Late Wittgenstein and the Mathematical Philosophy of Intuitionism
Abstract: This essay takes intuitionism as its guiding thread to explicate Wittgenstein’s late philosophy of mathematics, with the hope that it may contribute to a grasp of his late philosophy; on the other hand, it also cites Wittgenstein’s philosophical claims to provide support for intuitionism. Thus, this essay attempts both to sort out the mathematical philosophy of Wittgenstein’s later period and, at the same time, to offer a brief critical account of intuitionism. The author discusses separately the views of intuitionism and of late Wittgenstein on themes such as constructivism, empiricism, positivism, Platonism, logicism, formalism, conventionalism, the axiomatization movement, and the development of mathematical science, showing the striking similarities between late Wittgenstein and intuitionism in the philosophy of mathematics.
Keywords: intuitionism, law of excluded middle, constructivism, Platonism, logicism, formalism, conventionalism, axiomatization movement, language games
Introduction—Intuitionism and Wittgenstein’s Intellectual Turn… 1
I. Intuitionism—“Don’t think, but look!” 2
II. The law of excluded middle and actual infinity—“Truth is grounded in experience” 3
III. Platonism—“Mathematics is a language game” 7
IV. Logicism—“The mind cannot be grasped by a machine” 9
V. Formalism—“Meaning is not a dead thing” 13
VI. Conventionalism—“Language is connected with a form of life” 16
VII. The axiomatization movement—“Mathematics does not need foundations” 18
VIII. Pure mathematics—“Application is crucial” 20
Introduction—Intuitionism and Wittgenstein’s Intellectual Turn
Wittgenstein is undoubtedly one of the most influential philosophers of the twentieth century, and there is no counting the studies and commentaries on his philosophical thought at home and abroad. Yet I have found that people seem to pay rather less attention to Wittgenstein’s philosophy of mathematics.
It is true that Wittgenstein’s achievements in the philosophy of mathematics are highly controversial; some critics even accuse him of insufficient mathematical training, of self-contradiction in his views, and so on. However that may be, anyone studying Wittgenstein’s philosophy—especially his later philosophy—cannot afford to ignore his philosophy of mathematics:
The reason Wittgenstein’s philosophy of mathematics is unavoidable is not merely that he left behind a large number of remarks on mathematics throughout his life, but more importantly because mathematics was a key factor in the formation and transformation of Wittgenstein’s philosophical views—he first became interested in philosophy precisely through the study of mathematics, and the turning point between his early and later thought was likewise caused by the philosophy of mathematics.
It is said that “in 1929, at the earnest persuasion of the young mathematician F. P. Ramsey and stimulated by the lecture on the foundations of mathematics by L. E. J. Brouwer that he heard in Vienna, Wittgenstein finally returned to Cambridge,”[①] and after his return to the philosophical stage, whether in his discussions with the Vienna Circle or in the lectures and notes of that period, mathematics was always the most important topic. So to say that the philosophy of mathematics is the source of Wittgenstein’s later philosophy is by no means an exaggeration.
And compared with his earlier philosophy, which was more strongly influenced by the logicist school in mathematics, Wittgenstein’s later philosophy was clearly more deeply influenced by formalism and intuitionism, with intuitionism exerting the more far-reaching influence: we should note that the mathematician Brouwer, who reawakened Wittgenstein’s philosophical interest, was precisely the standard-bearer of intuitionism.
Of course, Wittgenstein’s contact with intuitionism was by no means as simple as attending a lecture. On another front, the other figure who prompted Wittgenstein’s return to Cambridge—the young mathematician Ramsey, whom Wittgenstein specifically thanks in the preface to Philosophical Investigations—though inclined toward logicism in his early years, gradually turned toward intuitionism in his final years[②]. Ramsey died young in 1930, so he did not complete his intellectual turn. But in the last two years of his life he was in very close contact with Wittgenstein, who “discussed my views with him in countless conversations” (see the preface to Philosophical Investigations[③]).
In addition, in Wittgenstein’s discussions with the Vienna group from 1929 to 1932, he would of course also have come into contact with many doctrines of intuitionism (as well as formalism), for example when Wittgenstein read Weyl’s (Hermann Weyl[④]） conference paper and his book Philosophy of Mathematics and Natural Science, and made many comments on them (see v.2, p.47 ff.).
Finally, the reason Wittgenstein perhaps inclined toward intuitionism may be that “the Kantian-Schopenhauerian influence present in his early mathematical thought was also an important inner incentive.”[⑤] Intuitionism also has deep connections with currents in philosophy such as transcendentalism, romanticism, and individualism, but I shall not expand on that here; instead, I will focus on the relationship between the mathematical philosophy of intuitionism and Wittgenstein’s later philosophy.
Wittgenstein’s style is highly unrestrained, making it difficult to discern any clear line of development in his work. For that reason, I choose to use intuitionism as the thread for explicating Wittgenstein’s late philosophy of mathematics, which may perhaps help us grasp his later philosophy; on the other hand, Wittgenstein’s philosophical claims in his later period also, from a different starting point, provide support for intuitionism. This essay therefore attempts both to sort out late Wittgenstein’s philosophy of mathematics and, at the same time, to offer a brief commentary on intuitionism.
I. Intuitionism—“Don’t think, but look!”
Before proceeding further, it is necessary to offer some basic introduction to intuitionism:
Intuitionism is one of the most influential currents in contemporary mathematics and logic, and also the most distinctive. Its intellectual roots can be traced back to Kant’s transcendental philosophy; if we look only at its opposition to actual infinity, its lineage can even be traced back to Aristotle. In the contemporary period, Leopold Kronecker and Henri Poincaré were pioneers of intuitionism, while Brouwer was the true founder of intuitionism as a clearly articulated school. Its claims won sympathy or acceptance from many celebrated mathematicians and logicians, including René Baire, Henri Léon Lebesgue, Arend Heyting, Weyl, and others.
As its name suggests, intuitionism’s original and most important position is its emphasis on “intuition.” This does not mean, of course, that it denies the logicality and rigor of mathematics. Everyone acknowledges that mathematics is the science that demands logical rigor most stringently, and intuitionists cannot deny this fact. But what intuitionists want to stress even more is the place of intuition, inspiration, and creativity in mathematics, things that were submerged and ignored in the tide of the axiomatization movement. Intuitionism reminds people that mathematics is not only the most exacting of sciences, but also the most creative.
Intuitionists appear to be very radical, even one might say very “reactionary”; they resisted and opposed almost every trend in contemporary mathematics and logic—axiomatization, logicization, formalization, isolation, and so on. Like “postmodernism,”[⑥] intuitionism is perhaps less a school than a rebellious current of thought: on the one hand, one can find abundant intuitionist influence even among those who are not intuitionists; on the other hand, among those clearly waving the banner of intuitionism, views and claims differ enormously, making it difficult to generalize in any simple way.
It is worth noting that the word “intuition” is also extremely ambiguous in intuitionism, and it is hard to say just what “intuition” actually is. In a certain sense, what intuitionists emphasize is precisely those unsayable elements in mathematical activity; if they could be stated clearly, they would no longer be intuition.
Poincaré once noted the ambiguity of the word “intuition”: “To construct arithmetic, … something besides pure logic is needed, and to name this something, we are forced to use the word intuition. But how many different meanings lie hidden behind this same word?”[⑦] As Poincaré says, among different intuitionists, and even for the same intuitionist in different contexts, what is stressed as “intuition” is not single in meaning. Perhaps precisely because of the ambiguity and indeterminacy of the word “intuition,” although Wittgenstein in certain contexts acknowledges its important significance (e.g. v.7, p.181, IV§44; v.4, p.275, etc.), he does not seem to like using the concept; he remarks that “intuition is only a superfluous embellishment” (Philosophical Investigations §213).
On another occasion, Wittgenstein says: “One might ask: do we need a new capacity for intuition at every step of a proof? (The individuality of mathematics) In some cases it may be something like this: if I get hold of a general (variable) rule, then I must always anew acknowledge: this rule can also be applied here (that is to say, it is applicable to this case too). Foreseeing the action does not free me from this activity of understanding. For the form in which the rule is applied is in fact a new form at every step. But the problem is not one of an activity of understanding, but rather one of a decision to act.” (v.4, p.280, §13) Yet perhaps what intuitionism emphasizes as the need for intuition at every step is not only an emphasis on understanding, but also on the human capacity for free decision.
Although Wittgenstein thought that introducing the word “intuition” was unnecessary or liable to cause misunderstanding, what he emphasized in many contexts may also appropriately be called “intuition”; for example, one commentator notes: “In the Tractatus Logico-Philosophicus, the bond between language and world is fixed by the relation between name and object. But now, according to Wittgenstein’s new view, those relations are composed and sustained precisely by certain rule-governed human activities, which perhaps can be called ‘custom’ or ‘intuition.’”[⑧] In other contexts, such as Wittgenstein’s aforementioned “a decision to act”; the “instinct” for “fixing a language game” (see v.7, p.173, IV§23); “knowing something but being unable to put it into words” (Philosophical Investigations §78); “showing itself in the proper light” (Philosophical Investigations §81); “‘He understands’ must involve more than: he thought of this formula. And equally more than: any process or outward expression accompanying understanding and more or less indicating understanding.” (Philosophical Investigations §152); and so on—these kinds of expressions or formulations can in fact all be summed up by “intuition.”
As I said above, “intuition” refers precisely to those things that cannot be conceptualized and thought or spoken. The apprehension of “intuition” does not come through analysis and inference, but rather—just as the Latin root of intuition suggests—through “seeing.” Wittgenstein’s famous saying “Don’t think, but look!”[⑨] (Philosophical Investigations §66) is the best interpretation of “intuition.”
“Intuition” emphasizes an activity or capacity of the individual mind, whereas late Wittgenstein emphasizes the impossibility of a “private language.” Is there a contradiction here? Quite the opposite: although late Wittgenstein rules out a private language, he does not completely abandon his earlier solipsism (see Philosophical Investigations §24), while what intuitionism stresses is precisely the individual’s “essentially nonlinguistic mental activity”[⑩] Intuitionists have also expressed the view that linguistic phenomena are products of culture and custom (I will mention this again later), so the part of mathematical activity belonging to the mathematician’s own creativity must be supralinguistic.
“Mathematics cannot be reduced to language—this is Brouwer’s central thesis—and from this it follows that language is merely a kind of auxiliary means, something that makes social organization possible.”[11] And a similar view—that language is only a kind of auxiliary means, a “ladder” that can be discarded once one has climbed it, and that only by transcending propositions does one correctly see the world—was already emphasized by Wittgenstein in his earlier period (the end of Tractatus Logico-Philosophicus). And this claim about the limits of language was still maintained by Wittgenstein in his later period.
In the discussion that follows, I will treat Wittgenstein as a uniquely styled intuitionist, or rather, as someone who, from the standpoint of a non-mathematician, provided distinctive support for intuitionist claims. I believe this does not distort Wittgenstein too much—why should we not say that Wittgenstein was an intuitionist? Some commentators have listed many differences between Wittgenstein and intuitionism, for example Tu Jiliang notes: “For instance, regarding the role of intuition, on the one hand he acknowledges that mathematics also needs intuition and that intuition can play a certain role in mathematics; on the other hand, he opposes, as intuitionists do, the infinite exaggeration of intuition’s role in mathematics. Again, on the one hand he opposes intuitionism’s attack on the validity of the law of excluded middle; on the other hand, he appropriately limits the scope of validity of the law of excluded middle. Again, he acknowledges that mathematics contains many difficult problems that are currently impossible to resolve completely, but he does not agree with Brouwer’s view that these problems are absolutely insoluble, and so on.”[12] However, as will be mentioned later, intuitionists do not in fact “infinitely exaggerate” the role of intuition; nor do they fail to acknowledge the validity of the law of excluded middle; moreover, Wittgenstein’s view of what counts as solving a problem is rather special… To put it simply, many of the differences between Wittgenstein and the intuitionists mentioned by commentators are based on misunderstanding, while many others are nonessential differences (there are often enormous differences even among different intuitionists).
If one were to say that Wittgenstein and the intuitionists have any important difference, one point is worth mentioning: compared with most intuitionists, who were mathematicians, Wittgenstein was a philosopher reflecting on mathematics from outside mathematics. The route by which he reached similar conclusions, and the way he expressed them, were quite different from the style of ordinary intuitionists. Even so, according to Wittgenstein’s doctrine of “family resemblance,” to classify Wittgenstein as an intuitionist is by no means excessive.
II. The law of excluded middle and actual infinity—“Truth is grounded in experience”
When people mention “intuitionism,” the first thing that probably comes to mind is its rejection of the “law of excluded middle” — “if something is not false, then it is true.” Does one really have to object to such an obvious, indeed so “intuitive,” logical principle? Added to this, intuitionism is often associated with romanticism and irrationalism, so it is no wonder that it leaves some people with the impression of something “unreasonable” or “heretical.”
First of all, it should be pointed out that one need not be too surprised at intuitionists’ reflection on the law of excluded middle. In fact, even from the standpoint of classical logic, not every “sentence” has a truth value. For example, do we think that statements like “bread is brave,” “electrons are sweet,” or “unicorns are less than zero,” and the like, are also “either true or false”? Rather than saying that such sentences are “false,” it is better to say that they are simply meaningless and unintelligible.
To give another example, intuitionism can acknowledge that the assertion “all crows are black” has a truth value, because the proposition “every crow in the world is either black or non-black” (ignoring ambiguous colors) is legitimate. This is because “all the crows in the world,” though countless, are after all finite; “the world,” though vast, is also finite; in principle we can comb the world through and locate every crow, then inspect their colors one by one. Although this is technically hard to do, the problem is after all a “real” one.
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 simply fictional, invented in a story, and Aesop never directly or indirectly provided any information that would allow us to judge its color.
Only if there is sufficient basis within Aesop’s story to judge its color—for example, if somewhere it says that the crow once dropped a white feather—would one be entitled to think that talking about that “crow’s” color is meaningful.
Wittgenstein also once mentioned that inquiries into the characteristics of fictional figures in novels would be indeterminate, because the author might reply: “I haven’t decided yet.”
The crux of the matter is just which statements are real, and which are illusory. Intuitionism does not directly oppose the law of excluded middle. For them, as with classical logic, any semantically legitimate, meaningful[13] statement is still either true or false. Michael Dummett pointed out: “For intuitionist logic, the double negation of the law of excluded middle is a valid semantic principle, just as bivalent logic takes the law of excluded middle itself to be valid: it is inconsistent to assert of any statement that it is neither true nor false.”[14]
Wittgenstein himself, as well as some commentators, misunderstood intuitionists’ claims about the law of excluded middle (perhaps because intuitionists’ rhetoric was too vehement), and thus came to think that Wittgenstein’s restrictions on the law of excluded middle were quite different from those of intuitionists.
For example, Tu Jiliang points out: “When Wittgenstein criticized Brouwer and other intuitionists for attacking the universal validity of the law of excluded middle, he did not absolutely or comprehensively affirm its universal validity, because he believed that for infinite sequences, whether the law of excluded middle applies is indeed problematic.”[15] Yet the position attributed here to Wittgenstein is precisely the intuitionists’ stance. Again, Wittgenstein thought that Brouwer maintained that “besides yes and no, there are also situations in which there is no decidability” (v.3, p.203, §174) — but this is not accurate. Intuitionists do not advocate three-valued logic (“many-valued logic” has another group of supporters, mostly within the logicist tradition), that is, besides the two truth values “yes” and “no,” there is a third coequal truth value, “undecidable.”
In fact, intuitionists and Wittgenstein held similar views: it is not that certain meaningful propositions can be neither true nor false; rather, certain statements are simply meaningless because of abuse of words, and of course then there is no question of their being true or false.
Wittgenstein noted: “When someone wants to make us firmly remember that the proposition of excluded middle cannot be escaped, — clearly, there is something inappropriate in his question. When someone puts forward the proposition of the law of excluded middle, it is as though he were offering us two pictures to choose from and saying that one of them must correspond to the facts. But what if the question is whether these pictures are applicable here?” (v.7, p.198, V§10) The key point is whether speaking of truth or falsity in relation to a certain proposition is “applicable.” So, under what circumstances is it appropriate to speak of truth and falsity? First, Wittgenstein believed that judgments of truth and falsity have nothing to do with “reality,” but are merely part of “language games” (see *Philosophical Investigations*, §136).
Wittgenstein’s anti-realism will be discussed later. Here, it is worth clarifying another basic philosophical stance shared by Wittgenstein and intuitionism, which is of course also closely related to anti-realism:
Although late Wittgenstein left the camp of logical empiricism or logical positivism, in essence what Wittgenstein really gave up was only “logicism,” not empiricism or positivism. Wittgenstein never completely abandoned the empiricist and positivist approach; rather, like intuitionists, he extended empiricist and positivist positions into the realm of mathematics.
One might say that intuitionists are the most thoroughgoing empiricists. Compared with them, so-called “logical empiricism” actually falls short. For logical empiricism is really “logic/empiricism” — they resolutely carry out empiricism in every domain, except that they make an exception in “logic” (and in mathematics, regarded as a derivative of logic). Intuitionism, by contrast, insists that mathematical truth is also empirical, and that logic and mathematics, like other empirical natural sciences, are not absolutely reliable. Brouwer said: “There is no non-empirical truth, and logic is not an absolutely reliable instrument for discovering truth. … Mathematics that, in strict accordance with this viewpoint, proceeds and derives theorems by means solely of the method of intuitive construction is called intuitionistic mathematics.”[16]
Empiricism and positivism are always closely related; naturally, intuitionism’s view of mathematics also tends toward positivism. Paul Benacerraf said: “Intuitionists in mathematics seem, after all, to be verificationists”[17]
Wittgenstein also held that the truth of mathematical propositions likewise depends on experience. For example, he mentioned: “‘There are 60 seconds in a minute.’ This is a proposition very much like a mathematical proposition. Does its truth depend on experience? … If all those connections that make the measurement of time meaningful did not exist, could we still speak of minutes and seconds? … Just as without a game of chess, checkmate would also be meaningless.” (v.7, p.295, VII§18) In Wittgenstein’s view, mathematics and logic, like other sciences, come from the language games of human life, and their truth, or validity, can in no way be something beyond experience.
Elsewhere, Wittgenstein noted: “What is the difference between ‘The book is somewhere on the table’ and ‘Something will happen at some moment in the future’? Clearly, the difference is that in one case we have a definite method by which we can verify whether the book is on the table, whereas in the other case we do not have such a method.” (v.4, p.242, §6) In Wittgenstein’s view, superficially similar sentences cannot be treated as grammatically identical if their verifiability, or the way in which they may be verified, differs. For words such as “exist” or “there is,” if in principle they cannot be verified by experience, then their use will be ambiguous and unclear.
Thus Wittgenstein agreed with some of constructivism’s (which intuitionists generally endorsed) claims. He mentioned: “There is thus the dispute whether a non-constructive proof of existence is really a proof of existence. That is to say, it asks: if I have no possibility of finding where it exists, do I understand the proposition ‘there is’?” (v.7, p.223, V§46)
By the way, under the constructivist approach, Wittgenstein’s views on the “continuum” are similar to those of the intuitionists. Both deny Cantor’s famous diagonal proof that “there are more real numbers than rational numbers.” Wittgenstein pointed out: it is impossible to lay out all the real numbers in a sequence ready-made; therefore Cantor’s argument that the real number generated by the diagonal method differs from any number in the sequence is invalid, because the sequence of numbers is being listed without end, and one must always wait until the next row of real numbers has been listed before one can add another digit to the end of the new diagonal number. At that point, whether the next number to be listed differs from the diagonal number is forever uncertain, because the next number has not yet been listed — “there is always one in the sequence whose difference from the diagonal sequence is uncertain. One can say: they follow one another, tending toward infinity, but it is always the original sequence that is in front.” (v.7, p.82, II§9)
Wittgenstein repeatedly emphasized that a straight line is by no means “a set of points,” which is precisely the point of Brouwer’s notion of the continuum — “The continuum is not intuited as a set of points, not something that just happens to present itself as line-like. It is a unity of multiplicity, arising from my awareness that I can endlessly insert numbers ‘between’ those things I have already constructed…”[18]. Wittgenstein believed that saying, for instance, that one draws a cross on a line is not to say “I drew a cross on this ‘point,’ and this point belongs to this line, therefore I drew a cross on the line.” In fact, before I construct it, there simply is no such “point” on the line. Wittgenstein also gave the example of shooting at a target: to say “hit anywhere on the target and you win” is not a proposition, but a general rule. “Can the point hit be marked in any way other than by being hit? Was that point already there on the target beforehand?” — the correct formulation is: “You hit the target, therefore …,” rather than “You hit here, and since here is on the target, therefore …” (see v.4, p.234, §3) A whole is not the sum of parts before we divide it, let alone the collection of “all” its parts. This claim, besides being close to that of the intuitionists, can also be traced back to Kant — “Suppose the whole were infinitely branched, that is quite inconceivable. It may nevertheless be assumed that the parts of matter can be divided to infinity in the analysis of that same matter…. One can determine in the whole a quantity of division, to the same degree as one advances in the regress of division.”[19]
Returning to the main point: once mathematical propositions are also regarded as empirical propositions, and the principle of verifiability is introduced into mathematics, one must inevitably be alert to the range of applicability of the law of excluded middle. Wittgenstein mentioned: “When Brouwer attacks the application of the law of excluded middle in mathematics, he is correct insofar as what he attacks is a process similar to the proof of empirical propositions.” (v.4, p.427, §39) Intuitionists believe that only those mathematical statements for which, in principle, a method can be constructed that can effectively decide them in a finite number of steps are legitimate. Wittgenstein agreed with these claims, and further pointed out: if a mathematical statement has no “use,” then it is likewise meaningless. For example, he mentioned: “One can form such an expression as: ‘classes of all classes which are numerically equal to the class of the infinite sequence,’ just as: ‘classes of all angels that are all at the location of a needle tip’; but as long as there is no use for this expression, it is empty. Such a use is not something still to be discovered, but something still to be invented.” (v.7, p.90, II§38)
From the above we can see that the attack on the law of excluded middle is inseparable from the whole philosophical stance of intuitionism and its understanding of mathematics. More specifically, the questioning of the law of excluded middle expresses intuitionists’ dissatisfaction with abuses of language in mathematics, and especially their criticism of the confusion in the use of words such as “exist,” “all,” and “infinite.”
Both intuitionists and Wittgenstein especially pointed out the abuse of words like “infinite” and “infinitude” in mathematics. Wittgenstein said: in mathematics one should avoid the word “infinite” — “it should be avoided where it seems to confer meaning on calculation rather than derive meaning from calculation.” (v.7, p.94, II§58) Wittgenstein repeatedly emphasized that infinity does not have the status of a number; the essence of infinity lies precisely in the fact that it is a possibility, not an actuality. “An infinite possibility has no size at all.” (v.3, p.144, §138) Elsewhere Wittgenstein noted: “To say that a technique is infinite does not mean that it carries on forever — its value increases without measure; it means that it has no mechanism of termination, … one may say that a playing field is boundless if the rules of the game do not prescribe boundaries.” (v.7, p.91, II§45)
And Wittgenstein believed that many people mistakenly understood infinity as a possibility to be some real, completed object. For example, he pointed out: “It is as when we say ‘this proposition applies to all numbers,’ and we believe that in our thoughts all numbers are already contained, like apples in a box.” (v.4, p.244, §6) “In his basic laws, Russell seems to be saying something like this about a proposition: ‘It has already come out — all I still need to do is infer it.’ Frege said the same thing somewhere: the straight line connecting two points is actually already there before we draw it; when we say that a transformation (say in the sequence +2) has actually already been done before we carry it out in verbal or written form — as though what we do is merely discover them,” (v.7, p.10, I§21) “You tended to express it in this way: ‘Even though I have not yet completed these steps in writing, speaking, or thought, they are, strictly speaking, already completed.’ As though they were in some unique way already determined, already anticipated — as if mere meaning alone could anticipate reality.” (*Philosophical Investigations*, §188) And so on. Wittgenstein believed that all of these were illusions caused by misuse of the word “infinite.”
In short, just as intuitionists accept “potential infinity” but reject “actual infinity,” Wittgenstein held that infinity merely indicates the absence of a limit and can in no way refer to something real. And Wittgenstein’s rejection of actual infinity, like that of intuitionism, was on the one hand due to anti-“Platonism” (to be discussed in detail later), and on the other hand also due to empiricism and positivism — talking about things that in principle cannot be experienced is meaningless — for example, Wittgenstein mentioned: “I could simply say: why infinitely many propositions cannot come from one proposition is that writing down infinitely many propositions is impossible (that is to say, saying such things is meaningless).” (v.4, p.232, §3) On the other hand, it should be noted in passing that Wittgenstein also often adopted a certain pragmatist strategy[20].
Of course, like intuitionism, while emphasizing that mathematical propositions, like ordinary science, all come from experience, Wittgenstein did not deny the place of other elements in mathematical activity. He said: “Experience certainly tells me how calculation arises, but I acknowledge not only experience.” (v.7, p.59, I§164)
III. Platonism — “Mathematics Is a Language Game”
As mentioned above, the thought of Wittgenstein and intuitionism both carry a distinct anti-Platonist or anti-realist tendency. Wittgenstein even said: “In philosophy, the most troublesome thing is not experience, but realism.” (v.7, p.246, VI§23)
“Platonism” has had a much greater impact on the whole of modern science than most people imagine. Even influential historians of science such as Edwin Arthur Burtt have pointed out that, throughout the entire process by which modern science rose from Copernicus to Newton, the revival of Platonism was always a crucial trigger and driving force[21]. But this “Platonism” that propelled the rise of modern science actually refers to the “mathematization” of the real world. In other words, modern Platonism broke down the boundary between the world of ideas and the real world, making it possible for our “ideal study” of the world of ideas to be used, quite naturally, to explain the real world. And in a later era, a world of ideas transcending the real world seems to have reappeared in science—namely, Platonism in mathematics. People discovered that the real world was after all not “ideal” enough, while the development of mathematics seemed once again to require support from more “ideal” objects; thus many mathematicians turned once more to the transcendent kingdom of ideas for help.
One reason for this situation is that a long-accepted belief has been overturned: “Mathematics is an abstraction from and reflection of the physical world; mathematical objects exist in corresponding real structures that they delineate.”
The overturning of this belief began, to some extent, with the birth of non-Euclidean geometry. Euclidean geometry had long been regarded as a true description of the physical world, but non-Euclidean geometry shattered this naive idea, and the “certainty” of mathematics began to shake. People started to accept that “multiple” mathematics might exist at the same time. Of course, there is only one real world; if physical space is non-Euclidean, then what, after all, does Euclidean geometry describe? Naturally, even before non-Euclidean geometry, negative numbers, complex numbers, infinitesimals, and so on had already been troubling mathematicians and philosophers for many years.
Another development came from the natural sciences: modern cosmology and quantum physics both eliminated “infinity” from the physical world. As David Hilbert said: “We have now established the finiteness of the universe in two ways, namely in the infinitely small and in the infinitely large.”[22] Most people agreed with Hilbert’s view: “If mathematics is to be independent of ambiguous empirical assumptions, then it must by no means base assertions about the existence of infinite structures on physical considerations.”[23]
Thus the question of how to “dispose of” 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 seemed to be the most convenient course. Yet such a mysterious world is puzzling.
The influence of Platonism often permeates people’s habits of thought and language unconsciously. For example, Paul Bernays pointed out: “An example of this way of building theory (the Platonistic one) 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, straight lines, and planes exists from the outset. Euclid assumes that one can connect two points by a straight line; Hilbert states the axiom that, given any two points, there always exists a straight line on which both points lie. … This example already shows the intention, which is becoming manifest, of severing altogether the connection between objects and the reflecting subject.”[24]
Georg Cantor, the founder of set theory, went even further and explicitly declared himself a Platonist[25]. He not only appealed to the “kingdom of ideas,” but also invoked mysticism and God to defend himself. He said: “The reality of mathematical objects does not lie in the real world, but in God’s infinite wisdom: the internal truth of mathematical objects, namely their logical consistency, guarantees that these objects are ‘possible,’ while God’s absolutely infinite essence guarantees the eternal existence of these ‘possible objects’ in God’s thought.”[26]
The extreme form of “Platonism” in mathematics may be stated as follows: “Mathematics consists of a body of propositions that deal with an independent reality composed of familiar mathematical objects, such as sets, numbers, functions, and spaces. Mathematical discovery consists in revealing, by deduction, truths about this independently existing reality, on the basis of axioms that we recognize as true by means of a special intuitive faculty distinct from sensory experience (which gives us only knowledge of the empirical world). Mathematical objects are independent of our thinking; unlike physical objects, they do not cause changes in the human mind through interaction with the body, thereby eventually leading to knowledge of them. But they must be postulated in order to account for the existence and growth of mathematical knowledge and of other knowledge (insofar as these depend on mathematical knowledge).”[27]
As Benacerraf said: “It is difficult to attribute this view exactly to anyone, although Gödel perhaps came closest to it since Plato. … Most mathematicians would probably reject this extreme form of Platonism; certainly not many contemporary philosophers or psychologists would consider acceptable the concept of a supernatural capacity to survey a realm of objects existing independently of space and time. But if one resists returning to such a view, then one cannot avoid certain problems raised by the mention of ‘intuition’ (which is encountered in the works of many philosophers and set theorists).”[28]
Even logicism or formalism also sought to avoid the extreme form of Platonism. The former tried to evade the problem by reducing mathematics to logic, but in fact it still had to run into difficulties when 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, tried simply to eliminate the “meaning” of mathematical objects altogether: mathematical objects do not need to place their meaning in the real world or the world of ideas, because mathematical objects do not need meaning in the first place; the significance of mathematics lies in the coherent, self-consistent system it forms. But these efforts largely avoid the issue, and their rejection of Platonism is not thoroughgoing. The claims of logicism and formalism will be discussed further below.
Without question, intuitionism’s rejection of Platonism is the most self-conscious and the most thoroughgoing. Heyting points out: “We do not ascribe to the integers or to any other mathematical object an existence independent of our thought, a so-called transcendent existence. Even if every thought of mine refers to an object which is thought to exist independently of it, we prefer to leave that an open question, and in any case such an object has no need to be wholly independent of human thought. Even if they must remain independent of the individual acts of thinking, mathematical objects are, by their nature, dependent on human thought. Their existence is secured only insofar as they can be determined by thought. They have properties only insofar as these properties can be recognized in them by thought. But this possibility of knowledge is disclosed to us only through the activity of knowing itself. Trust in transcendent existence, unsupported by concepts, must be rejected as an instrument of mathematical proof.”[29]
The intuitionists are not, however, directing their fire at the belief in a “world of ideas” as such. You may trust in transcendent existence, just as you may believe in God—that is your freedom (one thinks here of what Wittgenstein says: this is superstition, but not an error; see Philosophical Investigations §110). But that is a matter of faith, not of science. Science need not interfere with belief in transcendent objects; at the same time, such beliefs must not intrude into science, much less become tools of proof within science. As noted earlier, intuitionists place mathematics together with physics and the other natural sciences; mathematics is likewise an empirical, positive, fallible, natural science—only somewhat more abstract. Logic is even more abstract than mathematics, but neither can ever transcend experience in search of metaphysical or even theological things.
Incidentally, intuitionists firmly severed any connection between mathematics and a transcendent world, but on the other hand they tried to rebuild the almost-severed connection between mathematics and the real world; I shall expand on these tendencies below.
“The whole core of the intuitionistic standpoint is the view that undecidable mathematical statements do not, by virtue of a Platonistic stipulation of their truth conditions, legitimately possess meaning.”[30]—anti-Platonism can be said to be one of the main threads for understanding intuitionist thought.
Starting from empiricism, and emphasizing that “mathematical objects are not transcendent existences independent of human thought,” Wittgenstein is similar to intuitionism in these respects. The difference is that Wittgenstein opposes Platonism more from the standpoint of philosophy of language. Wittgenstein pointed out that “language” is not “some sort of non-spatial, non-temporal non-thing” (see Philosophical Investigations §108), and “‘Language (or thinking) is something unique’—this has proved to be a superstition produced by the deceptive power of grammar (not a mistake!).” (Philosophical Investigations §110)
In Wittgenstein’s view, not only does “language” not occupy an all-highest position, but within language there are also no concepts that enjoy any special privilege—“Indeed, as long as the words ‘language’, ‘experience’, ‘world’ have a use, this use must be as humble as that of the words ‘table’, ‘lamp’, ‘door’.” (Philosophical Investigations §97)
Since language is nothing but the product of real human activity, and any word, including mathematical concepts, is nothing but part of ordinary language, there is of course no such thing as a suprarreal kingdom of ideas either. Human linguistic activity is like play, and “the use of the words ‘true’ and ‘false’ may also be a component of this game” (Philosophical Investigations §136).
Here Wittgenstein seems close to formalism (formalism will be discussed in detail below), holding that truth and falsity do not consist in agreement with “reality,” but are judged within the internal rules of the established “game.” For example, he once mentioned: “The steps that do not become a problem are logical inference. But the reason they do not become a problem is not that they ‘certainly agree with truth’—or for some such reason. No, it is just because they are called ‘thinking,’ ‘saying,’ ‘inferring,’ ‘arguing.’ There is here not the slightest question of a correspondence between what is said and reality; rather, logic precedes such correspondence, in the same sense that, before the method of measurement is established, correctly or incorrectly stating lengths is impossible.” (v.7, p.57, I§156) Of course, in the prophecy game we “play,” the ascription of truth values does indeed have a special significance, because this is “a characteristic feature of the game we play with propositions,” just like “win” and “lose” in chess. But we can also imagine some “games very similar to chess, where pieces are moved too, but there is no winning or losing, or the conditions of winning and losing are different.” (v.7, p.73, I-Appendix III §2) Thus truth and falsehood, just like victory and defeat, depend on the rules of the game being played—“What is called ‘losing’ in chess may be winning in another game.” As we ask, “In which system is it ‘provable’?” we must also ask, “In which system is it ‘true’?” (v.7, p.75, I-Appendix III §8).
However, if mathematics is only a language game, then can its rules be laid down arbitrarily? If so, where do the certainty and reliability of mathematics come from? On this point, Wittgenstein differs from Brouwer, who advocated “basic intuition,” and is instead closer to the conventionalism of Poincaré, an early forerunner of intuitionism. The relation between Wittgenstein, formalism, and conventionalism will be discussed in detail below.
IV. Logicalism — “The Mind Cannot Be Grasped by a Machine”
I mentioned that Wittgenstein’s turn in thought stemmed from his mathematical-philosophical position shifting from logicism to intuitionism. Therefore, before discussing late Wittgenstein’s views on logicism, it is worth spending a little time on the relationship between logicism and intuitionism.
Logicism is probably one of the most influential schools in modern debates over the foundations of mathematics, and indeed in modern philosophy as a whole. It has almost dominated the development of Anglo-American philosophy throughout the entire twentieth century. In mathematics, its claim, put in the simplest terms, is this: mathematics can be derived from logic. More specifically, this includes: “1. Mathematical concepts can be derived from logical concepts by explicit definitions. 2. Mathematical theorems can be deduced from logical axioms by purely logical deduction.”[31]
As noted earlier, intuitionism can be said to represent empiricism and positivism in mathematics, and we know that most logicists were naturally also so-called “logical empiricists” or “logical positivists,” and so on. Seen from this angle, logicism and intuitionism, though mortal enemies, have quite a lot in common.
On the rejection of “metaphysics,” logicism and intuitionism are similar. Logicism may even be more resolute. Logical empiricism did not merely reject metaphysics in science; it despised metaphysics in its entirety. In their view, all traditional metaphysical problems were “pseudo-problems” and completely meaningless. Even questions such as “Is the external world real?”—since they are a typical metaphysical problem—were, in the initial phase of logical empiricism, simply meaningless. Compared with intuitionists, logical empiricists ought in principle to deserve the label “anti-realist” more fully (at least “non-realist”). The problem is that logical empiricists, precisely in logic and mathematics, no longer consistently carried through their empiricism, whereas intuitionists, though moderate on the question of the reality of the external world, mounted the strongest resistance to realism in the Platonistic sense, so much so that it was intuitionism that came to be more closely associated with the term “anti-realism.”
Many logicists also explicitly expressed support for the intuitionists’ anti-Platonism. For example, Rudolf Carnap said: “I think we should adhere to Frege’s dictum that in mathematics only those entities whose existence has been proved (meaning, for him, proved in finite steps) may be regarded as existing. I agree with the intuitionists that every logical-mathematical operation, proof, and definition requires finitude, not because of some accidental empirical fact, but because of the nature of the subject matter itself.”[32] “Like the intuitionists, we regard only those expressions as properties that are constructed, by means of definite rules of construction, from undefined primitive properties within a suitable domain, through a finite number of steps. The difference between us is this: we not only consider the rules of construction used by the intuitionists to be valid, but further allow the expression ‘for all properties’.”[33]
The relation between intuitionism and logicism seems very subtle. Just now I mentioned that intuitionism seems to emphasize “experience” more than logical empiricism; now we shall see that logicism seems to appeal to “intuition” more than intuitionism!
Kurt Godel mentioned: “The question of the objective existence of mathematical intuitive objects (incidentally, a precise 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 axiomatics of set theory and its extensions as an open series is enough to make the question of truth or falsity of propositions such as Cantor’s continuum hypothesis meaningful. But perhaps what proves more than anything else the legitimacy of accepting this criterion of truth in set theory is the fact that continual appeal to mathematical intuition is not only necessary for obtaining unambiguous solutions to transfinite set-theoretic problems, but is also necessary for solving finite number-theoretic problems (such as Goldbach’s conjecture),”[34]
Gödel’s “constantly appealing to mathematical intuition” is precisely what intuitionism demands be cut short! The difference lies in this: the logicists believe that human intuition can grasp and use concepts such as the “actual infinite,” whereas the intuitionists reject this outright.
Simply put, intuitionism uses the logicist’s own ideals—empiricism and positivism—to oppose the actual infinite; while logicism, by contrast, appeals to “intuition” to support the legitimacy of the concept of the actual infinite.
The intuitionist restriction on “intuition” is reasonable and internally consistent. No one would think that human intuition is always correct, or that everything correct is necessarily intuitive. Some critics object to intuitionism on the grounds that “some theorems in intuitionist mathematics are also too complicated to be grasped by intuition,” but this rests on an excessively simplistic, merely “literal” understanding of intuitionism. As Poincaré said: “Logic and intuition each have their necessary role. Neither can be dispensed with. Only logic can give us reliability; it is the instrument of proof, whereas intuition is the instrument of invention.”[35] Intuitionism has never denied the importance of logic.
By contrast, the restriction that logical empiricism places on “experience” is open to question. In fact, many logicists, “under the influence of a deeply entrenched absolutist view of logic, regard logic as purely formal and empty of content.”[36] They believe that the truths of deductive logic are absolutely necessary, and at most what is worth discussing is the question of the “justification” of inductive logic; they have never even seriously considered whether deductive logic also needs “justification.” But where, then, does logic’s “absolute necessity” come from? In fact, it is precisely because logic stands furthest from experience and bears only the most indirect relation to it that people get the illusion that “logic is absolute.” Logical rules, like other scientific laws and like common sense, “have an empirical origin; they come from the experience-based intuitions formed by people in the course of long-term social practice, and they are the result of people’s logical abstraction of everyday linguistic experience and thought experience.”[37]
The intuitionist position is based precisely on their acknowledgment of the “empirical” character of logic. From a linguistic perspective, logic is the result of people’s abstraction from ordinary language experience; and in mathematics specifically, the logic within mathematics is precisely an abstraction from the experience mathematicians have accumulated over long years of mathematical exploration and creation. Heyting said: logical theorems “are in no essential way different from mathematical theorems; they are simply more general, just as ‘addition of integers is commutative’ is a more general statement than ‘2+3=3+2.’ This is true of every logical theorem: it is merely 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.”[38] (Wittgenstein also held that logic is a part of mathematics and by no means its foundation; I will discuss this in detail later.)
We can see that the most fundamental disagreement between intuitionism and logicism is not whether to accept the “actual infinite,” but rather how to understand “what logic is.” As for the rejection of the actual infinite, that is merely a conclusion reached by intuitionists only after they have affirmed the “empirical” character of logic—since there is no actual infinite in experience, and since there is no transcendent world or supernatural power to provide assurance, how could our so-called “intuition” of the “actual infinite” possibly be legitimate?
As Brouwer put it: “Intuitionism on the one hand refines logic, and on the other hand attacks logic as a source of truth.”[39] Intuitionism does not fail to value logic. Just as we do not discard and despise physics simply because physics does not supply absolute truth, recognizing that physics gives us only relative truths is not a denigration of physics, but a sign of progress in human thought, and something necessary if physics is to continue developing healthily. In a similar way, intuitionism values logic, just as it values the empirical sciences. What intuitionism opposes is taking logic as a “source of truth” — human knowledge comes from experience, and if one wishes, one may also say that ultimately it comes from the “external world,” “ultimate reality,” “things in themselves,” “God,” or the like; but in any case: one can only say that science comes from reality, not that reality comes from science. Science is after all a historical human activity, and logic and mathematics are no exception!
After this brief survey of the relation between logicism and intuitionism, it may be easier to understand Wittgenstein’s philosophical turn. It should be said that the later Wittgenstein did not completely overturn many of the logicist claims he had accepted in his earlier period, nor did he move toward a rejection of logic. The later Wittgenstein still acknowledged logic’s important status; he remarked: “It is correct to say that mathematics is logic: it moves within the rules of our predicting. This gives it a special stability, a distinctive and unquestionable status.” (v.7, p.59, I§165) We also see that the later Wittgenstein still supported the effort to make expression more precise and rigorous, though he was wary of the tendency to become excessively fixated on precision. He said: “One could also say: by making our expressions more precise, we can eliminate some misunderstandings; but now it seems as if we were pursuing a particular state, a state of complete precision; as if that were the true goal of our inquiry.” (“Philosophical Investigations” §91)
Wittgenstein believed that this pursuit of “crystal-clear purity” in logic should not be the true goal of scientific inquiry, and that science should instead be grounded in ordinary language— “The more closely we examine actual language, the sharper becomes the conflict between it and our requirements. … We have got on to smooth ice where there is no friction, and so in a certain sense the conditions are ideal, but also, just because of that, we are unable to walk. We want to walk: so we need friction. Back to the rough ground!” (“Philosophical Investigations” §107)
In Wittgenstein’s view, even if the effort to bring mathematics into agreement with “crystal-clear” logic could in fact be achieved, it would still not be something to aspire to. For the logicalization of mathematics is at the same time the mechanization of human beings; people’s “intuition is oppressed by the dead weight of a written mode of expression” (see v.7, p.154, III§81).
Wittgenstein acknowledged that “experience tells us that calculations done by machines are more reliable than memory. Experience tells us that when we calculate with machines, our lives run more smoothly.” But he asked: “But is smoothness necessarily our ideal (is our ideal really that everything be wrapped in cellophane)?” (v.7, p.154, III§81)
Wittgenstein warned: “Logical machine — it will become an all-pervasive, supernatural apparatus. — We must be wary of this picture.” (v.7, p.45, I§119) Once people know only how to calculate like machines, to follow rules in a mechanical way, they make themselves into machines (see v.7, p.171, IV§20; v.7, p.331, VII§60). Yet “one cannot grasp spirit with a machine.” (v.7, p.154, III§81) Here Wittgenstein is in accord with the romantic or humanist temperament of the intuitionists.
In Wittgenstein’s view, the excessive pursuit of logical precision is not only something to be wary of; it is false as well. He said: “Through Russell and Whitehead, especially Whitehead, a false precision has entered philosophy, and it is the worst enemy of the precision of reality. The root of this error lies here: a calculation may be the mathematical foundation of mathematics.” (v.4, p.276, §12) (Wittgenstein’s claim that mathematics does not need a foundation will be discussed later.)
One of the most important reasons for saying that the precision sought by logicism is false rather than real has already been mentioned earlier: namely, that “logical inference is part of a language-game” (v.7, p.308, VII§30), and “the rules of logical inference are the rules of a language-game.” (v.7, p.312, VII§35) Thus the precision of logic amounts to nothing more than making the rules of this language-game especially standardized and strict; it has nothing to do with “reality.”
For Wittgenstein, mathematics is certainly normative, but “norm” (“Norm”) does not mean the same thing as “ideal” (“Ideal”) (see v.7, p.334, VII§61). Logic is a “normative science”; it offers a set of “games with fixed rules” — “but we cannot say that people using language are necessarily playing such a game” (“Philosophical Investigations” §81)
That is to say, logic provides some norms with a certain force of compulsion (see v.7, p.44, I§117), but it is still not a description of reality. The ideal pictures offered by logic and science are not descriptions of real language, thought, or real objects; rather, they are prescriptions of a certain kind. Wittgenstein said: “ ‘The description of the motion of a mechanism is based on the assumption that all its parts are perfectly rigid.’ On the one hand we acknowledge that this assumption does not accord with reality, while on the other hand no one doubts at all that perfectly rigid parts would move in this way. But does that yield certainty? The issue here is not really one of certainty, but of making a certain stipulation. We do not know whether objects, if they are rigid in such-and-such a standard, would move like this; but if, in certain cases, those parts moved that way, we would certainly call them ‘rigid.’ ” (see v.7, pp.46~47, §119~122)
Simply put, the idealized problems discussed by logic or science do not involve the question of the exactness of reality, but only the stipulation of certain words and grammatical rules. For people’s actual activities, these stipulations and rules are like “signposts” (see “Philosophical Investigations” §85); and as for signposts, the question is not one of truth or certainty, but only of appropriateness or effectiveness: “If a signpost serves its purpose under normal conditions, it is appropriate.” (“Philosophical Investigations” §87) And research into the setting up of signposts is after all not research into people’s actual walking.
Wittgenstein called the tendency to reduce mathematics to logic a “catastrophic intrusion” of logic into mathematics. With no small irony, he exclaimed: “One is almost tempted to say that making furniture consists in glue.” (v.7, p.209, V§24)
It is obvious that, with regard to this “catastrophic intrusion” of logic into mathematics, the attitude of the intuitionists and that of Wittgenstein are very much in agreement. Compared with them, Wittgenstein places more emphasis on arguments from the philosophy of language, which perhaps can be seen as a supplement to the intuitionist position.
V. Formalism — “Meaning is not a dead thing”
As mentioned above, Wittgenstein’s intellectual turning point was also influenced by the formalist school. Wittgenstein acknowledged that “Frege failed to see the rational core of formalism,” and this was precisely the important difference between him and Frege (see v.2, p.110). And Wittgenstein’s later philosophy, as Tasić put it, “seems to be trying to reach a compromise between the formalist and romantic viewpoints”[40] Tasić even declared that “it is precisely this wavering, the ghost of Wittgenstein’s split personality, that haunts and permeates most postmodern thought.”[41]
The situation may not be as complicated as Tasić makes it sound. In fact, formalism and intuitionism were already close in many respects. Some of the “rational core” of formalism that Wittgenstein admired may well have been precisely the part that intuitionism could accept. Before further discussing Wittgenstein’s attitude toward formalism, let us first make a brief assessment of the formalist school and its relation to intuitionism:
The founder of formalism was the celebrated Hilbert, and soon afterward his student John von Neumann also became one of its followers. These two can be said to be two mathematical giants ranked only behind Poincaré (though Poincaré is celebrated as “the last all-round mathematical genius”)! However, the number of formalist followers was not very large, and they were mainly mathematicians around Hilbert, though the influence of their views extended far beyond the world of mathematics.
As already mentioned above, Hilbert was the first to deny the reality of “infinity,” and he further objected to the logicists’ unrestricted acceptance of infinity in logic: “Statements such as ‘in a finite totality there exists an object with a certain property’ are perfectly in keeping with our finitistic methods. But statements such as ‘either p+1 or p+2 or p+3… or (up to infinity)… has a certain property’ are themselves an infinite logical product. Such a generalization to infinity is, without further explanation and precautionary measures, just as impermissible as the generalization from finite products to infinite products in the calculus. Therefore such generalization is usually meaningless.”[42] “Infinity is nowhere to be found in reality. It exists neither in nature, nor does it provide a legitimate basis for rational thought — that striking harmony between being and thought. Contrary to the previous efforts of Frege and Dedekind, we believe that to obtain scientific knowledge, certain intuitive concepts and powers of insight are necessary; logic alone is not enough. Operations with infinity become determinate only through finitism.”[43] “As a further prerequisite for the application and realization of logical deduction, there must, in concept formation, be something outside logic: certain concrete objects that are intuited as directly experienced things before all thought.”[44] “Substantive logical deduction is indispensable. Only when we make arbitrary abstract definitions, especially those that involve infinitely many objects, are we misled. In these cases, we have unlawfully used substantive logical deduction; that is to say, we have not sufficiently attended to the prerequisites necessary for the valid application of such deduction. In recognizing that these necessary prerequisities must be taken into account, we find ourselves in agreement with the philosophers, especially with Kant. Kant taught us — and this is a principal component of his doctrine — that the subject matter of mathematics is given independently of logic. Therefore mathematics can never be built up by logic alone. It follows that Frege’s and Dedekind’s attempts to construct mathematics in this way were doomed to failure.”[45]
From the lengthy passages quoted above, one might think he was simply stating the intuitionists’ views (which is also another reason I quoted so much at once)! It is clear, then, that in opposing logicism, formalism and intuitionism are indeed very similar. Thus, in Hilbert and his students’ gradual development, from 1920 to 1930, of what came to be called Hilbert’s proof theory or metamathematics for establishing the consistency of any formal system, principles that even intuitionists found acceptable were adopted[46]—and this is hardly surprising.
But the differences between formalism and intuitionism are also obvious:
Hilbert could not tolerate the “destruction” of the achievements of classical mathematics by intuitionists. He insisted on retaining “infinity” within mathematics. Although it could not exist anywhere outside the human mind, nor be legitimate even within “logic,” Hilbert said: “Infinity still very likely occupies a legitimate place in our thinking, serving as an indispensable concept.”[47]
If one wants to retain actual infinity as a legitimate concept, while knowing full well that its meaning cannot be “anchored” either in the empirical world of reality or in the transcendent world of ideas, and Hilbert also did not choose to throw himself into transcendentalism, then where exactly can the meaning of these mathematical concepts be “anchored”? Hilbert’s choice was: simply not to seek the meaning of mathematical concepts anywhere! Or rather, the meaning of mathematical concepts lies only in the mathematical system itself.
So Hilbert set about constructing formal systems that had been stripped of “meaning.” In Hilbert’s view, the intuitive connection that mathematical concepts—say, lines and points—might originally have to certain real objects was unimportant; “line” and “point” were merely symbols, and the meanings of these mathematical symbols could only be exhibited in the relations among the symbols themselves. For example, a mathematical statement such as “between any two points there exists a line” could be replaced by other symbols, such as “between any two chairs there exists a table,” and this would not alter the statement’s meaning in mathematics! — “Hilbert decided to express all accounts of logic and mathematics in symbolic form. These symbols, though they can express the perception of intuitive meaning, cannot find any interpretation in the formal mathematics he proposed. Hilbert hoped to include some symbols that could even represent infinite sets, but these had no intuitive meaning. These ideal elements, as Hilbert called them, are necessary for the construction of all mathematics, so their introduction is justified, although Hilbert believed that in the real world there exist only finitely many things, and things are composed of finitely many elements.”[48]
In other words, by stripping the whole of mathematics away from reality and intuitive meaning, Hilbert evaded the problem of the meaning of “the infinite.” Hilbert, like the intuitionists, held that the actual infinite has no meaning either in reality or in intuition; yet he accepted the actual infinite precisely by abstracting away the intuitive meaning of the whole of mathematics! As Brouwer put it: “For the formalist, the exactness of mathematics lies only in developing methods for relational series, and has nothing to do with the meaning one tries to assign to these relations or to the entities involved in them.”[49]
Thus each mathematical element is able to express itself only in its relations of mutual reciprocity with the other elements in the mathematical system—“these statements, theoretical in the mathematical sense, Hilbert calls ‘ideal elements’; he compares their introduction to the introduction of the ‘points at infinity’ in projective geometry: they are introduced as a convenience for making the theory of the things that really concern you simpler and more elegant. If their introduction does not lead to contradiction, and if they have these additional uses, then such an introduction is proved justified: hence one must seek a proof-achieving system for the complete system of first-order arithmetic.”[50]
That is to say, the introduction of actual infinite sets, like the introduction of points at infinity, serves only to make the whole theory more elegant and to make other operations with real significance more convenient. So the pressing problem for the formalists is to prove that introducing these meaningless, convenient concepts will not make the system inconsistent. Thus, once Hilbert had 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 in solving this problem. …”[51]
However, Hilbert’s bold claim that he had “answered the continuum problem” was never truly realized, and it would later be shown to be impossible to realize forever! Hilbert’s dream was doomed to end in failure.
I will not continue the discussion of the continuum problem here, because whether the consistency of a formal system can be proved is, in the dispute between intuitionism and formalism, fundamentally irrelevant.
Intuitionism holds that even if the consistency of Hilbert’s formalized mathematics were proved, this theory—this formalized mathematics—would still be meaningless. Weyl complained that Hilbert “saved” classical mathematics by means of “a complete reinterpretation,” that is, by formalizing it and thereby stripping away its meaning. “Thus it is in principle transferred out of the intuitive system and becomes a game of formulas conducted according to fixed rules.” “Hilbert’s mathematics may perhaps be a magnificent game of formulas, even more interesting than chess. Since its formulas do not possess any generally recognized real meaning by which they could be taken to express intuitive truth, what relation can it have to knowledge?”[52]
The intuitionists continued to stress that they relied on the meaning of mathematics to determine whether it was correct or not, whereas the formalists (and logicists) were dealing with an ideal or meaningless supernatural world. Brouwer said: “Arbitrary use of Aristotelian logic leads to formally valid but substantially meaningless assertions; classical mathematics, by abandoning the meaning in many logical constructions, abandons reality.”[53]
The founder of intuitionism, usually thought of as anti-realist, was actually accusing his opponents of “abandoning reality”! In fact, the intuitionist 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 “reality”!
Of course, intuitionist logic can also, and must, be “formalized”; however, this does not mean that intuitionism accepts formalism. Heyting, Brouwer’s student, was the first to formalize intuitionism and to establish a genuinely intuitionist logical system, yet Heyting himself explicitly pointed out: “It is also impossible to bring formalism and intuitionism into agreement by formalizing intuitionist mathematics. Admittedly, even in intuitionist mathematics, the completed part of a theory can be formalized. It would be very useful to investigate for the moment the meaning of this formalization. We may regard this formal system as the linguistic expression of mathematical thought in a particularly suitable language.”[54] For intuitionism, the fundamental function of a so-called “formal language” is exactly what its name says it is—“language.” The role of “language” is to facilitate communication of “thought” between people; a more perfect language can express thought more clearly and accurately, but what truly has meaning, what truly matters, is something beyond language. And Heyting goes on: “If we adopt this point of view, then we run headlong into 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, we can never guarantee mathematically that a formal system correctly expresses our mathematical thought. (My note: this claim is further supported by the Löwenheim–Skolem theorem.)”[55]
Formalization is a refinement of language as a tool of communication (logical rules of inference, and so on, can probably be likened to grammar); this is important. But the progress of language lags behind culture and thought, and introducing the alphabet into a primitive tribe will not greatly increase the knowledge it possesses. In the same way, the development of mathematics itself always runs ahead of formalization, and the progress of mathematics will ensure that formalization can never be finally completed. Heyting said: “Formalization can be carried out within mathematics, and thus become a powerful mathematical tool. Of course, one can never be sure 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 expand this formal system. … Intuitionistic activity is independent of formalization, and formalization can only follow behind mathematical construction.”[56]
In his opposition to logicism, Wittgenstein’s position is similar both to intuitionism and to formalism. And Wittgenstein agrees with intuitionism in opposing the extraction of the “meaning” of mathematics; in opposing the severing of mathematics from “reality”; and in holding that formal languages are also “languages,” while what is truly “meaningful” in mathematical activity is beyond language. All of the above has already been discussed earlier and need not be repeated here. In addition, Wittgenstein also maintained that the formalization of mathematics, or any comparable work of laying a foundation for mathematics, is merely a retrospective record of a continuously developing mathematical activity, and not a true “foundation” of mathematics; this point will be discussed in detail shortly.
For Wittgenstein, mathematics is indeed, but by no means only, a set of rules of the game. Just as with chess, it has “not only rules, but also a point.” (v.7, p.67, I-Appendix 1 §20) Mathematics does not investigate symbols, just as chess does not investigate wooden chess pieces (see v.4, p.270, §10)
Wittgenstein pointed out: “The sign seems dead by itself. What gives it life? In use it is alive. Does it breathe life into it in use? — Or is use its life? — The sign by itself is dead; use is the life of the sign.” (Philosophical Investigations §432)
Wittgenstein regarded formalist mathematics, or some activity similar to “certain symbols can be constructed from other symbols according to certain rules,” as what remains after mathematics has been stripped of content (see v.7, p.117, III §38). Such mathematics, stripped of meaning or content, is lifeless.
What is “meaning”? — On the one hand, Wittgenstein repeatedly emphasized that “the use of a word in language is its meaning.” (v.4, p.51, §23) But at the same time he also pointed out that “meaning,” as the “final interpretation,” is something uninterpretable and unsayable—“Every sign is in principle capable of interpretation; but meaning cannot be interpreted. It is the final interpretation.” (see v.6, p.46) One could say that, for Wittgenstein, “meaning” lies in “use,” but is not only use. — “We want to say: ‘Meaning is indeed essentially a mental process, a process of consciousness and life, not a dead thing.’” (v.4, p.139, I §100)
Here a certain romantic temperament, akin to that of the intuitionists, reveals itself unmistakably; on another occasion Wittgenstein even remarked that for some mathematicians to say that “mathematics is not a creation of the mind” is “ridiculous” (see v.5, p.388)!
Wittgenstein even compared mathematical activity to musical composition. He remarked: “Take a main theme similar to Haydn’s main theme, take a section of a variation by Brahms, and pose the task of constructing the second part in the style of the first. This is a problem of the same type as a mathematical problem.” (v.7, p.285, VII §11)
As Tasic says: “Even for those famous romantic mathematicians, identifying mathematics with art would be excessive. However, the main strategy of mathematical critiques of logical reductionism is to develop this analogy to a certain extent in detail. From this point of view—sharply opposed to the view of the logicians—mathematics is something more than mathematical texts or the ‘mechanical’ decoding of texts. It is one kind of human activity, and texts are at most an imperfect record of it, a guide to the creation of mathematical meaning.”[57] Here Wittgenstein’s strategy remains aligned with that of the intuitionists and the romantics.
“I want to show the rich variety of mathematics.” (v.7, p.129, III §48) — Wittgenstein said.
VI. Conventionalism—“Language is connected with a form of life”
The discussion of formalism above skipped over one question: what exactly is the “reasonable core” in formalism that Wittgenstein praised?
Wittgenstein remarked: “There is something right and something wrong in formalism. Taking every syntax as a system of rules of the game is the respect in which formalism is right. What might Weyl (that is, Weyl—translator) have meant when he said that the formalist takes mathematical axioms to be something like the rules of a chess match? I have thought about this. I want to say: not only mathematical axioms, but all syntax is arbitrary.” (v.2, p.67)
It seems that the core concept of Wittgenstein’s later philosophy—“language-game”—has much to do with formalism!
However, Wittgenstein did not really insist that “all syntax is arbitrary.” For Wittgenstein, people cannot formulate language-games entirely at will; people are not, as formalism claims, concerned with the internal consistency of the rules of the game, but determine language-games by “instinct” (see v.7, p.173, IV §23). And this instinct comes from people’s forms of life, or customs and culture—“Language is connected with a form of life” (v.7, p.255, VI §34); “What belongs to a language-game is the whole culture.” (v.12, p.331, §26); “It is impossible for someone to have obeyed a rule only once … Obeying a rule, giving a report, giving an order, playing a game of chess, are customs (uses, institutions)” (Philosophical Investigations §199)
As mentioned earlier, on the question of where the certainty and reliability of mathematics come from, Wittgenstein differs from Brouwer, who advocated a “basic intuition,” and is in fact closer to Poincaré’s conventionalism, a precursor of intuitionism. In fact, the idea that “mathematics or science is something like a set of rules for action, similar to chess” was not unique to Wittgenstein or to the formalists. The analogy between mathematics and chess had already been proposed at least as early as Poincaré and even earlier by the extreme conventionalist Le Roy. In Science and Value, written in 1907, Poincaré commented: “According to Monsieur Le Roy, science is nothing but rules for action. … For example, for amusement, one lays down rules for games, rules for games like chess, … As in science, the rules of the game are indeed rules for action; but can anyone compare them without seeing the difference? The rules of a game are arbitrary conventions; even if the opposite conventions were adopted, that would not matter. Scientific rules, by contrast, are fruitful rules for action.”[58]
It is worth noting that Poincaré, too, starting from the analogy with chess, expressed a view similar to Wittgenstein’s “chess has not only rules, but also a point.” He said: “If you are watching a game and want to understand a match, it is not enough to know the rules by which the pieces move. That would only enable you to recognize each move as conforming to these rules, and such knowledge indeed has little value. If the reader of a mathematics book were merely a logician, he too would do this. To understand a game of chess is entirely different; one must know why the player moves this piece rather than that one, what he can do without violating the rules of the game. One can discern the internal basis that makes this series of successive moves an organic whole. There is every reason to indicate that this is necessary for the player himself, and it is so for the inventor as well.”[59]
For Wittgenstein, mathematics and logic, “like other languages, rest on conventions.” (Philosophical Investigations §355) Without “mutual agreement,” there is no “acting according to rules,” and there is no arithmetic either (see v.7, p.262, VI §41; v.7, p.267, VI §45). “Of course we can play this language-game only when agreement exists.” (v.11, p.218, §430) However, Wittgenstein especially emphasized that “people’s agreement, as a prerequisite of logical phenomena, is not an agreement in opinion, much less an agreement in opinion about logical questions.” (v.7, p.270, VI §49) This “agreement” refers to people’s agreement in action—“language phenomena are grounded in regularity, grounded in agreement in action.” (v.7, p.261, VI §39) — and this agreement in the way of seeing things comes from the technical education people receive within their culture and customs (see v.7, p.178, IV §35)
In fact, the conventionalism emphasized by Poincaré’s conventionalism seems not to refer only to consensus at the level of “opinion.” Poincaré also explored in depth the relation between science and language, noting that science, too, is a certain kind of “language,” and that scientific activity is, in a sense, an activity of creating and translating languages—“All that the scientist creates in a fact is only the language in which he states that fact.”[60] “Scientific facts are only raw facts translated into convenient language.”[61] Then is this creation of language arbitrary? Could people—or organisms—living under different cultural backgrounds and ways of life create scientific systems that are completely untranslatable into one another? Poincaré asks: “Are there some things independent of these conventions? In other words, are there some things that can play the role of general invariants?” Poincaré further “imagines more bizarre beings, which would make the common part of the two systems of expression smaller and smaller”; then could this common part possibly shrink until there is none at all? Poincaré continues: “Even if there were neither interpreter nor dictionary, if Germans and Frenchmen, after living for several centuries in worlds isolated from one another, suddenly came into contact, could you believe that there would be absolutely nothing in common between the science recorded in German books and the science recorded in French books? The French and the Germans would no doubt eventually understand one another. … This is because there remains between the French and the Germans something in common, since both are human beings. We can also understand our hypothetical non-Euclidean geometers, though they are not human beings, if they are nonetheless some kind of human-like creatures; but in any case even the minimum of humanity is indispensable.”[62]
Simply put, Poincaré believed that different systems of expression would after all have some common ground, which enables people from different cultures eventually to understand one another. Poincaré traced this common ground to “humanity,” whereas Wittgenstein more accurately traced it to people’s similar “forms of life” — agreement in language comes from agreement in forms of life (see Philosophical Investigations §241).
Wittgenstein once imagined a peculiar society (see v.7, pp.55~56, I§148~151), where people’s understanding of “many logs” and “a little firewood” differed sharply from ours, making translation difficult for us. Wittgenstein pointed out that this difference in language use arose precisely because “their whole way of paying is completely different from ours,” and linguistic phenomena are the expression of culture and custom.
Wittgenstein then compared mathematics with anthropology, saying: “Are mathematical propositions anthropological propositions, about how we human beings reason and calculate? — Is a legal code a work of anthropology, telling us how this people deals with thieves, and so on? — Could one say: ‘The judge consults a book on anthropology and on that basis sentences the thief to a term of imprisonment’? Well, the judge does not use the legal code as if it were an anthropological handbook.” (v.7, p.137, III§65) Mathematics is a human activity; the rules and propositions of mathematics all reflect human habits, though people do not regard mathematics as anthropology, just as a judge does not treat the legal code as an anthropological handbook. In Wittgenstein’s view: “It is obvious that we can use mathematical works for anthropological research.” (v.7, p.141, III§72) In this regard, Brouwer had also declared that logic and science should be “classified as ethnography”[63]!
VII. The Axiomatization Movement — “Mathematics Does Not Need Foundations”
Whether it was logicism or formalism, whether Frege, Russell, Hilbert, or Zermelo, the “axiomatization movement” is the umbrella term for them. They all hoped, in the Euclidean manner, 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 holds that mathematical axioms are all products of logic, whereas formalism supports the choice of axioms by the internal consistency of formal systems; but these differences are comparatively so small that people often fail to distinguish logicism from formalism at all.
Intuitionism, however, stood outside the whole tide, and reflected upon and criticized the whole trend toward “rigorization” and “axiomatization.”
Intuitionists were not opposed to the rigorization of axioms and deduction itself. “Axiomatization,” in its own sense, is a long-standing mathematical tradition that began with Euclid; it is by no means a modern invention. The problem with modern people, however, is that they emphasized “axiomatization” too much, and obsessed too much over developing rigorous logical deduction, gradually forgetting some more important, or at least equally important, factors.
Heyting said: “The intuitionist mathematician proposes to regard mathematical activity as a natural function of his intellect, as a free, lively activity of thought. In his view, mathematics is a product of the human mind. He uses language, whether natural or formalized, only in order to communicate thoughts, that is, to make his own mathematical ideas understood by others or by himself. This linguistic accompaniment is not the representative of mathematics, much less mathematics itself.”[64]
The demand for rigor is reasonable; however, rigor concerns the language of mathematics. Mathematical discovery (or, as Wittgenstein and the intuitionists would say, “invention”—this makes no difference here) is not obtained through language itself, not through word games. As Poincaré said: “Intuition is the instrument of invention.” As for rigorous “proof,” its role is nothing more than to follow after mathematical creation, checking and correcting errors, ensuring the reliability of mathematical creation, and serving as a means for “demonstrating” results.
Many outstanding mathematicians were dissatisfied with people’s excessive reverence for “proof.” Godfrey Harold Hardy said with some irony: “Strictly speaking there is no such thing as mathematical proof; … in the end we merely indicate certain points; … Littlewood and I both call proof nonsense; it is a pile of grandiloquent words invented to impress certain people, pictures used in lectures for demonstration, a tool to stimulate the imagination of elementary schoolchildren.”[65]
Even Whitehead, the author of Principia Mathematica, and Russell did not have much affection for logic and “proof.” In a lecture, Whitehead said: “Logic is regarded as the adequate analysis of the development of thought, but in fact it is not so. It is an excellent instrument, but it still needs some common sense as a background…. The final form of philosophical thought cannot be established on the basis of the exact formulations that constitute the foundational form of a special discipline; all exactitude itself is fiction.”[66]
Even Russell, who followed the logical program to the letter, was not sparing in his mockery of logic. In Principia Mathematica he wrote: “One of the chief merits of a proof is that it gradually instills in us a certain doubt about the result proved.” He also said, “The attempt to found mathematics on a system of undefined concepts and propositions is precisely inferred from this essence of mathematics: conclusions may be contradicted by contradiction, but they can never be proved. Everything ultimately depends on intuitive understanding.”[67]
Lebesgue, a mathematician inclined toward intuitionism, said: “Logic can lead us to reject certain proofs, but it cannot lead us to believe any proof.”[68]
In the eyes of many mathematicians, “the development of mathematics does not rely on logic, but on correct intuition. As Jacques Hadamard pointed out, rigorization is merely the sanctioning of the spoils of intuition, or, as Hermann Weyl said, logic is the hygienic discipline by which the mathematician keeps his thought healthy and strong.”[69]
It is quite apt for intuitionism to liken logic to a “hygienic measure.” Intuitionism never denies the importance of logic; however, to use a metaphor: intuition is the blood of mathematics—it provides energy and momentum; experience is the food of mathematics, from which mathematicians draw nourishment; while logic is a “hygienic measure,” a health aid and medicine that strengthens the body and prevents and treats disease, helping mathematics toward stability, maturity, and perfection. Seen in such a metaphor, the relationships among the various elements are immediately clear: intuition gives mathematics life, experience makes mathematics grow, and logic makes mathematics strong. But if one forgets, or even abandons, the blood and the food, and relies only on medicine, then not only can strength not be maintained, life itself cannot be sustained.
In opposition to the axiomatization movement, Wittgenstein’s attitude was entirely in line with that of the intuitionists. He repeatedly emphasized that “there is no metamathematics” (e.g. v.4, p.271, §10; v.4, p.276, §12, etc.)
Wittgenstein said: “Why should mathematics need a foundation?! I believe mathematics does not need such a foundation, just as propositions concerning physical objects or propositions concerning sense impressions do not need analysis; however, mathematical propositions, like other propositions, do indeed need an account of their grammar. The mathematical problem about so-called foundations is, in our view, not the foundation of mathematics, just as a painted rock is not the foundation of a painted castle.” (v.7, p.291, VII§16)
The metaphor “a painted rock is not the foundation of a painted castle” calls to mind another metaphor by the famous historian of mathematics M. Kline[70]: “A fable aptly summarizes the state of progress in this century regarding the foundations of mathematics. On the banks of the Rhine there has stood for many centuries a beautiful castle. In the castle’s cellar lives a group of spiders, and suddenly a strong wind scatters a complicated web they had painstakingly woven, so they frantically repair it, because they believe that it is precisely the web that supports the entire castle.”[71]
Wittgenstein pointed out that “logic and mathematics are not based on axioms.” “Whether a system is based on initial principles, or whether it is merely derived and developed from them, are two different things. The following two situations are also completely different: whether it is like a house built on its lowest wall, or whether it is like a celestial body freely floating in the air above which we are beginning construction below, although we can also build elsewhere.” (v.4, p.277, §12) Wittgenstein believed that even if logic formulas could be made to accord with mathematics, “this fact by no means shows that mathematics is founded upon logic.”[72] For mathematical activity had long been developing freely before we began constructing the so-called logical foundations. Wittgenstein said, “I am examining the record of the mathematician’s activity” (v.4, p.275, §11). He placed greater emphasis on “observing mathematics from the standpoint of action” (v.7, p.278, VII§5)
Like the intuitionists, Wittgenstein believed that mathematical logic, or work such as axiomatization, formalization, and rigorization, at best is only content of mathematics (for example v.5, p.364, §1). As mentioned earlier, for Wittgenstein, mathematics is a continuously developing human activity, “mathematics is not a strictly delimited concept” (Vol.7, p.223, V§46).
The mathematician F. Klein put it well: “In fact, mathematics has already grown like a great tree, but it did not begin by growing from the thinnest root, nor did it grow only upward; rather, as its branches and leaves expanded, its roots dug ever deeper downward…. Then we can see that the foundations in mathematics have no final end, and, from another perspective, also no initial starting point.”[73]
Mathematics is something that grows and develops. “The foundations of mathematics” can be counted as a field or branch within mathematics, and it, too, like mathematics as a whole, will be developing and will have no final conclusion. Any effort to provide mathematics once and for all with an absolutely solid, fixed foundation and to define mathematics’ clear boundaries is nothing but fantasy; the continual development of mathematics will bring new problems to the so-called field of mathematical foundations. It is impossible to formulate all the rules of mathematics once and for all; as Wittgenstein said: “Do we not also have cases where we play and at the same time make the rules? And there are also cases where we play and at the same time alter the rules.” (Philosophical Investigations §83) As mathematical activity advances, the rules of the mathematical game will also keep changing.
Although in fact we see that intuitionism’s positions concerning mathematical foundations are more coherent and stable than those of other schools, intuitionists’ ambition to “provide mathematics with an unchanging, solid foundation” is by no means as strong as that of the other schools. Heyting agreed with the pragmatist view: “First an explorer, and only then a philosopher. And if we wish, we can leave the latter to others.”[74] The final scene in his dialogic essay “Counter-Argument” is this: the intuitionist leads everyone into the classroom and presents the “samples” of intuitionist mathematics—“A few lessons will give you a better understanding of it than a long discussion.”[75]
The intuitionists did not persist in providing an authoritative, dogmatic answer to “what mathematics is”; they never bogged themselves down in endless philosophical disputes. They always paid more attention to mathematical activity itself, while also attending to natural science, and ultimately even to everyday life, and they interpreted their position through their own practice. Heyting said: “In describing intuitionist mathematics, I am conveying ideas to my audience; these statements should be understood not in the sense of some philosophical system, but in the sense of everyday life.”[76]
This attitude of “should be understood in the sense of everyday life,” of course, is completely in line with the concern of the later Wittgenstein. Wittgenstein said: “Philosophy may in no way interfere with the actual use of language; it can therefore ultimately only describe the use of language. For it can also not lay a foundation for the use of language. It leaves everything as it is. It also leaves mathematics as it is; it cannot promote any mathematical discovery. For us, ‘the primary problem of mathematical logic’ is also a mathematical problem. Just like any other mathematical problem.” (Philosophical Investigations §124)
VIII. Pure Mathematics — “Application Is Crucial”
Like intuitionism, it never wanted to be hostile to “mathematics”; yet the rejection of the actual infinite and the restriction of the law of excluded middle led it to inflict tremendous damage on the “massive achievements” already obtained by mainstream mathematics. And this is in fact the real reason why many people reject intuitionism: at times they do not even have the patience to understand intuitionism’s claims and ideas, and merely seeing the achievements of “mathematics” thus “damaged” makes them feel it is intolerable.
But is intuitionism’s “damage” to mathematics really that important? Are the things that modern mainstream mathematics is unwilling to give up really so significant?
We need to begin from another notable trend in the development of modern mathematics: namely, the “isolation” of mathematics.
The trend toward isolation is interrelated with trends such as Platonization, axiomatization, and formalization. As mentioned earlier: the connection between mathematics and physical reality began to be severed. Whether Platonism attempted to entrust mathematics to a transcendent world, or logicism attempted to make mathematics rely on logic, or formalism attempted to regard mathematics as a self-sufficient symbolic system, their common effort was precisely to make mathematics no longer need to depend on empirical science.
This trend first began with the inversion of the status of number theory and geometry, and the whole process of this inversion is precisely bound up with the rise of modern science.
In the mathematical tradition from Plato to Blaise Pascal, “geometry” was unquestionably the “foundation” of all mathematics. Pascal once said: “Everything that goes beyond geometry surpasses my understanding.”[77] From Pythagoras and Euclid onward, geometry had always been the basis of arithmetic, not the other way around. The initial axiomatization of mathematics began with geometry. 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 they had no spatial meaning! In fact, Wittgenstein was closer to these classical mathematicians; he too believed that “the logical certainty of proof has not surpassed its geometric certainty.” (v.7, p.122, II§43)
But ever since Descartes invented analytic geometry, the situation has been completely reversed. People first accepted the independence of arithmetic, regarding the meaning of arithmetic as self-evident as well; for example, in Kant, arithmetic and geometric intuition occupied equal status. In the end, the rise of non-Euclidean geometry dealt the decisive blow—people seemed no longer able to “understand” geometry! Thus geometry, having lost the support of intuition, had no choice but to throw itself on arithmetic.
And the characteristic of arithmetic is this: it is the farthest removed from reality. In fact, before 1900, among all branches of mathematics, the only one that could truly count as “mathematics for mathematics’ sake,” or as the pursuit of “beauty,” was number theory. All the other branches of mathematics were not only always closely tied to the empirical sciences, but their creation often had originally been for the purpose of dealing with certain physical problems—even non-Euclidean geometry was no exception! The motive that led Gauss and others to propose non-Euclidean geometry was not “change this axiom and see what I can come up with,” but rather “Is physical space really Euclidean?”; for this purpose Gauss even specially used three mountain peaks to carry out field measurements.
As M. Klein said: “For the mathematics created before 1900, we may draw a general conclusion: there is pure mathematics, but there are no pure mathematicians.”[78] What is called “pure mathematics” is number theory; yet number theory was often merely a toy for great mathematicians to amuse themselves in their spare time. Even in other mathematical fields, the work was often not the mathematicians’ main profession; their occupations were often professorships in astronomy, physics, and so on.
Of course, Gauss once said, “Mathematics is the queen of the sciences, and number theory is the crown of mathematics,” and difficult problems in number theory were hailed as “the jewels on 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 captivating. But this does not mean that Gauss thought number theory was the source of mathematics, or mathematics the source of science. The fact is rather the opposite. Just as the whole of science does not originate from mathematics, this queen, but from nature, from experience; so too the foundation of mathematics lies in the natural sciences, not in number theory aloofly perched above everything else.
And in modern times, with the exchange in status between arithmetic and geometry, number theory became the “foundation” of the whole of mathematics. No wonder this queen was becoming ever more “proud” and “self-satisfied.”
The result was this: by the end of the nineteenth century, the prevailing view was that every axiom in mathematics was arbitrary, and that axioms were nothing more than the basis for the reasoning that derives conclusions. Since axioms were no longer truths about the concepts contained within them, there was no need to care about the physical meaning of these concepts either. When some connection arises between axioms and reality, that physical meaning can at most serve as a guide to discovery (truth). Even concepts abstracted from the physical world are like this.[79]
In modern times, the view that mathematics is independent of experience and detached from the natural sciences became the mainstream position in the mathematical world. M. Klein said ruefully: “Nowadays one often hears and reads mathematicians’ pronouncements about independence from science. Mathematicians can now say, without hesitation and casually, that they care only about mathematics itself and have no interest in science. Although there is no precise statistic, of the mathematicians active on today’s mathematical stage, about 90 percent ignore science and revel in this blissful condition. Despite historical evidence and some opposing voices. But the trend toward abstractionism, the trend toward generalization for its own sake, and the trend toward studying arbitrarily chosen problems are becoming ever more intense; claims that this is for the reasonable need to study an entire class of problems in order to understand more concrete cases, or for the reasonable need to abstract in order to grasp the essence of a problem, are nothing but excuses. Their purpose is simply to study generalization and abstraction.”[80]
As the leaders of the mathematical world at the end of the nineteenth century, Poincaré and F. Klein (Felix Klein) both foresaw this trend and were deeply worried by it. In 1895 F. Klein said: “In the rapid development of modern thought, we cannot help fearing that our science faces the danger of becoming ever more independent. Since the rise of modern analysis, the close connection between mathematics and the natural sciences, to the benefit of both, is in danger of being destroyed.”[81]
People may not easily understand why intuitionism paid 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. Unwillingness to cut itself off from nature was an important reason intuitionists resisted logicism and formalism. Poincaré said: mathematics “need not forever stare only at its own navel for its own sake; it is linked with nature, and there will necessarily come a day when it returns to nature. Then these purely verbal definitions will certainly have to be discarded, and we shall no longer be deceived by these empty words.” “Logicism must be corrected, and one does not know at all what can be retained; needless to say, I am referring here to Cantorianism and logicism; true mathematics always has practical ends, and it will develop according to its own principles, paying no heed to the storm raging outside, and it will step by step pursue its accustomed victories—this is certain, and it will never cease.”[82]
Another pioneer of intuitionism, Kronecker[83], once wrote to Hermann von Helmholtz (a German physicist and the proposer of the law of conservation of energy): “Your reasonable and practical experience, together with the wealth produced by interesting problems, will give mathematics a new direction and new stimulus—one-sided, introspective mathematical speculation leads people into barren lands.”[84]
Other master mathematicians of the modern era also never ignored the natural sciences. Before turning to problems of the foundations of mathematics, Hilbert was in fact immersed in problems of physics; and von Neumann likewise pointed out: “It is undeniable that some of the most remarkable inspirations in mathematics, the best inspirations in those departments of mathematics imagined to be purest of the pure, all come from the natural sciences.”[85]
However, the disciples of the masters are always less clear-sighted than the masters themselves. “Purification” and “isolation” are indeed the trend in the development of modern mathematics. Even among mathematicians, “applied mathematics” has practically become a term of reproach, used by those “pure” mathematicians to accuse those who are “neglecting their proper duties” and “going astray.” As an applied mathematician himself, with considerable attainments in electromagnetism, M. Klein felt this keenly; he complained: “Failure to take into account the objective objects it serves will inevitably lead to its own termination. Pure mathematics itself is not a blissful realm. The purpose of mathematics is to discover things worth knowing; but under the present circumstances, research leads to research, which in turn leads to research. In today’s mathematical temple, no one dares to ask about meaning and purpose anymore. Mathematics must not be contaminated by the mundane things of reality; the thick walls of the ivory tower block the sight of the scholars dwelling within it, and these minds cut off from the world are content with an isolated condition.”[86]
I have no intention of criticizing so-called “pure mathematics.” In other contexts I shall offer the highest praise to pure mathematics as mathematics for mathematics’ sake, just as we praise poetry and art—who can say that artistic activity is meaningless? The meaning of human existence need not necessarily be related to “utility.” However, to leave the natural sciences and the real world behind in order to indulge oneself, to delight in one’s own company in the ivory tower of pure mathematics—this tendency is indeed unhealthy. Because the connection between mathematics and nature is no longer so clear and certain, because mathematics can no longer provide absolute objective truth, to simply cease caring about their relationship is undoubtedly a kind of laziness and evasion.
Although Wittgenstein avoids discussing the issue of “nature,” his emphasis on the relation between mathematics and “reality” is obvious. Wittgenstein especially stressed the application of mathematics; for example, he emphasized: “The application which mathematics desires is crucial” (v.7, p.192, V§5) “It is conceivable that people would have applied mathematics without pure mathematics” (v.7, p.169, IV§5), and so on.
Several representative figures of intuitionism also attached great importance to applied mathematics, and Weyl in particular made a great many contributions to mathematical physics. He said: “How convincing and close to the facts are the heuristic arguments and then the systematic structure in Einstein’s general relativity, or in Heisenberg-Schrödinger quantum mechanics! A truly mathematical theory should, just like physics, be regarded as a branch of the theoretical structure of the real world, and we should approach the extension of its foundations with the same seriousness and caution as we do physics.”[87]
Weyl affirmed that mathematics reflects the order of nature. In a conversation, he said: “Nature has an inherent hidden harmony, and the images reflected in our minds are simple mathematical laws. That is why natural phenomena can be predicted through the combination of observation and mathematical analysis. In the history of physics, this concept of inner harmony, or rather this dream, has been confirmed again and again in ways we did not expect.” [88]
As mentioned earlier, intuitionism not only values the connection between mathematics and the natural sciences; it further emphasizes that mathematics itself is natural science, that all truth comes from experience, and that nature is the only, final arbiter of scientific theory.
Of course, the authority of nature has been supported by many thinkers. Quine once said: “We shall approach set theory and mathematics as a whole in the same way we approach the theoretical part of the natural sciences. These truths or hypotheses are supported not so much by pure reasoning as by insightful and systematic contributions to the organization of empirical data in the natural sciences.”[89] M. Klein in turn noted: “People now know that the uncertainty of the foundations of mathematics, and doubts about mathematical logic, can, if not resolved, at least be evaded by strengthening its application to nature.”[90] Gödel also thought of a related issue: “Besides mathematical intuition, there exists another (though only probable) criterion of the truth of mathematical axioms, namely their fruitfulness in mathematics, or perhaps, if one adds it, in physics.”[91]
So, on which side does the criterion of nature favor? It seems that emphasizing the applicability of mathematics is unfavorable to intuitionists, because in terms of the “richness” of results, intuitionist mathematics can by no means compare with classical mathematics.
However, it must first be pointed out that, in the aspect Gödel referred to as “fruitful in mathematics,” if one looks only at the results of pure mathematics itself, this is unquestionably extremely unfavorable to intuitionism, because almost all the results that intuitionist mathematics can obtain can also be derived by classical mathematics. But to rebut intuitionism by saying that it “damages the existing mathematical system too much” is not convincing. Think of the popularity of astrology—astrology has a longer history than astronomy, and today the number of “researchers” and “students” of “astrology” in the world is probably greater than that of astronomers (at least in the United States, astrologers overwhelmingly outnumber astronomers). Its doctrines are undoubtedly “highly developed” as well, and astronomy can also be fully encompassed within astrology. But scientists will regard all of that as having gone astray from the start: its most basic presupposition, “a person’s destiny is related to the stars,” is wrong! A theory developed under a false presupposition can only mean that the more results it produces, the more errors it contains.
If one denies the premise of astrology, the huge theoretical edifice painstakingly built over thousands of years will collapse, and countless astrologers will have “nothing to do”; what a sacrifice! — Is such a defense of astrology reasonable? I am afraid not. Then what about a defense of mathematics?
Intuitionists believe: intuitionist mathematics is certainly far more “meager” than classical mathematics, but it contains more truth!
And what about its application in physics? Objectively speaking, intuitionist applied mathematics does indeed fall short of the achievements of classical mathematics, but the reason is still simple: nearly all intuitionist mathematical results can be regarded as results of classical mathematics.
But are the results that can only be obtained by acknowledging actual infinity really very useful in practice? The calculus, which is widely used in physics, can be explained by potential infinity without appealing to actual infinity—this was precisely the insistence of the founders of calculus. As for other counterintuitive conclusions, such as “one can cut a sphere into finitely many pieces and recombine them by simple translation and rotation 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 such a technique of creating something from nothing, more enticing than alchemy, that would certainly be wonderful; but everyone knows that this is impossible. Many mathematicians do not even think this theorem poses any trouble; on the contrary, they often use it as an example to emphasize the wonder of mathematics: “Look, in the real world such absurd things occur, yet in the mathematical world they have been proved. How marvelous! How beautiful!” I admit that if one wants to show beginners some engaging cases in mathematics to stimulate their curiosity, paradoxes like the sphere-splitting conundrum are indeed excellent material. But if I am explaining to officials who allocate research budgets whether my mathematical research has any skill besides entertaining myself, then compared with “I can turn stone into gold!,” saying “I can make one become two!” is not really something to boast about.[92]
M. Klein, with the authority of a historian of mathematics, further reminds us: “We note that in many fields that were once pursued with vigor and enthusiasm, and praised by their supporters as the essence of mathematics, in fact they were nothing but temporary hobbies, or left only a slight influence on the whole march of mathematics; mathematicians of the first half of the century may have believed with confidence that their work was the most important, yet the place of their contributions in the history of mathematics still cannot be determined.”[93] It is still too early to be pleased with oneself over the rich and enormous “achievements” modern mathematics has obtained from actual infinity.
For intuitionists, giving up those seemingly beautiful results “not only did not entail a loss, but was an overwhelming victory, because logic, mathematics, epistemology, semantics, and metaphysics finally achieved harmonious consistency.”[94]
And the philosophy of Wittgenstein’s later period offers a new supplement to the “harmonious consistency” attained by intuitionists: intuitionists will discover that their claims have also found support in modern philosophy after the linguistic turn.
At the same time, the mathematics philosophy of the later Wittgenstein or of intuitionism also fits well with the development trend of philosophy of mathematics in the second half of the twentieth century:
According to Zheng Yuxin[95], under the influence of philosophy of science, philosophy of mathematics in the second half of the twentieth century underwent a series of important changes and turns, such as placing emphasis on “research in the history of science” (Wittgenstein’s emphasis on mathematics as a human activity), placing emphasis on the analysis of the “social-cultural nature” of mathematics (the customs and institutions emphasized by Wittgenstein), affirming the “empirical” character of mathematics, and confirming the “identity of mathematics with the other natural sciences,” moving beyond Platonism, placing emphasis on “mathematical methodology” (Wittgenstein’s emphasis on the “normativity” of mathematics and his emphasis on psychology/pedagogy[96]), placing emphasis on “practical mathematicians” (Wittgenstein’s insistence that one should “observe mathematics from the standpoint of action”), and so on.
We find that these so-called important changes are almost all related to Wittgenstein’s claims. As for the importance of Wittgenstein’s philosophy of mathematics, there is probably no need to say more.
Wittgenstein: Philosophical Investigations, trans. Chen Jiaying, Shanghai People’s Publishing House, 2005
Edited by Tu Jiliang: The Complete Works of Wittgenstein, Hebei Education Publishing House, 2003
Volume I: Tractatus Logico-Philosophicus, etc. (trans. Chen Qiwei)
Volume II: Ludwig Wittgenstein and the Vienna Circle (trans. Huang Yusheng, Guo Dawei)
Volume III: Philosophical Remarks (trans. Ding Donghong, Zheng Yiqian, He Jianhua)
Volume IV: Philosophical Grammar (trans. Cheng Zhimin)
Volume V: Wittgenstein’s Cambridge Lectures, 1930–1932; Cambridge Lectures, 1932–1935 (trans. Zhou Xiaoliang, Jiang Yi)
Volume VI: The Blue and Brown Books; A Philosophical Survey (The Brown Book) (trans. Tu Jiliang)
Volume VII: Remarks on the Foundations of Mathematics (trans. Xu Youyu, Tu Jiliang)
Volume VIII: Philosophical Investigations (trans. Tu Jiliang)
Volume XI: Vermischte Bemerkungen; Zettel (trans. Tu Jiliang, Wu Xiaohong, Li Jie)
Volume XII: Remarks on Frazer’s “Golden Bough”; Intuition and others (trans. Jiang Yi)
Tu Jiliang: Research on Wittgenstein’s Later Philosophical Thought, Jiangsu People’s Publishing House, 2005
[US] Jaakko Hintikka: Wittgenstein, trans. Fang Xudong, Zhonghua Book Company, 2002
[Canadian] Vladimir Tasic: The Mathematical Roots of Postmodern Thought, trans. Cai Zhongdai and Jianping, Fudan University Press, 2005
[US] Paul Benacerraf and Hilary Putnam, eds.: Philosophy of Mathematics, trans. Zhu Shuilin, Ying Zhiyi, Ling Kangyuan, Zhang Yugang; proofread by Chen Yihong and Wang Shanping, The Commercial Press, 2003
[US] M. Kline: Mathematics: The Loss of Certainty, trans. Li Hongkui, Hunan Science and Technology Press, 2003
[US] M. Kline: Mathematics and the Search for Knowledge, trans. Liu Zhiyong, Fudan University Press, 2005
[US] Morris Kline: Mathematical Thought from Ancient to Modern Times, trans. Deng Donggao, Zhang Gongqing, et al., Shanghai Scientific and Technological Literature Publishing House, 2002
[Polish] Andrzej Mostowski: Thirty Years of Research on the Foundations of Mathematics, trans. Guo Shiming, Chen Anjie, Xiu Qingyun; proofread by Kang Hongkui, Huazhong Institute of Technology Press, 1983
[French] Poincaré: The Value of Science, trans. Li Xingmin, Guangming Daily Press, 1988
[German] Kant: Critique of Pure Reason, trans. Li Qiuling; proofread by Yang Zutao, China Renmin University Press, 2004
[US] R. Courant, H. Robbins: What Is Mathematics?, trans. Zuo Ping, Zhang Yici, Fudan University Press, second edition, 2005
Zheng Yuxin: New Reflections on Philosophy of Mathematics, Jiangsu Education Press, 1990
Chen Bo: Philosophy of Logic, Peking University Press, August 2005
Hu Zuoxuan, Deng Mingli: “The Philosopher Who Crowned Mathematics — Frank Ramsey”, Journal of Dialectics of Nature, No. 3, 2000
Du Hansheng: “Mathematics and Wittgenstein’s Philosophy”, Journal of Hubei Normal College (Philosophy and Social Sciences), No. 4, 1998
Zheng Yuxin: “The Important Influence of Philosophy of Science on the Modern Development of Philosophy of Mathematics — Also on the Revolution in Philosophy of Mathematics”, Journal of Nanjing University (Philosophy, Humanities, and Social Sciences), No. 1, 1999
[①]Ye Chuang: “Mathematics: A Special Language Game — On Wittgenstein’s Later Philosophy of Mathematics,” Journal of Dialectics of Nature, No. 4, 1992
[②]See Hu Zuoxuan and Deng Mingli, “The Philosopher Who Crowned Mathematics — Frank Ramsey,” Journal of Dialectics of Nature, No. 3, 2000
[③]The translation of Philosophical Investigations quoted in this article is taken from Chen Jiaying’s translation (Shanghai People’s Publishing House, 2005), while all other works by Wittgenstein are quoted from The Complete Works of Wittgenstein, edited by Tu Jiliang (Hebei Education Publishing House, 2003); citations are given in the form “v.7, p.173, IV§23” (Volume 7, page 173, Part IV, Section 23).
[④]Also translated as Wei’er, Weiye, Waier, etc.; he was a student of Hilbert, the founder of formalism, and also a follower of intuitionism.
[⑤]Du Hansheng: “Mathematics and Wittgenstein’s Philosophy,” Journal of Hubei Normal College (Philosophy and Social Sciences), No. 4, 1998
[⑥]By the way, perhaps just as logicism had a decisive influence on Anglo-American analytic philosophy, intuitionism and formalism also exerted a profound influence on modern continental philosophy, and in a certain sense are also one of the “roots” of postmodern currents of thought. See [Canadian] Vladimir Tasic, The Mathematical Roots of Postmodern Thought, trans. Cai Zhongdai and Jianping, Fudan University Press, 2005.
[⑦] [French] Poincaré: The Value of Science, trans. Li Xingmin, Guangming Daily Press, 1988, p. 197
[⑧] [US] Jaakko Hintikka: Wittgenstein, trans. Fang Xudong, Zhonghua Book Company, 2002, p. 33
[⑨]Although this sentence has a specific referent in its context, it can indeed reflect the basic concern of Wittgenstein’s later philosophy.
[⑩]From Brouwer. See [Canadian] Vladimir Tasic, The Mathematical Roots of Postmodern Thought, trans. Cai Zhongdai and Jianping, Fudan University Press, 2005, p. 68
[11] [Canadian] Vladimir Tasic, The Mathematical Roots of Postmodern Thought, trans. Cai Zhongdai and Jianping, Fudan University Press, 2005, p. 72
[12]Tu Jiliang, Research on Wittgenstein’s Later Philosophical Thought, Jiangsu People’s Publishing House, 2005, p. 265
[13]The “meaning” here follows the traditional understanding; in fact, intuitionism simultaneously maintains that “meaning does not depend on truth conditions.”
[14]Michel Dummett: The philosophical basis of intuitionist logic: see Philosophy of Mathematics, p. 132
[15]Tu Jiliang, Research on Wittgenstein’s Later Philosophical Thought, Jiangsu People’s Publishing House, 2005, p. 270
[16] L.E.J. Brouwer: Consciousness, Philosophy and Mathematics: see [US] Paul Benacerraf and Hilary Putnam, eds., Philosophy of Mathematics, trans. Zhu Shuilin, Ying Zhiyi, Ling Kangyuan, Zhang Yugang; proofread by Chen Yihong and Wang Shanping, The Commercial Press, 2003, p. 104 (hereafter in the notes below this book is abbreviated as “Philosophy of Mathematics”)
[17]Paul Benacerraf: Introduction to Philosophy of Mathematics: see Philosophy of Mathematics, p. 26
[18] [Canadian] Vladimir Tasic, The Mathematical Roots of Postmodern Thought, trans. Cai Zhongdai and Jianping, Fudan University Press, 2005, p. 56
[19]Critique of Pure Reason [A526 B554], referring to Li Qiuling’s translation
[20]For example, in v.7, p.71, I-Appendix II§6–7, on “the difference between throwing dice and counting dots in a game”: “But in an emergency, couldn’t a simple-minded person, just like ordinary people, also use counting dots to make a decision in the manner of drawing lots? We note that in the process of making a choice, the result has already been secretly settled; what role does the thing that makes us notice this play?”
[21]See [US] Edwin Arthur Burtt, The Metaphysical Foundations of Modern Physical Science, trans. Xu Xiangdong, Peking University Press, 2003
[22]David Hilbert: On the Infinite: see Philosophy of Mathematics, p. 214
[23]Paul Benacerraf: Introduction to Philosophy of Mathematics: see Philosophy of Mathematics, p. 7
[24]Paul Bernays: On Platonism in Mathematics: see Philosophy of Mathematics, p. 301
[25]See [US] M. Kline, Mathematics: The Loss of Certainty, trans. Li Hongkui, Hunan Science and Technology Press, 2003, p. 203
[26]See Zheng Yuxin, New Reflections on Philosophy of Mathematics, Jiangsu Education Press, 1990, p. 61
[27]Paul Benacerraf: Introduction to Philosophy of Mathematics: see Philosophy of Mathematics, p. 34
[28]Ibid.
[29]Arend Heyting: The intuitionist foundations of mathematics: see Philosophy of Mathematics, p. 61
[30]Michel Dummett: The philosophical basis of intuitionist logic: see Philosophy of Mathematics, p. 137
[31]Rudolf Carnap: The logicist foundations of mathematics: see Philosophy of Mathematics, p. 48
[32]Ibid., p. 57
[33]Ibid., p. 60
[34]Kurt Gödel: What is Cantor’s continuum problem?: see Philosophy of Mathematics, pp. 560–561
[35] [French] Poincaré: The Value of Science, trans. Li Xingmin, Guangming Daily Press, 1988, p. 202
[36]Chen Bo, Philosophy of Logic, Peking University Press, August 2005, p. 60
[37]Ibid., p. 61
[38]Arend Heyting: Argumentation: see Philosophy of Mathematics, p. 82
[39] L.E.J. Brouwer: Consciousness, Philosophy and Mathematics: see Philosophy of Mathematics, p. 111
[40] [Canadian] Vladimir Tasic, The Mathematical Roots of Postmodern Thought, trans. Cai Zhongdai and Jianping, Fudan University Press, 2005, p. 195
[41]Ibid., p. 198
[42]David Hilbert: On the Infinite: see Philosophy of Mathematics, p. 222
[43] Ibid., p. 231
[44] Ibid., p. 220
[45] Ibid.
[46] See [U.S.] M. Kelein, Mathematics: The Loss of Certainty, trans. Li Hongkui, Hunan Science and Technology Press, 2003, p. 253
[47] David Hilbert, On the Infinite: in Philosophy of Mathematics, p. 214
[48] [U.S.] M. Kelein, Mathematics: The Loss of Certainty, trans. Li Hongkui, Hunan Science and Technology Press, 2003, pp. 250~251
[49] L.E.J. Brouwer, Intuitionism and Formalism: in Philosophy of Mathematics, p. 91
[50] Paul Benacerraf, Mathematical Truth: in Philosophy of Mathematics, pp. 470~471
[51] David Hilbert, On the Infinite: in Philosophy of Mathematics, p. 231
[52] [U.S.] M. Kelein, Mathematics: The Loss of Certainty, trans. Li Hongkui, Hunan Science and Technology Press, 2003, p. 256
[53] Ibid., p. 255
[54] Arend Heyting, Argumentation: in Philosophy of Mathematics, p. 81
[55] Ibid.
[56] Ibid.
[57] [Canada] Vladimir Tasic, The Mathematical Roots of Postmodern Thought, trans. Cai Zhong and Dai Jianping, Fudan University Press, 2005, pp. 46~47
[58] [France] Poincaré, The Value of Science, trans. Li Xingmin, Guangming Daily Press, 1988, p. 312
[59] Ibid., p. 201
[60] Ibid., p. 312
[61] Ibid., p. 319
[62] Ibid., p. 327
[63] See [Canada] Vladimir Tasic, The Mathematical Roots of Postmodern Thought, trans. Cai Zhong and Dai Jianping, Fudan University Press, 2005, p. 67
[64] Arend Heyting, The Intuitionistic Foundations of Mathematics: in Philosophy of Mathematics, p. 60
[65] [U.S.] M. Kelein, Mathematics: The Loss of Certainty, trans. Li Hongkui, Hunan Science and Technology Press, 2003, p. 323
[66] Ibid., pp. 323~324
[67] Ibid., pp. 324~325
[68] Ibid., p. 324
[69] [U.S.] Morris Kline, Mathematical Thought from Ancient to Modern Times (Vol. 4), trans. Deng Donggao, Zhang Gongqing et al., Shanghai Scientific and Technical Publishers, 2002, p. 99
[70] At the same time he was both an applied mathematician and a mathematics educator, and was relatively sympathetic to intuitionism.
[71] [U.S.] Morris Kline, Mathematical Thought from Ancient to Modern Times (Vol. 4), trans. Deng Donggao, Zhang Gongqing et al., Shanghai Scientific and Technical Publishers, 2002, pp. 283~284
[72] C. Diamond (ed.), Wittgenstein’s Lectures on the Foundations of Mathematics, 1976, p. 260, cited in Tu Jiliang, Research on Wittgenstein’s Later Philosophical Thought, Jiangsu People’s Publishing House, 2005, p. 261
[73] [U.S.] M. Kelein, Mathematics: The Loss of Certainty, trans. Li Hongkui, Hunan Science and Technology Press, 2003, p. 325
[74] Arend Heyting, Argumentation: in Philosophy of Mathematics, p. 88
[75] Ibid., p. 88
[76] Ibid., p. 88
[77] [U.S.] Morris Kline, Mathematical Thought from Ancient to Modern Times (Vol. 4), trans. Deng Donggao, Zhang Gongqing et al., Shanghai Scientific and Technical Publishers, 2002, pp. 98~99
[78] [U.S.] M. Kelein, Mathematics: The Loss of Certainty, trans. Li Hongkui, Hunan Science and Technology Press, 2003, p. 288
[79] [U.S.] Morris Kline, Mathematical Thought from Ancient to Modern Times (Vol. 4), trans. Deng Donggao, Zhang Gongqing et al., Shanghai Scientific and Technical Publishers, 2002, pp. 110~111
[80] [U.S.] M. Kelein, Mathematics: The Loss of Certainty, trans. Li Hongkui, Hunan Science and Technology Press, 2003, p. 311
[81] Ibid., p. 294
[82] Ibid., p. 228
[83] 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 ideas were neglected for even longer than Cantor’s! It was only when Cantor’s set theory had developed to maturity and even produced paradoxes that Kronecker was remembered again.
[84] [U.S.] Morris Kline, Mathematical Thought from Ancient to Modern Times (Vol. 4), trans. Deng Donggao, Zhang Gongqing et al., Shanghai Scientific and Technical Publishers, 2002, p. 113
[85] [U.S.] M. Kelein, Mathematics: The Loss of Certainty, trans. Li Hongkui, Hunan Science and Technology Press, 2003, p. 307
[86] Ibid., p. 310
[87] Ibid., p. 340
[88] Ibid., p. 359
[89] Ibid., p. 341
[90] Ibid., p. 338
[91] Kurt Gödel, What Is Cantor’s Continuum Problem?: in Philosophy of Mathematics, p. 561
[92] 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 with a non-constructive proof; after attracting attention, further research then found constructive proofs. Even if one says that non-constructive proofs are “illogical,” just as intuitionism itself emphasizes that logic is not everything and that intuition and creativity are of great significance in the development of mathematics, intuitionists can at least acknowledge the heuristic significance of classical mathematics. In fact, many leading intuitionists—for example Kronecker, Brouwer, and Weyl—in their actual mathematical research often, as Poincaré said, “forgot their philosophy”; many of the mathematical achievements they themselves obtained were precisely those 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 action: they took intuition as the driving force and foundation of research, and nature as the aim and destination of research; as for logic, they could use it at the very end to check things afterward.
[93] [U.S.] Morris Kline, Mathematical Thought from Ancient to Modern Times (Vol. 4), trans. Deng Donggao, Zhang Gongqing et al., Shanghai Scientific and Technical Publishers, 2002, p. 116
[94] Paul Benacerraf, Introduction to Philosophy of Mathematics: in Philosophy of Mathematics, p. 27
[95] See Zheng Yuxin, “The Important Influence of Philosophy of Science on the Modern Development of Philosophy of Mathematics—Also on the ‘Revolution’ in Philosophy of Mathematics,” Nanjing University Journal (Philosophy, Humanities and Social Sciences), No. 1, 1999
[96] Philosophy of psychology is also an important part of Wittgenstein’s later philosophy, but since it is rather remote from the theme of this article, it is not discussed here.
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Someone
2007-05-18 11:43:53 Anonymous 207.47.202.52
Heyting then went on to say: “If we adopt this view, then we run headlong into the obstacle that language is fundamentally ambiguous. Since the meaning of a word can never be fixed precisely enough to rule out all possibility of misunderstanding, in mathematics we can never guarantee that the formal system correctly expresses our mathematical thought. (My note: this claim is even more strongly supported by the Löwenheim-Skolem theorem.)”
＝＝＝＝＝＝＝＝＝＝＝＝＝＝＝I didn’t quite understand this passage. How does semantic imprecision get linked to the Skolem theorem?
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2007-05-18 15:09:02
Sorry, this passage should have been deleted…
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2007-05-18 15:25:18
Something I wrote about this before:
(But I’m not very familiar with mathematics in this area, and simply citing some statements doesn’t mean I myself necessarily understood them, so I didn’t use this in the paper this time.)This is “a line of research that began with Leopold Löwenheim in 1915 and was simplified and completed through a series of papers published by Thoralf Skolem between 1920 and 1933, revealing yet another defect of mathematical structure.”
The Löwenheim-Skolem theorem says that any satisfiable first-order theory (using a countable language) has a countable model.That is to say, when we choose a countable model for the language of set theory, there will appear some sets that are ‘uncountable’ in a relative sense,
From the perspective of one model, a set may appear “countable”; from the perspective of another model, it may turn out to be an uncountable geometry. As Skolem summed it up, “within the scope of 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 formal systems are unable to “capture.”
The foregoing introduction is somewhat technical, so M. Klein offers a more vivid analogy: “Suppose people intend to compile a list of distinguishing features and believe that it can characterize, and only characterize, Americans; but to everyone’s surprise, someone discovers an animal that has all the features listed on the sheet, yet is completely different from an American. In other words, it is in fact impossible to use an axiomatic system to describe a uniquely determined class of mathematical objects. … A set of axioms can allow far more interpretations than people expect, and these interpretations differ in essential ways.”
Skolem’s paradox is the mathematical version of ‘looking for a horse by following the picture’: following a horse-breeding manual that supposedly lays out the features of a fine steed, one ends up finding a toad that fits every item on the list! The Löwenheim–Skolem theorem means that no matter how carefully and thoroughly one writes this horse manual, no matter how meticulously one characterizes mathematical objects, one can never avoid such substantive ambiguities.
This powerfully illustrates the intuitionist claim: whatever is “language” cannot avoid relative “vagueness”; even mathematics or logic is 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 was opposed to taking the axiomatic method as the foundation of set theory. Even von Neumann said in 1925 that his own axioms, together with other axiomatic systems concerning set theory, all bore the “mark of not being real, … set theory cannot be axiomatized unconditionally. … Since arithmetic, geometry, and so on do not have axiomatic systems, and since no such assumption is made with respect to set theory, there can likewise be no unconditioned axiomatization of infinite systems.” This situation, he continued, “is for me yet another argument in favor of intuitionism.”
Putnam states that “the two extreme positions—Platonism and positivism—both seem able to draw comfort from the Löwenheim–Skolem paradox; only the ‘moderate’ position (which tries to avoid the mysterious power of ‘grasping’ ‘mathematical objects’ while retaining the classical concept of truth) is beset with difficulties.” “The surprising fact is that, in the eyes of mathematical intuitionists, the whole problem does not exist at all. This may not come as a surprise to Skolem: his conclusion was precisely that ‘what most mathematicians want mathematics to handle, in the end, are workable computational operations, and they do not want mathematics to consist of formal propositions about some objects or other.’” Putnam further acknowledges that even if intuitionists succeed in mastering a sufficiently rich language as the metalanguage of some theory T, and can then even define “truth in T” in the manner of Tarski and discuss “models” in T, and so on, the “Skolem” paradox still will not arise again! For in intuitionism: “reference is given through sense, and sense is given through verification procedures rather than through truth conditions. The ‘gap’ between theory and ‘objects’ simply disappears—or rather, it never really appeared in the first place.”
Putnam’s final choice is to move toward some so-called “liberalized intuitionism,” in which he tries both to preserve the main ideas of intuitionism and not to “destroy” classical mathematics. But can this dream of getting the best of both worlds be realized? I will not go into that here. In short, the result is this: the non-intuitionist moves closer to the intuitionist, while the intuitionist need only continue to reexamine his or her longstanding claims.
Translated from the Chinese original with AI assistance. The original text is authoritative.
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