First, it is necessary to note that “proof” is an ancient concept, whereas “mathematical foundations” is a much newer one; ever since ancient Greece, “proof” has been one of the central concepts in the Western mathematical tradition, whereas the topic or subdiscipline of “mathematical foundations” emerged mainly only in the late nineteenth and early twentieth centuries.
The demand for “proof” makes mathematics in Western culture different from arithmetic and surveying. Mathematics pursues not merely the obtaining of a valid result, but even more the process of rigorous deduction and argumentation. But even within the Western tradition, the understanding of mathematical proof has by no means been static. And the reason we say that “proof” is the core concept of “mathematical foundations” is precisely that the whole debate over the problem of mathematical foundations arose on the basis of certain understandings of “proof.”
What we should notice is this: if mathematics devoted to proof has a long history, why did topics such as “mathematical foundations” or “metamathematics” only become so prominent in the twentieth century? This is precisely because the development of the concept of “proof” itself—that is, the change in our understanding of the nature of mathematics—brought the problem of mathematical foundations to the fore.
Modern people tend to understand mathematical proof as “a concept similar to that of a building structure. It divides mathematical knowledge into the foundation and the edifice. Its characteristics may be simply stated as follows: mathematical propositions should be divided into two groups, one requiring the proof of other propositions, called theorems; and one that can prove other propositions while itself requiring no proof at all, called axioms. Axioms are the foundation of the whole body of mathematical knowledge, and theorems are the superstructure built on these foundations.” But this is only one particular understanding of mathematical proof. If such an understanding had truly been the mainstream from ancient times onward, then how are we to imagine that the problem of “mathematical foundations” and the axiomatic movement did not suddenly flourish until the nineteenth and twentieth centuries?
Although the concept of “axiom” had already been established in the time of Plato, it found its most consummate expression in the *Elements*. However, on the one hand, Euclidean geometry was not the whole of classical mathematics; on the other hand, Euclid’s axioms were not necessarily understood as the “foundation” of geometry as a whole. Before the theorems and postulates there is at least a long series of definitions, and these definitions provide an intuitive understanding of mathematical concepts. In the eyes of modern people, these definitions are entirely superfluous, and the “definition” of conceptual meaning is also expressed through axioms; but Euclid plainly did not think definitions were unnecessary. In the eyes of the ancient Greeks, mathematics probably still rested first and foremost on human intuitive concepts, on explanatory definitions rather than on axioms as its starting point.
Because modern people regard mathematical knowledge as a collection of “propositions,” and “proof” as something that links proposition to proposition, “proof” naturally became the core concept of “mathematical foundations.” But has mathematical knowledge always been understood as a collection of “propositions”? When we think of “what mathematics includes,” do we think only of one theorem after another? No. We first think of things like “algebra,” “trigonometric functions,” “analytic methods,” “calculus,” “group theory,” and so on. These mathematical “contents” are not first presented as propositions or theorems, but as special “methods.” The content of mathematics first appears as various methods for dealing with problems; only when a certain method is actually applied and developed do questions of propositions and logical deduction arise. The progress of mathematics is also first manifested in the expansion of methods—for example, using unknowns to set up equations, using algebra to deal with geometry, using the method of exhaustion to find areas, and so on. The great achievements of mathematicians lie in the invention of new methods, in the opening of new approaches and new perspectives; the proof of one theorem after another becomes possible only on the premise that a certain method and mode of thought have already been opened up. As for the axiomatization of an entire system of theorems, that is the most lagging-behind matter of all. Only when this method has matured to the point that it can be fixed in textbooks can its axiomatic system gradually be perfected—*Elements* is precisely such a textbook, and it established its axiomatic system only at the end, once the whole field had already basically taken shape and been brought to completion. As M. Kline put it: “Naturally, the advance of mathematics was fostered mainly by people of extraordinary intuition and not by those who were adept at producing rigorous proofs.” From the actual course of mathematical development, if axioms are the foundation of mathematics, then one must admit that the whole mathematical edifice was built upside down.
Many modern people belittle the aspect of mathematics as a creative method, and pay attention only to its side of rigorous argumentation; they take deduction or proof, as the linking of proposition to proposition, to be the entirety of mathematical work. Thus “axioms” gained the title of “mathematical foundations.” The demand for “axioms” is ancient, but treating axioms as the foundation of mathematics is rooted in modern people’s particular obsession with “proof.”
The comparison of mathematical knowledge to a building is itself inappropriate. This seems to treat mathematical knowledge as something static, already finished and sitting there. This is of course a Platonic epistemology. But even according to the Platonic line of thought, if truth is regarded as a solid, eternal, unchanging edifice in the world of Ideas, human knowledge is still something that develops. In Plato’s view, truth is not proved, but is something fixed in the world of Ideas and accessible only through the mind’s eye; “proof” is a process of descent. No Platonist, however extreme, would regard human mathematical knowledge in the actual world as a static, unchanging thing. So how does human mathematical knowledge grow and change? Does it do so like a building, continuously erected upward on a fixed foundation? Clearly not. F. Klein said it well: “In fact, mathematics has grown like a tree, but it did not begin growth from the thinnest roots, nor does it grow only upward; on the contrary, as its branches and leaves expand, its roots sink ever more deeply downward…. Then we can see that there is no final conclusion to foundations in mathematics; from another angle, there is also no initial starting point.”
Even if we compare the whole of mathematics to a building, we cannot concern ourselves only with its formal structure. If the axiomatic system is the formal cause of the mathematical edifice, then its material cause, efficient cause, and final cause are equally important. The construction of an axiomatic system depends on logic; “proof” uses logic, but mathematics does not depend on logic, otherwise, as Wittgenstein satirically put it: “One is almost tempted to say that the whole art of doing furniture consists in the glueing together of pieces.” As M. Kline said, “It is logic that depends on mathematics, not mathematics on logic.” He acknowledges that the result of the rigorization of mathematics was this: “Not a single theorem in arithmetic, or algebra, or geometry was changed, and the theorems of analysis were only stated more carefully as required. In fact, all that these new axiomatic structures and rigorifications accomplished was essentially already known to mathematicians in the past. Indeed, rather than saying that these axioms can deduce certain theorems, it would be more accurate to say that they can only acknowledge the theorems already in hand. All this means that mathematical development does not depend on logic, but on correct intuition. As Hadamard pointed out, rigorization is merely the approval of intuition’s spoils, or, as Weyl said, logic is the hygiene by which mathematicians keep their thought healthy and strong.”
It is quite apt for intuitionist mathematicians such as Weyl to compare logic to a “hygienic means.” Intuitionism never denies the importance of logic; yet, to put it this way: intuition is the blood of mathematics—it provides energy and momentum; while sense and experience are the food of mathematics, from which mathematicians draw nourishment from nature and from experience; logic, by contrast, is the “hygienic means,” the health-preserving products and medicines that strengthen the body and prevent and treat illness, helping mathematics become stable, mature, and complete. Seen through such a metaphor, the relative positions of the various elements become immediately clear: intuition gives mathematics life, experience makes mathematics grow, and logic makes mathematics strong. But if one forgets, or even discards, blood and food and relies only on medicine, not only will one fail to remain strong, one will not even be able to sustain life.
Intuitionist mathematicians have the clearest understanding of the status of “proof.” On the one hand, they attach the greatest importance to proof; indeed, one might even say that the intuitionist logical system simply replaces the status of Platonic “truth-value” with “provability.” At the same time, they also attach the least importance to proof: in their view, even the most rigorous formal system is nothing more than a kind of “language,” and the significance of language lies in its ability to communicate and exchange thought, not in forming a strict grammar dictionary. Heyting said: “The intuitionist mathematician recommends regarding mathematical activity as a natural function of his intellect, as a free and vivacious activity of thought. Mathematics is, in his view, a product of the human spirit. He uses language, whether natural or formalized, only for the purpose of communication of thought, that is, to make himself understood by others or by himself with regard to his own mathematical ideas. This linguistic accompaniment is not the representative of mathematics, much less mathematics itself.”
In short, in my view, the very expression “mathematical foundations”—that is, treating the question of establishing the axiomatic system on which proof rests as the “foundation” of the whole of mathematics—is misleading in itself. Proof is one of the core concepts of mathematics, and axiomatization is also a requirement of mathematical development, but it is not the only thing.
See Ji-laoshi’s handout
[US] M. Kline: *Mathematics: The Loss of Certainty*, trans. Li Hongkui, Hunan Science and Technology Press, 2003, p. 323
[US] M. Kline: *Mathematics: The Loss of Certainty*, trans. Li Hongkui, Hunan Science and Technology Press, 2003, p. 325
Tu Jiliang, ed.: *The Collected Works of Wittgenstein*, vol. 7, *On the Foundations of Mathematics*, trans. Xu Youyu and Tu Jiliang, Hebei Education Press, 2003, p. 209 (V§24).
[US] M. Kline: *Mathematics: The Loss of Certainty*, trans. Li Hongkui, Hunan Science and Technology Press, 2003, p. 238
[US] Maurice Kline: *Mathematical Thought from Ancient to Modern Times* (vol. 4), trans. Deng Donggao, Zhang Gongqing, et al., Shanghai Scientific and Technical Publishers, 2002, p. 99
Arend Heyting: The intuitionist foundations of mathematics: see [US] Paul Benacerraf and Hilary Putnam, eds.: *Philosophy of Mathematics*, trans. Zhu Shuilin, Ying Zhiyi, Ling Kangyuan, and Zhang Yugang, proofread by Chen Yihong and Wang Shanping, Commercial Press, 2003, p. 60
Latest Comments
- Yeziqiu
2010-07-02 09:26:24 Anonymous 10.8.0.2
Current mathematics teaching emphasizes proving a pile of theorems, and neglects the cultivation of mathematical methods
- Jizha
2010-07-03 20:53:26
I just bought *Mathematics: The Loss of Certainty* and haven’t started reading it yet… I haven’t finished *Mathematical Thought from Ancient to Modern Times*. I’m really lazy
Translated from the Chinese original with AI assistance. The original text is authoritative.
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