Dewey and Boydstun on Pure Mathematics

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.

9 thoughts on “Dewey and Boydstun on Pure Mathematics

  1. That’s very interesting. I guess I still don’t quite get Dewey’s overall view, but these are good leads to go on.

    Having read your post, though, I have to admit that there is at least an element of pedagogical truth in Dewey’s view of mathematics. I was, as a high school student, a very, very poor student of mathematics. I was good enough at arithmetic, algebra I, and geometry, but for some reason, once I got to algebra II/trigonometry, analytic geometry, and pre-calculus, I started to do very badly indeed. I’ve always wondered why. I may not have a very strong aptitude for math, but I’m not utterly unintelligent, either. Why is it that, faced with mathematics, I suddenly became so unintelligent?

    I’ve encountered a similar phenomenon as an observer or evaluator of college-level instructors. I’ve audited two undergraduate courses in statistics here at Felician, one in the math department and one in psychology, and recently observed a professor of mathematics teaching pre-calculus (the topic that day was graphing polynomials). It seemed a repeat of my high school experience: even students of above-average aptitude seemed unable to perform the simplest mathematical task. Oddly, not having taken pre-calculus since tenth grade, but motivated to learn, pre-calc suddenly seemed puzzlingly easy. Same with statistics, at least at the very basic level taught to Felician undergraduates.

    There are explanations out there for math anxiety, and I’m sure they’re part of the equation, so to speak:

    In my case (back in high school), what I felt was not precisely math anxiety but an overwhelming sense of boredom with math. Math took a substantial degree of intellectual effort and discipline, but (at the time) seemed to have literally zero payoff. The relationship between effort and payoff, or attention-required-of-me and perceived-relevance-to-my-life, seemed very, very unfavorable. Temperamentally, as someone unable to perform an activity simply from the motive of duty, with no payoff beyond that, I would just shut down.

    I don’t know that that quite explains the reactions of my students, but maybe it does, or something close to it does. It strikes me as a more plausible explanation of students’ incompetence at algebra than it does incompetence at statistics, given (what I regard as) the practical importance of statistics. But then again, both my high school belief as well as my students’ belief in the “irrelevance” of mathematics were radically misconceived. I happen to think that their belief is more misconceived than mine was (but then, I would), but however you slice it, my point is that a misconceived or partly misconceived belief in the irrelevance of mathematics might well make it attitudinally difficult to learn, even for students of relatively high aptitude.

    So I wonder whether mathematical pedagogy could use a dose of Dewey. Some students, a small minority, will be attracted to math because they just like doing math. But I suspect that the vast majority see math as a pointless burden. One can’t entirely accommodate an attitude of that sort, but I do think one can and should try to meet it half-way.

    Two reforms come to mind. One is that we probably ought to re-visit and re-vamp the high school mathematics curriculum. It makes no sense (to me) that students learn geometry, analytic geometry, and trigonometry but no statistics or probability theory. I suppose you need some very basic geometry to function in the world, of a kind you’d get from an ordinary arithmetic class. But though I did well in high school geometry, I don’t remember a thing I learned. Meanwhile, students come to college with their minds full of statistical rubbish, eager to major in disciplines like nursing, business, psychology, and criminal justice, but with no idea how to think about statistics or probability. It seems to me that a Deweyan pedagogy would change that–and it needs a change.

    Another reform: though professors of mathematics might hate me for saying this, I think we need to induce them to spend more than pro forma time explaining what mathematics has to do with “the real world,” with practice. They need to impress on students the real practical liabilities of innumeracy in just the way that we’ve succeeded (sort of) in impressing people with the real practical liabilities of illiteracy. Of course, one can’t really do this if the mathematics curriculum ends up being irrelevant, but once re-vamped for relevance, the job will become a little easier.

    I’ve always been depressed at the thought of the math I didn’t learn, and I now find myself frankly alarmed at my students’ belligerently strident, amnesiac innumeracy. They’ll take the basic statistics class here, but a semester later not remember anything about basic data description, probability, normal distribution, standard deviation, confidence intervals, variance, proportion….anything. They’ll then major in subjects like criminal justice with a view toward going into the police academy and into a local police department, where they’ll hear talk (aimed at them) about “structural racism,” while having literally no idea how to read or evaluate a single study on the subject. I somehow think that your buddy JD could have helped out here, and still could.

    Liked by 1 person

    • Thanks for sharing this information. Yes, there is much more to Dewey’s labors in that one chapter of the LOGIC than I’ve mentioned here.

      I’m inclined to agree with you, Irfan, about learning some probability and statistics in high school. I didn’t learn anything of that until I took a course in college, in the Biology department, on human heredity. That Prof had written the text for our class, but he first had us learn probability calculations from a little paperback on it. If I had gone far enough in graduate Physics, I’d have come to a need for statistics in experimental data. Later in life, I took a course in probability and statistics oriented towards Business students. That was really neat. All of the problems in the text were from business situations. That reminds me now of another neat practical mathematics course I took later in life: that was a course called Engineering Economics. That is where you are performing calculations to decide whether this or that well-specified physical outlay should be undertaken under expected utility.

      I don’t think elementary geometry is needed in most peoples’ practical life. Geometry is merely the most holy of holies. Do keep this in your heart: three points determine a plane. Geometry is terribly important in physics, of course. For Newton, Einstein, and more.

      I once used trigonometry to answer a question from Jerry on what length he needed for a certain portion of a drapery he was making for us. I’ve used trig in my life mostly for one of my hammers for pounding the anvil against Rand’s definition of logic as the art of non-contradictory identification. When you are proving trigonometric identities in your trigonometry class, you show that the purported identity holds by reducing it (by elementary algebraic manipulations of the equation expressing the identity) to 1=1. It is purely deductive, and the principle of non-contradiction is not used.

      It has seemed to me that students who like math and do pretty well in it have a considerable economic asset. I mean if you can get training in something that requires competence in some mathematics, such as engineering, it seems likely you will make pretty good money.

      Liked by 1 person

      • Make that “Three non-colinear points determine a plane. Also, you may have noticed the little popsy nutshell philosophy I’ve spread around posting sites for many years now: “The most important thing in the world is love. The most important thing about the world is mathematics. The most important thing upon the world is the human mind.”

        Liked by 1 person

  2. It has seemed to me that students who like math and do pretty well in it have a considerable economic asset. I mean if you can get training in something that requires competence in some mathematics, such as engineering, it seems likely you will make pretty good money.

    Very true–will make you good money, and maximize your chances of getting a good job in the first place. But it’s a very unpopular and under-enrolled college major despite that. And in my experience, it’s never sold to students as an economic asset, even in schools (like mine) that pride themselves on their quasi-vocational and pre-professional orientation. Not entirely sure why.

    This is the Mission Statement of our math department:

    The mathematics curriculum is designed to promote the student’s understanding of mathematical concepts and their interrelations and applications; and provide a symbolic language as a tool for precise reasoning, expression, and computation. Mathematics major graduates will demonstrate the ability to reason critically and logically through problem solving, the ability to communicate effectively through oral and written presentations of solutions, and the ability to apply mathematical knowledge to novel situations. Such abilities lay a foundation for graduate work in mathematics and/or related fields, and prepare the student for employment in a wide range of math-science related fields such as business, computer science, education, insurance, and industry.

    Lines of boilerplate followed by the bland, pro forma recognition that “business, computer science, education, insurance, and industry” are “math-science related fields.” The truth is that, other things being equal, someone with good math skills has a huge competitive advantage in the market over someone who doesn’t. But I guess boilerplate is what accreditation agencies want to see, so that’s what we give them. “Education as Appeasement,” to paraphrase Rand.

    Liked by 1 person

    • Or, rather than resign yourself to appeasement, you could reach out to your Math colleagues and suggest changes regarding specifically something they’ve either “missed” or they simply need to update. I don’t believe it’s Middle States that’s overseeing our departments’ mission statements. That is in our hands….


      • I know better than to do that. If your assumption were true, I would start by throwing all the boilerplate out of my own syllabi, and write them the way I wanted–no Learning Outcomes, no adherence to Bloom’s Taxonomy, etc. But I write my syllabi as I do because I know it’s what Middle States et al want, not because I would write them that way if it were left up to me. And I’m sure the same is true of the Math Dept’s Mission Statement, or any other official curricular statement, whether in our catalog or just about anywhere else. Yes, I know that Middle States claims to adopt a hands-off attitude, but I don’t believe them. We’ve all internalized their demands because ultimately we have no real idea what they want, and yet can’t we afford to give them anything but what they’re looking for. It becomes clear enough over time how to adhere to the demands they’re not making.

        Anyway, changing a Mission Statement is bureaucratically more trouble than it’s worth. The real underlying problem is how to transform the traditional curriculum as taught in an old-fashioned liberal arts college into something more pragmatic, and attuned to the job market. Hard to do, whether in math or in philosophy, but the only way for either discipline to survive.


  3. Highly relevant–focused primarily on the damage done to math education by Common Core:

    Mike Sanford, one of my colleagues in the Math Dept here, points out on Facebook that one of the biggest problems with math education is the failure to connect math with creativity and the imagination. There’s a conception of math as nothing but a set of algorithms to follow, along with the (very unDeweyan) demand for standardized “assessment” and “accountability.”

    That’s something that all of our comments so far have missed, and I’m not sure that Dewey addresses it, either. We tend to think of math as having purely instrumental value, but forget that it can be connected back to creativity and the imagination. Given the current climate of opinion, I’m not sure whether there’s space anywhere to make that connection.

    Liked by 1 person

    • This geometry portion of the common core questions, linked below, is a real dumb-dumb down from what I took in geometry in that very same grade, tenth grade. And what centuries of what students learned from Euclid’s text. Our text was a modernized course developed at MIT, but it was Euclidean geometry. The whole end point is to be able to prove things on your own, having the definitions and postulates and your previously proven theorems. That is creative. That is a reality-power of human mind. The folks at MIT who developed the course were creative to accomplish that. This is synthetic geometry—no coordinates or equations of curves. Did one learn this art of proof required for synthetic geometry? If not, then, were I teaching a course for beginning epistemology/metaphysics, we’d first need see to it that everyone get far enough into this geometry to see what is going on there. Archytas (c. 428-347) determined by purely by synthetic methods (which is all they had) a certain widely sought (for the thrill of it all) mean proportional by a construction calling for the intersection of three surfaces of revolution: a cone, a cylinder, and a tore. That is mathematical creativity at the peaks. Original mathematical discovery is only through creativity (and prior training and subculture). It is, all the same, discovery. The objectivity-pudding is in the proof. It’s not like the poems I write.

      Liked by 2 people

      • I see that now, but then, I’m 49 rather than 14. I didn’t see it in ninth grade, when I learned geometry, and I can’t imagine how I would have seen it. I was actually a good geometry student–I got a B–and enjoyed it somewhat, but it didn’t capture my imagination the way literature did. I have a feeling that I had a sort of anxiety at the time that literature calmed but math exacerbated. Now that I’m older, I can appreciate the creative element in math, but I can’t imagine trying to persuade a math-phobic student of the virtues of math in terms of anything but its instrumental value.

        Not that that works, either. I tell my criminal justice students that in order to be effective police officers, they need mathematical skills:

        1. Police work requires a basic understanding of some academic criminology.
        2. But understanding of academic criminology requires basic competence with statistics.
        3. So police work requires basic competence with statistics.

        They don’t really contest either claim, and can’t quarrel with the logic, but they insist dogmatically that math is irrelevant to their would-be careers in policing.

        True story: I encountered an aspiring police cadet today who told me that academic study as such was irrelevant to policing because most academics are not cops: only cops know what needs to be known about policing, and only cops can judge the activities of cops. So policing is a self-enclosed body of knowledge accessible only to cops, and outside of the ken of non-cops. Cop-knowledge aside, all other knowledge is irrelevant to policing and a waste of time.

        I asked him what he thought about the judicial branch of government, which judges the activities of cops but doesn’t consist of cops. He said that ideally, courts, judges, jurors, etc. wouldn’t exist, but that their existence was a concession to a corrupt world.

        If prosecutors and judges are irrelevant to policing, I imagine that mathematicians are, too. “Let no one who has studied geometry enter here.”

        What are the chances of successfully teaching such a student metaphysics or epistemology via geometry? I leave it as an exercise.

        Liked by 1 person

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