In my paper on Dewey’s 1915 book on German philosophy and WWI, I had quoted a general epistemological viewpoint maintained by Dewey: There are in truth “no such things as pure ideas or pure reason. Every living thought represents a gesture made toward the world, an attitude taken to some practical situation in which we are implicated.”
Irfan questioned whether I thought that correct when it comes to mathematics.
“Maybe it’s true of some parts of mathematics, but is it true of all of mathematics? Do professors of mathematics, or even college math majors, go into mathematics because it represents ‘an attitude taken to some practical situation in which we are implicated’”? Irfan inclined to think Dewey’s general position either implausible or as involving a very odd conception of “practical situation in which we are implicated.” He rather thought that math-folk got on with it due to an enjoyment of math-thought and perhaps, contra Dewey, a desire to escape from practical concerns. In any event, “it’s hard to make out what Dewey is trying to say.”
In a 2012 issue of the Journal of the International Society of the History of Philosophy of Science, there is a fine paper on “John Dewey’s Logic of Science,” authored by Matthew Brown. In a footnote, Brown writes that Dewey’s attempt to apply his philosophy of science to theory of formal disciplines like mathematics is somewhat obscure and difficult. Yes, although digging into it, I now think Dewey was onto a pretty good vein concerning mathematics and that he could get by with a little help from this friend (me).
By the lights of his 1938 Logic: The Theory of Inquiry, Dewey would affirm the kind of interest Irfan mentioned that mathematicians have in their discipline. Dewey would say that attractions to the formal disciplines, like attractions to the sciences, are attractions to varieties of rational inquiry. All of these disciplines are inquiries that have continually discovered improved methods for success of the discipline.
Dewey jettisons the old talk and thought of the “a priori” in logic and mathematics and mathematical logic. He applies the replacement “warranted assertability” in formal disciplines, as in science. This pattern of replacement is followed by Philip Kitcher in his book The Nature of Mathematical Knowledge. Like Dewey, Kitcher takes up the challenge of showing how pure mathematics is built ultimately from our perceptual experience of the world. (Kitcher does not talk of assertability, only warrant.)
Kitcher does not burden himself with the further Dewey burden of showing that pure mathematics consists of “attitudes taken in some practical situation in which we are implicated.” However, advances in empirical sciences do strongly tend to beget new technology that does bear on practical situations, at least from the nineteenth century forward. It is not the (usually unforeseen) technology, of course, that usually drives the affiliated mathematical advance, but the science driver. So at one remove from practical life, we can acknowledge the “practical” driver of advances in pure mathematics that is mathematical utility in scientific theory and in scientific empirical techniques. That driver has been enormous, but as Irfan intimated, mathematical advances have also been attained without that empirical motivation. However much we would decline Dewey’s extension of “the practical,” there remains the question related to the practicality one, the enduring question for Dewey as for every philosopher musing on mathematics: How does the pursuit straightforwardly of new pure mathematics fit into a general account of inquiry?
Dewey takes axioms in mathematics to be postulates “neither true nor false in themselves,” but as having “their meaning determined by the consequences that follow from their implictory relations to one another” and as being worthy provided merely “they be rigorously fruitful of implied consequences” (18).
At our home, I’m comfortable making measurements to do carpentry, but my husband Walter is especially suited to keeping the financial balance sheets. Dewey takes the former to be “performed upon existential conditions; the latter upon symbols. But the symbols in the latter case stand for possible final existential conditions while the conclusion when it is stated in symbols, is a pre-condition of further operations that deal with existences” (22). Dewey would liken logic and mathematics to Walter-work.
Dewey argues that the postulates of mathematics and logic “are not arbitrary or mere linguistic conventions. They must be such as control the determination and arrangement of subject-matter with respect to achieving enduringly stable beliefs” (23). Kin of Peirce. Dewey maintains a stiff autonomy of logic, it is inquiry into inquiry, and it “does not depend upon anything extraneous to inquiry” (28). Therefore, logic does not rest on a priori intuition of first principles, on metaphysical or epistemological presuppositions, or on psychological principles.
Dewey recognized that the theory of his Logic “has a twofold task of doing justice to the formal character of the certification of mathematical propositions and of showing not merely the consistency of this formal character with the comprehensive pattern of inquiry, but also that mathematical subject-matter is an outcome of intrinsic developments within that pattern” (391). Mathematical discourse “is conducted exclusively with reference to satisfaction of its own logical conditions . . . the subject-matter is not only non-existential in immediate reference but is itself formal on the ground of freedom from existential reference of even the most indirect, delayed and ulterior kind” (393).
Let me help out my buddy JD. The history of mathematics shows development of pure mathematics out of practical mathematics. That development is by certain methods, themselves evolving a bit up to the last century. I say that pure mathematics lifted off elementary practical mathematics goes on to preserve truth from a certain field of formality present in the practical mathematics. And more, the further creative and logical development of pure mathematics captures further truth of that field of formal structure. Then the further developments can turn out to have application, even very direct ones, to new empirical phenomena. There is just one main problem. What is that certain field of formality peculiar to mathematics? And that invites the question What is the way in which mathematics and logic stand to each other? Developments in pure mathematics are not merely, entirely logical deduction of formal structure residing in ancient, elementary practical mathematics. Developments in pure mathematics require new elements taken from a reservoir of what pure mathematics is. Working mathematicians may proceed with an intuitive grasp of what that reservoir and formal field is. Philosophers have offered definitions of mathematics as the science (organized discipline) of quantity and structure. What sort of formal structure together with quantity carves the field of pure mathematics to this point of its development? In my metaphysics, I’m trying out my category N in the link below. Pretty sure it’s just right. With a little more life, I’ll let you know for sure.