Analysis and Mathematics

Analysis and Mathematics

Mathematics is an analytical discipline (a branch of it is even called Analysis). It analyzes shapes (geometry), numbers (arithmetic), populations (statistics), motion (calculus), logical relations (mathematical logic), chance (probability theory), games (game theory). The OED gives us two definitions of “analysis”: “a detailed examination of something in order to explain or interpret it” and “the process of separating something into its constituent elements”. Putting these together, we could say that analysis is the detailed examination of something in order to separate it into its constituent elements. Mathematics does this in a rigorous and precise way: it breaks entities into their parts and specifies the relations that constitute wholes. It isn’t interested in the meaning of these entities in human life, or their psychological significance, or their poetic resonance, or their historical influence; it wants to know their constituent structure, pure and simple. It takes an abstract domain and dissects it, probes it, disassembles it. It looks for primitive entities and relations and investigates how they combine to produce more complex entities. A mathematics text can always be called A Mathematical Analysis of X. But is all analysis mathematical analysis? That is by no means clear; it depends on our notion of analysis and our interpretation of “mathematical”. Chemical analysis is clearly mathematical: it uses numbers and geometric shapes, and it employs rigorous methods (the OED defines “mathematical” as “(of a proof or analysis) rigorously precise”). Economic analysis is typically quantitative and mathematically formulated (e.g., the laws of supply and demand). Anatomical analysis specifies bodily parts in numerical and geometric terms (size and shape). But what about psychoanalysis or textual analysis or conceptual analysis—are these mathematical?

It might be said that psychoanalysis is mathematical in that it divides the psyche into precisely three parts—the id, ego, and superego. It also specifies quantitative laws of motivation expressed hydraulically. It aims for rigor and precision (it tries to be mathematical). The mathematics is crude and simple, but it is still mathematical in form—it resembles a typical scientific theory. Psychoanalysis is a bit like evolutionary science in this way: it too is concerned with quantitative structural relations—the probability of survival given certain genetic and environmental conditions. Some biology is aptly described as mathematical biology. Textual analysis can certainly be mathematical, even if not all is—word frequencies, inscriptional forms, etc. Astronomy also analyzes the physical universe mathematically, though some is not mathematically formulated. Linguistics is partly mathematical, though not wholly. Logic is certainly analytical and has attained the status of a mathematical discipline. Genetics is pretty mathematical, as is psychology. The concept of the mathematical is broad and inclusive; it doesn’t need to approximate the equations of advanced theoretical physics. The human mind is quite mathematical, thinking habitually in terms of quantity and number—counting, measuring, computing. It is perfectly possible to be mathematical about the arrangement of your room or the structure of a piece of music or the shapes in a painting. We shouldn’t overintellectualize mathematics or equate it with what we learn in school in a mathematics class. Other animals have rudiments of it, as do pre-school children. Mathematical competence is arguably innate (remember the Meno). Mathematical thought is more pervasive than might be supposed. We think in terms of sets and their relations. We recognize the magnitude of multiplicities. We add, subtract, divide, and count. We symbolize and generalize, abstract and mathematize. We have mathematical minds, and we like to impose mathematical methods on reality.

I say all this in order to prepare the ground for a somewhat startling thesis: conceptual analysis is mathematical analysis. Analytic philosophy is mathematical philosophy. In analyzing concepts, we mathematize them, as we have so much else. We can approach the matter both historically and intrinsically. Historically, we have the figures of Frege, Russell, Wittgenstein, Carnap, Tarski, Quine, Kripke, Montague, and many others—all mathematicians by training and eager to apply its methods and attitudes to philosophy. Russell wrote Introduction to Mathematical Philosophy and Montague wrote Formal Philosophy. The shape of so-called analytical philosophy was certainly influenced by mathematics, modelled on it, inspired by it. Tarski’s theory of truth is a prime example. Carnap’s Aufbau is in this tradition. These philosophers wanted philosophy to be rigorous and precise, and mathematics was their paradigm. This conception of philosophy contrasts with other conceptions prevalent in the intellectual tradition: historical and cultural (Vico and Collingwood), psychological (Locke and Hume), religious (Aquinas and many others), linguistic (Austin and later Wittgenstein). It is ahistorical, anti-psychologistic, secular, and contemptuous of ordinary speech. It likes proof not poetry, dissection not description, abstraction not particularity. It is Pythagorean not Hegelian or phenomenological. It is about logic not life, entailment not lived experience. It prefers science to stories. It deals with the form of thought not the dynamics of action (the “deed”). It is formalistic not humanistic. It prefers rigor to uplift. It reveres Plato more than Aristotle, and has no time for Nietzsche. It is pure not applied. It is anti-biological. It is above politics. It is contemplative not practical. It deals in necessities not contingencies, essences not empirical facts. It is more rationalist than empiricist (Russell’s first book was on Leibniz). Thus, analytical philosophy is mathematically oriented, mathematically imbued; it grows from mathematics.

Viewed intrinsically, we can note that conceptual analysis is what the name implies—analytical; it breaks concepts into their more primitive elements. It reduces unity to multiplicity. This multiplicity has a number—the number of necessary conditions that constitute the concept. Thus, knowledge has three necessary conditions, according to the classical analysis—belief, truth, and justification. These conditions are recognized in their numerical dimension: this is a three-component concept. Each component is added to the others to produce the concept being analyzed. We could represent the analysis in the form of an equation: knowledge equals belief plus truth plus justification. This appeals to our mathematically inclined mind. We could say the same about the causal analysis of perception, Grice’s analysis of speaker meaning, Suit’s analysis of the concept of a game, or any other attempt at conceptual analysis. We could divide concepts into three-component concepts, two-component concepts (a bachelor is an unmarried man), and one-component concepts (primitive concepts like the good, according to Moore). That is, we could dwell on the numerical attributes of concepts. Concepts have a mathematics, according to the tenets of analytical philosophy, construed as methodologically mathematical. We already do this with regard to other points of conceptual analysis: we speak of one-place, two-place, and three-place predicates; we distinguish first- and second-order quantifiers; we call existence a second-level concept; we talk regularly of the finite and the infinite. Of course, we have a philosophy ofmathematics, as well as a generally mathematical philosophy. We are well aware that concepts have mathematical properties. Above all, we approach the task of analytical philosophy in a mathematical spirit—rigorously, precisely, as mathematically as we can. Some analytical philosophers even go so far as to apply mathematical model theory to philosophical issues (e.g., Montague). Such philosophers want philosophy to approximate to a mathematical science, and they are not wrong to be so motivated, because conceptual analysis fits the paradigm. This is why Russell’s theory of descriptions was so applauded by philosophers in the analytical tradition: it provides a mathematical analysis of the definite article (symbols, quantifiers, formality). Similarly, for Tarski’s theory of truth and Davidson’s attempt to convert it into a theory of meaning. If only we could employ calculus to resolve the problems of space and time, or the statistics of neurons to solve the mind-body problem! It is an entrancing vision. It beats windy pronouncements about the course of history, or the problematics of the text, or the puny joys of “experimental philosophy”. Mathematical philosophy sounds like the kind of philosophy a serious student might get behind. Okay, it hasn’t quite lived up to the hype, but its heart is in the right place—some distance from the actual heart (unless you are Plato or Pythagoras). It carries better credentials than merely descriptive ordinary language philosophy, which threatens to turn into a kind of localized anthropology. In any case, the analysis of concepts looks like a bona fide case of mathematical analysis: it is precise, abstract, systematic, and susceptible of proof.

How does the mathematical method arise in the human mind? We don’t know, but here is a promising suggestion: it arises from the language faculty. For the language faculty is itself mathematically structured: it consists of a finite array of primitive elements (words) that combine in a rule-governed way to generate an infinity of linguistic strings (sentences). We might plausibly regard the mathematical faculty as wholly or partially derived from the language faculty, arising at some point in evolutionary history. Without going into the matter further, we can entertain the hypothesis that our capacity to conduct conceptual analysis ultimately derives from our capacity for language; crudely, conceptual structure and composition mirror linguistic structure and composition, as seen through the lens of mathematics. The analytic-mathematical turn in philosophy owes its origin to the linguistic-mathematical turn in evolution: the human brain went linguistic and mathematical long ago and then much later it occurred to people to investigate the concepts involved in philosophical problems mathematically. Mathematics grew out of language millennia ago, developed in the course of human history, and then suggested itself as the royal road to philosophical enlightenment. According to this hypothesis, creatures without a language faculty will not develop a mathematical faculty, and will therefore not develop what we call analytical philosophy. This is a new type of “linguistic turn” in philosophy, happening perhaps millions of years ago not in the middle part of the twentieth century. First, we have the mathematical turn in philosophy that did happen in the twentieth century, but this turn depended on the prior existence of mathematics in the human mind, going back into deep evolutionary history, some time after the appearance of language in humans. From language to mathematics to analytical philosophy—the long curve of human intellectual history.

Mathematical-analytical philosophy was conceived as exclusionary—we should do that kind of philosophy and not the other kinds of philosophy. But this is not compulsory: we can let the mathematical kind of philosophy roam over its domain of operations but we can also make room for the historical, psychological, anthropological, spiritual types of philosophy. We can have existentialist philosophy as well as essentialist philosophy—the human condition and necessary and sufficient conditions. The humanistic and the formalistic. For example, we can philosophize about the role of games in human life as well as analyze the concept of a game in quasi-mathematical style. Collingwood can coexist with Russell, Sartre with Carnap. Doing analysis does not preclude investigating Dasein. We can talk about the structure of our concepts and the meaning of life. We can have analytical philosophy and the other kind (it has no accepted name)—historical, hermeneutical, humanistic, humid. We can have the dry and the moist, the desiccated and the succulent. By all means let’s take the gustatory turn! We can be both mathematical and musical (bacchanalian even). Punk philosophy is not out of bounds (it belongs with existentialist philosophy). We can talk about money and sexual perversion as well as knowledge and the form of the good. You could happily describe yourself as an analytical-humanistic philosopher. What you don’t want to be is a bullshit philosopher (either pedantically so or pretentiously so). There is more than one way of being good at philosophy.[1]

[1] Unfortunately, few people are good at both types of philosophy: if you are good at one, you tend not to be good at the other. Practically, I would encourage graduate schools in philosophy to educate students in both traditions, while allowing that students will generally be better at one sort of philosophy than the other. There is, however, no excuse for the dismal prose style of the typical product of an American philosophy graduate program. Someone should really do something about this.