The Natural History of Constructivism

Kazimir Malevich, from http://happyfamousartists.com/blog/2014/07/malevich-revolutionary-of-russian-art-tate-modern/

On the Genealogy of Formal Thought

In my post Constructive Moments within Non-Constructive Thought I made a speculation about the origins and development of constructivism. In that post I wrote:

“…without having made any special commitment to a constructive methodology, with the consequent focus on how a proposition is proved rather than that a proposition is proved, one might, through repeated efforts to prove a difficult theorem, have tried every imaginable avenue in pursuit of the proof, and, after much effort and much experience of the formal conceptual space in which the theorem is located, have acquired a constructive, step-by-step, bottom-up view of the theorem and its proof. This is what I will call a constructive moment within non-constructive thought. Were it not for these constructive moments within non-constructive thought, the classical tradition might have wandered even farther from constructivism. Indeed, the argument could be made that constructivism has its origins in the constructive moments within non-constructive thought.”

Later, thinking more about this, I realized that the natural history of constructivism was itself a worthwhile focus of inquiry that should be investigated on its own terms. What is the natural history of constructivism? Is there a single natural history of constructivism, or are there many histories of constructivism, corresponding to the many forms of constructivism known today? Is constructivism one or many? Is its history one or many?

If the many forms of constructivism are woven from many constructivist traditions, do all these traditions take place against a background of non-constructive thought (or, perhaps, against a background that does not bother to distinguish between constructive and non-constructive methods), or is there some subtle, underlying constructive intuition or intuitions that have always been just below the surface of rational thought, even if they were never brought to full and explicit consciousness until the twentieth century?

Simply from asking a number of question of this kind, it is obvious that the history of constructivism and constructivist thought could be a substantial research project that could consume many years of honest toil; the question of the natural history of constructivism is not going to be settled in a blog post. Nevertheless, a few general remarks can be made on the natural history of constructivism that might someday be followed up by more extensive work.

In Salto Mortale I discussed Greek conceptions of the infinite, which I now see point to the origins of constructivism in the origins of the western intellectual tradition. The Greek concern with order, proportion, and harmony as both intellectually and aesthetically satisfying issused in a distinction between the peras (περας) and the apeiron (ἄπειρον), the former being the limited, which limitation makes possible order, proportion, and harmony, while the latter is the unlimited, from which follows the disordered, the grotesque, and the inharmonious. For the Greeks, aesthetic intuitions were co-equal with formal intuitions, so that it was natural for the two to reinforce each other. Indeed, for the two to reinforce each other was another source of harmony.

Much of Greek thought, then, constitutes finitism in practice, though not in principle. There was no Greek philosopher who explicitly argued for finitism as a doctrine, although the Epicharmus fragment quoted in Salto Mortale — A mortal should think mortal thoughts, not immortal thoughts — comes very close to an explicit advocacy of finitism.

Medieval Scholasticism took Aristotle’s works as foundational texts, and as Aristotle had completely embodied the implicit finitism of the Greeks, medieval thought followed this Aristotelian lead. But Scholasticism, no less than any intellectual movement, embodied tensions, and the tensions in Scholasticism arose from Aristotelian finitism on the one hand, and on the other hand the interest of the Scholastics in being able to give a rational account of the infinity of God, which was ever-present in their thoughts. This tension is expressed in the catalogues of insolubilia drawn up by medieval philosophers.

The Scholastics cut their teeth on Aristotelian logic, and much of the trivium was taken up with a detailed if not exhaustive study of Aristotle’s logical works (known as the Organon). This careful study of logic made these medieval philosophers acutely aware of logical paradoxes, which they collected and categorized. These insolubilia were not primarily concerned with the dialectic of the finite and the infinite, but rather those perennial logical paradoxes that also (like the problem of the infinite) can be traced back to ancient Greece, as, for example, Epimenides’ paradox of self-reference concerning a liar who says, “I am lying to you.”

Descartes, often called the Father of Modern Philosophy, and responsible for the epistemological turn on modern philosophy — how can I know what I know? — was ambivalent about the infinite, and tells us in his Principles of Philosophy that we ought only to invoke the indefinite because the human mind cannot know the infinite. This follows in an obvious way from the Cartesian epistemological turn, in so far as epistemology is really about how finite human beings can come to know anything. I would consider the matter closed at this point had I not read Alexander Koyré’s essays on Descartes, in which he argues that, “Descartes’ position concerning the infinite has often been misunderstood by historians…” (Cf. Appendix K, “Descartes on the Infinite and the Indefinite” in Koyré’s Newtonian Studies) There is much scope for further research on the early modern conception of the infinite and its reaction to the Scholastic tradition.

If Descartes is the Father of Modern Philosophy, Immanuel Kant is the favorite son of modern philosophy, who takes the epistemological work of the father far beyond what the father could have imagined. Kant’s insistence upon what I would call an anthropocentric epistemology has similar consequences as the Cartesian epistemic concern for restricting human knowledge to what is possible for human beings. Kant takes this and runs with it; indeed, one of Kant’s themes is the possibility of experience, and if we are going to limit ourselves to the possibility of experience, we are going to rule out the infinite and logical paradoxes, which confound human experience. Not also the Kantian interest in placing definite limits on human thought, which, we will see below, is one of the few unifying features of constructivism.

In Kantian Non-Constructivism I noted that Kant’s conception of exhibition in intuition is the fons et origo of construtivism, but that there are nevertheless non-constructive aspects to Kant’s thought — what might be called non-constructive moments within constructive thought, in contradistinction to constructive moments within non-constructive thought.

This brings us to the late nineteenth and early twentieth centuries, when constructivism came into full flower and was explicitly formulated. Partly this was in reaction to Cantor’s set theory and transfinite numbers (elsewhere I have called Brouwer’s intuitionism the first post-Cantorian philosophy of mathematics), and partly it was a result of the possibility of giving a rigorous form to perennial logical paradoxes that had been known and discussed since the Greeks. This not only proved to be a major challenge to set theory, but also resulted in a happy proliferation of logically rigorous paradoxes. After Russell formulated his eponymous paradox he realized that it belonged to a class of paradoxes, and, given that insight, he was in a position to formulate an infinitude of paradoxes (I’m citing Russell from memory, as I don’t recall the exact source of this idea, and I am sure that Russell expressed this idea in very different language).

The most important figure in the flowering of constructivism was L.E.J. Brouwer and his intuitionism. In The Two Philosophies of Mathematics I wrote:

“Brouwer’s formal aspect blossomed into the embarras de richesses that is contemporary constructivism, which by a multiplicity of methods seeks to restrict the methods of mathematics. But the constructivists turned out to be the truly parsimonious school of thought in the long run, always seeking to double down on the profusion of mathematics, while the material aspect of logicism was transformed into contemporary Platonism, notwithstanding the fact that the origin of this is to be found in Russell, who was the greatest Ockhamist of his generation.”

Constructivism may, from an historical point of view, be compared to the Protestant Reformation, with which it shares many surprising parallels. Before the explicit emergence of constructivism with Brouwer, the tradition of classical eclecticism was similar to the internal diversity of the Catholic church, that is to say, the universal church. Classical mathematics and Catholicism were both superficially unified, simply because there was no tradition of thought outside these universal institutions (unless we go quite far afield, into another civilization), but this unity was the unity of a “big tent” that subsumed enormous internal diversity. Luther was to the Protestant Reformation as Brouwer was to the constructivist revolution (Weyl, at least, called it a revolution; I would call it a reaction).

Luther and Brouwer both believed that the tradition in which they had been educated had become corrupt and needed to be reformed. Luther nailed 95 theses to the door of the Wittenberg Castle church; Brouwer published academic papers that questioned the law of the excluded middle. While both saw the need to reform an institution becoming progressively more corrupt, both also saw themselves as the definitive and final answer to this corruption; neither of them realized that they were in fact opening Pandora’s box, and that their innovations would be followed by a thousand narrow sects, each concerned with its own particular bête noire of corruption. Far from stemming heresy, both inadvertently encouraged a proliferation of heresy, and, with that proliferation, the unity of the universal institution vanished in a puff of smoke.

After Brouwer, the most important figure is Gödel, but as I have written so many posts on Gödel’s relationship to constructivism (cf. “Further Resources, below) I won’t make a separate exposition of Gödel here. (Gödel, 1906–1978, was a younger contemporary of Brouwer, 1881–1966; most of their lives overlapped.)

Today we have a vibrant constructivist tradition of formal thought, though as splintered as contemporary Protestantism. There are intuitionists, finitists, strict finitists, ultrafinitists, homotopy type theorists, and others. Is there any principle that unifies these forms of constructivism? In virtue of what do we call all these schools of thought “constructivist”? I have thought about this many times, but the only unity I can find is not a principle, but a simple fact about formal practice: all forms of constructivism limit the acceptable practices of classical mathematics. In other words, constructivism is the rejection of eclecticism in formal thought.

The next step beyond a natural history of constructivism would be to attempt to map out the future of constructivism. What hath Brouwer wrought? Will constructivist thought continue to diverge into ever more luxuriant forms, or will the constructivist tradition converge upon a few core principles and coalesce into a unified tradition within formal thought? Will another form of formal thought emerge from constructivism, something different from both constructivism and non-constructivism or classical eclecticism? Will constructivism fall out of fashion and eventually become the remit of fascinated antiquaries? The poet Paul Valéry speculated that this would be the future of Cantorism; I can just as well suggest that it may be the future of constructivism.

As we began with a number of questions, we end with a number of questions — suggestive of future research, both for the directions in which the answers to these questions might take us, and also for the assumptions incorporated into these questions, which might be brought to the surface and made explicit.



Originally published at geopolicraticus.tumblr.com.