R. Hirsch. Mathematics - the pure science? In Science and Society Spring 1996 issue, Vol. 60, No. 1, pp58 -79. http://www.cs.ucl.ac.uk/staff/R.Hirsch/R.Hirsch/papers/maths/maths.html Warning: the PDF file is defective.
A quarter century ago I thought about the philosophy of mathematics, but I have not had the time to pursue it over the past two decades. Hirsch says, as I thought way back when, that mathematics presents a special problem for materialists, seeing that its domain consists of purely abstract objects, not physical objects and processes. Reviewing the problems encountered in the foundations of mathematics, Hirsch adumbrates two fundamental positions--Platonism and intuitionism--and criticizes them both. As an alternative, he turns to "the Marxist approach". _THE_ Marxist approach? "Here mathematics is not viewed as a purely objective reflection of the world nor as a purely subjective human construction but it is produced by a dialectic interaction between the material world and human consciousness: mathematics is based on human _experience_." I'm pretty much all right with this up until the last word--"experience" is not a clear term. But we shall see what develops.
The passing reference to Trotsky immediately following is not encouraging. The shortcomings of Engels in this are then summarized, but then what remains valid in Engels is highlighted. Mathematics comes from the material world and the ability to abstract from it. Mathematics doesn't come necessarily directly from nature, but also from itself. One could say, though Hirsch doesn't put it in these words, that mathematics is the abstraction of the process of abstraction. But ultimately, like all ideas, mathematics arises from experience, says Hirsch. I don't exactly disagree, but this term "experience" requires a precise delineation.
The production of conceptual objects at the highest level of abstraction and systematization is analysis. But then follows the necessity of synthesis, re-inserting particular events back into the totality, and accounting for becoming, process, change. "The abstraction of this wider view is _dialectics_." With some nitpicking, I can go along with this. Naturally, as a mathematician, Hirsch does my nitpicking for me, emphasizing that mathematics (and by implication logic, though he doesn't say this), is not limited to a static view of the world, as boilerplate diamat propaganda would have it. But then, he sats that there is an inherent temptation to stasis when one reduces everything to fixed formulas, which then prove to be inadequate to changing realities. While this doesn't thrill me, Hirsch then says: "Dialectics provides the philosophical framework for reasoning about change and interaction and the process of synthesis." This I agree with.
Hirsch then addresses the question: is mathematics and experimental subject? This involves some complications. Hirsch discusses the development of science in practice and the relation between the development of theories and testing them. "Indeed in the Marxist view the whole notion of objective truth can be defined as the correspondence between theory and practice." I don't think this sentence is well-formulated.
Then, of course, we pass to the 'problem' of abstraction. "At its most abstract level, mathematics deals with objects and structures which cannot correspond directly with any real practice." (I'll add that Engels acknowledged this at times, but the Stalinists and Maoists screwed with mathematicians in unacceptable ways.) How would the various philosophies of mathematics handle the continuum hypothesis? Hirsch acknowledges the difficulties of a materialist conception, and even more so with the question of practical utility, but then argues that the bulk of mathematics, if not its most abstract problems, does indeed intersect with the real world or we never would have gotten anywhere with it. The issue to address is how mathematics developed.
Hirsch provides a historical materialist sketch of the conditions under which mathematics developed and the practical problems stimulating its development. He adds that coping with its own internal logical contradictions also spurs its development. Finally, he references Lakatos viz. a dialectical approach to the development of mathematical proof.
I have no problem with any of this, and many of the purest mathematicians would acknowledge this. It's been a quarter century since my encounter with Saunders MacLane, but I remember a diagram he drew on a blackboard pertaining to the development of mathematics out of certain activities. Not a big deal. But note that Hirsch's hesitations in pinning down his subject means that he shifts back and forth between the domains (though this is a crude way to put it) of the historical materialist perspective--or how ideas get to be produced--with the ontological perspective--or how an assemblage of entities fit into the world picture. A truly dialectical world picture of mathematics has to account for the ontological status of its entities, perhaps constructed out of 'experience' in the broadest and vaguest sense of the term, and situated somewhere between a platonist and a subjectivist world-picture. There are other dialectical views out there of conceptual objects if not strictly mathematics ones, such as Markovic's dialectical theory of meaning.
Hirsch concludes that formal logic and dialectics complement one another. While in some sense this may be correct, Hirsch reverts to the usual characterizations of both and their relation to one another. This is rather unfortunate. I would hope philosophy of mathematics both mainstream and marxist would have evolved beyond this by now. Perhaps it has, but here I see no thinking beyond my encounters of a quarter century past, and then there were indications of philosophical developments in the works, such as F. William Lawvere in category theory. One problem with Hirsch, that in trying to be both politically "relevant" and do justice to his field, he demonstrates the social inertia which inhibits people in his position, i.e. the philosophical underdevelopment of both the mainstream bourgeois and marxist milieux.
So the object lesson of this piece and Hirsch's other essay is not entirely a constructive one; it is a historical one. And, abstractly, another ironically dialectical lesson in how Marxism both does and does not apply to various sciences. The lesson is not entirely negative, i.e. that Marxists should keep their ignorant hands off science, math, and logic, and not entirely positive, i.e. that they have done a creditable job in handling these matters. Rather, the difficulties in coming to terms with these subjects are also important social facts, and not just because of the retarding influences of Stalinism and Trotskyism, but because the ways in which we make sense of our world are integral to the marxist world picture, even though 'marxism' is not competent to pass specific judgment on all matters.
_______________________________________________ Marxism-Thaxis mailing list [email protected] To change your options or unsubscribe go to: http://lists.econ.utah.edu/mailman/listinfo/marxism-thaxis
