In last week’s assignment I wrote about “proof” and “the foundations of mathematics.” Since it was a response to a set prompt, I wasn’t very satisfied with how it turned out, and now I want to add a few more remarks.
For a long time I have admired the concept of “proof” as the fundamental spirit of mathematics, or rather, “proof” is the core of the “mathematical spirit.” The demand for “proof” is what distinguishes mathematics from the technologies of calculation and geodesy, which solve practical problems; it represents an attitude of freedom, reason, and disinterestedness, and one cannot overstate the significance of “proof.”
But this emphasis does not mean becoming excessively fixated on the form of proof itself (logicism, formalism). In fact, rather than saying that logicism values proof, it would be more accurate to say that it actually neglects “proof.” It treats “proof” as a formal deductive process among propositions, as a chain of connections between proposition and proposition, but in doing so it misses the fact that “proof” is first and foremost a human, creative “activity.”
Looking only at a strict proof process, it really does resemble a machine: propositions are linked and interlocked, running from input to output in exact and orderly fashion, and the axioms and premises fed in at the beginning undergo one rigorous mechanical operation after another until the proposition to be proved is finally produced. But the question is: within all this, what is “proof”?
When we turn the word proof into a noun, we say “give a proof,” and then we easily take the machine just mentioned itself—the one that takes axioms and premises as input and outputs the proposition to be proved—as “proof.” But proof is first a verb, an activity; it refers to how human beings build that machine in the first place.
For that machine to operate, it first begins with the input items, namely the initial conditions or axioms. So when we regard proof as the core of mathematics and then regard such a formal machine as proof itself, it is only natural that we take the axioms as the “foundation” of the whole of mathematics. Intuitionism places greater emphasis on “proof,” but what it emphasizes is not that machine as the product of proof; rather, it focuses on the human activity of proving.
It is human beings who do the proving, not machines or God. Once people discover a method for proving a general proposition, of course they can mechanize proof, so that specific propositions within the relevant range of generality can all have their proof procedures mechanically produced. But in the fundamental sense, it is still human beings who prove—just as the answer key at the back of a textbook can also provide a proof, yet the one fundamentally doing the proving is still the person who wrote the answers. Once a mathematical proof has been constructed, its input-to-output operation really is mechanical; however, that mechanical operation itself is not the main activity of mathematics. The main activity of mathematics is to creatively discover the method for producing such a machine.
Mathematicians are like a group of mechanics: what they care about is not what the machine is supposed to output, but the making of the machine itself.
This tradition of mathematics as a free art originated in ancient Greece, but it is not necessarily the tradition of Platonism. Plato’s world of Forms is also a certain fetishized understanding of mathematics, similar to how logicism treats the formal machine that strings propositions together as proof itself. Platonism, too, takes mathematics to be something self-subsistent, detached from human activity. This is of course a view of mathematics, a metaphysical conviction, but it is not necessarily the only expression of the mathematical spirit.
The tradition of mathematical proof clearly predates Plato, and after Plato it seems to have developed independently as well. A typical example before Plato is the tradition of compass-and-straightedge construction—the rationality, rigor, disinterestedness, and other features embodied in compass-and-straightedge construction are all marks of the mathematical spirit, yet it still emphasizes the notion of “doing,” pursuing a “method” for “making” something, rather than seeking a “discovery” of ready-made truth.
Proof is something human beings do; God does not need proof. Even for Plato, mathematics of proof is only a “way downward,” not a road to truth. Mathematical practice merely helps people apprehend the world of Forms; the true truth is not the result of proof, but something that can be intuited only through the eye of the soul, freed from the body. And modern Platonists go even further than Plato, abolishing even this dimension of transcendent intuition, so that it seems as though “truth” is what appears as the output of proof, while the process of proof itself is merely something waiting to be discovered.
Of course, there are all kinds of views of mathematics; I am not saying that the Platonic view of mathematics is detached from the mathematical spirit, only that it understands it in a somewhat fetishized way. And has intuitionism not been fetishized as well? But what I want to emphasize here is this: it is not necessarily only that one understanding that expresses the mathematical spirit. A rebellion against axiomatization, logicism, and Platonism in the philosophy of mathematics does not mean a rebellion against the mathematical spirit of freedom, rigor, and disinterestedness.
July 18, 2010
Translated from the Chinese original with AI assistance. The original text is authoritative.
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