The City University of New York
Abstract. In this paper I analyze two problems associated with the notion of proof in Dummett’s theory of meaning for mathematical statements. In the first place, I will claim that the rule-following paradox posses a serious challenge to Dummett’s claim that the knowledge of the meaning of a mathematical sentence is closely tied to the ability to recognize what counts as a proof of it. I will consider Dummett’s replies to the sceptic paradox and conclude that neither of them constitutes an appropriate response. In the second place, I will maintain that there is a tension between the notion of provability in principle and Dummett’s formulation of the acquisition and manifestation arguments. I will claim that, in trying to avoid this problem, mathematical anti-realism approaches the view it intends to reject, namely, a platonist conception of meaning.
The term ‘anti-realism’ was introduced by Michael Dummett in the late ’50 to denote certain kind of reservation about realism. Unlike realism, which constitutes a definite doctrine, the anti-realist view is not a specific philosophical doctrine but the rejection of a doctrine. For this reason, it can be seen as an opposition to realism. However, the kind of reservation that anti-realism has regarding realism presents a definite character, since it objects what constitutes, according to Dummett, the cardinal thesis of realism: the unrestricted acceptance of the principle of bivalence, the principle that states that every meaningful sentence is determinately true or false. The realist idea that reality is thoroughly determined independently of our knowledge of it seems to Dummett well expressed by the semantic principle that every sentence is true or false. It is the rejection of this principle and, as a consequence, the rejection of the idea that reality is fully determined independently of us, what characterizes semantic anti-realism.
Dummett maintains that the rejection of the principle of bivalence, and with it the rejection of realism, leads to the revision of classical logic. The source of this thesis, essential in Dummett’s proposal, is mathematical intuitionism. As indicated by Dummett, those who first realized that rejecting realism entailed rejecting classical logic were the intuitionists of the school of Brouwer. They considered, based on the assumption that mathematical entities are mental constructions, that a mathematical sentence is true only if we are capable of proving it. Then, they affirmed that there is no reason to consider that every mathematical sentence is subject to the principle of bivalence. They claimed that classical logic, which underlies classical mathematics, imports realist ontological assumptions that become evident in the acceptance of that principle. Once that the principle of bivalence, together with the metaphysical assumptions, is rejected, there is no reason to assume the law of excluded middle, although the validity of this law does not depend absolutely on the principle of bivalence. Thus, while rejecting this principle, they not only rejected mathematical realism, but also classical logic.
Dummett considers the intuitionistic account of the meaning of mathematical sentences as a prototype for a sustainable version of anti-realism in other areas of discourse. This suggestion is possible because Dummett’s presentation of the case for intuitionism is based on considerations about meaning and understanding that are not restricted to any particular area of discourse. Thus, since the arguments supporting a generalized version of anti-realism come from mathematical discourse, the analysis of Dummett’s arguments in this context is especially relevant.
According to Dummett, the critical disagreement between a realist and an intuitionistic view of mathematics is essentially a semantic dissent about meaning and mathematical truth. The realist view, usually called platonism, is depicted by Dummett as sustaining that mathematical sentences possess an objective truth-value, independently of our means of knowing it. Hence, for the platonist, the meaning of mathematical sentences is not dependent on our possession of evidence for them. In contrast, the anti-realist view rejects the objectivity of truth and maintains that a mathematical sentence can be true only in virtue of something we could know and could count as evidence for its truth. Hence, for the anti-realist, the meaning of mathematical sentences is to be taken as given by what counts as a proof of them and, in consequence, only those sentences that can be proved are true.
Dummett’s view of the disagreement between platonism and anti-realism brings about certain changes in relation to the usual formulation of the dispute, since the dispute was traditionally about the ontological status of mathematical objects. The main difference is that Dummett believes, in agreement with Kreisel’s dictum, that the problem is not the existence of mathematical objects, but the objectivity of mathematical sentences. Consequently, he thinks that the fundamental disagreement is not addressed by any metaphysical question about mathematical objects. In addition, he maintains that the disagreement about the nature of mathematical objects can only be expressed by different metaphors, one metaphor in which mathematical objects are independently existing objects and another metaphor in which mathematical objects are creations of the human mind. The problem is, according to Dummett, that neither of these images is particularly adequate to characterize mathematical objects and, as a consequence, it is not possible to justify the selection of one of these images over the other. Thus, he considers that it is pointless, at least at the beginning, to formulate the conflict in ontological terms, and proposes to set aside the metaphysical facet of the dispute and start with the disagreement over the correct theory of meaning for mathematical sentences. He maintains that once the selection of a particular theory of meaning has been justified, the theory will determine a particular image of the metaphysical character of mathematical reality. This image will also be justified, since it will only be an extension of the theory of meaning selected.
The general aim of this paper is to analyze certain problems that arise when Dummett’s theory of meaning is applied to mathematical discourse. For this purpose, I will focus on some problems associated with the role played by the notion of proof in Dummett’s theory of meaning. On one hand, I will claim that the ability to recognize what counts as a proof of a mathematical sentence does not assure that the requirement of full manifestation of knowledge of meaning posed by Dummett can be satisfied. For this claim the rule-following paradox will play an important role, since it shows the problems that arise when from a finite set of uses of a sentence, we seek to derive the meaning of the sentence. On the other hand, I will maintain that there is a conflict between the notion of provability in principle and the acquisition and manifestation arguments, at least if we keep insisting with the full manifestation requirement, as Dummetts suggests. I will consider the strict finitist view, a radical version of anti-realism that sustains that the notion of provability in principle must be rejected, and I will analyze the difficulties that the anti-realist position faces when trying to dismiss this view.
I shall divide this paper into four parts. First, I shall expound Dummett’s theory of meaning for mathematical sentences, and I shall present Dummett's main arguments against a realist conception of meaning: the acquisition argument and the manifestation argument. Then, in the second part, I shall argue that the ability to recognize what counts as a proof of a mathematical sentence does not guarantee that the full manifestation requirement can be met. Third, I shall argue that there is a tension between the notion of provability in principle and the acquisition and manifestation arguments. And fourth, I shall present the conclusions.
Dummett’s proposal in favor of intuitionistic reasoning in mathematics stems from a general theory of meaning according to which the meaning of a mathematical sentence consists exclusively in the role that it plays as an instrument of communication between individuals. This commitment to the public character of meaning is expressed by saying that the meaning of a mathematical sentence is exhaustively determined by its use. Accordingly, Dummett maintains that, if two individuals agree about the use of a certain sentence, then they also agree about its meaning.
This theory of meaning ought also to be, according to Dummett, a theory of understanding; that is, a representation of what it is known when an individual knows the meaning. However, that knowledge of meaning cannot always be verbalisable, that is, knowledge consisting in the ability to set the rules according to which the expression is used. The problem is that to explain all knowledge as explicit knowledge would be circular, since any such explanation presupposes what is to know the meaning of some sentences. As a consequence, knowledge of meaning has in the end to be explained as implicit knowledge, that is, in terms of some practical ability with respect to the use of the language. But, in order to ascribe this implicit knowledge meaningfully, we must be able to recognize the manifestation of this knowledge in the use: there must be an observable difference between someone who has this knowledge and someone who does not. Thus, Dummett maintains that to know the meaning of a sentence must consist in the capacity to use or respond to its uses correctly.
Dummett also makes another demand: a theory of meaning for mathematical sentences has to be based on some general feature of these sentences. In Dummett’s opinion, the selection of a particular feature of these sentences would allow us to describe uniformly every other feature of the use of the sentences in terms of that feature. Thus, the meaning of an individual sentence must then be explained in terms of this central feature. One notion that, according to Dummett, is excluded as a central notion for a theory of meaning for mathematical sentences (because it violates the principle that use exhaustively determines meaning) is the notion of truth. He affirms that this notion, considered as a feature that every mathematical sentence possesses or not independently of our capacity to recognize its truth-value, cannot be the central notion for a theory of meaning. The problem is, in Dummett’s view, that the acceptance of the principle of bivalence makes impossible to give an account of the required connection between the condition for a sentence to be true and the use of the sentence. In the next section I will examine Dummett's arguments against a realist conception of meaning based on the notion of truth and on the unrestricted acceptance of the principle of bivalence.
On the basis of the preceding considerations, Dummett proposes to replace the notion of truth as the central notion of a theory of meaning for mathematical sentences by a notion compatible with the principle that use exhaustively determines meaning. Since he considers that the notion of proof is compatible with this principle and with the requirement that we must be able to recognize knowledge of meaning in the ability to use or respond to the use of a sentence, he takes the notion of proof as the best candidate for a central notion of a theory of meaning for mathematical sentences.
Thus, we can say that an individual understands the meaning of a sentence if he is able to recognize a proof of it. Furthermore, we can say that he grasps the meaning of any smaller expression than a sentence if he knows how its presence in a sentence contributes to determine what counts as a proof of it. Therefore, it is clear, based on what we stated above, that Dummett’s account of understanding expresses an insistence on the idea that every mathematical sentence must have a determined individual content and rejects a holistic interpretation of mathematics. It is also clear that this commitment to molecularism goes together with a commitment to the principle of compositionality, according to which every sentence has a content that belongs to it in agreement with the way its constituents contribute to determine it.
In short, Dummett’s theory of meaning in favor of mathematical intuitionism is based, first, on the principle that use exhaustively determines meaning; second, on the idea that a theory of meaning is a theory of understanding; third, on the defense of a molecular vision of mathematical language where every sentence must have a determined individual content, though a grasp of this content cannot in general consist in verbalisable knowledge, but it must be capable of being fully manifested by means of the use of the sentence; fourth, on the principle of compositionality; and fifth, on the selection of the notion of proof as the central notion of his theory of meaning.
Dummett develops, with the aim of questioning those conceptions of meaning based on the notion of truth and the unrestricted acceptance of the principle of bivalence, two main arguments: the acquisition argument and the manifestation argument. To these arguments I turn next.
The argument of acquisition intends, on the one hand, to make more plausible Dummett’s theory of meaning and, on the other hand, to question those views that maintain that the meaning of mathematical sentences transcends their use. It intends to show that learning mathematics consists exclusively in learning how to use the sentences of this language. Thus, according to Dummett, this process of learning just consists in learning how to determine when certain sentences can be established by computation, in determining the role played by these sentences in mathematical proofs, in establishing from what they may be inferred and what may be inferred from them, etc. Dummett considers that this is all that we are shown when we learn the meaning of the mathematical expressions, because it is all we can be shown. Similarly, our competence in making the correct use of these expressions is all that other individuals have in order to judge whether or not we have acquired a grasp of their meanings. Therefore, a grasp of the meaning of mathematical sentences can only consist in the capacity to make a correct use or respond properly to the use of these sentences. Dummett affirms that, when we learn mathematics, we just learn to recognize what determines that a certain sentence is true or false. Thus, the argument gives support to his theory of meaning, since it shows that the way we learn mathematics is compatible with it.
Furthermore, Dummett maintains that, if we accept that there is an ingredient of meaning that transcends the use, it will be possible to conceive that an individual behaves like someone who understands this language without actually understanding it. Thus, the argument concludes that this supposition cannot be considered an adequate explanation of what is to know the meaning of mathematical sentences, since it implies, on the one hand, that meaning is ineffable and, on the other hand, that nobody can be sure of being understood by another individual, since anyone could attach to his sentences a different meaning from that he assigns to them.
By means of the manifestation argument, Dummett challenges the fundamental principle of realism, namely, the principle that every meaningful sentence has a determined truth-value, independently of our means to come to know that value. Thus, the argument goes against the principle of bivalence: it tries to establish the incoherence of all-embracing this principle. In the context of mathematics, Dummett seeks to apply the manifestation requirement, according to which the knowledge of the meaning of a sentence must be fully manifested in the capacity for use or respond to its uses, with the purpose of calling into doubt the platonist account of mathematics based on the unrestricted acceptance of the principle of bivalence.
According to the platonist view, the principle of bivalence holds for every mathematical sentence. For the anti-realist, on the other hand, it may hold or fail. The principle's failure is not due to vagueness or ambiguity, because these features of language are not considered at all. One way to justify the bivalence is to show that the discourse under consideration is decidable, that there is an effective mechanical method that allows us to determine, given any sentence of this discourse, its truth-value. So, when a discourse is decidable, it is bivalent as well: each of its sentences is either true or false. But the problem is that it is not always possible to determine the truth-value of a sentence in this way. As pointed by Dummett, in every mathematical theory of interest we find sentences for which we do not have a mechanical means of decision that can effectively tell us whether the sentence is true or false. Goldbach’s Conjecture -every even number greater than 2 can be expressed as the sum of two primes- is an example, because in this case we do not have any method that leads to a proof of either its truth or falsity.
In the case of undecidable sentences, where the truth-value transcends any means of provability, we can only guarantee that each sentence is either true or false if our notions of truth and falsity transcend our capacity to recognize the truth-value of these sentences, that is, if we can give rise to a conception of what means for a sentence to be true or false independently of our capacity to recognize it as true or false. This conception is rejected by Dummett. However, both the platonist and the anti-realist agree that undecidable sentences are understood. The problem, then, does not emerge in relation with the supposition that these sentences are understood, but in connection with the way in which understanding must be explained.
In the platonist account the notion of truth is conceptually divorced from the ability to recognize a sentence as true. Dummett maintains that, for this reason, the theory has difficulties in order to give an account of understanding. The platonist account is, in this sense, vulnerable to those conceptions that sustain that the meaning we ascribe to the sentences only can be associated with our ability to recognize the truth-values of those sentences.
We have seen that, in the context of mathematical discourse, we do not have an effective decision procedure for determining for every sentence whether it is true or false. If we assume, with the platonist, that the principle of bivalence holds in this context, we will have in this context undecidable sentences that are determinately true of false. But, if we assume, for example, that one of this undecidable sentences is true, then the state of affairs that has to take place for this sentence to be true is a state that we will not be capable of recognizing as obtaining whenever it obtains. Consequently, the truth of this undecidable sentence transcends our capacity to recognize it as true. Therefore, our knowledge of what is for the sentence to be true cannot be, in fact, fully manifested in our use of the sentence. In this sense, we are going against the requirement of manifestation, since our grasp of the meaning of the sentence cannot be fully manifested in our practical capacities. If, in order to avoid that, we suppose that the sentence is not true, the objection arises again: if the negation of this sentence is true, then its truth transcends our capacity to recognize it. Nonetheless, this is again contrary to the manifestation requirement.
The manifestation requirement cannot be met because in both cases we are unable to recognize that the truth condition obtains when it obtains and, as a consequence, the knowledge that constitutes the grasp of the meaning cannot be fully manifested by the way in which the sentence is used. This leads Dummett to say that in the platonistic conception of meaning it is not clear what is to know the truth condition of a sentence, since this condition cannot always be acknowledged as obtaining. Thus, Dummett concludes that this conception cannot be a theory in which the meaning of a sentence is completely determined by its use. But then, the problem that arises is that, if we accept that there is an ingredient of meaning that transcends the use, it will not be possible to offer an adequate explanation of what is to know the meaning of a sentence, since the knowledge that constitutes the grasp of the meaning will be ineffable. Hence, it will be possible for and individual to behave as if he understood the mathematical language without actually understanding it. Thus, this view cannot assure communication and cannot satisfactorily explain our grasp of the meaning of mathematical sentences.
On the basis of this argument it is possible to understand why the dispute between realism and anti-realism does not arise in the context of decidable discourses. If we have an effective decision procedure to determine the truth-values of the sentences, then even the platonist is capable of fully manifesting his understanding of the meaning of any sentence. Given a particular sentence, all he needs to do is to apply this effective procedure. It will lead him to a determined answer about the truth-value of the sentence. If it is true, he will be able to recognize its truth. If it is false, he will be able to recognize its falsity. In any case, the platonist will be capable of fully manifesting his understanding of the truth conditions of the sentence. Thus, the anti-realist will not be able to criticize the realist in the case of a decidable discourse. As a matter of fact, the anti-realist himself will in this case accept the principle of bivalence and, as a consequence, will consider classical logic justified in connection with this discourse.
Now, what can be concluded from Dummett’s arguments? It is evident that the arguments do not demonstrate that there is no ingredient of meaning that transcends the use. In any case, what they show is that, if this ingredient exists, it has to be ineffable, since it cannot be manifested in our use of the language. Therefore, these arguments do not establish that the only way we have to give meaning to mathematical sentences is in relation with the way in which we use these sentences. Rather, what they show is that, if there were an ingredient of meaning that transcends the use, it would be unknowable and incommunicable. We then see that these arguments express an epistemological problem, and that they do not arrive at an ontological conclusion. Even if we assume that such ingredient is ineffable, we are not allowed to conclude that it does not exist.
However, even though the arguments do not establish that semantic facts that are not manifested in the use do not exist, we may think that it is not necessary for an anti-realist theory of meaning to demonstrate that. In this sense, all that an anti-realist theory needs to show is that it constitutes a better explanation of meaning than that provided by a platonist theory. Thus, to show that the anti-realist theory provides a better explanation of meaning would lead us to abandon those conceptions based on the supposition that there is an ingredient of meaning that transcends the use. But, what would turn this theory into a better explanation of meaning?
I consider that it is possible to say that Dummett's theory constitutes a better explanation meaning if we can show that it assures our grasp of meaning. The acquisition argument and the manifestation argument show that a platonist account of understanding is inadequate. Then, it seems reasonable to sustain that an anti-realist theory can only be considered a better explanation if it provides a satisfactory account of our knowledge of meaning. If this theory did not meet this requirement, then there would be no relevant difference between these theories about their competence for giving an account of understanding. Thus, the selection of one of these theories instead of the other would be, just as the selection of an ontological view instead of the other, an arbitrary decision without justification.
We may then ask the following question: how can Dummett's theory provide us with a satisfactory account of understanding? It seems that, if his theory were capable of complying with the requirement of full manifestation of knowledge of meaning, it would be possible to assert that it gives a satisfactory account of understanding. In other words, if in Dummett's theory every aspect of our grasp of the meaning of a mathematical sentence can be fully manifested in the ability to recognize what counts as a proof of it, the theory would constitute a satisfactory account of understanding, since knowledge of meaning and communication among individuals would be assured.
As already explained, it is possible, in accordance with Dummett’s theory of meaning, to ascribe knowledge of meaning to a person only if he is capable of fully manifesting that knowledge. Otherwise, the ascription would be empty. Thus, knowing the meaning of a sentence is described as consisting in a possession of some practical ability. In the mathematical case, a grasp of the meaning of a sentence consists in the ability to recognize what counts as a proof of a sentence and, as a consequence, to attribute to someone the capacity to recognize what counts as a proof is to attribute to him understanding of the sentence.
Now, a proof proceeds according to certain logical principles or rules of inference. Hence, this ability to recognize what counts as a proof of a sentence will consist in the end in the ability to recognize the rules according to which the sentence is used. That is the case because, in order to follow a proof and be able to recognize it as a proof, we have to recognize different transitions as applications of the rules of inference. For this reason, grasp of meaning has to be manifested either in the ability to state the rules according to which the proof proceeds, or, given that knowledge of meaning cannot in general consist in verbalisable knowledge, in the ability to recognize and follow the rules according to which the proof proceeds. In the latter case, however, we must be capable, upon reflection, to acknowledge the rules as correct when they are put to us. As a consequence, knowledge of meaning consists, in both cases, in an implicit or explicit knowledge of the rules according to which the proof proceeds.
These considerations leave us with the following problem: given that only by recognizing the rules that govern the use of mathematical sentences we are able to acknowledge a proof of them, ¿is it possible to guarantee a grasp of the meaning of mathematical sentences from the ability to use those sentences? In other words, can someone’s understanding of a mathematical sentence be fully manifestable in the ability to recognize what counts as a proof of it?
In what follows I will try to show that the answer to this question is negative. I will claim that the relation between knowledge of meaning and the ability to recognize what counts as a proof of a mathematical sentence can be called into doubt on the basis of the sceptical paradox originated in Wittgenstein’s considerations about rules and developed by Kripke in Wittgenstein on Rules and Private Language.
Kripke’s sceptic challenge begins with the observation that, given any rule that we follow or apply, that rule has been followed or applied only finitely often in the past. Let’s suppose that, on the basis of this fact, I state the following normative demand: computing 68 + 57 ought to yield the result 125. Let’s suppose also that I have been computing sums of numbers less than 57 satisfactorily, but I have never before dealt with numbers grater than 57. However, I am certain that 68 + 57 ought to yield the result 125. Against this claim, ‘Kripke’s sceptic’ suggests that I might just as well respond 5, and then he poses as an alternative possibility: perhaps by plus I always mean ‘quus’, an operation compatible with addition until we reach numbers grater than 57, at which point the answers are not the sum of those numbers but always 5. About this we are told:
For the sceptic holds that no fact about my past history –nothing that was ever in my mind, or in my external behavior- establishes that I meant plus rather than quus (…) But, if this is correct, there can of course be no fact about which function I meant, and if there can be no fact about which particular function I meant in the past, there can be none in the present either.
Kripke’s example of quaddition intends to show that the mathematical rule for addition can be reinterpreted in a way compatible with all available facts concerning our use of the language. As a consequence, different speakers might be attaching different meanings to their expressions without showing their divergent behavior. And their different interpretations will have equally good reasons for being the correct interpretation.
According to Kripke, the difficulty is that it is not possible to uniquely determine from some finite number of uses of an expression, the rule applied. And, given that every finite selection of uses leaves room for deviant interpretations of the rule applied, no present application of the rule can be justified. Hence, the meaning attached to an expression cannot be determined.
It is possible to consider that the sceptical argument illustrates an epistemological problem about facts of meaning and rules to which we cannot have access. However, according to Kripke, this is not the case. The reason is that the sceptic’s challenge is not restricted to our actual capacities. Instead, we are allowed to have perfect recall and access to all aspects of our former behavior and mental life. Since even with and extension of our capacities we are still unable to come up with any fact that constitutes our meaning plus rather than quus, Kripke’s sceptic concludes that there is no fact of the matter about our meaning one thing rather than another. There are no facts of the matter as far as rules and meaning are concerned.
Now, if this is so, what about our original problem? Is it possible to satisfy the full manifestation requirement and assure, in this way, our understanding of mathematical sentences?
As already explained, Dummett considers that a model of understanding is a representation of what it is known when an individual knows the meaning. And, given that the meaning of a mathematical sentence is exhaustively determined by its use, knowledge of meaning is granted if we are able to recognize a proof of it. But, as we said before, the ability to recognize what counts as a proof of a sentence consists in an explicit or implicit knowledge of the rules that govern the uses of the sentence. Thus, in accordance with the sceptical argument, we must admit that, even when these rules had been explicitly formulated and accepted by us, it is not possible to determine which rule we should derive from such uses of the sentence. There is nothing in our formulation of the axioms and of the rules of inference and nothing in our minds that enable us to determine which rule we are following. In this sense, given that a proof can be reinterpreted in a deviant way, it will be impossible, from the ability to recognize a proof of a sentence, to know the meaning of that sentence. Thus, the sceptical challenge calls into doubt Dummett’s idea that we know the meaning of a mathematical sentence if we can recognize a proof of it.
As a consequence, it will not be possible to sustain that if two individuals agree about the use of a certain sentence, they also agree about its meaning. The problem is that it is conceivable that both individuals recognize something as a proof of a sentence and, at the same time, disagree about the meaning of the sentence. Hence, it will not be possible to establish, from a finite collection of the uses of a particular sentence, how these individuals understand the sentence.
If this situation is conceivable, Dummett’s requirement of full manifestation cannot be satisfied and, therefore, the ability to recognize what counts as a proof of a sentence cannot help us to explain how we grasp meaning. Let me say again that the problem is that no finite collection of uses of a sentence can show beyond all doubt that an individual knows the meaning of the sentence. As a consequence, the requirement that every aspect of the individual’s grasp of the meaning of a sentence must be such that his knowledge of it can be shown by his use of the sentence, cannot be meet, since the individual’s uses are, inevitably, going to be finite. For this reason, there can never be conclusive evidence for attributing him any specific understanding of the sentence. Hence, on the basis of the sceptic argument, we can sustain that there is no way in which the possession of a practical ability can be fully manifested. And we can conclude that Dummett’s requirement that a grasp of meaning must be able to be fully manifested cannot be reasonably demanded.
Now, we said that Dummett’s objections to the platonist conception of meaning are based on the idea that this conception cannot adequately explain how we understand the meaning of mathematical sentences. The manifestation argument intends to refute any conception of meaning as only partially communicable demanding that every aspect of our understanding should be manifested. But, if the above argument is correct, that demand cannot be met even in a theory that takes the notion of proof as the central notion. In this sense, the anti-realist and the platonist theories of meaning face analogous problems.
But if both theories face similar problems while giving an account of understanding, then the argument to the best explanation cannot be applied in this case. In other words, we said that an anti-realist theory would only constitute a better explanation in case it can provide a satisfactory account of our knowledge of meaning. For that aim it is essential that the theory be able to satisfy the full manifestation requirement. But, given that it cannot do so, there is no relevant difference between the platonist and the anti-realist account of understanding, since both theories seem to be incapable of meeting the full manifestation requirement. Therefore, there will be no ground that can help us decide between the rival conceptions of understanding advocated by platonists and anti-realists. As a consequence, it seems that we made no progress by setting the dispute between platonism and anti-realism from a meaning theoretical starting point: the decision for or against one of these conceptions of meaning seems to be, just as the selection of one ontological view instead of the other, unjustified.
In “Wittgenstein’s Philosophy of Mathematics” and in “Reply to Penco” Dummett discusses Wittgenstein’s considerations about rules and poses some objections. I shall consider these two responses in turn in order to determine whether they constitute an adequate answer to the problem presented in this section.
Dummett’s attempted responses to the rule-following paradox
In “Wittgenstein’s Philosophy of Mathematics” Dummett examines Wittgenstein’s considerations about rules and suggests an objection. He says:
Consider a favourite example of Wittgenstein’s: you train someone to obey orders of the form ‘Add n’ with examples taken from fairly small numbers, then given him the order ‘Add one’ and find that he adds two for numbers from 100 to 199, three for numbers from 200 to 299, and so forth. Wittgenstein says that there need have been nothing either in what you said to him during the training or in what ‘went in your mind’ then which of itself showed that this was not what you intended. This is certainly true, and shows something important about the concept of intention (it is a very striking case of what Wittgenstein means when he says in the Investigations that if God had looked into my mind, he would not have been able to see there whom I meant). But suppose the training was not given only by example, but made use also of an explicit formulation of the rule for forming from an Arabic numeral its successor. A machine can follow this rule; whence does a human being gain a freedom of choice in this matter which the machine does not possess?
Dummett’s account implies that a machine is capable of following the rule of addition without leaving room for divergent interpretations. From that, he intends to trace an analogy between human beings and machines in order to suggest that the practice displayed by machines can be extended to human beings. This answer has been identified by Kripke as a variant of the dispositional response to the sceptical paradox. In what follows, I will examine this response and explain why it does not work.
The dispositional response attempts to respond to Kripke’s sceptic by appealing to our dispositions in order to justify our attaching some determinate meaning rather than another to a particular expression. In this sense, the dispositionalists claim that when we are asked, for example, to compute the sum of two numbers, to mean addition by ‘+’ is to be disposed to answer with the sum of those numbers. Thus, the claim is that our dispositions are what allow us to determine our meaning plus rather than another divergent function such as quus. Therefore, the dispositionalists maintain that being disposed to answer one way rather than another settles our meaning plus.
Kripke’s main objection to the dispositional response is that it loses the normative element of meaning. The dispositionalist only offers a description of the relation between our meaning addition by ‘+’ and the problem of what we should say in some future case. All they say is that if by ‘+’ we meant addition, the answer will be the sum of the numbers. Thus, the dispositional account only tells us how we would respond but not how we ought to respond. But a descriptive account is not enough because it does not answer the sceptic’s claim that we have no justification for our giving one answer rather than another. These considerations lead Kripke to conclude that the dispositionalist simply equates correctness and performance. Since the relation between the meaning of ‘+’ and the intention to future action is normative, the dispositional account cannot solve the sceptical paradox because it does not explain what we should respond in some future case.
Kripke’s view that appealing to machines is a variant of the dispositionalist response is due to the assumption made by this account that human beings can be interpreted as machines embodying the rule that computes the function. As a consequence, Kripke’s attitude toward both is essentially the same: these responses are inadequate because they lose the normative element of meaning.
As Kripke points out, the term ‘machine’ may refer to the actual physical machine or to an abstract program. In the second case, the problem is that also the program can be interpreted in a divergent way. Therefore, such a move is pointless. Now, if we refer to the concrete machine, Kripke considers that two problems arise. In the first place, the machine is finite and, as a consequence, it accepts only finitely many numbers as inputs and gives only finitely as outputs. Thus, it is possible to interpret the answers given by the machine in a divergent way. In the second place, the machine can malfunction and, in order to determine its accuracy, we will have to rely on the program. But, in this case, the previous response applies: a program can be interpreted in a divergent way.
Hence, it seems that this attempt to avoid the rule-following paradox is doomed. The attempt to answer any question about how we ought to respond in some future case by reference to a concrete machine or an abstract program only reproduces the problems of the dispositionalist account. Therefore, the problem posed in the previous section remains unsolved: our disposition to accept a proof of a mathematical sentence given to us cannot determine how we ought to understand that sentence. As well, our understanding of, for example, the symbol ‘+’, cannot be determined by reference to a machine: machines seem to be unable of fully manifesting how they compute a mathematical function such as addition, since their responses can be reinterpreted in a divergent way.
Before going on with the other response given by Dummett to the sceptical paradox, I want to mention that Tennant has defended a dispositional account that establishes an analogy between machines and human beings. I will not discuss the details of his proposal here, but let me say that Tennant’s account intends to shift the burden of proof on to the sceptic by rejecting the possibility of a global reinterpretation. Tennant maintains that, in order to consistently sustain the claim that we might just as well respond with ‘5’ when asked about the sum of the two numbers 68 and 57, the sceptic needs to provide a wholesale reinterpretation about our usage of mathematical language. But, according to Tennant, that will lead Kripke’s sceptic to undertake an extremely implausible reinterpretation. Thus, given that Kripke’s sceptic has not shown how a global reinterpretation can be coherently sustained, Tennant considers that we can reject this possibility until a proper proof of it has been provided.
However, even if we concede this point, Tennant’s claim is insufficient to block the possibility that two speakers may have divergent understandings of a mathematical sentence without revealing their divergence in the ability to recognize what counts as a proof of it. Even if one of these divergent understandings cannot be coherently sustained in general, it is possible to think that its inconsistency could never be manifested in the speaker’s ability to use the sentence. In this case, there could never be conclusive evidence for attributing him any specific understanding of the sentence and, for this reason, we would not be entitled to equate his ability to recognize a proof of a mathematical sentence with his grasp of the meaning of the sentence. Consequently, Tennant’s strategy does not explain how a finite collection of uses can constitute a full manifestation of the speaker’s understanding of the sentence.
Lets consider Dummett’s second attempt to answer the sceptical paradox. In “Reply to Penco” Dummett says:
The man who continues the sequence 2,4,6,…,1000,1004,1008,… cannot explain to us why he finds natural to do this; nor, unless he had happened to think in advance what he would do when he reached 1000, could he have said anything to us at the outset that would have revealed to us what was going to happen when he reached 1000. For us it is just a brute fact that he finds it natural to continue 1000,1004,…, and for him it is just a brute fact that we find it natural to continue 1000,1002,…It does not follow that these are brute facts for God: it does not follow that God could not have told what he was going to do.
He goes on to say
I have little to pit against them [the considerations’ about following a rule], indeed, save the conviction that, while they appear epistemologically irrefutable, they do not entail what I have called their metaphysical corollary [that there is no determinate thing that would be a correct application of the rule], which I find myself unable to believe.
In contrast with Kripke, Dummett thinks that the rule-following paradox poses an epistemological problem, but not a metaphysical problem about the claim that there are no facts of the matter as far as rules and meaning are concerned. He maintains that, even if we cannot discern which rule we are following, that does no imply that nothing can constitute a correct application of the rule. In this sense, Dummett seems to be claiming that there is a gap in the argument between our impossibility to determine the rule we are following and, hence, to determine the meaning we are attaching to it, and the conclusion that there are no facts that can determine the correct application of the rule. Therefore, according to Dummett, the paradox only introduces a problem about our impossibility to know and to manifest to other speakers the rule we are following and the meaning that we are attaching to it, because it has not properly proved that there are no facts about rules and meaning.
Setting aside the debate about the extension of the rule-following argument, let’s suppose that Dummett is right in his claim. Can we say then that this response solves our previous problem? I believe that it does not. This response would make meaning and rules ineffable: if from the ability to recognize what counts as a proof of a sentence we cannot determine the rule we are following, we will be unable to determine the meaning of the sentence and, as a consequence, we will be also unable to communicate grasp of meaning to other speakers. Hence, it is evident that the claim that the rule-following paradox constitutes only an epistemological problem will not help us to make sense of the manifestation requirement, since if our grasp of meaning cannot be determined, is difficult to see how we can claim that it is possible for every aspect of someone’s understanding to be fully manifested. Therefore, Dummett’s refusal to accept the metaphysical conclusion is not sufficient because, even if we accept his claim, it seems that it is not possible to make sense of the full manifestation requirement.
Based on what we stated before, we can say that Dummett’s attempts to avoid the rule-following paradox cannot provide us with an adequate answer to the objection that the ability to recognize what counts as a proof of a mathematical sentence cannot satisfy the full manifestation of knowledge of meaning requirement. In spite of this, the possibility to find an adequate response to this objection remains open. Such response will have to stress the objectivity of mathematical proofs in order to justify the claim that the meaning of a sentence can be taken to consist in what counts as a proof of it. But, if the objectivity of mathematical proofs is, as the rule-following paradox suggests, illusory, then it will not be possible to determine the meaning of a sentence by reference to its proof. Thus, while the objections to the objectivity of mathematical proofs continue unanswered, the anti-realist’s claim that knowledge of meaning is closely tied to the ability to recognize what counts as a proof of a mathematical sentence will be unwarranted. In addition, that response will have to be compatible with the deep constraint that the anti-realist acknowledges concerning manifestation of grasp of meaning. It seems to me highly implausible that such constraint could in the end be met. However, it is of fundamental importance for the anti-realist to do so, because it is by reference to the full manifestation requirement that he is not only able to criticize the platonist conception of meaning and differentiate his own position from a platonist account, but also to defend a revision of classical logic and mathematics.
In the next section I will consider another problem that arises in Dummett’s theory of meaning for mathematical sentences. This problem is also associated with the role played by the notion of proof in Dummett’s theory; in particular, it is connected with the way in which this notion has to be understood in order to give meaning to mathematical sentences. For that aim, I will take into account an extreme version of anti-realism called strict finitism, I will explain its differences with Dummett’s account, and I will consider the problems that this radical position poses to a moderate anti-realist such as Dummett.
3. Anti-realism and Strict Finitism
In accordance with the anti-realist perspective, mathematical proofs, in terms of which we give meaning to mathematical sentences, can be constructions that we are capable of effecting only in principle. This view does not require the proofs to be carried out in practice and, therefore, some of these proofs may be too complex or too lengthy for a human being to get them done.
A much more radical view is that according to which the principle of bivalence is accepted only in connection with those sentences that can be proved or disproved in practice. This view, known as strict finitism, insists on the claim that we can only give meaning to mathematical sentences and learn how to use them on the basis of the proofs that we can carry out in practice. Strict finitism was suggested by Wittgenstein and developed by Yesenin-Volpin, who attempted a reconstruction of mathematics along strict finitist lines.
Strict finitism only accepts finite proofs of mathematical sentences. It rejects those proofs that, because of their length and complexity, can only be carried out theoretically and maintains that those methods intended to accomplish proofs in connection with infinite sets (like the sieve of Eratosthenes that allows us to determine, given a natural number, whether it is prime or not) must be rejected. Thus, this view establishes as a requirement to avoid any reference to infinite collections. Strict finitism, nevertheless, must not be taken as the same as Hilbert's finitism, since they disagree on central issues. According to strict finitism, only feasible numbers are well defined. Hilbert's finitism, on the other hand, considers as well defined those numbers that are potentially realizable. This is the reason why, in order to differentiate strict finitism from other more moderate views, P. Bernays and G. Kreisel labeled this view "strict finitism".
It has been maintained that the same arguments used by anti-realism to promote a revision of classical logic and mathematics lead this position to strict finitism. However, many anti-realists consider that strict finitism is incoherent. Thus, semantic anti-realism not only rejects platonism as a viable option for the philosophy of mathematics, but also strict finitism.
Semantic anti-realism differs from strict finitism because it accepts constructive proofs of infinite collections. Dummett considers that the kind of constructivism required for strict finitism is much more extreme than the one needed for his own proposal. An anti-realist may say, for example, that every natural number is or is not prime because we have a constructive method to settle this issue. But we have seen that the strict finitist view rejects those methods intended to prove sentences about infinite collections. The problem here is that there are numbers so big that we cannot determine whether they are prime or not. Thus, strict finitism requires a much more radical revision of mathematical practice than intuitionism, since, if we accepted the restrictions posed by this view, mathematical discourse would be limited to practical possibility.
Mathematical anti-realism claims that the platonist theory of meaning must explain, since in that theory the notion of truth transcends our means to prove a certain sentence, how a grasp of what it is for a sentence to be true can be communicated. It also considers that this theory should explain how someone who has understood this notion of truth that transcends our capacities could manifest, by means of the use of the sentence, this understanding. Strict finitism agrees with anti-realism on two issues: (i) it is necessary for the platonist to offer such explanations; (ii) these explanations cannot be satisfactorily given by the platonist theory of meaning. Nonetheless, it maintains that this claim for explanation also arises to everyone who accepts that it is not required for a sentence to be proved or disproved in practice in order to consider it as determined in its truth-value. Thus, the strict finitist sustains that a moderate version of anti-realism must provide an explanation about the way in which we acquire the knowledge that constitutes the grasp of the meaning and about the way in which we manifest this knowledge by reference to a notion of proof that only demands, in order to ascribe this knowledge, that an individual be capable in principle of recognizing what counts as proof.
Anti-realism, while committing itself to a notion of provability in principle, attributes to us a notion of truth that is in practice inaccessible. For this reason, it does not require that, for every true sentence, it be the case that, in suitable circumstances, an individual could know the sentence’s truth by reference to the cognitive capacities that the individual has. According to Tennant, this requirement (that is, the strict finitist’s requirement) would bind too strongly what is true to what an individual knows. Therefore, anti-realists propose to abstract from the limitations of the actual individuals and their tendency to error. What they require in order to consider a certain sentence as determined in its truth-value is that a finite extension of our cognitive capacities would allow us to recognize a proof or a disproof of it.
The strict finitist considers, on the other hand, that it is impossible to explain, since we learn a language through practice and we manifest our understanding of mathematics through practice, that we have genuinely acquired an understanding of what it is for a sentence to be true without being able in practice to recognize it as true. In this sense, strict finitism argues against anti-realism in a similar way to that used by anti-realism against platonism. Therefore, we can entertain the possibility that the arguments of acquisition and manifestation are sliding the anti-realist into the strict finitist view.
According to Wright, the anti-realist has two options in order to address the objections posed by the strict finitist: he may either answer the objections giving the required explanation, or he may question the legitimacy of this demand. As we have seen, many anti-realists consider that strict finitism is incoherent. Hence, it is possible to think that one of the ways in which we can question this demand consists in pointing out the incoherence of this position, since that would allow the anti-realist to justify his unwillingness to give to his own arguments such a radical extension.
The discussion about the viability of strict finitism has aroused a controversy. Certain authors such as Dummett believe that this position is vulnerable to the sorites paradox and, as a consequence, they maintain that it is incoherent. Wright, however, has presented his doubts about the correctness of Dummett's conclusions.
I will not examine this controversy here. Nevertheless, I want to point out that the debate about the viability of strict finitism in the philosophy of mathematics does not exclude the possibility that an anti-realist position would in the end collapse into a strict finitist view. Even if strict finitism turns out to be an incoherent position, the problem about what resources the anti-realist has in order to avoid this view would remain open. In this sense, if strict finitism is incoherent, and if it is not possible to show that the anti-realist’s arguments do not need to entail the acceptance of this position, then it could be contended that anti-realism is not a viable position in the philosophy of mathematics either. Thus, the main problem is, beyond the debate about the incoherence of strict finitism, whether mathematical anti-realism can or cannot avoid strict finitism.
It is then crucial for mathematical anti-realism to attain a sustainable notion of provability in principle compatible with the acquisition and manifestation arguments.
Anti-realism and the notion of provability ‘in principle’
According to Dummett, we are allowed to consider a sentence as determined in its truth-value if the sentence can in principle (that is, theoretically) be proved. However, strict finitism insists that a rigorous implementation of the acquisition and manifestation arguments would lead us to the conclusion that only those sentences that can be proved in practice have a determined truth-value. Nevertheless, anti-realism intends to avoid that conclusion. But, as we have already seen, to claim that strict finitism is incoherent does not provide us with a sufficient reason for rejecting that conclusion. Then, we may wonder how it is possible to do so.
It seems that, in order to dismiss the strict finitist's objections, the anti-realist must explain how it is feasible to grasp the meaning of mathematical sentences and to learn how to use them, even though it is not possible in practice to recognize what counts as a proof of them. That is, the anti-realist must show that through the notion of provability in principle we can give an adequate account of understanding and learning.
Hence, it is obvious that, in order to dismiss the strict finitist’s objections, the anti-realist must elucidate the notion of provability in principle and also show that his theory of meaning, based on this notion, is not threatened by the arguments of acquisition and manifestation.
Let's see in the first place how it is possible to characterize the notion of provability in principle. We have seen that, given a certain sentence, we may say that this sentence has a determined truth-value if it is possible in principle to recognize a proof or a disproof of it. It is irrelevant whether this proof is too complex or too long for a human being to recognize it as a proof. But, what does it mean to say that someone is capable of recognizing a proof in principle? Clearly, it entails an idealization of our cognitive powers, since if we were restricted to our actual cognitive capacities, we would be incapable of recognizing those proofs that are too complex or too long. But this does not imply that these proofs have to be recognized by an omniscient and omnipotent god, since what it is relevant here is what we are in principle capable of recognizing. Otherwise, it would be difficult to explain how we could be able to recognize these proofs. Thus, all we need is a finite extension of our actual cognitive capacities.
One criticism that this view has aroused is that we cannot give a precise account of what this extension is supposed to entail. In this sense, Kripke has maintained that the proposed idealization of our cognitive capacities generates a problem, because we ignore what consequences this extension might bring about. Moreover, he considers that, even if we tried to give a more precise account of the consequences that this extension could produce, such specifications would only be a mere speculation. Wright, similarly, has argued that it is not at all clear what this extension of our cognitive capacities could mean. He maintains that, since this hypothesis should be entertained independently of our present scientific theories, we would lack any account of what can justify the claim that such an increase had taken place. Thus, he concludes that it is doubtful whether an anti-realist point of view can consistently give content to this hypothesis.
Even though these objections constitute a serious challenge to mathematical anti-realism (inasmuch as they question the possibility of giving a precise account of the notion of provability in principle), I consider that there is an even more serious problem that the anti-realist perspective faces in connection with this notion.
Let's suppose that, in order to avoid strict finitism, we insist that the arguments of acquisition and manifestation just require that the truth of a given sentence could in principle be proved. However, when confronting this insistence, we must be sure that this is certainly the case. We must then show that the notion of provability in principle is compatible with the idea that learning mathematics consists exclusively in learning how to use the sentences of this language. It is also necessary to show that the knowledge that constitutes the grasp of the meaning of a sentence can be fully manifested by means of our capacity to recognize, in principle, a proof of it.
In the case of the manifestation argument the problem is that, if we assume that our capacity to recognize in principle what counts as proof of a sentence enables us to meet the manifestation requirement, then it is not clear how it would be possible to fully manifest a grasp of the meaning of that sentence. It is evident that we will have to concede that, in some cases, a grasp of meaning can be attributed even when knowledge of meaning cannot be fully manifested. If we have to accept that some sentences possess a determined truth-value, even though only a finite (but wide enough) extension of our cognitive capacities would allow us to recognize it, then in such cases we would not be able to fully manifest our grasp of the meaning in our ability to recognize what counts as a proof of them. At best, we will only ideally be able to recognize those proofs that are too long or too complex, but that idealized ability cannot be showed in practice. Thus, it is unclear on what criterion a grasp of meaning could be attributed to someone, since knowledge of meaning will not be fully manifested by means of his capacity for use or respond to the use.
Similarly, in the case of the acquisition argument, if what we acquire when we learn mathematics is an ability to recognize, in principle, what constitutes a proof or a disproof of a sentence, then it is not clear how we can learn to grasp the meaning of those sentences the proofs of which are too complex or long. It is evident that in this case it will be necessary to recognize that sometimes a grasp of the meaning can be attributed even when knowledge of meaning cannot be fully manifested. Since only with an extension of our cognitive capacities we would be able to recognize these proofs, and since only by reference to our proficiency in making the correct use other individuals may decide whether we have obtained a grasp of meaning, it will be possible to attribute to someone mastery of the practice even when it is not possible for him to fully manifest a grasp of meaning.
Based on what we stated above, we may say that a weakening of the full manifestation requirement is needed. It seems that only demanding a weaker manifestation, the tension between the notion of provability in principle and the manifestation and acquisition arguments can be avoided. Thus, we will have to admit that a grasp of meaning cannot always be fully manifested through our capacity for use a sentence or respond to its uses. Nevertheless, by doing so we will be weakening these arguments as well.
But, the problem that arises here is that the distinction between the platonist and anti-realist views gets blurred. The arguments of manifestation and acquisition express the main anti-realist objection upon a platonist conception of meaning. Thus, if the anti-realist accepts that a grasp of meaning can be attributed even when it cannot be always fully manifested, then the arguments that he presents against the platonist will lose the required bite. When challenged with the anti-realist’s arguments that intend to show that a platonist theory of meaning cannot satisfactorily explain our grasp of the meaning of mathematical sentences because it is unable to meet the manifestation requirement, the platonist may answer that this demand is baseless. They can claim that this demand does not need to be satisfied, since the anti-realist theory is supposed to be an adequate model of understanding, even though a grasp of meaning cannot always be fully manifested in that theory. Therefore, if we accept a weakening of the full manifestation requirement, it will become difficult to explain what is the difference between anti-realism and platonism.
By taking as its central notion the notion of provability in principle, anti-realism seems to approach its own conception of meaning to the platonist conception. What will then prevent us from considering, paradoxically, that anti-realism is just a variant of realism? It seems that it will not be easy for the anti-realist to answer this question. The notion of provability in principle will not allow us to fully manifest our understanding of the meaning of a sentence by means of our capacity for use or respond to its uses. Then, it is not clear what we gain by taking this notion instead of the notion of truth, since both notions seem to have similar problems.
From the preceding considerations it is possible to assert that the rejection of strict finitism is not an easy task for anti-realism, since the same characteristic arguments of this position seem to lead us to accept it. Furthermore, when trying to reject strict finitism, mathematical anti-realism approaches the other view it intends to reject, that is, a platonist conception of meaning. I then consider that, on the basis of what I presented so far, it is possible to conclude that the equilibrium that anti-realism tries to hold between these two positions is extremely weak.
I have attempted to show that Dummett’s theory of meaning for mathematical sentences faces two main objections, both of them associated with the role played by the notion of proof in that theory. The first one casts doubts on the anti-realist idea that every aspect of our grasp of the meaning of a mathematical sentence must be fully manifested in the ability to recognize what counts as a proof of it. I have argued that a dummettian theory of meaning faces some problems in order to meet its own manifestation requirement. In particular, I have claimed that the challenge posed by the rule-following paradox, that states that no finite collection of uses of a sentence can show beyond all doubt that an individual knows the meaning of the sentence, constitutes an objection to the anti-realist view. These considerations allowed me to maintain that, whereas the skeptical paradox remains unanswered, Dummett’s manifestation requirement cannot be reasonably demanded. In addition, I have also analyzed Dummett’s attempts to elude the rule-following paradox and I have concluded that neither of them provides an appropriate response to the objection that the ability to recognize what counts as a proof of a mathematical sentence cannot satisfy the requirement of full manifestation of knowledge of meaning.
The second objection comes from the strict finitist perspective. According to this view, an accurate implementation of the acquisition and manifestation arguments is incompatible with the notion of provability in principle defended by the anti-realist. Therefore, the strict finitist argues that we can only give meaning to mathematical sentences and learn how to use them on the basis of the proofs that we can carry out in practice. In order to avoid this perspective, the anti-realist must show that the notion of provability in principle is sufficient in order to make sense of these arguments. But we have seen that the finitist´s claim is not baseless, and that it is not at all clear how the anti-realist could do that. On the other hand, if the anti-realist attempts to elude this criticism by weakening the full manifestation requirement, allowing understanding to be attributed even when a grasp of meaning cannot always be fully manifested through our capacity for use a sentence or respond to its uses, the distinction between the platonist and anti-realist views will become unclear.
I have tried to show that the acquisition and manifestation arguments constitute, in Dummett’s hands, a double-edged sword. These arguments allow the anti-realist to criticize the platonist conception of meaning and to differentiate its own position from the platonist account, since the latter cannot meet the demands that these arguments pose in order to give an account of understanding and learning. But, on the other hand, it is on the basis of these same arguments that the anti-realist theory of meaning is open to the objections of the rule-following paradox and the strict finitist view.
V I thank Eduardo Barrio and Alberto Moretti for their comments on a previous version of this paper.
 Dummett’s proposal in favor of intuitionism differs from that of Brouwer. Brouwer maintained that the meaning of a mathematical sentence is essentially private and incommunicable, and insisted that intuitionistic mathematics should be a private study of mathematical constructions. These ideas led Brouwer to say: “To begin with, the First Act of Intuitionism completely separates mathematics from mathematical language, in particular from the phenomena of language which are described by theoretical logic, and recognizes that intuitionistic mathematics is an essentially languageless activity of the mind (…)” (“Historical background, principles and methods of intuitionism”, South African Journal of Science, 49, (1952), 139-146). In contrast, as we shall see, Dummett’s argument for intuitionism stresses the public and communicable character of the meaning of mathematical sentences.
 Kreisel, G., (1959), “Wittgenstein’s Remarks on the Foundations of Mathematics”, British Journal for the Philosophy of Science, 9, 135-158.
 See specially Dummett, M., (1973), “The Philosophical Basis of Intuitionistic Logic”, in Truth and Other Enigmas, London, Duckworth, pp. 216-217; (1976), ‘What is a Theory of Meaning? (II)’ in The seas of language, Oxford, Oxford University Press, pp. 34-93; (1977), Concluding Philosophical Remarks, in Elements of Intuitionism, Oxford, Clarendon Press; (1991), The Logical Basis of Metaphysics, Cambridge, Harvard University Press.
 Cf. Dummett, M., “The Philosophical Basis of Intuitionistic Logic”, p. 217. See also Wright, C., (1986), “Truth Conditions and Criteria”, in Realism, Meaning and Truth, Oxford, Blackwell.
 Some formulations of this argument may be found, for example, in Dummett, M., “The Philosophical Basis of Intuitionistic Logic”, pp. 217-218; and in Wright, C., Realism, Meaning and Truth, pp. 13-16, 122-123.
 Some formulations of this argument may be found, for example, in Dummett, M., (1973), “Can Truth be Defined?”, in Frege: Philosophy of language, London, Duckworth, p. 466; “The Philosophical Basis of Intuitionistic Logic”, pp. 223-225; “What is a Theory of Meaning? (II)”, pp. 46, 92; (1979), “What does the Appeal to Use Do for the Theory of Meaning?”, in The Seas of Language, p. 116; (1991), The Logical Basis of Metaphysics, pp. 314-316.
 As a matter of fact, the sceptical paradox not only questions Dummett’s theory in the context of mathematics, but also its extension to any other area of discourse. However, I will only examine here the problems aroused by this paradox in the case of mathematics.
 Wittgenstein, L., (1953), Philosophical Investigations, Blackwell, Oxford; Wittgenstein, L., (1978), Remarks on the Foundations of Mathematics, 3rd edition, Blackwell, Oxford.
 Kripke, S., (1982), Wittgenstein on Rules and Private Language, Harvard University Press, Cambridge, Mass.
 Kripke, S., op cit., p. 13.
 Cf. Dummett, M., “The Philosophical Basis of Intuitionistic Logic”, p. 217.
 To say, instead, that our understanding of a sentence must be manifestable, will not help, since not only our uses of a sentence are finite, but also our dispositions.
 Dummett, M., (1959), “Wittgenstein’s Philosophy of Mathematics”, in Truth and Other Enigmas, pp. 166-185.
 McGuinness, B. and Oliveri, G. (eds.), (1994), The Philosophy of Michael Dummett, Dordrecht, Kluwer Academic Publishers, pp. 308-317.
 Dummett, M., “Wittgenstein’s Philosophy of Mathematics”, pp. 171-172.
 We can also harness machines in parallel in order to increase their reliability, but the problem is that both machines can malfunction. Besides, if the machines divert at some point, we will need the program in order to determine which one went wrong.
 Tennant, N., (1997), The Taming of Truth, Oxford, Oxford University Press, pp. 91-142.
 McGuinness, B. and Oliveri, G., op. cit., p. 314.
 Wittgenstein, L., Remarks on the Foundations of Mathematics.
 Yesenin-Volpin, A., “Le programme ultra-intuitioniste des fondements des mathematiques”, in Mostowski, A. (ed.), (1961), Infinitistic Methods, pp. 201-223; “The Ultra-Intuitionistic Criticism and the Anti-Traditional Program for the Foundations of Mathematics”, in Kino, A., Myhill, J. and Vesley, R. (eds.), (1970), Intuitionism and Proof Theory, pp. 3-45.
 Bernays, P., (1935), “Sur le platonisme dans les mathematiques”, in L’ Enseignement Mathematique, 34, pp. 52-69.
 Kreisel, G., (1959), op.cit., 135-58.
 Dummett, M., (1970), “Wang’s Paradox”, in Truth and Other Enigmas, pp. 248-249; Wright, C., (1982), “Strict finitism”, in Realism, Meaning and Truth, pp. 105.
 Tennant, N., op.cit., p. 143.
 Wright, C., op. cit., p. 111
 Cf. Dummett, M., “ Wang’s Paradox”.
 Cf. Wright, C., op. cit..
 Kripke, S., op.cit., p. 27.
 Wright, C., op. cit., p. 124.