My term paper for my class on logical paradoxes has now been decided: I’m going to write on intuitionism and the foundations of mathematics. At first I had been dead set on writing about inductive logic, and I read a lot of books, but the more I read, the fewer ideas of my own I seemed to have left; in the end they were almost completely buried. Writing an article that simply surveys the various schools of thought isn’t very interesting either, and I also seem unable to sort out any ideas of my own in such a short time, so in the end I had no choice but to look for another way out… To be honest, it wouldn’t be hard to cobble something together, but Bo Bo is fairly familiar with me, and has even praised me before; turning in something that merely muddles through would really feel too embarrassing to hand over, so I must try to write it as well as possible. But after all, time is limited, and I don’t know what shape it will finally take…
As for intuitionism, I do have a few ideas of my own. The thing that influenced me most was reading M. Kline’s Mathematics: The Loss of Certainty (that book is written so well! And his more classic books, Ancient and Medieval Mathematics and Mathematics: The Search for Knowledge, are also splendid), which gave me my first real understanding of the debates surrounding the foundations of contemporary mathematics. Intuitionist logic is the most distinctive branch of non-classical logic; unlike other technical variant logics, it has a solid philosophical foundation and is backed by a group of very eminent scientists. To be frank, I have personal sympathy for intuitionism, and the original reason is that it is associated with the names of the philosopher I admire, Kant, and the mathematician I admire, Poincaré. Of course, just as the axiomatization movement in fact divides at least into three schools—the logicism led by Cantor, the formalism led by Hilbert, and the axiomatic set theory initiated by Zermelo—the schools and claims within intuitionism are also very complex, and almost every advocate of intuitionism is different in his or her own way. I do not have much affection for a kind of intuitionism that sinks into mysticism.
That said, in fact I think it is not quite accurate to say in general terms that intuitionism is mysticism, supernaturalism, or at least anti-realism. One issue I want to analyze is: who exactly is the naturalist? Who exactly is the realist? First of all, one must make clear what “natural” and “real” mean. In fact, when intuitionists are called anti-realists, and axiomatic mathematicians are called realists, the “real” in question is a kind of Platonic realism. Plato is actually the hidden sovereign behind contemporary mathematics. I feel that when one is advocating the objectivity and truth of mathematics, there are at least three possible approaches to the question of what guarantees the truth of mathematics: the transcendental, the empirical, and the a priori. One line of thought can be traced back to Plato and Pythagoras, as well as to the rationalist tradition: it holds that the mathematical world is a real world beyond the world of experience, and Cantor himself admitted that he was a follower of Plato; another line holds that mathematics is an abstraction derived from experience, from the objective real world, and that the reason mathematics is so “effective” in the natural sciences is not strange at all, because it originally comes from the natural sciences; the third is an approach that can be traced back to Kant, but may also be influenced by the Platonic or Aristotelian tradition. Intuitionists are usually regarded as Kantian, which is apt. But Kant’s philosophy, like Plato’s or that of any great philosopher, can be interpreted and developed from many radically different perspectives. Is Kantianism anti-realism? I think not. Kant, in his own distinctive way, was precisely defending realism. In a kind of intuitionism that in some way upholds realism, mathematics, like other knowledge, originates in nature, and its certainty is guaranteed by nature outside human will. Yet pure nature is inaccessible to human beings, creatures of finite reason; humans can only describe nature through their own language. Even if this language for describing nature is precisely “mathematics,” it is still a human language, and therefore mathematics too is fallible and finite. Mathematics is reliable because of the certainty of nature, and mathematics is less than fully reliable because it remains, after all, a human invention. So I do not support the claim that the mathematical world constructed by human beings is the truly real world. Human beings can create mathematics, but they cannot create reality. What ultimately plays the decisive role is the real world, not the world of mathematical constructs. Opposing “actual infinity” is in fact a very natural claim—the real world does not in fact contain actual infinity; physics rejects infinity; what is called actual infinity exists only in the world constructed by mathematics. Intuitionism does not directly oppose the principle of bivalence; it does not think that there is a third case besides true and false, or rather, that this third case is “unknown,” which is not a new “truth value” but an honest acknowledgment of the limits of human cognition. Intuitionism merely rejects treating infinity as a completed “reality”; this is entirely naturalistic, because in nature there is no such reality as a completed infinity.
In contemporary mathematics, there is a tendency to sacralize “pure mathematics” while disparaging “applied mathematics.” Many mathematicians insist that mathematics naturally stands above any empirical science, that it has a world all its own. This attitude is not necessarily a bad one, but in fact mathematics has never been able to detach itself from the real world. There is no need to mention the great mathematicians of antiquity (there were few full-time mathematicians then, and almost all of the great masters of mathematics in fact devoted more of their energy to physics or astronomy); among the few great masters of the modern era—Poincaré, Hilbert, von Neumann, and the like—not one failed to attach great importance to the applications of mathematics. From the perspective of the separation between pure mathematics and applied mathematics, intuitionism can also be said to represent such a kind of rebellion: a rebellion against isolating mathematics, turning it into a completely “self-sufficient,” “self-contained” logical axiomatic system that cuts mathematics off from human creativity and from the real world. Even if this logical axiomatic system were built to be flawless and invulnerable, what would be the point? Moreover, this system is riddled with loopholes; the emergence of all kinds of paradoxes, as well as Gödel’s incompleteness theorems, and so on, all provide some support for intuitionist claims.
The earliest debates on the foundations of mathematics began with Euclid’s parallel postulate: which reflects the reality of this world, Euclidean geometry or non-Euclidean geometry? The advent of relativity provided strong support for the latter (although in fact the spacetime of relativity can still be explained using Euclidean geometry; it is just mathematically a little more concise to use non-Euclidean geometry). This makes me wonder whether the rise of quantum mechanics provides support for intuitionism. In fact, the EPR paradox I discussed can be more satisfactorily explained with intuitionist logic; throughout, it remains naturalistic rather than sinking into mysticism, whereas traditional classical realism, left unmodified, really cannot explain it no matter what.
However, although my paper will offer some defense of intuitionism, I myself am not an intuitionist. In fact, I also admire the detached “pure mathematics” that stands beyond the world; wandering through the pure world of mathematics is indeed an incomparably sacred and joyful thing. As a certain famous mathematician said: mathematicians are poets. Mathematical creation has incomparable aesthetic value. There is no true “infinity” to be found anywhere in the real world; “infinity” can only be found in religion, art, or mathematics. Precisely for this reason, mathematics is not merely a natural science; mathematics is mathematics, and she is unique.
The above is only my preliminary thinking; when I actually write the paper, it may well change.
August 7, 2006
Translated from the Chinese original with AI assistance. The original text is authoritative.
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