious about logic than even the most rigorist mathematicians: he needed not merely to adhere implicitly to the rules and standards of logical definition and deduction, but also to analyze explicitly those very rules and standards themselves. Such self-conscious analysis of definition and deduction was a topic that had, since antiquity, traditionally belonged to philosophy rather than mathematics. Frege needed to carry out a revolution in this philosophical subject, one that would bring it much closer to mathematics, and would bring progress to a field that Kant had described as having advanced not a step beyond the state in which it was left by its founder, Aristotle. (The description is slightly exaggerated, but essentially correct, in that each step forward in the two millennia after Aristotle had been followed by a step back.) It was Frege’s new logic, detached from its original role as part of a special project in foundations of arithmetic and applied to quite diverse subject matters, that was to become the single most important general instrument for philosophical analysis in the twentieth century. Indeed, to a large degree philosophical analysis simply is the logical analysis of philosophical rather than mathematical notions, carried out with the aid of Frege’s broad new logic, or still broader extensions of it introduced by his successors. It was by the creation of this general instrument of a new logic, rather than the specialized application he made of it to the philosophy of mathematics, that Frege became the grandfather of analytic philosophy. And the novelty in Frege’s logic was directly inspired by novel developments in mathematical analysis, as he himself emphasized.

In an article entitled “Function and concept,” Frege describes the broadening of the notion of function as follows (in the translation by Peter Geach and Max Black):

Now how has the reference of the word “function” been extended by the progress of science? We can distinguish two directions in which this has happened. In the first place, the field of mathematical operations that serve for constructing functions has been extended. Besides addition, multiplication, exponentiation, and their converses, the various means of transition to the limit have been introduced—to be sure, without people’s being always clearly aware that they were thus adopting something essentially new. People have even gone further still, and have actually been obliged to resort to ordinary language, because the symbolic language of Analysis failed, e.g., when they were speaking of a function whose value is 1 for rational and 0 for irrational arguments. [This is a famous example of DIRICHLET [VI.36].] Secondly, the field of possible arguments and values for functions has been extended by the admission of complex numbers. In conjunction with this, the sense of the expressions “sum,” “product,” etc. had to be defined more widely.

Frege adds at the end, “In both directions I go still further.” For it was the broadening of the notion of function by mathematicians that provided Frege with the clue he needed to develop a logic broader than Aristotle’s.

Before one can appreciate the advance represented by Frege’s logic, one must understand something of Aristotle’s. Though it is a pretty poor achievement if it is considered as the best the human race could do in this area in a couple of thousand years, it is a brilliant one when considered as the work of a single individual in the course of a career devoted to many other projects. For Aristotle created from nothing the science of logic, whose aim is to distinguish valid from invalid inferences of conclusions from premises. Here an inference is valid if its form alone, regardless of the material truth or falsehood of premises and conclusions, guarantees that if the premises are true, then the conclusion is true. Equivalently, the inference is valid if in all inferences of the same form in which the premises are true, the conclusion is true. Thus, to adapt an example of Lewis Carroll, the inference from “I believe whatever I say” to “I say whatever I believe” is not valid, because there are inferences of identical form in which the premise is true and the conclusion false, such as the inference from “I see whatever I eat” to “I eat whatever I see.”

The scope of Aristotle’s logic is limited by the limited range of forms of potential premises and conclusions he recognizes. In fact, he recognized only four: the universal affirmative “All A’s are B’s,” the universal negative “No A’s are B’s,” the particular affirmative “Some A’s are B’s,” and the particular negative “Some A’s are not B’s” or “Not all A’s are B’s.” The premise “I believe whatever I say” amounts to “All things that I say are things that I believe,” and hence is a universal affirmative. The invalidity of the inference in the Lewis Carroll example exemplifies the invalidity of the inference from “All A’s are B’s” to “All B’s are A’s.” The validity of the inference from the two premises “All Greeks are human beings” and “All human beings are mortal” to the conclusion “All Greeks are mortal” exemplifies the validity of the inference from “All A’s are B’s” and “All B’s are C’s” to “All A’s are C’s,” traditionally called the “syllogism in Barbara,” for reasons that need not concern us here. Aristotle’s logic was in part inspired by the practice of deduction in philosophical debate (“dialectic”) and in part by the practice of deduction in mathematical theorem-proving (“demonstration”), and he offers in his Posterior Analytics an account of a deductive science that is presumed to be based on the practice of the contemporary geometer Eudoxus, in the same sense and to the same degree in which his account in the Poetics of tragedy is based on the practice of the contemporary playwright Euripides. But, in fact, Aristotle’s logic is inadequate for the analysis of mathematicians’ actual arguments, because he makes no provision for forms of argument involving relations. He cannot, for instance, analyze properly the valid argument from “All squares are rectangles” to “Anyone who draws a square draws a rectangle,” because he has no way of representing adequately the form of the conclusion.

By contrast, if you open any present-day introductory logic text, you will find instructions on how to represent symbolically the forms of arguments involving relations. The example just given would appear textbook-style as follows:

∀x(Square(x) → Rectangle(x))

∴ ∀y(∃x(Square(x) & Draws(y, x)) →

∃x(Rectangle(x) & Draws(y, x))).

In words this would amount to the following. For every x, if x is a square, then x is a rectangle. Therefore, for every y, if there is an x such that x is a square and y draws x, then there exists an x such that x is a rectangle and y draws x. (Thus “→” means “if . . . , then . . . ,” “∀” means “for every,” and “∃” means “there is.”) This style of logical analysis is the invention of Frege.

Underlying it is a notion of a “concept” as a special kind of function, a function that (generalizing the mathematical notion in one direction) need not be given by any kind of mathematical description, and that (generalizing the mathematical notion in another direction) need not have as arguments any kind of numbers. A concept for Frege is a function whose argument or arguments may be any objects at all, and whose values are Truth and Falsehood. Thus, the concept Wise applied to the argument Socrates produces the value Truth, since Socrates is wise (at least to the extent of recognizing that he lacked perfect wisdom), while the concept Immortal applied to Socrates produces Falsehood, since Socrates was not immortal but died of drinking hemlock. Frege is able to handle relations because he follows the mathematical analysts who allowed functions of two or more arguments. Thus the two-argument concept or relation Taught applied to Socrates and Plato, in that order, produces Truth, since Socrates taught Plato, while applied to Plato and Socrates, in that order, produces Falsehood, since Plato did not teach Socrates. Aristotle’s simple “All A’s are B’s” becomes, for Frege, the more complex “For all objects x, if A(x), then B(x).” At the price of such extra complexity, he is able to logically analyze arguments turning on relations, as Aristotle was not.

Aristotle analyzed the concept Human Being in terms of the concepts Animal and Rational in the sense of “language-using.” In present-day textbook notation (writing “↔” for “if and only if”), this would be

Human(x) ↔ Animal(x) & Rational(x).

But Aristotle, with no theory of relations, was unable to analyze the notion of Mother (respectively, Father) in terms of Female (respectively, Male) and Parent. For Frege, Mother is analyzed as follows:

Mother(x) → Female(x) & ∃y Parent(x, y).

A mother is a female who is someone’s parent, and analogously for a father. Frege was even able to analyze the concept Ancestor in terms of the concept Parent, though this analysis is beyond the scope of the present sketch. Later philosophical analysis would have been unthinkable without Frege’s broadening of logical analysis beyond Aristotle’s, and Frege rightly saw his broadening of logical analysis as a direct extrapolation from the nineteenth-century mathematical analysts’ broadening of the notion of function they had inherited from their eighteenth-century predecessors.

3 Mathematical Analysis and Russell’s Theory of Descriptions

Like Frege, Russell found in mathematics both a source of problems and a source of methods. For the purposes of a specialized investigation of problems in the philosophy of mathematics, he created an instrument, his theory of descriptions, and a more general method, that of contextual definition, which his successors took up and applied to many other problem areas. Indeed, it was not merely Russell’s successors who applied these ideas to areas outside philosophy of mathematics, since Russell himself did so in his first publications on the subject. Thus it is not apparent from Russell’s still widely read “On denoting,” published in 1905 and even today a key item on the syllabus of students of analytic philosophy, that the theory of descriptions originated in the course of studies in foundations and philosophy of mathematics. Rather, this is a fact mentioned in Russell’s autobiographical writings and known to historians of twentieth-century philosophy. The degree to which the method of contextual definition, which the theory of descriptions exemplifies, was inspired by the nineteenth-century rigorization of analysis is perhaps not sufficiently appreciated even by such specialists.

A principal puzzle Russell addresses in “On denoting” is that of so-called negative existentials, such as “The king of France does not exist.” In superficial grammatical form this statement resembles “The queen of England does not agree,” and to that extent it appears to involve picking out an object (in this case, a person), and then attributing a property to him (or her, as the case may be). Thus it seems that in order to say that someone or something does not exist, one must assume that in some sense there is such a person or thing, to whom or which the property of nonexistence may be ascribed. Russell cites Alexius Meinong (a student of Husserl’s teacher Franz Brentano) as a philosopher committed to such a view. For Meinong had a theory of “objects beyond being and nonbeing,” exemplified by The Golden Mountain and The Round Square. But as Scott Soames reveals, in his Philosophical Analysis in the Twentieth Century, volume I: The Dawn of Analysis, Russell himself had briefly held a similar view in the first days of his and Moore’s joint rebellion against absolute idealism. It was through the development of his theory of descriptions that Russell was able to free himself from anything like commitment to Meinongian “objects.”

According to that theory, to say that a Golden Mountain exists is to say that there is something that is both golden and a mountain: ∃x(Golden(x)&Mountain(x)). To say that the Golden Mountain exists is to say that there is one thing that is both golden and a mountain and no other such thing:

∃x(Golden(x) & Mountain(x) & ~ ∃y(Golden(y) & Mountain(y) & y ≠ x)).

(Here “~” represents “it is not the case that.”) This is logically equivalent to saying there is something such that a thing is both golden and a mountain if and only if it is identical with that thing:

∃x∀y(Golden(y) & Mountain(y) ↔ y = x).

To say that the Golden Mountain does not exist is simply to deny this:

~∃x∀y(Golden(y) & Mountain(y) ↔ y = x).

To say that the king of France is bald is, similarly, to say that there is something such that a thing is king of France if and only if it is identical with that thing, and that thing is bald:

∃x(∀y(King-of-France(y) ↔ y = x) & Bald(x)).

This is not the place to go into the subtleties of Russell’s theory, whose main point should be clear from these few examples: when the logical form is properly analyzed, using the new logic, the phrase “the Golden Mountain” or “the present king of France” disappears. With it vanishes any appearance that we must acknowledge such an “object” as the Golden Mountain or king of France even in order to deny that any such object exists. The examples illustrate in miniature two lessons: first, that the logical form of a statement may differ significantly from its grammatical form, and that recognition of this difference may be the key to solving or dissolving a philosophical problem; second, that the correct logical analysis of a word or phrase may involve an explanation not of what that word or phrase taken by itself means, but rather of what whole sentences containing the word or phrase mean. Such an explanation is what is meant by a contextual definition: a definition that does not provide an analysis of the word or phrase standing alone, but rather provides an analysis of contexts in which it appears.

Russell’s distinction between grammatical and logical form, and his claim that the former may be systematically misleading, was to prove immensely influential, even among nontechnically oriented philosophers, such as the Oxford ordinary-language school, who saw no need to use special symbols to represent logical forms, and objected to details of Russell’s specific application of the distinction in his theory of descriptions. But Russell’s notion of contextual definition is one implicit already in the practice of Weierstrass and other leaders of the nineteenth-century rigorization of analysis, and familiar to Russell from his undergraduate mathematical studies, so that even the antitechnical ordinary-language school of philosophical analysts are being influenced at one remove (and, so to speak, in spite of themselves) by mathematical analysis.

Contextual definition was the tool the rigorizers used to dispel the mysteries surrounding the notions of infinitesimals and infinities in the calculus. The followers of Leibniz had, for instance, written df(x)/dx for the derivative of a function f(x), wherein dx was supposed to represent an “infinitesimal” change in the argument, and df(x) a corresponding “infinitesimal” change f(x + dx) − f(x) in the value when the argument changes from x to x + dx. (Leibniz claimed that this was all just a figure of speech, but his followers seem to have taken it literally.) These infinitesimals could be treated as nonzero in some circumstances—in particular, one could divide by them, as one cannot divide by zero—and yet treated as zero and neglected in other circumstances. Thus the derivative of the function f(x) = x² was computed as follows:

Here dx is treated as nonzero at the next-to-last step, and zero at the last step—the kind of procedure that outraged critics like Berkeley. In the course of the nineteenth-century rigorization, the infinitesimals were banished: what was provided was not a direct explanation of the meaning of df(x) or dx, taken separately, but rather an explanation of the meaning of contexts containing such expressions, taken as wholes. The apparent form of df(x)/dx as a quotient of infinitesimals df(x) and dx was explained away, the true form being (d/dx)f(x), indicating the application of an operation of differentiation d/dx applied to a function f(x).

Similarly, such an expression as lim_x→0 1/x = ∞, or “the limit of 1/x as x goes to zero is infinity,” was explained as a whole, without requiring any explanation of “∞” or “infinity” taken separately. The details, which now appear in any freshman calculus textbook, need not detain us. What is important historically is that the notion of contextual definition employed in Russell’s theory of descriptions was an idea that would have been familiar to him as a student of mathematics. To acknowledge this is, needless to say, not to deny that there is a certain genius involved in extracting such an idea from its original context of mathematical analysis and employing it to resolve philosophical puzzles. To acknowledge the germs of Russell’s ideas in ideas of Weierstrass is merely to indicate more precisely what kind of genius Russell, like Frege before him, was bringing to bear on philosophical issues: a kind of philosophical genius informed by knowledge of mathematics.

4 Philosophical Analysis and Analytic Philosophy

Anyone who acquires a new tool is in some danger of behaving like the proverbial man with a hammer to whom everything seems to be a nail. There is no denying that some of the first people to apply the new methods of Frege and Russell were overenthusiastic about what such methods could accomplish. Russell himself, having established to his own satisfaction that mathematics could be reduced to pure logic once one had a sufficiently rich and powerful logic, went on to conclude that every science apart from mathematics could be reduced to logical compounds of statements about immediate sensory impressions—“sense data” as they were called. The logical positivists reached a similar conclusion, and were ready to ban any statement that did not admit such a reduction, from the assertions of Hegelian or absolute idealist metaphysicians on, as a “pseudo-statement,” or mere nonsense.

Conscientious attempts to work out just how science, even the parts concerned with theoretical entities not directly observable (such as quarks and black holes in the science of today), could be reduced logically to statements about sense data, or at least to statements about everyday observable objects (such as meter readings), failed. Hence the positivists were forced to acknowledge that their program could not succeed, and (since they did not wish to dismiss large parts of modern science as mere pseudo-statements) that their standards of meaningfulness were too rigid. But as Soames emphasizes, this very acknowledgment of failure was a kind of success, because few if any philosophical schools before the positivists had even stated their aims with sufficient clarity to make it possible to see that they were unachievable. The new logical resources provided by Frege and Russell had both tempted the positivists to conjecture more than they could prove and made it clear to them that proof of their conjecture was impossible.

With experience the scope and limits of the new methods gradually came to be better understood. Russell’s theory of descriptions had been hailed by his student F. P. Ramsey as “a paradigm of philosophical analysis,” which indeed it is. But it came to be appreciated that the kind of application Russell made to the issue of negative existentials, where a philosophical problem was completely dissolved by philosophical analysis, would seldom be possible. Analysis, in general, is only a preliminary, a process that makes it clearer what the real problems are, and not a panacea, exposing all apparent problems as mere pseudo-problems.

As analytic philosophy has developed, enthusiasm has been replaced by dedication: recognition of the limitations of Frege’s and Russell’s methods has led not to the abandonment of the goal of clarity, which was the underlying motive of the great pioneering figures, but rather to firmer adherence to it. Today, when one can read large tracts of philosophy in the analytic tradition without encountering a single explicit analysis, let alone one expressed in special logical symbolism, one still finds almost everywhere a clarity of prose style that instantly distinguishes writing in this tradition from the writings of Continental philosophers (to say nothing of the Continentalizing philosophasters to be found in certain humanities departments in universities in the English-speaking world). This clarity—found, to be sure, already in the mathematician–philosopher Descartes, the first truly modern philosopher, but lost in many of his successors—is the ultimate influence and legacy which the pioneers of analytic philosophy transmitted from mathematics to their philosophical heirs.

Further Reading

I recommend Philosophical Analysis in the Twentieth Century (Princeton, NJ: Princeton University Press, 2003) by Scott Soames for those wishing to read more about this subject. Each of the two volumes of this work contains substantial lists of primary and secondary sources at the end of each of its several parts.