Abstract: From the perspective of defending intuitionism, the author explains intuitionism’s claims in the philosophy of mathematics and the philosophy of logic. The basic position of intuitionism is that mathematics is a creation of the human mind. Intuitionism rejects assigning any mathematical object a transcendent existence beyond mind and matter; intuition and creation, rather than proof and deduction, are the most important mathematical methods, the life of mathematics; the meaning of mathematics, like that of the other sciences, lies in discovering problems worth studying, touching the mysteries of nature, and ceaselessly pursuing truth. In the first half of this essay, the author sorts out intuitionism’s resistance to Platonism and to the various modern trends in mathematics associated with the axiomatic movement, including logification, formalization, and purification (of mathematics), among others. Then the author lists some developments in mathematics and science that support intuitionist claims, including the rise of non-Euclidean geometry, various logical paradoxes, Gödel’s theorem, the Löwenheim-Skolem theorem, and, in particular, quantum mechanics. Finally, drawing on the discussion of quantum mechanics and leading into the relation between mathematics and physical reality, the author puts forward his own thoughts on how physical reality should be understood.
Keywords: intuitionism foundations of mathematics logicism formalism Platonism axiomatization paradox existence and construction
Contents:
I. The Basic Claims of Intuitionism
II. Intuitionism and Its Opponents
III. Developments Favorable to Intuitionism
3. Gödel’s Theorem and the Continuum Hypothesis
4. The Löwenheim-Skolem Theorem
IV. Existence and Construction
Conclusion: What Is Mathematics?
Intuitionism is a current of thought that has had a tremendous influence in contemporary mathematics and logic, and is also the most peculiar one. Its intellectual sources can be traced back to Kant’s transcendental philosophy; if we speak only of its opposition to actual infinity, it can be traced back even further to Aristotle. In modern times, Leopold Kronecker and Henri Poincaré were forerunners of intuitionism, while L. E. J. Brouwer was the true founder of intuitionism as a clearly defined school. Its claims have won the sympathy or acceptance of many famous mathematicians and logicians, including René Baire, Lebesgue (Henri Léon), Arend Heyting, Hermann Weyl, and others.
As the Polish mathematician Mostowski put it: “Among the various non-classical logics, intuitionistic logic occupies a unique position: it is the only logic ever established that is actually being used by a rather large group of competent scientists. It is also the only logic that has been extended beyond propositional logic and predicate logic and used to describe certain parts of mathematics. Łukasiewicz, who was the first to invent an unusual logic, hoped that one day there would appear several logics that, like non-Euclidean geometry, would be put to practical use. Most of the non-classical logics invented so far, however, have not been used in practice. Although several of them have been studied in a metamathematical way on the basis of two-valued logic. It seems that intuitionistic logic is the only one that still has a chance to realize Łukasiewicz’s program. At the same time, this logic is built on an original and consistent view of mathematics. These two facts explain why intuitionistic logic aroused strong interest from the moment of its creation.”[①]
The influence of intuitionism on the development of contemporary mathematics and logic needs no further elaboration. The purpose of this essay is not to introduce the history of intuitionism, but rather, through sorting out its views and positions, to explore the various problems and predicaments encountered in contemporary “foundations of mathematics” and in logic, and to draw inspiration from intuitionism’s radical standpoint.
Intuitionism is indeed radical, and one might even say extremely “reactionary”: it resists and opposes almost every trend in contemporary mathematics and logic—axiomatization, logification, formalization, isolation, and so on. Studying such a rebellious current of thought will help us more comprehensively discover and reflect on the various problems and predicaments that may be lurking in the present development of mathematics and logic.
Unfortunately, although intuitionism has had a huge influence in the West—though perhaps it does not have many followers, very few scholars can avoid it when discussing related issues—it seems to have long been neglected in Chinese academia. This is probably because intuitionism is often associated with terms like “transcendentalism” and “anti-realism,” and these terms have traditionally been looked down upon by Chinese academic circles. That may lead many scholars to assume, prejudicially, that intuitionism can only produce error and need no longer be taken seriously. But in fact, although all the other schools in the Western fields of foundations of mathematics and logic have been attacked by intuitionism, none of them have escaped its influence. Even those who are not intuitionists, including intuitionism’s fiercest opponents, will often, in one way or another, accept some of its claims. It is virtually impossible to examine and discuss questions concerning the foundations of mathematics while leaving intuitionism out.
As noted above, intuitionism is less a school than a current of thought: on the one hand, one can find abundant intuitionist influence even among non-intuitionists; on the other hand, among self-declared intuitionists, views and claims vary tremendously, making any sweeping generalization difficult.
In view of this, I feel that if one wants to present intuitionism’s positions and ideas well, adopting an objective description from the standpoint of an observer is not the best choice. Moreover, what I want to do is not historical exposition, but to draw inspiration from intuitionist thought. The best way is for me myself to take the intuitionist perspective, and, from within intuitionist lines of thought, compare and argue with various other positions; only then is it easiest to bring out intuitionism’s distinctiveness and its “reactionary” character.
Perhaps I really am an intuitionist, perhaps not; and perhaps my sorting out of intuitionist views does not accord with the intentions of those intuitionists themselves. But these are secondary matters. What matters is to explore the inspiration intuitionism brings to the various problems of the foundations of mathematics, logic, and even philosophy as a whole, as well as to people’s modes of thinking.
I. The Basic Claims of Intuitionism
Before comparing it with other positions, it is still necessary to briefly introduce intuitionism’s basic claims and dispel some misunderstandings.
First, I think that the research method of “slapping on labels” before thorough understanding is highly harmful. For example, taking “transcendentalism,” “anti-realism,” “idealism,” “subjectivism,” and so on as preconceived impressions of intuitionism is hasty.[②] Intuitionism is indeed deeply indebted to Kant, but Kant’s philosophy was mainly of “inspiration” significance for intuitionism. Intuitionism is not necessarily fully transcendental; as will be mentioned later, intuitionists can almost be counted as the most persistent “empiricists” or “positivists” in mathematics. Moreover, even with Kant himself, it is a major problem to simply classify him as an idealist or anti-realist. Rash “labeling” easily causes serious misunderstandings; of course, Kant’s philosophy is not the topic of this essay.
As the name suggests, intuitionism’s original and most important position is its emphasis on “intuition.” This of course does not mean that intuitionists deny the logical rigor of mathematics. Everyone acknowledges that mathematics is the science with the highest demand for logical rigor, and intuitionists cannot deny this fact. But what intuitionists want to emphasize more is the status of intuition, inspiration, and creativity in mathematics, and these things were submerged and neglected in the tide of the axiomatic movement. Mathematics is not only the most rigorous of sciences, but also the most creative.
As Poincaré said: “Logic and intuition each have their necessary role. Neither can be dispensed with; only logic can give us certainty, it is the instrument of proof; intuition, however, is the instrument of invention.”[③]
The relation between intuition and logic will be further explored later in the opposition between intuitionism and the axiomatic movement as a whole.
The emphasis on intuition is directly related to intuitionism’s view of mathematics: intuitionism believes that mathematics originates in the structure of the human mind, and that mathematicians are more inventors than discoverers (Wittgenstein also held such a view, for example). It must be noted that to see “… originates in the human mind” and immediately classify it as transcendentalism or idealism is absurd, unless one asserts that the human mind cannot create anything at all; otherwise, whatever it creates will in some sense be its “product.” If one claims that “the certainty of human knowledge about the external world originates in the structure of the human mind,” that is roughly transcendentalism; if one claims that “all human knowledge of the external world itself originates in the human mind,” that is probably a kind of (epistemological) idealist transcendentalism; while if one claims that “the external world itself originates in the human mind,” that is a particularly extreme (ontological) idealism (and no longer transcendentalism at all). These three claims are very different. Intuitionism merely claims that “mathematics” originates in the human mind, and on the one hand does not rule out the possibility that the “certainty of mathematics” comes from the external world, nor has it yet undertaken any discussion of the external world.
Rather than classifying intuitionism as transcendentalism, it would be better to regard it as the most thoroughgoing empiricism. By comparison, so-called “logical empiricism” is actually somewhat less so. For logical empiricism is really “logic/empiricism”ism: it adheres to empiricism in all fields, with the sole exception of “logic” (and mathematics, taken as a derivative of logic). Intuitionism, by contrast, insists that mathematical truth is also empirical, and that logic, mathematics, and the inductive method already identified by Hume in the other natural sciences are not absolutely reliable. Brouwer said: “There exists no non-empirical truth, and logic is not an absolutely reliable instrument for the discovery of truth. … Mathematics, carried out strictly according to this view and using exclusively the method of introspective construction to derive theorems, is called intuitionistic mathematics.”[④]
Empiricism and positivism are always closely related. Naturally, intuitionism’s view of mathematics also tends toward positivism. Paul Benacerraf said: “The intuitionist seems to be a positivist in mathematics, though not necessarily in other disciplines.”[⑤]
Empiricism, positivism, and related issues will be discussed later. The reason for bringing them up now is mainly to correct some possible misunderstandings—namely, that intuitionists are strange, unlike anyone else, utterly incomprehensible, and so on. In fact, intuitionism is similar to, and continuous with, the thought currents regarded as “mainstream,” such as logicism.
One of the main reasons intuitionism is easily taken for some incomprehensible “heretical doctrine” is probably its opposition to the law of excluded middle. A thing is either true or false—what could be wrong with that? Intuitionists actually refuse to accept such a belief, which is so consonant with “intuition”; they must be crazy!
In fact, there is no need to be overly surprised by this. First, even in the classical logical approach, not every “sentence” has a truth value. For example, do we really think that statements like “bread is brave,” “electrons are sweet,” or “unicorns are less than zero” are each either true or false? Rather than saying these sentences are “false,” it is better to say that they are meaningless, nonsensical.
Intuitionism does not directly oppose the law of excluded middle. For intuitionists, just like in classical logic, any semantically valid, meaningful[⑥] statement is still either true or false,
Michael Dummett pointed out: “For intuitionistic logic, the double negation of the law of excluded middle is a valid semantic principle, just as two-valued logic regards the law of excluded middle itself as valid: it is inconsistent to assert that any statement is neither true nor false.”[⑦]
The problem is that not all statements that are valid in classical logic are meaningful in intuitionism. What intuitionism mainly objects to is that certain objects are meaningless—for example, talking about actual completed reality as “actual infinity” is, for intuitionism, meaningless. Intuitionism holds that only those mathematical statements for which, in principle, a procedure can be constructed that can decide them effectively in finitely many steps are legitimate.
To use another analogy, intuitionism accepts the assertion “all crows under heaven are black” as having a truth value, because the proposition “every crow under heaven is either black or non-black” (leaving aside ambiguous colors) is legitimate. This is because “the crows under heaven,” though countless, are after all finite; “under heaven,” though vast, is also finite; in principle, we can search for all the crows in a systematic sweep and inspect their colors one by one. Although this is technically difficult, the problem is after all “realistic” in principle.
However, the following statement is meaningless: “The crow in the story of ‘The Crow and the Fox’ in Aesop’s Fables is either black or non-black.” For that crow is purely a fictional creature in a story, and Aesop never directly or indirectly provided any information by which to judge its color.
Only if there is sufficient basis within Aesop’s story to determine its color—for example, if somewhere it is said that the crow once dropped a white feather—would one be entitled to regard talking about the color of that “crow” as meaningful.
The crux of the dispute lies in where “mathematical objects” are actually located. Clearly, they do not exist in the real world; at least, “actual infinity” is nowhere to be found in the real world. Then on what grounds are statements about those mathematical objects meaningful? Platonism holds that mathematical objects exist in some “world of ideas” that is beyond the real world and independent of the human mind. But what on earth is this mysterious world? Intuitionists (and formalists as well) think that this view is not merely too deeply mired in metaphysics, but positively entangled with theology!
I hope the analogy with Aesop’s Fables does not give rise to the misunderstanding that intuitionism regards mathematics as nothing more than fictional stories. Formalists may indeed really think this way: what they require is that the entire story be rigorously self-consistent and able to stand on its own. But intuitionists believe that mathematical construction is by no means arbitrary; rather, it has an inner objectivity and definiteness, which may be the main aspect of transcendentalism within intuitionism.
Not all intuitionists “believe” that “Goldbach’s conjecture is either true or not true,” or that “there are either infinitely many twin primes or there are not,” and so on. Heyting said: “In fact, all mathematicians, even intuitionists, are convinced that mathematics in some sense is related to eternal truths, but when one tries to specify that sense precisely, one gets lost in a fog of metaphysical difficulties. The only way to avoid these difficulties is to exclude them from mathematics.”[⑧]
Belief is belief, but once one begins to ask, from a scientific point of view, about the “meaning” of these statements, the matter is no longer so simple. Intuitionists hold that mathematics is a construction of the human mind, while perhaps also arising from the real world, but in any case it is not some mysterious thing transcendent of mind and matter — “mathematics ought not to rely on such concepts.”[⑨]
Since the meaning of mathematical objects is difficult to derive from experience drawn from the real world, there is only one way to judge the meaning of such abstract propositions, namely: “existence must be constructed” — “in the study of mental mathematical constructions, ‘existence’ and ‘being constructed’ must be synonymous.”[⑩]
Only what has been constructed, what can be reached by the finite human mind, has meaning. As for the likes of actual infinity, which cannot be constructed — because both the human mind and the material world are finite — things that exist neither in the real world nor in the structures of the mind are meaningless, and propositions involving them of course have no truth or falsity to speak of, much like “Was the crow in Aesop’s fables black?” or “Was Sherlock Holmes’s great-grandfather left-handed?” and so on. We easily understand that these questions are meaningless because the objects they refer to simply do not “exist.” So where does actual infinity exist? Weyl said: “The law of excluded middle may perhaps be valid for God, who can inspect the infinite sequence of natural numbers all at once; but this is something impossible for human logic.”[11]
In short, the main disagreement between intuitionism and the other schools arises from their understanding of the nature of mathematics — what is mathematics? What are its objects, its methods, and its meaning? Intuitionism holds that mathematical objects derive from the constructions of the human mind, but at the same time are closely connected to the objective world, while having nothing to do with a transcendent world or the supernatural; intuition and creation, rather than proof and deduction, are the most important mathematical methods, the very life of mathematics; the meaning of mathematics is also like that of the other sciences: discovering problems worth studying, touching the mysteries of nature, and ceaselessly pursuing truth. Further, precisely on the basis of this view of mathematics, and while modestly acknowledging that the powers of the human mind are limited, a series of concrete claims is produced. For example, resistance to the law of excluded middle, and so on. Therefore, in order to understand intuitionism more truly, one should not begin with “opposition to the law of excluded middle,” this particular instance at the very last link in the chain of intuitionist reasoning, but rather start from the most basic philosophical ideas of intuitionism itself.
II. Intuitionism and Its Opponents
1. Platonism
The foregoing has in fact already touched on intuitionism’s objection to so-called Platonism — mathematical objects do not exist in an independent kingdom beyond mind and matter. Conversely, the position that to a greater or lesser extent tacitly accepts such a “world of ideas” may be called “Platonism.”
The influence of “Platonism” on modern mathematics, indeed on modern science as a whole, is far greater than most people imagine. Even some influential historians of science such as Edwin Arthur Burtt have pointed out that throughout the entire process of the rise of modern science from Copernicus to Newton, the revival of Platonism was always a crucial factor of trigger and impetus.[12]
The influence of Platonism often seeps unconsciously into people’s habits of thought and language. For example, Paul Bernays pointed out: “An example of this kind of way of building theory (the Platonic way) can be found in Hilbert’s axiomatization of geometry. If we compare Hilbert’s axiomatic system with Euclid’s axiomatic system, … we see that Euclid speaks of figures to be constructed, whereas for Hilbert the system of points, lines, and planes exists from the outset. Euclid assumes that one can join two points with a straight line; Hilbert states as an axiom: given any two points, there always exists a straight line on which both points lie. … This example already shows the intention of completely severing the connection between objects and the reflecting subject.”[13]
Georg Cantor, the founder of set theory, explicitly declared himself a Platonist[14]; he not only appealed to the “kingdom of ideas,” but also invoked mysticism and God in his defense. He said: “The reality of mathematical objects does not lie in the real world, but in the infinite wisdom of God: the internal truth of mathematical objects, namely logical consistency, guarantees that such objects are ‘possible,’ while the absolutely infinite essence of God guarantees the eternal existence of these ‘possible objects’ in God’s thought.”[15]
The extreme form of “Platonism” in mathematics may be described as follows: “Mathematics consists of a body of propositions that discuss an independent reality composed of familiar mathematical objects such as sets, numbers, functions, and spaces. Mathematical discovery consists in revealing, by deduction from axioms taken to be true on the basis of a special intuitive faculty different from sensory experience (which gives us only knowledge of the empirical world), truths concerning this independently existing reality. Mathematical objects are independent of our thinking; unlike physical objects, they are not known through interactions with the human organism that bring about changes in the brain and thereby ultimately lead to knowledge of them. But they must be posited in order to account for the existence and growth of mathematical knowledge and of other knowledge (to the extent that these rely on mathematical knowledge).”[16]
As Benacerraf said: “It is hard to ascribe this view to anyone in all its purity, though Gödel may be the closest there has been to a Platonist since Plato. … Probably most mathematicians would reject this extreme form of Platonism; surely no contemporary philosopher or psychologist would regard as acceptable the notion of a supernatural faculty for surveying the realm of objects existing independently in space and time. But if one rejects going back to that view, one cannot avoid some of the problems that arise from the mention of ‘intuition’ (which is encountered in the works of many philosophers and set theorists). So what, after all, is ‘intuition’? If the Platonist view introduced above is wholly mistaken, how can such a faculty possibly exist?”[17]
The dilemma mathematics encounters here is similar to the dilemma natural science encountered in Hume — if the “God” on which, in Descartes, the connection between mind and matter depended does not exist, then what force guarantees that “induction” is so “fruitful”? Likewise, if one removes that world of ideas beyond mind and matter, how can one explain the “certainty” of mathematics? As Hilary Putnam asked: “If there is no explanation of the fact that most of mathematics is true, how can our theories be consistent if all we do is arbitrarily write down meaningless symbols (including trial and error)? How can they be so fruitful?”[18]
No one denies that mathematics is “fruitful,” indeed very “reliable,” and both intuitionism, logicism, and formalism seek to provide reasons, in different ways, for why mathematics is so “reliable.” Naturally, disagreement over “what mathematical objects are” becomes the focal point of the debate.
First of all, it should be noted that the reason Platonism revived in modern mathematics, to the point that modern people seem almost closer to Plato than Euclid did, lies in large part in the overturning of the long-accepted belief that mathematical objects are abstractions from and reflections of the physical world, and that for mathematical objects there exist corresponding real structures that they depict.
This overturning of the belief began to some extent with the birth of non-Euclidean geometry. Euclidean geometry had long been regarded as a true depiction of the physical world, but non-Euclidean geometry shattered this naive idea; the “certainty” of mathematics began to waver, and people came to accept that “multiple” kinds of mathematics may exist simultaneously. Of course, there is only one real world. If physical space is non-Euclidean, then what is Euclidean geometry depicting? Naturally, even before non-Euclidean geometry, negative numbers, complex numbers, infinitesimals, and so on had long troubled mathematicians and philosophers for many years.
Another development came from natural science: modern cosmology and quantum physics have both eliminated the “infinite” from the physical world. As David Hilbert said: “We have established in two respects that the universe is finite, namely in the infinitely small and in the infinitely large.”[19] Most people agree with Hilbert: “If mathematics is to be independent of ambiguous empirical assumptions, then it must by no means ground assertions of the existence of infinite structures on physical considerations.”[20]
Thus, how to “deal with” the “infinite” in mathematics became the focus of controversy. Since one cannot find “infinite structures” in the real world, allowing them to exist in some transcendent world of ideas seems the most convenient solution. Yet we have already pointed out that such a world, being neither material nor human mind, is puzzling.
Even logicism or formalism hopes to avoid extreme Platonism. The former tries to evade the problem by reducing mathematics to logic, but in fact it ultimately still runs into difficulties in explaining why logic is reliable — after all, is the truth of logic guaranteed by a priori intuition, by a transcendent world, or by experience? Formalism, meanwhile, tries to eliminate outright the “meaning” of mathematical objects: mathematical objects need not entrust their meaning to the real world or the world of ideas, because mathematical objects need no meaning at all; the meaning of mathematics lies in the consistent, self-contained system it forms. But these efforts are to a large extent ways of sidestepping the issue, and their rejection of Platonism is not thoroughgoing. These matters will be discussed further below.
There is no doubt that intuitionism’s rejection of Platonism is the most conscious and the most thorough. Heyting pointed out: “We do not ascribe to the integers, or to any other mathematical objects, an existence independent of our thinking, i.e. a transcendent existence. Even if every one of my thoughts involves an object thought to exist independently of it, we leave this question open as much as possible; in any case, such an object need not be completely independent of human thought. Even if they must remain outside the individual acts of thinking, mathematical objects are by their nature dependent on human thought. Their existence is guaranteed only insofar as they can be determined by thought. Their having properties is also only meaningful insofar as these properties can be recognized by thought from them. But this possibility of knowledge is shown to us only by the activity of knowing itself. Trust in a transcendent existence unsupported by concepts, as an instrument of mathematical proof, must be rejected.”[21]
Intuitionists are not aiming their arrows at belief in a “world of ideas” itself. You may trust in transcendent existence, just as you may believe in God; that is your freedom. But that is a matter of faith, and has nothing to do with science. Science need not interfere with belief in transcendent things; at the same time, such beliefs must not intrude into science, much less become tools of proof within science. Intuitionists place mathematics together with physics and the other natural sciences: mathematics too is an empirical, positive, fallible, natural science, only more abstract; while logic is even more abstract than mathematics, yet they still can never transcend experience to seek things that neither matter nor mind can provide.
By the way, intuitionists have firmly severed all connection between mathematics and the transcendent world, but on the other hand, they try to rebuild the nearly severed connection between mathematics and the real world; I shall elaborate on these tendencies below.
“The entire core of the intuitionist position is the view that undecidable mathematical statements do not legitimately possess meaning by virtue of a Platonist specification of their truth conditions.”[22] — anti-Platonism can be said to be one of the main threads for understanding intuitionist thought. Only up to this point has this article truly completed its “introduction.” Below, I shall explain the intuitionist viewpoint through some more specific questions.
2. Logicism
Logicism is probably one of the schools that has had the greatest influence in modern debates over the foundations of mathematics, indeed in modern philosophy as a whole. It has virtually dominated the development of Anglo-American philosophy throughout the twentieth century. In the field of mathematics, its claim can be summed up in the simplest language as: mathematics can be deduced from logic. More specifically, this includes: “1. Mathematical concepts can be derived from logical concepts by explicit definition. 2. Mathematical theorems can be derived from logical axioms by purely logical deduction.”[23]
As mentioned above, intuitionism can be said to be the empiricist and verificationist position in mathematics, and we know that most logicists are naturally also so-called “logical empiricists” or “logical positivists,” and so forth. From this perspective, although logicism and intuitionism are mortal enemies, they actually have quite a lot in common.
In their rejection of “metaphysics,” logicism and intuitionism are similar. Logicism may even be more resolute. Logical empiricism not only rejects metaphysics in science, but despises metaphysics altogether. In their view, all traditional metaphysical problems are “pseudo-problems” and utterly meaningless. Even questions like “Is the external world real?” because they are typical metaphysical questions, were in the eyes of early logical empiricism entirely meaningless. Compared with intuitionists, logical empiricists should arguably deserve the title of “anti-realists” more fittingly, or at least “non-realists.” The problem is that logical empiricists do not strictly carry through their empiricism in logic and mathematics, whereas intuitionists, though mild on the question of the reality of the external world, resist with the greatest force realism in the Platonic sense, to such an extent that it is intuitionism that is more often associated with the term “anti-realism”…
Many logicists have also explicitly expressed support for the anti-Platonism of intuitionists. For example, Rudolf Carnap said: “I think we should adhere to Frege’s famous dictum that in mathematics only those things whose existence has been verified (that is, proved within finite steps) can be regarded as existing. I agree with the intuitionists: every logical-mathematical operation, proof, and definition requires finitude, not because of some accidental empirical fact about human beings, but because of the nature of the subject itself.”[24] “Like intuitionism, we regard as properties only those expressions that are constructed from undefined primitive properties within a suitable domain in accordance with determinate rules of construction by means of finitely many steps. The difference between us is that we not only regard the rules of construction used by the intuitionists as valid, but also further allow the expression ‘for all properties’.”[25]
The relation between intuitionism and logicism seems to be very subtle. Just now, we mentioned that intuitionism seems to stress “experience” more than logical empiricism; now we shall see that logicism seems to appeal to “intuition” more than intuitionism does!
Gödel (Kurt Goder) noted: “The question of the objective existence of mathematical intuitive objects — and, incidentally, it is an exact replica of the question of the objective existence of the external world — is not decisive for the problem discussed here. The mere psychological fact that there exists a sufficiently clear intuition capable of generating the axioms of set theory and their extensions in an open-ended series is enough to make meaningful the question of the truth or falsity of propositions such as Cantor’s continuum hypothesis. But perhaps more than anything else, what proves that it is reasonable to accept this criterion of truth in set theory is the fact that continual recourse to mathematical intuition is not only necessary for obtaining unambiguous answers to problems in transfinite set theory, but also necessary for solving problems in finite number theory (such as the Goldbach conjecture),”[26]
Gödel’s “continual recourse to mathematical intuition” is precisely what intuitionism demands be cut off! The difference is that logicalists believe human intuition can grasp concepts such as the “actual infinite,” whereas intuitionists reject this outright.
Simply put, intuitionism uses the ideas of logicalism — empiricism and verificationism — to oppose the actual infinite; whereas logicalism appeals to “intuition” to support the legitimacy of the concept of the actual infinite.
Intuitionism’s restriction of “intuition” is reasonable, and internally consistent. No one would think that human intuition is always correct, or that everything correct is intuitive. Some writers object to intuitionism on the grounds that “certain theorems in intuitionistic mathematics are also too complicated to be grasped by intuition,” but this rests on a merely “literal” and overly simplistic understanding of intuitionism. Intuitionism does not deny the importance of logic; it simply differs in emphasis as to what comes first. Intuitionism stresses the finitude of human mental capacities, and therefore rejects the claim that human beings can legitimately master the concept of the “actual infinite.”
By contrast, the restriction of “experience” in logical empiricism is debatable. In fact, many logicalists are “influenced by a deeply rooted absolutist view of logic, and regard logic as purely formal and contentless.”[27] They hold that the truths of deductive logic are absolutely necessary, and that at most the only issue worth discussing is the “justification” of inductive logic; they have never even seriously considered whether deductive logic also needs “justification.” But where does this “absolute necessity” of logic come from? In fact, it is precisely because logic stands at the greatest distance from experience, and has the most indirect relation to it, that people get the illusion that “logic is absolute.” Logical rules, like other scientific laws and common sense, “have an empirical origin; they come from the experience-based intuitions formed by people in the course of long-term social practice; they are the result of a logical abstraction of people’s experience of everyday language and of thought.”[28]
The claims of intuitionism are based precisely on their acknowledgment of the “empirical” character of logic. From a linguistic perspective, logic is the result of abstracting from people’s experience of everyday language; and specifically in mathematics, the logic of mathematics is precisely an abstraction from the experience mathematicians have accumulated in long-term mathematical exploration and creation. Heyting said that logical theorems “do not differ essentially from mathematical theorems; they are merely more general, just as ‘addition of integers is commutative’ is a more general statement than ‘2+3=3+2.’ The same is true of every logical theorem; it is simply a mathematical theorem of extreme generality. That is to say, logic is a part of mathematics and can by no means serve as its foundation.”[29]
We can see that the most fundamental disagreement between intuitionism and logicalism is not whether to accept the “actual infinite,” but how to understand “what logic is.” As for the rejection of the actual infinite, that is only a conclusion intuitionists draw after they have argued for the “empirical” nature of logic — since there is no actual infinite in experience, and no transcendent world or supernatural force to provide guarantees, how could our so-called “intuition” of the “actual infinite” possibly be legitimate?
As Brouwer said: “Intuitionism on the one hand refines logic, and on the other attacks logic as a source of truth.”[30] Intuitionism does not disregard logic. It is like how we do not discard or despise physics simply because physics does not provide absolute truth; realizing that physics offers us only relative truths is not to belittle physics, but rather a progress in human thought, and a necessity if physics is to continue developing healthily. Similarly, intuitionism values logic, just as it values the empirical sciences. What intuitionism opposes is treating logic as a “source of truth” — human finite minds can never grasp “absolute truth”; human beings will never stop “seeking truth.” Human knowledge arises from nature, that is to say: science arises from truth, not truth from science, and logic is no exception!
3．Formalism
Formalism can hardly be counted as a philosophical school, but its position on the foundations of mathematics is extremely important. The founder of formalism was the illustrious Hilbert, and later his student John von Neumann also became one of its followers. These two may be said to be, after Poincaré, two of the few mathematical giants in history — although Poincaré was hailed as “the last universal mathematician”! By comparison, most followers of logicalism were mathematical logicians (even Gödel, who may count as the greatest mathematical logician, made his principal mathematical achievements basically in number theory alone), and it is difficult to compare them with mathematical masters like Hilbert, who excelled in multiple branches of mathematics and even mathematical physics (for Poincaré, in almost all branches), and who, as leaders of the mathematical world, directly and substantively propelled the development of modern mathematics.
As the most outstanding mathematicians, Hilbert and his followers perhaps had more authority on the question “what is mathematics” — that is, relative to logicalism; intuitionism need not feel ashamed, since the mathematicians who supported intuitionism were no less impressive.
As mentioned earlier, Hilbert was the first to deny the reality of “the infinite,” and he then objected to logicalists’ unrestricted acceptance of infinity within logic: “A statement such as ‘in a finite totality there exists an object with a certain property’ is entirely in accord with our finitist method. But a statement such as ‘either p+1 or p+2 or p+3 … or (and so on, to infinity) … has a certain property’ is itself an infinite logical product. Such a generalization to the infinite, without further explanation and precautionary measures, is just as impermissible as the generalization from finite products to infinite products in calculus. Therefore such a generalization is usually meaningless.”[31] “The infinite is nowhere to be found in reality. It does not exist in nature, nor does it provide a legitimate basis for rational thought — the striking harmony between being and thinking. Contrary to the earlier efforts of Frege and Dedekind, we believe that in order to obtain scientific knowledge, certain intuitive concepts and insights are necessary; logic alone is not enough. Operations carried out with the infinite can become determinate only through finitism.”[32] “As a further precondition for applying logical deduction and carrying out logical operations, there must be something in concept formation, namely certain concrete objects outside logic that are intuited as directly experienced prior to all thinking.”[33] “Substantive logical deduction is indispensable. It is only when we make arbitrary abstract definitions, especially those involving infinitely many objects, that we are deceived. In such cases, we illegitimately use substantive logical deduction; that is to say, we have not paid sufficient attention to the preconditions required for the valid application of such deduction. Recognizing that these necessary preconditions exist, we find ourselves in agreement with philosophers, especially with Kant. Kant taught us — and this is a main component of his doctrine — that the subject matter of mathematics is given independently of logic. Thus mathematics can by no means be founded on logic alone. It follows that Frege’s and Dedekind’s attempts to found mathematics in this way are doomed to fail.”[34]
Judging from the lengthy passages quoted above, this is simply a statement of intuitionist claims as well (which is another reason I quoted so much in one breath)! In fact, in their opposition to logicalism, formalism and intuitionism are indeed very similar. Thus, in the period from 1920 to 1930, when Hilbert and his students gradually developed what was called Hilbert’s proof theory or metamathematics in order to establish the consistency of any formal system, they adopted principles that even intuitionists felt could be accepted[35], and this is hardly surprising.
But the disagreement between formalism and intuitionism is also obvious:
Hilbert could not tolerate intuitionists’ “destruction” of the achievements already attained by classical mathematics. He insisted on retaining “the infinite” in mathematics, even though it could not exist anywhere outside the human mind, and was not even legitimate within “logic.” Hilbert said: “The infinite still very likely occupies a legitimate place in our thinking, functioning as an indispensable concept.”[36]
To preserve the actual infinite as a legitimate concept, while knowing full well that its meaning could not be “lodged” in the empirical world of reality or in a transcendent world of ideas, Hilbert also did not choose to side with apriorism. So where, exactly, could the meaning of these mathematical concepts be “lodged”? Hilbert’s choice was: simply not to seek the meaning of mathematical concepts anywhere at all! Or rather, the meaning of mathematical concepts exists only in the mathematical system itself.
Thus Hilbert set about constructing a formal system stripped of “meaning.” In Hilbert’s view, the intuitive connection that mathematical concepts — such as line and point — might originally have with certain real objects was not important. “Line” and “point” are just symbols; the meaning of these mathematical symbols can only be manifested in the relations among the symbols themselves. For example, a mathematical statement such as “between any two points there exists a line,” if replaced by other symbols, say “between any two chairs there exists a table,” would not change the statement’s meaning in mathematics at all! — “Hilbert decided to express all logical and mathematical statements in symbolic form. These symbols, although they can express the perception of intuitive meaning, cannot find interpretation in the formal mathematics he proposed. Hilbert hoped to include symbols that could even represent infinite sets, but they have no intuitive meaning. These ideal elements, as Hilbert called them, are necessary for building all of mathematics, so their introduction is justified, even though Hilbert believed that in the real world only finitely many things exist, and things are composed of finitely many elements.”[37]
In other words, Hilbert avoided the problem of the meaning of infinity by stripping the whole of mathematics of reality and intuitive meaning. Like intuitionism, Hilbert held that the actual infinite has no meaning either in reality or in intuition; yet he accepted the actual infinite by abstracting away the intuitive meaning of mathematics as a whole! As Brouwer said: “For formalists, the precision of mathematics lies only in developing methods for series of relations, and has nothing to do with the meaning one attempts to give to these relations or to the entities involved in them.”[38]
In this way, each mathematical element expresses itself only through its mutual relations with the other elements in the mathematical system — “these statements, which are ‘theoretical’ in the mathematical sense, Hilbert calls ‘ideal elements,’ and compares their introduction to the introduction of the ‘points at infinity’ in projective geometry: they are introduced as a convenience that makes the theory of the matters you are truly concerned with simpler and more elegant. If their introduction does not lead to contradiction, and if they have these additional uses, then the introduction is justified: thus one seeks a proof to carry out for the complete system of first-order arithmetic.”[39]
That is to say, the introduction of actual infinite sets is just like the introduction of points at infinity: it is merely for the sake of making the whole theory more elegant, and for making other operations that have real significance more convenient. Therefore, the urgent problem formalists must solve is to prove that introducing these meaningless convenience concepts does not create inconsistency in the system. Accordingly, once Hilbert founded formalism, he immediately threw himself into research on the continuum problem. He remarked: “The theory I have developed provides a solution to the continuum problem. Proving that every mathematical problem is solvable is the first and most important step toward solving this problem. …”[40]
However, Hilbert’s grand claim that he had “solved the continuum problem” was never truly realized, and was later to be shown impossible forever! Hilbert’s dream was destined to end in failure.
At this point I shall not continue the discussion of the continuum problem, not only because questions concerning the consistency and completeness of formal systems will also be addressed later, but more importantly because: whether the consistency of a formal system can be proved is, in the disagreement between intuitionism and formalism, fundamentally irrelevant.
Intuitionism holds that even if the consistency of Hilbert’s formalized mathematics were proved, such a theory — that is, such formalized mathematics — would still be meaningless. Weyl complained that Hilbert “saved” classical mathematics through “a radical reinterpretation,” that is, by formalizing it, while in fact stripping away its meaning. “This thereby transfers it in principle out of the intuitionistic system, forming a formula game played according to fixed rules.” “Hilbert’s mathematics may perhaps be a wonderful formula game, even more interesting than chess. Since its formulas do not possess an accepted objective meaning by which they can represent intuitive truth, what relation does it have to knowledge?”[41]
Intuitionists continued to emphasize that they rely on the meaning of mathematics to determine whether it is correct, whereas formalists (and logicalists) are dealing with an ideal or meaningless supernatural world. Brouwer said: “The arbitrary use of Aristotelian logic leads to formally valid but substantively meaningless propositions; classical mathematics, by renouncing the meaning in many logical structures, has renounced reality.”[42]
Who, after all, is the “anti-realist”? The founder of intuitionism, usually regarded as anti-realist, is actually accusing his opponent of “renouncing reality”! If we had already pinned the label of “anti-realist” on intuitionism in advance, then at this point we would surely have difficulty understanding Brouwer’s line of thought. But in fact, intuitionism’s position is quite consistent: what intuitionists oppose is Platonic realism, and one major reason they oppose mathematical realism so strongly is precisely that they value and emphasize the connection between mathematics and (natural) reality! Of course, intuitionism’s claim is that the “precision” of mathematics exists within the human mind, rather than, as formalism would have it, “existing on paper.”[43] Yet intuitionism does not claim that “the human mind has no relation to reality”; on the contrary, it is precisely trying to emphasize the inseparable connection between mind and reality. Therefore, to say that “mathematics needs to be connected with reality” is not at odds with saying that “mathematics originates in the human mind.”
The unwillingness to sever the relation between mathematics and reality is precisely the main reason intuitionism opposes formalism. Of course, intuitionistic logic can also be, and needs to be, “formalized”; however, this does not mean that intuitionism accepts formalism. Heyting, Brouwer’s student, was the first to formalize intuitionism and to truly establish a system of intuitionistic logic; yet it was Heyting himself who clearly pointed out: “It is also impossible to reconcile formalism and intuitionism by formalizing intuitionistic mathematics. Indeed, even in intuitionistic mathematics, the completed part of a theory may be formalized. It would be very useful to investigate for the moment the significance of this formalization. We may regard this formal system as the linguistic expression of mathematical thought in a specially appropriate language.”[44] For intuitionism, the fundamental role of so-called “formal language” is exactly what the name says — “language.” The function of “language” is to facilitate communication between people of “thought”; a more perfect language can express thought more clearly and accurately, but what truly possesses meaning and what truly matters is always thought itself. Moreover, Heyting went on: “If we adopt this point of view, then we crash violently against the obstacle that language is fundamentally ambiguous. Since the meaning of a word can never be fixed precisely enough to exclude the possibility of all misunderstandings, in mathematics we can never guarantee that a formal system correctly expresses our mathematical thought. (Author’s note: This claim is further supported by the Löwenheim–Skolem theorem; see later.)”[45]
Formalization is an improvement of language as a tool of communication (logical deduction rules and the like can probably be likened to grammar), and this is important; however, the progress of language lags behind thought. Introducing an alphabet to a primitive tribe will not greatly increase the knowledge they possess. Likewise, the development of mathematics itself always precedes the progress of formalization; the progress of mathematics will make it forever impossible for formalization to be finally completed. Heyting said: “Formalization can be carried out within mathematics, thereby becoming a powerful mathematical tool. Of course, one can never be certain that this formal system completely represents any domain of mathematical thought; at any moment, the discovery of new methods of reasoning will force us to enlarge this formal system. … Intuitionistic activity is independent of formalization, whereas formalization can only follow behind mathematical construction.”[46]
4．The Axiomatization Movement
Whether it was logicism or formalism, whether it was Gottlob Frege, Bertrand Russell, Hilbert, or Ernst Zermelo, the “axiomatization movement” was the common denominator. They all hoped, in the manner of Euclid, to establish a solid and stable foundation for the whole of mathematics. The difference between logicism and formalism lies in the fact that the former held mathematical axioms to be products of logic, while formalism supported the selection of axioms by appealing to consistency within a formal system; but compared with each other, these differences were so small that people often failed to distinguish logicism from formalism at all.
Intuitionism, however, stood outside the entire current and reflected on and criticized the whole trend toward “rigorization” and “axiomatization.”
Intuitionists were not opposed to axioms and the rigorization of deduction as such. “Axiomatization,” in its own sense, is a long-standing mathematical tradition that began with Euclid and was by no means a modern invention. The problem with modern people, however, is that they overemphasized “axiomatization,” became excessively entangled in the development of rigorous logical deduction, and gradually forgot some more important, or at least equally important, factors.
Criticism of the axiomatization movement was by no means limited to intuitionists. In fact, many distinguished mathematicians felt uneasy about this trend.
Richard Courant, a distinguished member of the Göttingen school who later moved to the United States (obviously because the Göttingen school had disintegrated under Nazi persecution), said with evident concern: “The present fashion of over-emphasizing the axiomatic-deductive character of mathematics seems to be in danger of becoming the prevailing one; in fact, the elements of creation and invention, the intuitive factors which play the role of guidance and propulsion, although they often cannot be formulated by simple philosophical formulas, are nevertheless the core of any mathematical achievement, even in the most abstract fields. If perfection of deductive form is the goal, intuition and construction are at least a driving force. There is one view that constitutes a serious threat to science itself: it asserts that mathematics is nothing else than a set of conclusions derived from definitions and axioms, and that these definitions and propositions, apart from the requirement that they be free of contradiction, can be arbitrarily created by mathematicians according to their will, …”[47]
Hilbert was precisely the leader of the Göttingen school in those years. Although he was the founder of formalism, as is often the case in intellectual history — the founder of a certain “-ism” is not necessarily its fervent believer — “Hilbert was certainly not a fanatical believer in natural formalism: thus when his questions concerned the syntactic properties of formal systems, the answers were given by intuitively correct reasoning, and at the same time he explicitly held that any formalization of such reasoning was unnecessary.”[48]
Hilbert’s outstanding achievements in mathematics were by no means limited to “formalism.” His academic career was almost a stage-by-stage specialization in one field after another: in sequence, invariant theory, the theory of algebraic number fields, the foundations of geometry, integral equations, and physics, and only then did he turn to the study of the general foundations of mathematics. Hilbert’s own academic trajectory is itself an excellent illustration of “mathematics precedes formalization.” Hilbert’s “23 Mathematical Problems” are also famous throughout the world. It can be said that, compared with solving any single difficult problem, Hilbert’s act of organizing these problems had a far greater guiding effect on the development of twentieth-century mathematics. He said: “As long as a branch of science can put forward a large number of problems, it is full of vitality; the lack of problems presages the decline and cessation of independent development.”
Axiomatization is only one of the “problems” in mathematics; it may be extremely important, but it is not necessarily the most important, much less the only important one, and there is certainly no problem of the sort that “if the foundational problem is not properly solved, mathematics can hardly advance.”
Under the wave of axiomatization, logicization, and formalization, many people did not retain the clear-headedness that those great mathematicians still possessed. They gradually conflated the image of “mathematical creation” with that of “axiomatic systems” and “logical deduction.” They knew only that mathematics was the most rigorous of all human intellectual activities. This is not wrong; however, they forgot that mathematics is at the same time also one of the most creative, the freest, and the most vigorous of disciplines.
Nowadays, similar misunderstandings about mathematics are widespread. In people’s minds, the image of a typical mathematician is probably this: slightly bald, wearing glasses, buried in piles of papers, slovenly, with one black sock and one white sock, writing at full speed while carrying out calculations… In short, mathematics has gradually become associated with a certain “stiff” image.
Heyting said: “The intuitionistic mathematician recommends the mathematical work as a natural function of his intellect, as a free and lively activity of thought. In his view mathematics is a product of the human spirit. He uses language, whether natural or formalized, only in order to communicate thoughts, that is, to make himself or others understand his own mathematical ideas. This linguistic accompaniment is not a representation of mathematics, let alone mathematics itself.”[49]
The demand for rigor is reasonable; however, rigor is directed at the language of mathematics. Mathematical discovery (or invention, which makes little difference here) is not obtained through language itself, through manipulating verbal games. As Poincaré said, “Intuition is the tool of invention.” As for rigorous “proof,” its function is nothing more than to follow after mathematical creation, checking and correcting errors, ensuring the reliability of mathematical creation, and serving as a means of “demonstrating” results.
Many distinguished mathematicians were dissatisfied with people’s excessive reverence for “proof.” Godfrey Harold Hardy said with no little irony: “Strictly speaking, there is no such thing as a mathematical proof; … In the end we merely point out certain salient features; … Littlewood and I both use the word proof in this way of something which is emphatically not proof, but a robust salesmanship, a kind of propaganda used in lectures and other places to tempt the young into mathematics.”[50]
Even Whitehead and Russell, the authors of Principia Mathematica, did not have much affection for logic and “proof.” In a lecture, Whitehead said: “Logic has been considered an adequate analysis of the development of thought; in fact, it is not. It is a marvellous instrument, but it needs to have some common sense in the background… The final form of philosophy cannot be based upon the exact expression which is the foundation of special sciences; all exactitude itself is fictitious.”[51]
Even Russell, who followed the program of logicization in full, spared no effort in mocking logic. In Principia Mathematica he wrote: “One of the chief merits of proofs is that they instill in us, gradually, a certain scepticism as to the result proved.” He also said that people’s attempts to found mathematics on a system composed of undefined concepts and propositions are precisely inferred from this very essence of mathematics: conclusions may be denied by contradiction, but they will never be proved. Everything ultimately depends on intuitive understanding.[52]
The mathematician inclined toward intuitionism, Henri Lebesgue, said: “Logic can make us reject certain proofs, but it cannot make us believe any proof.”[53]
Morris Kline, the author of Mathematical Thought from Ancient to Modern Times, a distinguished historian of mathematics, applied mathematician, and mathematics educator, said of logic: “Logic is not a reliable tool for discovering truth; truths that cannot be obtained by other methods cannot be deduced by logic either. … Logic is nothing more than a magnificent edifice of language; the most important advances in mathematics are obtained not through the perfection of logical forms but through the transformation of its basic theories. It is logic that depends on mathematics, not mathematics that depends on logic.”[54] He acknowledged that the result of the rigorization of mathematics was this: “No arithmetic, algebraic, or geometric theorem was altered, and theorems in analysis were simply stated more carefully as required. In fact, all that these new axiomatic structures and rigor accomplished was something mathematicians had already known in essence. Indeed, rather than saying that these axioms can derive some theorems, it is more accurate to say that they can only acknowledge the theorems already in hand. All this means that mathematical development does not rely on logic but on correct intuition. As Jacques Hadamard pointed out, rigorization merely ratifies the spoils of intuition, or, as Hermann Weyl (an intuitionist) said, logic is the hygiene by which mathematicians keep their thought healthy and strong.”[55]
Intuitionism’s comparison of logic to a “hygienic means” is quite apt. Intuitionism has never denied the importance of logic; however, to use a metaphor: intuition is the blood of mathematics — it provides energy and momentum; sense and experience are the food of mathematics, from which mathematicians draw nourishment from nature and experience; while logic is the “hygienic means,” the health product and medicine that strengthen the body and prevent and cure disease, helping mathematics to become stable, mature, and complete. Seen through such a metaphor, the relations among these elements become perfectly clear: intuition gives mathematics life, experience makes mathematics grow, and logic makes mathematics strong. But if one forgets, or even abandons, blood and food, and relies only on medicine, then not only can strength not be maintained, life itself cannot be sustained.
That said, Courant and the others did not seem to need to be overly worried: the tendency toward excessive reverence for deduction in mathematics unleashed by the axiomatization movement seems to have had influence only among philosophers or the public, while it has scarcely disturbed the most distinguished mathematicians, the very people who lead the tide of mathematical development — much as the difficulties empiricism encountered in foundational questions about the inductive sciences, and the philosophers’ endless disputes, have had almost no effect on the pace of the natural sciences.
As Morris Kline put it, “Of course, the forward movement of mathematics has been largely driven by those with extraordinary intuition, rather than by those skilled in producing rigorous proofs.”[56] This was true in ancient times, it remains true in modern times, and it is unlikely to change in the future.
As for the impact of the formalization movement on mathematics? Let us not forget that the philosophers’ “formalization” of logic originally began as a borrowing from mathematics; the great success of mathematical logic was originally built on the basis of “imitation” of mathematics! Formalization itself originally originated from within mathematics. In short, mathematicians seem never to have paid much heed to opinions coming from philosophy.
Morris Kline joked: “A parable aptly summarizes the state of progress concerning the foundations of mathematics in this century. On the banks of the Rhine stands a beautiful castle that has been there for many centuries. In the basement of the castle lives a group of spiders. Suddenly a violent wind scatters a complex web they have painstakingly spun, and so they hasten to repair it in panic, because they believe that it is the web that supports the entire castle.”[57]
Intuitionists still need not feel guilty. The “philosophical current” that truly had a substantive impact on the development of mathematics was precisely intuitionism. Intuitionism directly led mathematics to divide into the two major domains of constructive mathematics and non-constructive mathematics. Quine said: “Those mathematicians who tolerate and use non-constructive methods also admit that, if a constructive proof is found for a theorem that has previously been proved non-constructively, that is a step forward.”[58]
5．Pure Mathematics
Is mathematics itself also an “opponent” of intuitionism? Obviously, intuitionism never wished to be hostile to “mathematics”; rather, it hoped to reexamine and rebuild mathematics. Yet the prohibition of actual infinity and the restriction of the law of excluded middle led it to inflict enormous damage on the large number of achievements already obtained by mainstream mathematics. And this is in fact the real reason many people reject intuitionism. They sometimes do not even have the patience to understand intuitionism’s claims and ideas; they merely see “mathematics” being so “damaged” and find it intolerable.
But is intuitionism’s “damage” to mathematics really so important? Are the things modern mainstream mathematics refuses to give up truly that significant?
We need to begin with another notable development trend in modern mathematics: its “isolation.”
The trend toward isolation is interconnected with the trends toward Platonization, axiomatization, and formalization. As mentioned earlier, the connection between mathematics and physical reality began to be severed. Whether it was Platonism trying to entrust mathematics to a transcendent world, or logicism trying to make mathematics take refuge in logic, or formalism trying to regard mathematics as a self-sufficient symbolic system, their common effort was precisely to make mathematics no longer need to depend on the empirical sciences.
This trend first began with the inversion of the relative status of number theory and geometry, and the entire process of this inversion was precisely related to the rise of modern science.
In all mathematical traditions from Plato to Blaise Pascal, “geometry” was unquestionably the “foundation” of the whole of mathematics. Pascal once said: “Everything that is beyond geometry surpasses my power of understanding.”[59] Since Pythagoras and Euclid, geometry had always been the foundation of arithmetic rather than the other way around. The original axiomatization of mathematics began with geometry. This was because geometry was directly related to the physical world, a mathematical abstraction of physical space and physical entities; people understood arithmetic through the “geometric meaning” of expressions. For example, for a long time mathematicians were unwilling to accept powers above the third, because those had no spatial meaning!
Since Descartes invented analytic geometry, the situation was completely reversed. People first accepted the independence of arithmetic, and regarded the meaning of arithmetic as self-evident as well; for example, in Kant, arithmetic and geometric intuition were of equal status. Finally, the rise of non-Euclidean geometry delivered the decisive blow — people became unable to “understand” geometry any longer! Thus geometry, having lost the support of intuition, had no choice but to take refuge in arithmetic.
And the distinguishing feature of arithmetic is this: it is the farthest removed from reality. In fact, before 1900, among all branches of mathematics, the only one that really deserved to be called “mathematics for mathematics’ sake,” or “pure mathematics” pursued for the sake of “beauty,” was number theory. All the other branches of mathematics, moreover, were not only always closely linked with the empirical sciences, but the very creation of these branches was often originally intended to deal with certain physical problems—even non-Euclidean geometry was no exception! The train of thought that led Gauss and others to propose non-Euclidean geometry was not “Hey! Let’s replace this axiom and see what I can come up with!”, but rather “Is physical space really Euclidean?” For this reason Gauss even made use of three mountains to conduct field measurements.
As M. Kline says: “For the mathematics created before 1900, we may draw the general conclusion: there is pure mathematics, but there are no pure mathematicians.”[60] So-called “pure mathematics” is number theory; yet number theory was often merely a hobby for those great mathematicians, a toy for their leisure time. In fact, even in other areas of mathematics, mathematics itself was often not the mathematicians’ main profession; their occupations were often professor of astronomy, professor of physics, and so on.
Of course, Gauss once said that “mathematics is the queen of the sciences, and number theory is the crown of mathematics,” and the difficult problems in number theory were hailed as “pearls in the crown,” waiting to be plucked by the greatest mathematicians. What does this symbolize? — The “queen” symbolizes “nobility” and “supreme beauty”; the nobility of number theory is intoxicating and enthralling. But this does not mean that Gauss thought number theory was the root of mathematics, or mathematics the root of the sciences. The fact is rather the opposite. Just as the whole of science does not arise from mathematics as queen, but from nature, from experience; so the foundation of mathematics lies in the natural sciences, not in the lofty number theory above.
And in modern times, because the status of arithmetic and geometry has been reversed, number theory has become the “foundation” of the whole of mathematics, and no wonder modern mathematics is becoming more and more “arrogant” and “self-satisfied.”
The result was this: by the end of the nineteenth century, the prevailing view was that every axiom in mathematics is arbitrary, and that axioms are nothing more than the basis of the reasoning from which conclusions are derived. Since axioms are no longer truths about the concepts contained in them, there is then no need to care about the physical meaning of those concepts. When some connection between axioms and reality does arise, that physical meaning can at most serve as a guide to discovery (of truth). This is true even of concepts abstracted from the physical world.[61]
In modern times, the view that mathematics is independent of experience and stands above the natural sciences became the mainstream in the mathematical world. M. Kline said with resignation: “One now frequently hears and reads of the mathematicians’ talk about independence from science. Mathematicians now can say, without hesitation and rather casually, that they are concerned only with mathematics itself and have no interest in science. Although there is no exact statistic, about 90 percent of the mathematicians active on the mathematical stage today ignore science and revel in this blissful state. Despite the support of history and a few dissenting voices. But the trend toward abstractionism, toward generalization for the sake of generalization, and toward the study of arbitrarily chosen problems is becoming more and more intense; to say that this is justified as a way of studying a whole class of problems in order to know more about concrete cases, or as a justified need for abstraction in order to get at the essence of a problem, is nothing but an excuse. Their purpose is simply to study generalization and abstraction.”[62]
As the leader of the mathematical world at the end of the nineteenth century, Poincaré and F. Klein (Felix Klein) both foresaw this tendency and were deeply worried by it. In 1895 F. Klein said: “In the rapid development of modern thought, we cannot but fear that our science faces an ever greater danger of becoming isolated. Since the rise of modern analysis, the close connection between mathematics and the natural sciences, beneficial to both sides, is in danger of being destroyed.”[63]
People may not easily understand why intuitionism should pay such close attention to the relation between mathematics and nature. In fact, this stance has been consistent from the very beginning, starting with the pioneers of intuitionism. Their unwillingness to be severed from nature is precisely one of the important reasons why intuitionists opposed logicism and formalism, perhaps the most important motivation. Poincaré said: mathematics “need not forever stare at its own navel merely for its own sake; it is connected with nature, and it will surely one day return to nature. Then these purely verbal definitions will surely have to be discarded, and one will no longer be deceived by these empty words.” “Logicism must be revised, and one does not know at all what can be retained; needless to say, what is meant here is Cantorism and logicism; genuine mathematics always has its practical purpose, and it will continue to develop according to its own principles, paying no heed to the raging storms outside, and it will certainly, step by step, continue to pursue its accustomed victories, and this will never stop.”[64]
Another pioneer of intuitionism, Kronecker[65], wrote to Hermann von Helmholtz (German physicist and proposer of the law of conservation of energy): “Your sound practical experience and your wealth of interesting problems will give mathematics new directions and new stimuli—one-sided, introspective mathematical reflection leads people into barren lands.”[66]
Some other master mathematicians of modern times have never neglected the importance of the natural sciences either. As mentioned above, Hilbert was, before turning to foundational questions in mathematics, engrossed precisely in problems of physics; and von Neumann also pointed out: “There is no denying that some of the most remarkable inspirations in mathematics, the very best inspirations in those supposedly purest departments of mathematics, have all come from the natural sciences.”[67]
Yet disciples of the masters are always less clear-headed than the masters themselves. “Purification” and “isolation” are indeed the trend in the development of modern mathematics, so much so that among mathematicians, “applied mathematics” has practically become a word of reproach, used by those “pure” mathematicians to denounce people who “neglect their proper work” and “go astray.” On this point, M. Kline himself, as an applied mathematician with considerable attainments in electromagnetism, felt it keenly; with some complaint he said: “Failure to take account of the objective objects it serves is bound to lead to its own self-termination. Pure mathematics itself is not a blissful realm. The purpose of mathematics is to discover things worth knowing, but, as matters now stand, research leads to research, and that in turn leads to research. In today’s temple of mathematics, no one dares ask about meaning and purpose anymore. Mathematics must not be contaminated by the vulgarities of reality; the thick ivory tower blocks the vision of the scholars living deep inside it, and these minds cut off from the world are also content with their isolated condition.”[68]
I have no intention of condemning so-called “pure mathematics.” On other occasions I will offer the highest praise to pure mathematics pursued for mathematics’ sake, just as we praise poetry and art; who can say that artistic activity is meaningless? The meaning of human existence does not necessarily have to be related to “utility.” Yet to leave the natural sciences and the real world behind, and to take solitary delight in oneself within the ivory tower of pure mathematics for the sake of self-satisfaction and self-intoxication—this tendency is indeed unhealthy. Because the connection between mathematics and nature is no longer so clear and definite, and mathematics can no longer provide absolute objective truth, one simply stops caring about their relationship; this is undoubtedly a kind of laziness and evasion.
I have mentioned that intuitionism does indeed hold that mathematics is a construction of the human mind, but it has never claimed that the human mind is unrelated to nature. The a priori intuition of “number” comes from “time”; tracing it back to the source, in the final analysis all science still comes from nature. Science must ultimately return to nature as well.
Several representative figures of intuitionism all attached great importance to applied mathematics, and Weyl in particular made many contributions to mathematical physics. He said: how convincing and how close to the facts are the heuristic arguments and the systematic structure of Einstein’s general theory of relativity, or Heisenberg-Schrödinger quantum mechanics! A genuine mathematics should, like physics, be regarded as a branch of the theoretical structure of the real world, and we should approach the expansion of its foundations with the same seriousness and caution as we do physics.”[69]
Weyl affirmed that mathematics reflects the order of nature. In a conversation, he said: “There is an inherent harmony in nature, and what is reflected in the image in our minds are simple mathematical laws. This is why natural phenomena can be predicted through a combination of observation and mathematical analysis. In the history of physics, this concept of an inner harmony, or rather this dream, has been confirmed again and again, unexpectedly for us.”[70]
We often call intuitionism “a priori theory,” and strictly speaking that is not wrong. But the problem is this: saying that “the concept of an integer is a priori” does not mean that everything is a priori. Moreover, intuitionism never claimed that “the a priori” is absolute or necessary. Just as empiricism acknowledges that experience is fallible, it is not surprising that intuitionism emphasizes the finitude and fallibility of intuition.
In his book Philosophy of Mathematics and Natural Science, Weyl added: “If there is not supported by a faith in truth and reality that is prior to experience, if there is not a continuous interaction between facts and structure and the image of thought, then science will wither and die.”[71] This perhaps makes clear just where intuitionism appeals to the a priori—it is not that they believe the “a priori” can provide absolutely necessary truth; the fact is that the “a priori” provides “science.” Without a priori intuition, intuition, and belief, “science” cannot easily be established. For example, the a priori intuition of integers is the basis on which mathematics can be established; intuition of the law of non-contradiction is the basis of logic; the belief in the “reality” of nature is the basis of science, and so on. But what is provided by the “a priori” is not anything absolutely necessary either. Intuition itself also originates in nature, so nature is the final judge. This means that intuition is also fallible, and intuitionists further point out on this basis that the “intuition” of a principle like the “law of excluded middle” likewise comes from empirical things; this “intuition” arises from the long-term induction of human experience with finite things, yet when the object changes from finite to infinite, this habit of thought no longer works. We can see that the claims of intuitionism have always been tightly interconnected.
As already mentioned above, intuitionism not only values the connection between mathematics and the natural sciences, but also emphasizes that mathematics itself is natural science, that all truth comes from experience, and that nature is the sole and final judge of scientific theory.
Of course, the authority of nature has the support of many thinkers. Quine said: “We will take set theory and mathematics as a whole in the way we take the theoretical part of the natural sciences. These truths or hypotheses are less supported by pure reasoning than they are insightful, systematic contributions to the organization of empirical data in the natural sciences.”[72] M. Kline mentioned: “People now know that the uncertainty of the foundations of mathematics and the doubts about mathematical logic, even if they cannot be resolved, can be bypassed by strengthening its application to nature.”[73] Gödel also thought of a related question: “Aside from mathematical intuition, there exists another criterion—though only a probable one—for the truth of mathematical axioms, namely that they are fruitful in mathematics, and perhaps also in physics.”[74]
Then which side does nature’s criterion favor? It seems that emphasizing the applicability of mathematics is disadvantageous to intuitionists, because in terms of the “richness” of results, intuitionistic mathematics cannot compare with classical mathematics.
However, first it must be pointed out: if one considers only the results of pure mathematics itself in the aspect Gödel referred to as “fruitful in mathematics,” then this is undoubtedly extremely unfavorable to intuitionism, because almost all the results that intuitionistic mathematics can obtain can also be obtained by classical mathematics. But to refute intuitionism on the grounds that it “damages the existing mathematical system too much” is not convincing. Think of the current craze for astrology—its history is almost as long as that of astronomy, and today the number of “researchers” and “students” of astrology in the world probably exceeds that of astronomers (at least in the United States, astrologers vastly outnumber astronomers); its theories are undoubtedly “highly developed” as well. But scientists would think that all of that went wrong from the very start, and that its most basic presupposition—“a person’s fate is related to the stars”—is mistaken! A theory developed under false presuppositions can only mean that the more results it produces, the more falsehoods it contains.
If one denies the premises of astrology, the enormous theoretical edifice painstakingly built over thousands of years would collapse, and countless astrologers would have “nothing to do”; the sacrifice would be too great! — Is such a defense of astrology reasonable? Then what about a defense of mathematics?
Intuitionists believe: intuitionistic mathematics is indeed far poorer than classical mathematics, but it contains more truth!
Then what about applications in physics? Objectively speaking, intuitionistic applied mathematics really cannot compare with the outstanding achievements of classical mathematics, but the reason is still simple: nearly all the mathematical results of intuitionism can be regarded as results of classical mathematics. Perhaps, as I will mention below, intuitionism will enjoy certain advantages in quantum physics, but let me concede here in advance: these possible advantages are mainly in thought and understanding, not in practical utility.
But are there really many practical applications that can only be obtained by acknowledging actual infinity? The calculus, which is widely used in physics, can be explained by potential infinity without having to appeal to actual infinity—this was precisely the insistence of the founders of calculus. As for other counterintuitive conclusions, such as “a sphere can be cut into finitely many pieces and reassembled by simple translations and rotations into two identical spheres”—that is, the Hausdorff-Banach-Tarski paradox—once the axiom of choice is accepted, this becomes a mathematical theorem—could it possibly have any practical application at any time? If one could really develop a technology of creating something out of nothing that is even more tempting than alchemy, that would indeed be wonderful, but everyone knows that this is impossible. Many mathematicians do not even think this theorem presents any trouble at all; on the contrary, they often use it to emphasize the marvel of mathematics: “Look, in the real world such absurd things, yet in the mathematical world they have been proved. How magical! How wonderful!” I admit that if one wants to show laypeople some fascinating cases in mathematics to arouse their curiosity, paradoxes like the division of the sphere are indeed excellent material; but if one wants to explain to the officials who allocate research funding whether my mathematical research has any ability beyond self-entertainment, then compared with “I can turn stone into gold!”, “I can make one become two!” is not really something to brag about either.
M. Kline, with the authority of a historian of mathematics, further reminds us: “We observe that many fields which in the past were energetically and enthusiastically pursued, and which were praised by their adherents as the very essence of mathematics, were in fact nothing more than temporary hobbies, or left only a slight influence on the entire mathematical journey. The confident mathematicians of the first half of the century may have believed that their work was the most important, yet the place of their contributions in the history of mathematics is still uncertain today.”[75] As for the rich and enormous “achievements” that modern mathematics has obtained from actual infinity, it is too early to be smug about them.
For intuitionists, giving up those seemingly beautiful results “not only involves no sacrifice, but is a great victory, because logic, mathematics, epistemology, semantics, and metaphysics have finally attained harmonious consistency.”[76]
Finally, it is worth pointing out that the contradiction between intuitionism and mainstream mathematics is not irreconcilable. According to the intuitionist position, classical mathematics is also worth preserving in another sense. Many valuable theorems were first discovered through a non-constructive proof, which drew attention; after further research, a constructive proof was found. Even if one says that non-constructive proofs are “illogical,” intuitionism itself emphasizes that logic is not everything, and that intuition and creativity have great significance in mathematical development. Intuitionists can at least acknowledge the heuristic significance of classical mathematics.
In fact, many advocates of intuitionism, such as Kronecker, Brouwer, and Weyl, in their actual work of mathematical research often, as Poincaré put it, “forgot their philosophy”; and many of the mathematical achievements they themselves attained were precisely things that did not conform to intuitionistic mathematics. We should not accuse them of saying one thing and meaning another. In fact, they were still consistent in thought and congruent in word and deed: they took intuition as the motive force and foundation of research, nature as the pursuit and destination of research, and as for logic, let it be used at the very end to check things afterward.
III. Developments Favorable to Intuitionism
1. Non-Euclidean Geometry
As mentioned above, the establishment of non-Euclidean geometry in a certain sense was precisely what triggered the modern axiomatization movement, the fuse that brought out the dispute over the foundations of mathematics. In fact, understood in different ways, non-Euclidean geometry can also be regarded as favorable to logicism or formalism. Yet non-Euclidean geometry is often seen as a fatal blow to a priori theory; for intuitionists sympathetic to Kant, this would seem to be an unfavorable factor?
If one distorts the intuitionist claim into “truth is provided by intuition,” then the emergence of non-Euclidean geometry is undoubtedly a blow to that claim. In fact, it is precisely this understanding that led many commentators to conclude that Kant’s a priori theory was a failure. Of course, in Kant himself, he probably did indeed hope to find in intuition a support for absolute truth that was certain beyond doubt, and in Kant’s time it was taken for granted that space was Euclidean. Kant’s defense of the necessity of Euclidean geometry was also understandable. However, is it not too easy to refute a priori theory by saying “space is not Euclidean, therefore Kant was wrong”?
Kant’s contribution lies in this: he thought that the “certainty” of knowledge could also be provided “a priori.” If the argument goes: “Kant thought that the concepts of geometry came from the ‘a priori,’ and Kant thought that the only correct Euclidean geometry was in fact not absolutely correct—it was merely an approximation to real space—so to claim that the concepts of geometry come from the ‘a priori’ is wrong!”—if such a rebuttal can hold, then the following rebuttal can also hold: “In the past people thought that physical knowledge came from ‘experience,’ and in the past people thought that the only correct Newtonian mechanics was in fact not absolutely correct—it was merely an approximation to the real world—so to claim that physical knowledge comes from ‘experience’ is wrong!” Why is it that fewer people agree with the latter set of claims, while many people think that the former set of claims really does refute a priori theory?
Non-Euclidean geometry cannot be used as a powerful refutation of a priori theory, let alone as a threat to intuitionism.
In fact, non-Euclidean geometry can instead become support for intuitionism. Intuitionism emphasizes intuition, emphasizes that mathematics is a creation of the human mind, but it also emphasizes that mathematics and logic, like the natural sciences, are fallible! Precisely because they are creations of the human mind, they must be fallible, mutable, and developing. Non-Euclidean geometry is exactly such an example: it shows that certain things long regarded by people as beyond doubt are in fact alterable, and even incompatible with physical reality. In particular, the subversion brought about by non-Euclidean geometry occurred within mathematics itself, which, compared with revolutions in physics or astronomy, has a more special significance.
People learned that natural science does not provide absolute truth, yet still fantasized that mathematics remained certain beyond doubt, absolute, and unchangeable; but now people have learned that mathematics is not absolute or unique either. The last group of people chose to place their faith in logic, still fantasizing that absolute certainty could be found in logic. But why not be more honest, more humble? Why not admit that with a finite human mind one can forever grasp only the relative, and cannot grasp absolute truth? Absolute truth is nowhere to be found; this is determined by human finitude. Why must mathematics or logic be regarded as “exceptions”?
Although intuitionism does not question the law of excluded middle for the following reason, we can indeed make this association: just as the parallel postulate is not absolute, neither is the law of excluded middle, because both derive from abstraction from experience.
By the way, since we are mentioning it: has relativity already proved that real space is non-Euclidean? Actually, no. Poincaré had already grasped all the intellectual preparations for constructing relativity before Einstein did (Einstein is just like Newton, a synthesizer of many achievements). He had long since discovered that treating physical space with non-Euclidean geometry might be more “convenient”; however, this does not mean that Euclidean geometry cannot be used to deal with spacetime in conformity with relativity. So long as one assumes that objects (for example, rulers used to measure distance) will contract under the action of gravity, one can equivalently describe non-Euclidean space with Euclidean geometry; the difference between the two lies only in the convenience of calculation. The questions of “equivalent description” and of the relation between mathematics and physical reality will be discussed further below.
2. Various Logical Paradoxes
It was precisely the discovery of Russell’s paradox that triggered the so-called “crisis” of the foundations of mathematics, and the various syntactic and semantic paradoxes associated with it plunged logicians, mathematicians, and philosophers into deep difficulty. Type theory, axiomatic set theory, the NBG system, Quine’s NF and ML systems, and so on: scholars exhausted every trick in the book to eliminate paradoxes. These efforts are all quite meaningful. But what we are discussing here is the special status of the intuitionist solution.
First, as Benacerraf pointed out: “One must consider that these truncated forms of paradox arise by exploring the consequences of various demands of absolute Platonism.”[77] Set-theoretic paradoxes are indeed related to a Platonist understanding of mathematical objects, and intuitionism, through its constructive mathematics, eliminates Platonism in the cleanest and most thorough way. By prohibiting any form of “actual infinity,” it very easily dissolves set-theoretic paradoxes. Compared with other theories that more or less carry a certain degree of ad hocness—that is, theories that seem to have been specially built merely to cope with paradox—the advantage of intuitionism is obvious. No one can deny the “clean and decisive” way intuitionism resolves set-theoretic paradoxes; people simply reject intuitionism’s solution to paradox from the standpoint that “the destruction is too great.”
Moreover, what many people have not noticed is that intuitionism does not merely resolve set-theoretic paradoxes, that is, various syntactic paradoxes; it also once and for all prevents various semantic paradoxes.
What are called semantic paradoxes are mainly problems posed in logic and language, such as the liar paradox and its various variants. Its most concise form is: “This sentence is false.”
Restricting the law of excluded middle leaves semantic paradoxes nowhere to stand. It should be noted that modifying the law of excluded middle may not even solve semantic paradoxes. The facts show that “all sorts of schemes for dealing with paradox using many-valued logic and the theory of truth gaps are wishful fantasies, because none of them can escape various kinds of ‘strengthened liar paradoxes,’ such as …… ‘This statement is not true.’ ”[78]
Many-valued logic is the true opponent of the law of excluded middle, yet intuitionistic logic is not many-valued logic. It remains binary logic. As stated earlier, its special feature is only that it raises stricter requirements for “what kinds of sentences it makes sense to talk about truth values for?”
Thus intuitionism accepts reductio ad absurdum, but does not accept proof by contradiction; that is, it accepts “if P, then not-not-P,” but does not accept “if not-not-P, then P.” Here, with respect to “This sentence is false”: suppose this sentence is true, and we get that this sentence is not true, a contradiction. From this, it is permissible to say that this sentence is not true. However, if from the premise that this sentence is not true one then substitutes and again derives a contradiction, this can only show that “this sentence is not true” is wrong, and cannot be pushed backward to conclude that this sentence is true. So what is ultimately obtained is only “this sentence is not true,” “this sentence is not not true,” “this sentence is not not not true”…… and one will never arrive at a contradiction. The key point is this: when intuitionism says “this sentence is not true,” it is not asserting that the sentence “this sentence is not true” is itself true. In fact, “this sentence is not true” means that the sentence is either false or “undecidable.” But once one declares “P is undecidable,” this sentence itself can also perfectly well be undecidable. When one admits that the sentence “P is undecidable” is undecidable, that is precisely to say that “P” is indeed undecidable. Likewise, the sentence “‘P is undecidable’ is undecidable” can also be undecidable. If this goes on, it merely leads to a series of sentences of the form “ ‘P is undecidable is undecidable is undecidable……’ is undecidable,” all of which are undecidable, but this never encounters a contradiction anywhere.
For those accustomed to classical logic, it is natural to underestimate intuitionism’s power to resolve paradoxes. Putnam had a similar misunderstanding; in passing, he once pointed out a “contradiction” in constructivist thought:
In “Mathematics Without Foundations” he wrote: As a side remark, I want to point out that the following two principles, which seem to be accepted by many, can be shown inconsistent by Gödel’s theorem:
(I) Even if some arithmetical (or set-theoretic) statements lack truth values, any statement to the effect that an arithmetical (or set-theoretic) statement has (or lacks) a truth value is itself always either true or false (i.e. this statement either has a truth value or lacks one);
(II) All and only decidable statements have truth values.
For the statement that a mathematical statement S is decidable may itself be undecidable. So according to (II), the statement “S is decidable” lacks a truth value. But according to (I), the statement “S has a truth value” has a truth value (in fact false, because according to (II), if S has a truth value, then S is decidable, and if S is decidable, then “S is decidable” is also decidable). Since the statement “S has a truth value” is equivalent to “S is decidable,” “S is decidable” must also be false. But the statement “S is decidable” lacks a truth value. Contradiction.[79]
First of all, it is obvious that in exposing this “contradiction,” Putnam is using classical logic from beginning to end; extracting intuitionist claims and then analyzing them with classical logic can of course lead to contradiction. On the one hand, intuitionists, unlike logicists, do not place the decidability of logic at the highest level, believing that everything correct is derived from logic; intuitionist claims, such as “only decidable statements have truth values,” do not require being situated within their logical system. “Truth value” is a concept in logic, but logic does not contain all truth, and the meaning of sentences is not provided by their truth conditions either. We should regard “only decidable statements have truth values” as reasonable, but there is no need to assign it a truth value, let alone bring this proposition into the system of logical deduction. On the other hand, Putnam’s analysis does not restrict the law of excluded middle. When he argues that “S has a truth value” has a truth value, he merely assumes in advance that it has the truth value “true,” derives a contradiction, and then declares that it has the truth value “false.” But intuitionism holds that the law of excluded middle is valid only under the condition of “having a truth value” (being decidable). As already pointed out above: even if “P is undecidable” is itself undecidable, this statement is still valid.
Even according to intuitionist thinking, many newly discovered kinds of paradox, such as the “surprise exam paradox” and other pragmatic paradoxes, can also be easily dissolved. There is no need to expand on that here. The basic trick remains the restriction on proof by contradiction. Similarly, intuitionists do not require assigning a truth value to “I do not know (can infer) P”; “I do not know that I do not know P” and “I do not know P” are not contradictory.
Chen Bo said: “The emergence of paradox is related to three factors, namely self-reference, negative concepts, and totality and infinity. Although one cannot say that these three factors necessarily lead to paradox, paradoxes generally contain these three factors.”[80] If “totality and infinity” really are one of the necessary conditions for the production of paradox, then intuitionism has already eliminated any form in which this condition might obtain. It is not at all incredible to think of intuitionism as a “one-stop solution” for resolving various paradoxes. On the contrary, once proof by contradiction is restricted in this way, and once a statement reduces to being “undecidable” and no longer gives rise to conflict, what paradox could still arise? That is difficult to imagine.
In short, while other logicians, mathematicians, and philosophers were frantically scrambling to cope with paradoxes, intuitionism won almost effortlessly and overwhelmingly!
3. Gödel’s Theorem and the Continuum Hypothesis
It was mentioned earlier that the rigorization, logicization, axiomatization, formalization, and so on of mathematics—“all that these new axiomatic structures and rigor have done is, in essence, what mathematicians already knew in the past. Indeed, rather than saying these axioms can deduce some theorems, it is better to say they can only acknowledge the theorems that are already there.”[81] But this statement now seems inaccurate. The axiomatization movement did indeed bring about some things mathematicians did not know before, and even things of the most intellectually shocking and revolutionary kind—for example, the world-famous Gödel incompleteness theorem, which probably counts as the highest achievement in the field of mathematical foundations. But this seems even more like a kind of irony……
In 1931, in an article titled “On Formally Undecidable Propositions of Principia Mathematica and Related Systems,” Gödel revealed his astonishing conclusion.
Gödel’s incompleteness theorem shows that if a formal theory is strong enough to include number theory and is consistent, then it must be incomplete. That is to say, if a formal system is not self-contradictory, then there must be meaningful statements in the system that cannot be proved within that system. This directly destroys Hilbert’s dream that “all problems are solvable.” At the same time, a corollary of this theorem shows that the consistency of any mathematical system encompassing the integers cannot be established by the logical principles adopted by the several foundational schools of that time—including logicism, formalism, and axiomatic set theory.
This result may also have come as a surprise to intuitionism. In earlier criticism of formalism, intuitionism held that even if your formal system were indeed proved consistent, that would not mean much. But now intuitionism’s advantage is undoubtedly even greater.
To put it more technically, according to Gödel’s theorem, “a consistency proof of a formalized theory cannot be represented in the formal system under consideration. —…… It is impossible to prove the consistency of a formalized theory that can express an arithmetic proposition by elementary combinatorial methods.”[82]
And intuitionism posits the a priori character of the integers; their claim is that integers, rather than logic, are the most basic thing, and intuitionistic constructive mathematics is undoubtedly more reliable. This made intuitionism a way out for supporting the consistency of mathematics. Benacerraf noted: “If this is so, then we reach the conclusion that a more powerful method than elementary combinatorial methods is needed to prove the consistency of axiomatic number theory. A new discovery by Gödel and Gentzen leads us toward just such a more powerful method. They have shown (independently of each other) that the consistency of intuitionistic arithmetic implies the consistency of axiomatic number theory. This result was obtained using Heyting’s formalization of intuitionistic arithmetic and logic, and the argument is carried out by elementary methods in a rather simple way. To draw the conclusion that axiomatic number theory is consistent from this result, it is enough to assume the consistency of intuitionistic arithmetic.”[83]
Of course, even without the help of intuitionism, people were all convinced that mathematics could not possibly fail to be consistent, and so the question became: mathematics must be incomplete!
A concrete case of incompleteness was later proven: that was the “continuum problem” which Hilbert had thought he had already solved. Through the efforts of Gödel and others, it was shown that whether one accepts or rejects the continuum hypothesis and the axiom of choice (the continuum hypothesis implies the axiom of choice), both are consistent within the original mathematical system. In other words, the continuum hypothesis and the axiom of choice are undecidable; like Euclid’s parallel postulate, one can either accept the continuum hypothesis, or not accept it in a different way, and in either case construct a consistent mathematical system.
What is the continuum hypothesis? Simply put, it is: “There does not exist an infinite set of real numbers that is equivalent neither to the set of integers nor to the set of all real numbers.” In other words: there does not exist a set whose elements are more numerous than the set of integers but fewer than the set of real numbers. Of course, in intuitionism this problem does not exist at all, because they simply do not recognize the legitimacy of “the set of real numbers” as a completed infinite set; at most, they conditionally acknowledge discussion of “all integers.”
But for classical mathematics, the continuum problem is of major significance. In classical mathematics, the formulation of the continuum problem is entirely legitimate; it is built on the classical mathematical realism that treats concepts such as “the set of all sets of real numbers,” “the set of all real numbers,” and so on as completed sets. Moreover, this does not involve self-reference or universal sets that might lead to paradoxes. According to the classical view, the Continuum Hypothesis has a truth value: it is either true or false. The Axiom of Choice is the same. If one could handle infinitely many sets all at once, as God might, then it ought to hold: the Axiom of Choice says that if a set is divided into a number of parts, one can pick an element from each part to form a new set. This is obviously valid when the “number” is merely finite. For example, we can divide the students of a school into a number of classes, and then select the class monitor from each class to form a “set of all class monitors.” But when the number of classes is infinite, this is no longer something that can be taken for granted within the mathematical system. Denying this axiom does not produce a contradiction. On the contrary, accepting the axiom does lead to some seemingly absurd conclusions, such as the Banach–Tarski paradox mentioned earlier, the bizarre sphere-doubling theorem.
To deny it is to violate intuition and nature; to accept it is to be free of contradiction and valid in derivation—these are precisely the customary reasons formalism gives for defending its chosen “axioms.” Yet now, in the case of the Axiom of Choice, one may also say: accepting it leads to things that violate intuition and nature, while denying it is contradiction-free.
Because the Axiom of Choice, though widely used in classical mathematics (and often used unreflectively before Zermelo made it explicit), does indeed bring many puzzles and oddities, it has become the second most discussed axiom after Euclid’s parallel postulate. Opposition to the Axiom of Choice is by no means limited to intuitionists. However, the intuitionists’ opposition to the Axiom of Choice is natural and consistent with their position. Before the Axiom of Choice and the Continuum Hypothesis had even left other scholars scrambling, before these issues had yet to be noticed, intuitionism had already preemptively guarded against them. The successive discoveries of set-theoretic and logical paradoxes, Gödel’s theorem, the Continuum Problem, and the Löwenheim–Skolem theorem to be mentioned below, can be said to have left mathematicians and philosophers of various foundational schools “too busy to cope.” Although one cannot say that the other strategies were all unsuccessful, there is always a sense of post hoc repair and regrouping. In terms of the coherence and consistency of thought and doctrine, no school can yet compare with intuitionism, which may be said to “meet all changes with one’s unchanged stance.”
4. The Löwenheim–Skolem Theorem
This is “a line of research begun by Leopold Löwenheim in 1915 and simplified and completed through a series of papers published by Thoralf Skolem between 1920 and 1933, revealing another defect in mathematical structures.”[84]
The Löwenheim–Skolem theorem says that every satisfiable first-order theory (in a countable language) has a countable model.[85]
In other words, when we choose a countable model for the language of set theory, some sets will appear that are uncountable in a “relative” sense.
A set that is “countable” from the standpoint of one model may, from the standpoint of another model, be a geometrically uncountable set. As Skolem summed it up, “Within axiomatic set theory, even concepts such as ‘finite,’ ‘infinite,’ and ‘simple infinite sequence’ turn out to be relative.” … The “intuitive concept of a set” is something that a formal system cannot “capture.”[86]
The above exposition is a bit specialized; M. Klein offers a more vivid analogy: “Suppose that people intend to draw up a table of characteristics and believe that it can characterize, and only characterize, Americans, but to everyone’s surprise someone discovers a kind of animal that possesses all the characteristics listed in the table, yet is completely different from an American. In other words, it is in fact impossible to describe a unique mathematical object by means of an axiomatic system. … A set of axioms can permit far more interpretations than people expect, and these interpretations are essentially different.”[87]
Skolem’s paradox is the mathematical version of “following the map to catch the horse”: according to a horse-identification manual that was believed to describe the features of a thousand-li horse, one ends up discovering that a toad actually satisfies all the characteristics listed in the book! The Löwenheim–Skolem theorem means that no matter how carefully and exhaustively one writes this horse-identification manual, no matter how meticulously one characterizes mathematical objects, one can never avoid such substantive ambiguities.
This powerfully illustrates the intuitionist claim: every “language” must unavoidably have a relative “vagueness,” and even mathematics and logic are no exception!
Even those who oppose intuitionism can hardly deny that this is a result supportive of intuitionist claims. After repeatedly reconsidering his own conclusions, Skolem stated in a 1923 paper that he opposed taking the axiomatic method as the foundation of set theory. Even von Neumann, in 1925, expressed agreement with the view that his own axioms and all other axiom systems concerning set theory bear “the mark of unreality, … set theory cannot be axiomatized unconditionally. … Since there is no axiomatic system for arithmetic, geometry, and so on, and since no such assumption is made for set theory, there must likewise be no unconditional axiomatization of infinite systems.” He continued: “This situation, for me, is yet another argument in favor of intuitionism.”[88]
Putnam says, “The two extreme views—Platonism and positivism—seem both to be able to draw comfort from the Löwenheim–Skolem paradox; only the ‘moderate’ view, which tries to avoid the mysterious ability to ‘grasp’ ‘mathematical objects’ while retaining the classical concept of truth, has serious difficulties.”[89] “The astonishing fact is that, to the mathematical intuitionist, the whole problem simply does not exist. This may not have been a surprise to Skolem: his conclusion was precisely that ‘most mathematicians hope that mathematics ultimately deals with feasible computational operations, rather than being composed of formal propositions about certain objects.’”[90] Putnam further acknowledges that even when intuitionists succeed in mastering a sufficiently rich language as the metalanguage for some theory T, and can even define “true in T” in the Tarski manner and discuss “models” in T, and so on, the “Skolem” paradox still will not arise again! For in intuitionism: “Reference is given through meaning, and meaning is given through verification procedures rather than through truth conditions. The ‘gap’ between theory and ‘objects’ simply disappears—or rather, it never appeared in the first place.”[91]
Putnam’s final choice is to move toward some so-called “broadened intuitionism,” in which he tries both to preserve the main ideas of intuitionism and to avoid “damaging” classical mathematics. But can this dream of having it both ways be realized? I will not elaborate further here. In short, the result is this: the non-intuitionists are moving closer to the intuitionists, while the intuitionists need only keep re-examining their own consistent claims.
Nature is the final judge. We see that only when Einstein proposed relativity, showing the significance of non-Euclidean geometry for dealing with physical space, did non-Euclidean geometry truly become firmly established. So, is there any sign that limitations on the law of excluded middle have also been “confirmed” in physics?
First, one should acknowledge the statement cited by Heyting: “It is too early to stress the few and weak indications that it may be of some use in physics,”[92] and I do not intend to “stress” anything either. But there are indeed some “indications” that at least intuitionist philosophical ideas have a certain significance in understanding the “paradoxes” of quantum physics. In this section, I will first offer the simplest introduction to some problems in quantum mechanics, and in the next chapter, when discussing the relation between mathematics and physical reality, I will continue to examine these examples.
Quantum mechanics was a true physical “revolution” of the twentieth century, and its subversive impact was even more profound than that of relativity. Or rather, relativity does not really count as “subversive” at all. Just as Newton was not a subversive revolutionary but rather a great synthesizer and systematizer of thought, Einstein’s status is similar. Therefore, relativity caused hardly any controversy among physicists (though the opposite was true among the general public), whereas the shock of quantum mechanics was obviously much greater. Einstein was precisely the first to stir up the controversy, and that controversy has still shown no sign of becoming clarified to this day.
The difficulty of understanding quantum mechanics is universally acknowledged. Niels Bohr, one of the founders of quantum mechanics, said: “Anyone who is not shocked by quantum theory has not understood it.” Richard Feynman, a physicist who in the second half of the twentieth century could be ranked alongside Einstein in the first fifty years, said: “If you think you understand quantum mechanics, you don’t understand quantum mechanics.” John Wheeler said: “The quantum is the deepest mystery we know. I have never been more puzzled than I am today.” In fact, quantum mechanics is so astonishing precisely because there, classical logic seems to fail completely!
To save space, I will mention here only the most representative experiment: the double-slit interference experiment:
“Wave-particle duality” is probably at least a concept known to all and sundry among intellectuals. It says that elementary particles, such as electrons and photons, are “both waves and particles.” The following simple experiment can display this remarkable duality:
The double-slit interference experiment is a simple experiment encountered already in middle school physics, and it reveals the wave properties of light: if a beam of light is sent through two parallel slits, and a screen is properly placed behind the slits, one can observe alternating bright and dark interference fringes. The dark fringes are caused precisely by the cancellation of the crests and troughs of the coherent waves passing through the two slits. Interference fringes can also be observed with electrons, proving that electrons are waves as well.
Yet we also know that energy is discontinuous; light is made up of countless tiny, indivisible “photons,” and electrons are of course the same. So what if one emits only one particle at a time? If it passes through the double slit, it will leave a small dot on the screen, showing its particle nature.
The following thought seems obvious: if a particle passes through the double slit and reaches the screen, then it “either passed through slit A or passed through slit B.” Just as a particle can leave only one dot on the screen, it cannot pass through both slits at the same time!
Then, if we release 100 particles one after another onto the screen, should the result not be equivalent to first closing slit A and letting 50 particles pass through slit B, then closing slit B and letting 50 particles pass through slit A?
The above analysis is entirely logical. Yet the fact is: although one particle striking the screen leaves only a single dot, enough dots on the same screen will reveal an overall pattern. The result of the experiment is that if both slits remain open and enough particles pass through one by one, the pattern formed on the screen is still an interference pattern!
We know that interference fringes are a wave phenomenon, the result of waves passing through both slits simultaneously and interfering with each other. But how does a single particle produce interference after passing through the double slit? Does it interfere with itself? Can one particle pass through both slits at the same time?
However, if we try to check how the particle passes through the double slit—for example, by installing a detector at one of the slits—the result is still: the particle “either passed through slit A or passed through slit B.” But once this observation is made, the final result on the screen can no longer be an interference pattern; it becomes a simple superposition, that is, “first close slit A and let 50 particles pass through slit B; then close slit B and let 50 particles pass through slit A”!
Perhaps even more puzzling is that an improved experiment—for example, replacing the double slit with semitransparent mirrors—the so-called “delayed-choice experiment”—can postpone the decision about whether to perform observation midway until after the particle has already passed through the double slit! In other words: whether interference ultimately occurs, that is, “how the particle passes through the double slit,” can be decided by the experimenter only after the particle has in fact already passed through the double slit!
This is not fantasy, nor hypothesis, but a fact confirmed by experiment. Feynman said: “Quantum electrodynamics theory, from the standpoint of common sense, describes nature as absurd. But it agrees completely with experiment. Therefore, I hope you will accept nature as it is—absurd.”
Einstein could not tolerate the collapse of the traditional realist world, and during his lifetime he unremittingly objected to quantum mechanics. Yet all his objections ultimately failed, especially the thought experiment he proposed to expose the absurdity of quantum mechanics—the EPR paradox—which was finally given undeniable experimental confirmation by Aspect in 1982! In other words, Einstein hoped to refute quantum mechanics by means of the absurd result of a thought experiment, much as Galileo’s thought experiment about dropping balls from a tower, but the result favorable to quantum mechanics was actually verified in practice—indeed, the absurdity of quantum mechanics was itself confirmed. Although someone like Abraham Pais declared that even if Einstein had taken up fishing after 1925 it would not have caused any loss to physics, objectively speaking, Einstein’s resistance was highly significant in promoting the development of quantum mechanics. But a considerable number of philosophers’ resistance to quantum mechanics was extremely poor. They were repelled by the absurdity of quantum mechanics and eager to recover the lost classical realist world. Yet their opponents were no longer philosophers “discussing strategy on paper,” but scientists, who possessed a vast and difficult theory that was nonetheless extraordinarily successful, and who had solid experimental data as their basis. These were not things philosophers could compare themselves with.
However, intuitionism is the most special case (I have seldom seen intuitionists discuss the wave-particle paradox, so from here on the relevant views are mine, offered as a temporary intuitionist). It does not need to confront scientists head-on—because it need not oppose any conclusion of quantum mechanics—yet it can still eliminate the “logical” absurdity of the wave-particle paradox while remaining far from the supernatural and mysticism.
It is very simple: “either passed through slit A or passed through slit B” — this is the use of the law of excluded middle. Intuitionism maintains that the law of excluded middle is not an absolute truth, but something induced and distilled from experience and habits of thought; it has always been valid in all the experiences we have encountered before, but this does not mean it remains valid when we encounter a problem never seen before. For example, the law of excluded middle valid for finite things cannot be directly extended to the infinite; and, similarly, the law of excluded middle valid in the classical world cannot be directly extended to the quantum world!
Then why say that the law of excluded middle cannot be simply applied to the quantum world? The reason is also very simple—and entirely consistent with intuitionism’s longstanding thought—“existence must be constructed”!
As mentioned above, intuitionism is “constructivist” in mathematics: to say that a mathematical statement has a truth value, it must be, in principle, possible to decide it. Now we extend this line of thought to physics: to say that a physical statement has a truth value, it must be, in principle, possible to decide it.
What does “in principle” mean? For example, the following assertions are all supported: “The center of the Milky Way either has a black hole or it does not.” “The sun either will perish in five billion years or it will not perish.” Even if human technological power makes it difficult to verify things a hundred million light-years away, and even if the human lifespan makes it impossible to wait for billions of years, a light-year and a meter, a hundred million years and a day, are not essentially different, just as in mathematics there is no insurmountable boundary between 10 and 10100, but the difference between finite and infinite is substantive. Therefore, the above assertions are all decidable “in principle,” and the law of excluded middle has not failed.
Then, are there things in the physical world that humans cannot decide even “in principle”? People long believed there were not, just as people long did not believe there existed mathematical assertions that could not be decided. But those optimistic beliefs are now in the past. The uncertainty principle in the quantum world mentioned above reveals matters that are undecidable in the physical world:
First, the “uncertainty principle,” or the so-called “indeterminacy principle,” states that any act of “observation” must disturb the observed object. For example, to know where a particle is, one must use “light” to shine on it in order to find out. The effect of photons on macroscopic objects is negligible, but for an elementary particle, a single photon is enough to greatly disturb its state.
Therefore, observing the particles midway will inevitably affect the final result; and if we want to know: “When obtaining some result that can emerge only if the particles are left undisturbed along the way (for example, interference fringes), what exactly is the state of the particles on the way (for example, did they pass through slit A or slit B)?” — this is illegitimate. For to speak of a condition in which there has been no disturbance along the way means that, in principle, we cannot even determine the state of the particle along the way in such a condition! And according to the intuitionist claim, it is illegitimate to assign a truth value to a problem that is in principle undecidable. For example, since in principle one cannot determine whether the particle passed through slit A, it is illegitimate to assert that the particle “either passed through slit A or passed through slit B”!
Once we accept the line of thought that “existence must be constructed,” the wave-particle paradox instantly vanishes without a trace! Perhaps, as Wheeler said: “In the real world of quantum physics, basic phenomena are not phenomena until they are recorded.”
This line of thought is undoubtedly, in one sense, “anti-realism.” Just as in mathematics, intuitionism holds that mathematical concepts are human “language” and must therefore possess a certain ambiguity, so too in physics, physical concepts are human creations; physical “constructs” are not nature itself. These concepts undoubtedly ultimately derive from nature, but their immediate source is the construction of our minds. As James put it: “All the immense achievement of mathematical and physical science… has come from our invincible desire to cast the world into more reasonable images in our minds than those that are merely thrown at us in chaotic disorder by our experience.” — Concepts such as “wave,” “particle,” “light,” “electron,” and so on are all images cast out of chaotic experience in the human mind. The formation of these images is certainly related to nature, but they are always formed within the finite human mind; thus these concepts are undoubtedly marked by human fabrication and uncertainty.
People nowadays are probably no longer unfamiliar with all kinds of “ambiguous figures”; for example, one way of looking at a picture yields a young girl, another way yields an old woman, and Wittgenstein’s classic image, which looks like a duck from the front and a rabbit from the side, is also memorable. In fact, the simplest example is that “6” becomes “9” when turned upside down…
So, here is a sheet of white paper with only a “6” drawn on it. If we say there “exists a 6” on the paper, is that correct? What about saying there “exists a 9”? Clearly, 6 is not 9, a rabbit is not a duck, an old woman is not a young girl. So, on this paper, is it 6 or 9? What is the objective answer?
There is no objective answer. Whether what is on the paper is 6 or 9 cannot be answered apart from the knowing subject: on the one hand, it depends on the subject’s “perspective”; on the other hand, it also depends on the subject’s prior concepts. Only someone who knows Arabic numerals will recognize a 6 or a 9, while someone with a different conceptual basis may offer a completely different interpretation. So what, then, is absolutely objective existence, detached from anyone’s subjective perspective and concepts? We may say that there is indeed something there, but it is unsayable. Even saying “those are some ink marks,” “that is a picture,” “that is a sheet of paper”… “ink marks,” “picture,” “paper,” and so on are still human concepts and human language! Even the “existence” we usually speak of — if it is not understood as the “thing-in-itself” — is probably also a human concept. Nature itself is likely a mass of chaos; the division between matter and emptiness, the segmentation of time and space, and so on cannot possibly escape the imprint of human participation. Ultimately, these concepts are all words created by human beings; language can achieve relative precision, but it can never attain absolute precision.
Is it a particle or a wave?
Particles and waves, like 6 and 9, like rabbits and ducks, are also concepts constructed by human beings. Is that “thing” ultimately a wave or a particle? This is just like asking whether it is 6 or 9; no answer can be given apart from the knowing subject — you must tell me from what perspective it is to be seen: checking along the way, or only looking at the final screen? Unlike ambiguous figures, with physical reality we shall never be able to observe it from all perspectives; indeed, in the quantum world, observing from this perspective will cause one to lose forever the chance to observe from that other perspective. Therefore, any meaningful physical statement must incorporate the factor of perspective, and for questions involving physical reality one must tell me the observer’s state before a valid answer can be given. Questions about what state something is in when it can never possibly be observed are meaningless; insisting on assigning a truth value to them is precisely what gives rise to paradox.
Einstein pointed out: “Physical concepts are free creations of the human mind, and are not, however it may seem, uniquely determined by the external world. In our striving to understand reality we are somewhat like a man trying to understand the mechanism of a sealed watch. He sees the face and the moving hands, even hears its ticking, but he has no way of opening the case. If he is ingenious he may form some picture of the mechanism which could be responsible for all the things he observes; but he can never be sure that his picture is the only one which can explain his observations. He can never compare his picture with the real mechanism, and he cannot even imagine the possibility or the meaning of such a comparison.” In *The Meaning of Relativity*: “The world of our experiences can be theoretically understood as a creation of the mind of man, and without this creation science would not be possible.” Even so, “this world of ideas is to a small extent independent of the nature of our experiences, just as clothes are to a certain extent independent of the form of our bodies.”[93]
We must have a clear understanding: physical reality itself is certainly not a human creation, but our ideas about physical reality, our physics, are creations of our minds. This is again the intuitionist position — mathematics is a creation of the human mind, and mathematics and logic are not separated by any insurmountable divide from the natural sciences. Science is a creation of the human mind, and precisely because of this — precisely because of the finitude of the human mind — we must soberly recognize the relative vagueness of scientific concepts, and never arrogantly and complacently take any idea to be absolute.
Conclusion: What Is Mathematics?
In 1901, Russell said, “The chief achievement of modern mathematics has been to discover what is really mathematics.”[94]
But have philosophers really understood “what is really mathematics”? Or, to put it differently, does any school in the philosophical debate over the foundations of mathematics have the standing to provide an authoritative answer to the question “what is mathematics”?
Suppose, for example, someone wants to know “what is classical music?” The answer may go on at length, from history and figures to musical characteristics, and even statistical analyses of rhythm and melody, but would that satisfy the questioner? The simplest and most effective “answer” is to take him to a concert and let him listen to a Mozart piano piece — “There, this is classical music.”
Mathematics is similar. Mathematicians do not attach much importance to philosophers’ explanations of their work. Courant said: “Fortunately, creative thinking goes on in spite of certain doctrinaire philosophical beliefs, and it would be hindered in producing constructive achievements if it submitted to such beliefs; for the expert as well as for the layman, the only answer to the question ‘What is mathematics?’ is not philosophy but the living experience in mathematics itself.”[95]
The philosopher Putnam also said: “I hope to convince you that all varieties of systems in the philosophy of mathematics, without exception, do not need to be taken seriously.”[96]
F. Klein put it well: “In fact, mathematics has grown into a great tree, but it did not grow from the thinnest root, nor has it grown only upward; rather, as its branches and leaves expanded, its roots dug ever deeper downward…. Thus we can see that there is no final conclusion to the foundations of mathematics; from another point of view, neither is there an original starting point.”[97]
Mathematics is something that grows and develops. “The foundations of mathematics” can count as one field or branch within mathematics, and, like mathematics as a whole, it too will develop and will have no final conclusion. Any effort to provide mathematics once and for all with an absolutely solid, immovable foundation is nothing but fantasy; the continual development of mathematics will bring new problems to the field of foundations.
Although in fact we see that the coherence and stability advocated by intuitionism are incomparable with those of other schools, intuitionists’ ambition to “provide mathematics with a firm and unshakable foundation” is far less intense than that of the other schools. Heyting agrees with the pragmatist view: “First a discoverer and then a philosopher. And if we like, the latter can be left to others.”[98] The final scene of Heyting’s dialogue “Disputation,” in which he explains the differences and relations among intuitionism, classicism, formalism, pragmatism, symbolism, and so on, is that the intuitionist leads everyone into a classroom and presents the “samples” of intuitionist mathematics — “Rather than lengthy discussions, a few lessons will give you a much better understanding of it.”[99]
The intuitionists did not cling to providing an authoritative, dogmatic answer to “what is mathematics.” They never became mired in endless philosophical wrangling. They have always been more concerned with mathematical activity itself, while also attending to the natural sciences, and even, in the end, to everyday life, interpreting their position through their own practice. Heyting said: “In describing intuitionistic mathematics, I am conveying ideas to my listeners; these statements should not be understood in the sense of some philosophical system, but in the sense of daily life.”[100]
[U.S.] 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
[U.S.] M. Kline: *Mathematics: The Loss of Certainty*, trans. Li Hongkui, Hunan Science and Technology Press, 2003
[U.S.] M. Kline: *Mathematics and the Search for Knowledge*, trans. Liu Zhiyong, Fudan University Press, 2005
[U.S.] Morris Kline: *Mathematical Thought from Ancient to Modern Times*, trans. Deng Donggao, Zhang Gongqing, et al., Shanghai Scientific & Technical Publishers, 2002
[Poland] Andrzej Mostowski: *Thirty Years of Studies on the Foundations of Mathematics*, trans. Guo Shiming, Chen Anjie, and Xiu Qingyun, proofread by Kang Hongkui, Huazhong Institute of Technology Press, 1983
[France] Poincaré: *The Value of Science*, trans. Li Xingmin, Guangming Daily Press, 1988
[Germany] Kant: *Critique of Pure Reason*, trans. Deng Xiaomang, proofread by Yang Zutao, People’s Publishing House, 2004
[U.S.] R. Courant and H. Robbins: *What Is Mathematics? An Elementary Approach to Ideas and Methods*, reviewed by I. Stewart, trans. Zuo Ping and Zhang Yici, second edition, Fudan University Press, 2005
Zheng Yuxin: *New Theory on the Philosophy of Mathematics*, Jiangsu Education Press, 1990
Chen Bo: *Philosophy of Logic*, Peking University Press, August 2005
Zhang Jianjun: *An Introduction to the Study of Logical Paradoxes*, Nanjing University Press, 2002
Department of Philosophy, Peking University
August 15, 2006
[①] [Poland] Andrzej Mostowski: *Thirty Years of Studies on the Foundations of Mathematics*, trans. Guo Shiming, Chen Anjie, and Xiu Qingyun, proofread by Kang Hongkui, Huazhong Institute of Technology Press, 1983, pp. 12–13
[②] As for the terms “intuitionism,” “logicism,” and “formalism” used in this article, the author is also merely understanding them as certain intellectual currents as wholes. In discussion one may say that a person’s certain views represent this or that “ism,” but I will avoid forcibly subsuming any person under a particular “ism” in advance, unless he himself acknowledges it.
[③] [France] Poincaré: *The Value of Science*, trans. Li Xingmin, Guangming Daily Press, 1988, p. 202
[④] L. E. J. Brouwer: Consciousness, Philosophy, and Mathematics: see [U.S.] 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. 104 (This book will hereafter be abbreviated in the notes as *Philosophy of Mathematics*)
[⑤] Paul Benacerraf: Introduction to *Philosophy of Mathematics*: see *Philosophy of Mathematics*, p. 26
[⑥] The “meaning” here follows the traditional understanding; in fact, intuitionism simultaneously maintains that “meaning does not depend on truth conditions”
[⑦] Michael Dummett: The Philosophical Basis of Intuitionist Logic: see *Philosophy of Mathematics*, p. 132
[⑧] Arend Heyting: Disputation: see *Philosophy of Mathematics*, p. 79
[⑨] Ibid.
[⑩] Ibid.
[11] [U.S.] M. Kline: *Mathematics: The Loss of Certainty*, trans. Li Hongkui, Hunan Science and Technology Press, 2003, p. 239
[12] See [U.S.] Edwin Arthur Burtt: *The Metaphysical Foundations of Modern Physical Science*, trans. Xu Xiangdong, Peking University Press, 2003
[13] Paul Bernays: On Platonism in Mathematics: see *Philosophy of Mathematics*, p. 301
[14] See [U.S.] M. Kline: *Mathematics: The Loss of Certainty*, trans. Li Hongkui, Hunan Science and Technology Press, 2003, p. 203
[15] See Zheng Yuxin: *New Theory on the Philosophy of Mathematics*, Jiangsu Education Press, 1990, p. 61
[16] Paul Benacerraf: Introduction to *Philosophy of Mathematics*: see *Philosophy of Mathematics*, p. 34
[17] Ibid.
[18] See Zheng Yuxin: *New Theory on the Philosophy of Mathematics*, Jiangsu Education Press, 1990, p. 77
[19] David Hilbert: On the Infinite: see *Philosophy of Mathematics*, p. 214
[20] Paul Benacerraf: Introduction to *Philosophy of Mathematics*: see *Philosophy of Mathematics*, p. 7
[21] Arend Heyting: The Intuitionistic Basis of Mathematics: see *Philosophy of Mathematics*, p. 61
[22]Michel Dummett: The philosophical basis of intuitionistic logic: See “Philosophy of Mathematics,” p. 137
[23]Rudolf Carnap: The logicalist foundations of mathematics: See “Philosophy of Mathematics,” p. 48
[24]Rudolf Carnap: The logicalist foundations of mathematics: See “Philosophy of Mathematics,” p. 57
[25]Ibid., p. 60
[26]Kurt Gödel: What is Cantor’s continuum problem?: See “Philosophy of Mathematics,” pp. 560~561
[27]Chen Bo: “Philosophy of Logic,” Peking University Press, August 2005, p. 60
[28]Ibid., p. 61
[29]Arend Heyting: Argumentation: See “Philosophy of Mathematics,” p. 82
[30]L.E.J. Brouwer: Consciousness, philosophy, and mathematics: See “Philosophy of Mathematics,” p. 111
[31]David Hilbert: On the infinite: See “Philosophy of Mathematics,” p. 222
[32]Ibid., p. 231
[33]Ibid., p. 220
[34]Ibid.
[35]See [US] M. Kline, “Mathematics: The Loss of Certainty,” translated by Li Hongkui, Hunan Science and Technology Press, 2003, p. 253
[36]David Hilbert: On the infinite: See “Philosophy of Mathematics,” p. 214
[37][US] M. Kline, “Mathematics: The Loss of Certainty,” translated by Li Hongkui, Hunan Science and Technology Press, 2003, pp. 250~251
[38]L.E.J. Brouwer: Intuitionism and formalism: See “Philosophy of Mathematics,” p. 91
[39]Paul Benacerraf: Mathematical truth: See “Philosophy of Mathematics,” pp. 470~471
[40]David Hilbert: On the infinite: See “Philosophy of Mathematics,” p. 231
[41][US] M. Kline, “Mathematics: The Loss of Certainty,” translated by Li Hongkui, Hunan Science and Technology Press, 2003, p. 256
[42]Ibid., p. 255
[43]L.E.J. Brouwer: Intuitionism and formalism: See “Philosophy of Mathematics,” p. 90
[44]Arend Heyting: Argumentation: See “Philosophy of Mathematics,” p. 81
[45]Ibid.
[46]Ibid.
[47][US] R. Courant, H. Robbins, revised by I. Stewart, “What Is Mathematics? An Elementary Approach to Ideas and Methods (Revised Edition),” translated by Zuo Ping and Zhang Yici, Second edition, Fudan University Press, October 2005, p. 3
[48]George Kreisel: The Hilbert program: See “Philosophy of Mathematics,” p. 241
[49]Arend Heyting: The intuitionistic foundations of mathematics: See “Philosophy of Mathematics,” p. 60
[50][US] M. Kline, “Mathematics: The Loss of Certainty,” translated by Li Hongkui, Hunan Science and Technology Press, 2003, p. 323
[51]Ibid., pp. 323~324
[52]Ibid., pp. 324~325
[53]Ibid., p. 324
[54]Ibid., p. 238
[55][US] Morris Kline, “Mathematical Thought from Ancient to Modern Times” (Vol. 4), translated by Deng Donggao, Zhang Gongqing, et al., Shanghai Science and Technology Press, 2002, p. 99
[56][US] M. Kline, “Mathematics: The Loss of Certainty,” translated by Li Hongkui, Hunan Science and Technology Press, 2003, p. 323
[57]Ibid., pp. 283~284
[58]See Zhang Jianjun, “An Introduction to the Study of Logical Paradoxes,” Nanjing University Press, 2002, p. 78
[59][US] Morris Kline, “Mathematical Thought from Ancient to Modern Times” (Vol. 4), translated by Deng Donggao, Zhang Gongqing, et al., Shanghai Science and Technology Press, 2002, pp. 98~99
[60][US] M. Kline, “Mathematics: The Loss of Certainty,” translated by Li Hongkui, Hunan Science and Technology Press, 2003, p. 288
[61][US] Morris Kline, “Mathematical Thought from Ancient to Modern Times” (Vol. 4), translated by Deng Donggao, Zhang Gongqing, et al., Shanghai Science and Technology Press, 2002, pp. 110~111
[62][US] M. Kline, “Mathematics: The Loss of Certainty,” translated by Li Hongkui, Hunan Science and Technology Press, 2003, p. 311
[63]Ibid., p. 294
[64]Ibid., p. 228
[65]We often accuse Kronecker of being harsh toward his student Cantor. Indeed, as a teacher Kronecker was unworthy, but we should also note that, as a thinker, his thought was neglected for even longer than Cantor’s! It was only when Cantor’s set theory had developed to maturity and paradoxes had begun to appear that Kronecker was remembered again.
[66][US] Morris Kline, “Mathematical Thought from Ancient to Modern Times” (Vol. 4), translated by Deng Donggao, Zhang Gongqing, et al., Shanghai Science and Technology Press, 2002, p. 113
[67][US] M. Kline, “Mathematics: The Loss of Certainty,” translated by Li Hongkui, Hunan Science and Technology Press, 2003, p. 307
[68]Ibid., p. 310
[69]Ibid., p. 340
[70]Ibid., p. 359
[71]Ibid.
[72]Ibid., p. 341
[73]Ibid., p. 338
[74]Kurt Gödel: What is Cantor’s continuum problem?: See “Philosophy of Mathematics,” p. 561
[75][US] Morris Kline, “Mathematical Thought from Ancient to Modern Times” (Vol. 4), translated by Deng Donggao, Zhang Gongqing, et al., Shanghai Science and Technology Press, 2002, p. 116
[76]Paul Benacerraf: Introduction to “Philosophy of Mathematics”: See “Philosophy of Mathematics,” p. 27
[77]Paul Bernays: On Platonism in mathematics: See “Philosophy of Mathematics,” p. 304
[78]Chen Bo: “Philosophy of Logic,” Peking University Press, August 2005, p. 117
[79]Hilary Putnam: Mathematics without foundations: See “Philosophy of Mathematics,” p. 357
[80]Chen Bo: “Philosophy of Logic,” Peking University Press, August 2005, p. 118
[81][US] Morris Kline, “Mathematical Thought from Ancient to Modern Times” (Vol. 4), translated by Deng Donggao, Zhang Gongqing, et al., Shanghai Science and Technology Press, 2002, p. 99
[82]Paul Bernays: On Platonism in mathematics: See “Philosophy of Mathematics,” p. 314
[83]Ibid.
[84][US] M. Kline, “Mathematics: The Loss of Certainty,” translated by Li Hongkui, Hunan Science and Technology Press, 2003, p. 277
[85]Hilary Putnam: Models and reality: See “Philosophy of Mathematics,” p. 490
[86]Hilary Putnam: Models and reality: See “Philosophy of Mathematics,” p. 491
[87][US] M. Kline, “Mathematics: The Loss of Certainty,” translated by Li Hongkui, Hunan Science and Technology Press, 2003, p. 277
[88]See [US] M. Kline, “Mathematics: The Loss of Certainty,” translated by Li Hongkui, Hunan Science and Technology Press, 2003, pp. 278~279
[89]Hilary Putnam: Models and reality: See “Philosophy of Mathematics,” p. 493
[90]Ibid., p. 510
[91]Ibid., p. 511
[92]Arend Heyting: Argumentation: See “Philosophy of Mathematics,” p. 86
[93]See [US] M. Kline, “Mathematics and the Search for Knowledge,” translated by Liu Zhiyong, Fudan University Press, 2005, pp. 241~242
[94][US] M. Kline, “Mathematics: The Loss of Certainty,” translated by Li Hongkui, Hunan Science and Technology Press, 2003, p. 283
[95][US] R. Courant, H. Robbins, revised by I. Stewart, “What Is Mathematics? An Elementary Approach to Ideas and Methods (Revised Edition),” translated by Zuo Ping and Zhang Yici, Second edition, Fudan University Press, October 2005, p. 5
[96]Hilary Putnam: Mathematics without foundations: See “Philosophy of Mathematics,” p. 343
[97][US] M. Kline, “Mathematics: The Loss of Certainty,” translated by Li Hongkui, Hunan Science and Technology Press, 2003, p. 325
[98]Arend Heyting: Argumentation: See “Philosophy of Mathematics,” p. 88
[99]Ibid., p. 88
[100]Ibid., p. 88
Latest comments
- apostar
2008-01-01 19:17:02 Anonymous 210.32.189.188 http://blog.sina.com.cn/apostar
Teacher Zhang:
Sorry, something major happened in my family recently, so I haven’t been online and didn’t see your message. I had read this article before, but I didn’t know this was your blog. I rather agree with the views in your article. I myself think that perhaps I am someone who, in philosophical views, tends toward intuitionism. It’s just that I insist on this position: any resolution of paradox that ultimately cannot be reduced to natural-language expression must have a loophole. Recently I’ve been reading Einstein’s *Relativity* (Chongqing Publishing House), and what surprised me is that he also reduced many profound philosophical problems to natural language. The way formal logic resolves paradoxes is itself a violation of Gödel’s incompleteness theorem; it’s just that most logicians nowadays have forgotten this point. Formal logic cannot solve many problems brought about by the empirical world, and knowledge cannot be obtained through pure deduction. This is actually a simple fact. - Gu Dwa
2008-01-01 20:50:04
Thanks, but there is no Teacher Zhang here……..
You can call me Little Gu classmate……
I didn’t see the meaning of “just…”; what you said about opposing formalization and axiomatization is also a claim of intuitionism. Of course, apart from intuitionists, many others also hold similar views. If science has any foundation, then that foundation is everyday life; it should not be turned around so that scientific theory is taken as the foundation of the lifeworld.
Translated from the Chinese original with AI assistance. The original text is authoritative.
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