us to draw nonâtrivial conclusions from them and to relate them logically to other theories .... Time to conclude: Mathematics in philosophy can be awesome.
Scientific Philosophy, Mathematical Philosophy, and All of That Hannes Leitgeb LMU Munich Recently, we changed the subtitle of Erkenntnis from “An International Journal of Analytic Philosophy” into “An International Journal of Scientific Philosophy”. In a sense, this was old news: “Scientific philosophy” had been amongst the terms of art that characterized the Logical Empiricist movement—exemplified by the work of Rudolf Carnap and Hans Reichenbach, the founders of the journal—from the start. But it was also good news: “scientific philosophy” expresses the attitude of philosophizing that is associated with the journal more adequately than the less specific “analytic philosophy”. And: I am convinced that one important way of doing philosophy in the future will be scientific. In this sense, having changed Erkenntnis’ subtitle accordingly should serve also as an indication of things to come. Let me explain. What is scientific philosophy? I see at least three different ways of understanding this term in the relevant literature (all of which, and more, have a place in Erkenntnis): First of all, there is a scientific philosophy in the sense of philosophy for the sake of science. This is very close to what the Vienna and Berlin circles had in mind, and especially Carnap: philosophy as an auxiliary discipline that reflects, on the metalevel, science proper with the aim of reforming and improving the language and logic of science wherever appropriate. In the last couple of years, Michael Friedman has taken up this line of reasoning again when he argues that philosophy has a crucial innovating and guiding role to play at those revolutionary stages of science at which one Kuhnian paradigm is being abandoned in favour of another and where philosophy may furnish even radical scientific change with rationality. Then there is philosophy as part of science: Willard van Orman Quine and his naturalized epistemology programme constitute the most famous proponents of scientific philosophy in this sense; more recently, Penelope Maddy, and also James Ladyman and Don Ross have argued for a similar naturalization of parts of philosophy. Philosophers should be working hand in hand with, and on the same object language level as, the scientists themselves. Epistemology, philosophy of mathematics, and metaphysics ought to be extensions of cognitive psychology, set theory, and physics, respectively, so that the former areas address some of the more general and foundational questions that arise in the latter ones. Thirdly, scientific philosophy can be understood as philosophy done (partially) by scientific methods. This is perfectly compatible with philosophy being a discipline of its own right, which possesses its own concepts, questions, problems, and hypotheses; with philosophy not necessarily being pursued, whether on the metalevel or on the object level, with the aim of facilitating scientific progress (though it is nice if this is a by‐product); and with philosophers working on the same traditional topics as their ancient Greek ancestors, such as truth, knowledge, existence, morality, consciousness, and so on, for the sole reason that we can’t help it, perhaps even by “human nature”—we simply have an urge to understand these concepts and what is denoted by them. But still philosophy might be scientific in the sense that amongst the methods by which these philosophical topics are being addressed, one finds some of the methods that are also used in science. It is scientific philosophy in this third sense that many young philosophers are doing these days. And it is also what I predict to become even more important in the future. So let me focus now on scientific philosophy of this last, methodological brand.
What are scientific methods? And which of them are such that we might want to apply them in philosophy? Answering this in general is just as difficult as solving the notorious demarcation problem of “scientific vs. non‐scientific”, but at least it is easy to give some paradigm case instances. Clearly, mathematical methods count as scientific methods, and many of them are used in philosophy: quite often these are logical methods (which may be subsumed under the more general term “mathematical methods”), such as methods from propositional logic, first‐order logic, higher‐order logic, proof theory, model theory, computability theory, set theory, non‐classical logic, modal logic, conditional logic, dynamic logic, possible worlds semantics, and so forth; but also some methods that belong to more mainstream parts of mathematics, such as methods from the calculus, probability theory, graph theory, game theory, category theory, etc., populate scientific philosophy. Following common usage, we may call such applications of mathematical methods to philosophical questions and problems “mathematical philosophy” or “formal philosophy”. For instance, Vincent Hendricks’ and John Symons’ Formal Philosophy volume (Automatic Press, 2005), in which various senior formal philosophers have been interviewed about their work, gives an excellent impression of the great range of philosophical topics that are approached by mathematical means these days. Secondly, computational methods: Paul Thagard’s Computational Philosophy of Science is a good example (MIT Press, 1993), or Branden Fitelson’s and Ed Zalta’s “Steps Towards a Computational Metaphysics” (Journal of Philosophical Logic, 2007), or all the recent applications of computer simulations in formal epistemology, formal ethics, formal social philosophy, and the like (for an instance, see Igor Douven’s “Simulating Peer Disagreements“, Studies in History and Philosophy of Science, 2010). Thirdly, experimental methods: whether they can be applied qua philosophical method is hotly debated these days, e.g., in a recent exchange in the New York Times (cf. http://opinionator.blogs.nytimes.com/2010/09/07/experimental‐philosophy/). Sometimes these methods mutually presuppose or strengthen each other: computation usually presupposes formalization; and empirically operationalizing and testing people’s inferences on the acceptability of conditionals (more about which below) is greatly helped by formal accounts (see, e.g., Niki Pfeifer and Gernot Kleiter on “The Conditional in Mental Probability Logic”, in Oaksford and Chater, eds., Cognition and Conditionals: Probability and Logic in Human Thought, Oxford University Press, 2010). I should add that there are many further kinds of scientific methods, and some of them might prove important for philosophy in the future, too. Presumably, mathematical and computational philosophy should not be quite as difficult to swallow for philosophical traditionalists as experimental philosophy, at least in so far as proofs and computations yield, in principle, a priori insight, whereas experimental results do not. And mathematical philosophy isn’t exactly the new kid in town either: what else than mathematical philosophy is to be found in Aristotle on logic, Leibniz on metaphysics, and Carnap and Reichenbach on philosophy of science? But that does not mean that mathematical philosophy isn’t still a no‐go for some of our colleagues. Why is that? For what reasons do some philosophers still deny that mathematical methods can play an important role in philosophy? Here is one (rare case of explicit) argument: Kant in the Critique of Pure Reason. According to Kant's Transcendental Doctrine of Method, philosophy cannot be developed along the lines of the definitions‐axioms‐proofs scheme that is known from mathematics, and this is for the following reason: mathematics is based on pure intuition, while
philosophy is not. For instance, when a geometrical concept, such as triangle, is defined in Euclidean geometry, it is always supplied with a corresponding construction method, such as a construction algorithm for triangles in general or maybe for general triangles. Such constructions presuppose our geometrical intuition of the Euclidean plane in order to be applicable at all. Accordingly, the axioms of geometry are justified in virtue of these geometrical intuitions, and proofs in Euclidean geometry typically involve geometrical constructions by which some geometrical objects are constructed from others in order to justify to ourselves that a particular theorem holds in the Euclidean plane. None of this can be done in philosophy, or so Kant argues, simply because our abstract philosophical concepts do not exhibit the same kind of intuitive content. Hence, philosophy cannot be done—not even in parts—in the style of mathematics. In contrast, physics can be done—in parts—mathematically, because some physical concepts do have, e.g., geometrical content. Considering the stage to which mathematics had developed by Kant’s time, this is not a bad argument at all: for mathematics did seem to rely on geometrical intuition (and possibly other forms of intuition) just as Kant had been claiming. From our modern point of view, however, Kant’s argument has been defeated: due to the arithmetization of analysis in the 19th century and the emergence of set theory as a quasi‐logical foundation for mathematics at the beginning of the 20th century, we know now that mathematics does not actually presuppose intuition in the way required by Kant’s argument. (Even though I doubt that intuition can be expelled from mathematics completely.) In the meantime, mathematics has developed into a theory of abstract structures in general, and the progress in modern logic shows that even the “space of concepts and propositions” itself has an intricate mathematical structure. So this is one argument against mathematical philosophy gone with the wind. More importantly, in the meantime, philosophers have accumulated a great lot of positive evidence for the thesis that applications of mathematical methods in philosophy can be very useful. I won’t argue for this thesis here—just take a look at the numerous entries in the Stanford Encyclopedia of Philosophy in which such applications of formal methods are being mentioned. Or consider all the sophisticated research done at the various centers for formal epistemology, mathematical philosophy, and the like, which have emerged in recent years. Or check out the great number of excellent papers that are being presented at the corresponding workshops and conferences. And so on. This shows quite conclusively that any argument to the contrary thesis must be unsound. However, usually, those of our colleagues who worry about the mathematization of (parts of) philosophy do not so much put forward arguments but really express a feeling of uneasiness or insecurity vis‐à‐vis mathematical philosophy. Maybe this is but an indication of a misleading view of mathematics? (Like: “Mathematics is about calculating numbers”, which is false—mathematics is concerned with giving proofs and studying structure as such.) Bad maths teachers in school? A little bit of laziness about learning the bits of mathematics that would be needed to appreciate paradigm cases of mathematical philosophy? Whatever the reason, such feelings are unfounded: mathematics can be very useful in philosophy. In fact, I want to claim more: Sometimes the application of mathematical is even required in order to make philosophical progress. Why is that? Let me explain this in terms of one important type of philosophical endeavour (among many others): the explication of philosophical concepts. That is: the construction of a new concept C’ which replaces a given, often pre‐theoretic concept C of philosophical interest, so that the extension of C’ coincides with that of C in the clear‐cut and uncontroversial
cases, but where C’ is permitted to, and indeed ought to, improve upon C in terms of exactness, fruitfulness, and simplicity in all the unclear or fuzzy cases. (Carnap in the introductory part of his Logical Foundations of Probability explains this in great detail. He also thinks that explications must be to the benefit of science—but as explained before, we might just as well be satisfied if they are to the benefit of philosophy.) Here are three paradigm case examples from the 1930s, 1950s, and 1970s, respectively: Alfred Tarski’s explication of truth, Carnap’s own explication of confirmation of hypotheses by evidence, and Ernest Adams’ explication of the acceptability of conditionals. Each of these explications has been extremely successful: today no philosopher could reasonably develop a theory of truth for sentences or propositions, or a theory of confirmation in science, or a theory of the subjective acceptability of if‐then sentences in natural language, without comparing their theories with these classical explications. Which does not mean that they could not be superseded by even better ones: in fact, precisely that has happened, as witnessed by Kripke’s theory of truth and the modern literature on formal theories of truth that emerged from it, contemporary work on the plurality of confirmation measures and their virtues and shortcomings (as by Branden Fitelson), and the suppositional theory of conditionals in its fully developed form (as defended by Dorothy Edgington). But what is more important for my purposes here: none of these classical explications, nor their revisions and improvements, would have been possible without the application of formal methods. Tarski’s explication needed second‐order logic or set theory, Carnap’s relied on the theory of logical or (in its modern guise) subjective probability, and Adams’ explication required subjective probabilities again (even though Isaac Levi, Peter Gärdenfors, and others have developed a qualitative theory of the acceptability of conditionals in which the formal theory of belief revision replaces probability theory). And this is no coincidence: for the desiderata of exactness and fruitfulness will always “pull” explication towards the application of mathematical methods. In many cases only the language of mathematics will allow philosophers to make their explicata more precise (“exactness”) than the corresponding explicanda, and very often only their mathematical background theories will supply explications with the right deductive structure to enable us to draw non‐trivial conclusions from them and to relate them logically to other theories in science or philosophy (“fruitfulness”). For instance: do conditionals express propositions and have truth conditions? Let us assume Adams’ account of measuring the degree of acceptability of “if A then B” in terms of the subjective conditional probability of B given A: then it is only due to the exactness and deductive proper of probability theory that David Lewis was able to prove that the answer to our question must be “no!” (given some plausible auxiliary assumptions—see Lewis, “Probabilities of Conditionals and Conditional Probabilities”, Philosophical Review, 1976). As Dorothy Edgington once said, this was a bombshell—hardly anyone would have guessed. E.g., Robert Stalnaker had assumed precisely the contrary in some of his (excellent!) papers prior to Lewis’ theorem. (Of course, some might take this to be an argument against Adam’s explication: but that’s just the usual modus ponens vs. modus tollens issue.) So we find that there are philosophical endeavours, such as the explication of philosophical concepts, which typically require mathematics. I haste to emphasize that not every philosophical paper deals with the explication of a concept, of course. But then again almost every philosophical paper includes arguments. In many cases, these arguments will be such that philosophers simply “see” that the argument’s premises
cannot be true without its conclusion being true as well. But, in some cases, the argument from some given philosophical premises to the intended philosophical conclusion might be really complex; the deductive “distance”, as it were, between premises and conclusion might become so large that we crave for a helping hand: then, just as in science, mathematical proof might be needed again in order to bridge the gap, even when the argument in question is not based on any explication. Just think of Frege’s derivation of arithmetic from Hume’s Principle and second‐order logic, on which the whole program of modern Neo‐Logicism about mathematics is based: there is no way to just see that arithmetic follows from all of that. Or, for some philosophical purposes, we aim to give an argument that is inductively strong rather than necessarily truth‐preserving: then we need to show that the premises make the conclusion at least likely. How can we do so? One salient way of pulling this off is to build a formal toy model that captures prototypical features of the problem in question, in which additionally the premises are clearly satisfied, and in which we can then, hopefully, also verify the conclusion. For instance, Luc Bovens’ and Stephan Hartmann’s Bayesian Epistemology (Oxford University Press, 2003) includes lots of illustrative examples of so‐called Bayesian networks that are used as such “prototypical” models by which more general claims about reliability, confirmation, underdetermination, testimony and the like can be supported. These are models in the sense of scientists, not of logicians; and they are not explications in the Carnapian sense either: for they are not models of concepts; and, by idealization, they distort reality (though in a good way), where explicata are supposed to be the new and better conceptual reality. Either way, mathematical methods supply us with ways of extracting information from philosophical assumptions and converting this information into support for philosophical theses, which in the case of complex arguments might be the only way of building feasible bridges between assumptions and theses. Obviously, mathematics won’t create the philosophical content, but it might be necessary for transforming the philosophical content. (Much as mathematics does for physics when it converts empirical law hypotheses and initial conditions into empirical predictions.) Other than enabling us to give explications of philosophical concepts and to construct and validate complex philosophical arguments, mathematical theorizing in philosophy comes with many additional benefits: it forces us to put our cards on the table, that is, to make tacit presuppositions explicit; it helps us to separate the essential from the accidental by making transparent what exactly is needed to make an argument go through; if two areas of philosophy share enough mathematical structure, it may allow us to translate arguments in the one area into arguments in the other; it functions as a means by which we can put some of our “intuitions” to the test and correct our epistemic biases (as highlighted and discussed by Catarina Dutilh Novaes in her forthcoming monograph on Formal Languages from a Cognitive Perspective, Cambridge University Press); it facilitates the illustration of abstract circumstances by means of diagrams (think of the illustrative power of possible worlds semantics concerning abstract concepts such as necessity or knowledge); it allows us to compare two philosophical theories in terms of the aesthetic appeal of their formal structures and to use this as a (highly tentative and defeasible) method of theory choice; it forges unexpected connections to scientific areas in which mathematical methods are accepted as a standard anyway; and if two ways of making a philosophical concept or thesis mathematically precise lead to the same philosophical conclusion, then this adds to the robustness of that conclusion. About the last two points: A while ago when I counted how often Lewis’ original paper on his impossibility result concerning conditionals and conditional probability was cited in
computer science journals, I counted not less than 78 citations. And even after replacing probability theory by belief revision theory, Gärdenfors was still able to prove that conditionals could not express propositions—just as Lewis had done before by probabilistic means—once the corresponding qualitative version of Adams’ theory of acceptability of conditionals was in place (see Gärdenfors, “Belief Revision and the Ramsey test for conditionals”, Philosophical Review, 1986). Time to conclude: Mathematics in philosophy can be awesome. Scientific philosophy can be awesome. And we are still at the beginning—the best is yet to come. This being said, some qualifications are due. Let me explain them for the case of mathematical philosophy again: Everyone who works as a mathematical philosopher knows some papers in which philosophers hide behind the symbols—where mathematical clothing is meant to conceal lack of philosophical content. Or where mathematics does not do much but to complicate some states of affairs that could, and should, have been described in more elementary terms. Or where a mathematical method is applied blindly without any awareness of its potential limitations. In other words: just as all others tools, also mathematical methods can be abused. But clearly that should not stop us from putting them to good use in philosophy. Accordingly, there are philosophical tasks for which mathematical philosophy does not pay off, or where it does not pay off as yet. Just as all other tools, also mathematical methods cannot be useful for each and everything. But one should not be too quick either in thinking that a particular area of philosophy would be beyond mathematical methods in principle—it might just need time until an area develops to the stage at which mathematical methods become applicable; when its basic concepts have stabilized and matured enough to be ready for explication, or when its arguments have become complex enough to cry out for the helping hands of logic and mathematics. Finally, as with the application of tools more generally, also the application of mathematical methods in philosophy needs training: in the traditional philosophy curricula, logic courses went some way towards achieving this. In the future, a combination of education in logic, a little bit of discrete mathematics, the fundamentals of probability theory, and a review of some examples of mathematical philosophy in epistemology, philosophy of science, metaphysics, philosophy of language, philosophy of mathematics, ethics, and so forth should suffice. I do recommend scientific philosophy to new graduate students as a way of doing philosophy that has a lot of potential. By this I do not want to say that scientific philosophy should be the only way of doing philosophy. Erkenntnis does not just publish articles in scientific philosophy either! As long as some minimal criteria are satisfied that ought to be characteristic of all kinds of academic research—to speak as clearly as possible, to defend some theses, to put forward arguments, to give concrete examples, to systematize one’s thoughts, to compare one’s view with rival ones, to stay critical of one’s theory, and so on—let many flowers bloom, also and especially, in philosophy. It is just that scientific philosophy is a particularly beautiful flower; and it is blossoming.