On difficulties concerning intuition and intuitive knowledge – response to James Page, this issue, p. 223

Charles Parsons

“Intuition” as understood in contemporary discourse (not only in philosophy) is primarily a source of belief or putative knowledge. It is natural to regiment the notion as a propositional attitude, so that one would speak of an agent as having the intuition that p, or perhaps as intuiting that p. There is, however, a notion of intuition of objects that has roots in the history of philosophy, particularly in Kant. Kant insists on the fundamental role of such intuition in mathematical knowledge, but he is not as explicit as one might expect in affirming the existence and role of intuition of mathematical objects, for want of a theory of such objects. In our time of greater explicitness about ontological commitments, a Kantian conception of intuition would naturally be developed so as to include intuition of mathematical objects. There is, however, much room for ambiguity both in the notion of intuition and in the concept of mathematical object.

A further problem would arise however the idea is further developed: how is intuition of objects related to propositional knowledge? In Kant, intuition is connected with knowledge of objects, but since such knowledge consists of judgments, it appears that intuition is a component of or at least gives rise to propositional knowledge. My own attempts to develop a conception of intuition are in the Kantian tradition, in particular in that it is primarily a notion of intuition of objects that I undertook to explicate. Then the question of the role of such intuition in propositional knowledge is a natural one.

It is on this point that James Page criticizes my views.(2) He concedes that there is intuition of abstract objects, at least of the kind on which I have focused most closely. The questions he raises concern the relation of this intuition to knowledge. I have used the term “intuitive knowledge” to describe knowledge that is based on intuition in an appropriate way.(3) Some of the objections Page makes were probably encouraged by the fact that I did not offer a definition or characterization of intuitive knowledge. I am not sure an exact characterization is even feasible. One might compare the related notion of perceptual knowledge, or knowledge by perception: the project of an “analysis” of knowledge in the sense of a philosopher’s definition has run into formidable difficulties, and concerning the further problem just what the role of perception has to be for knowledge to count as perceptual, one would not expect very sharp boundaries.

It should, however, be possible to explain the general idea and to state some of the general properties one would expect a notion of intuitive knowledge to have. One condition is surely that intuition should play an essential role in it. I stated as a necessary condition for intuitive knowledge that it should “rest on” the intuition of objects (1986, p. 215). Page, citing this remark, writes

Our knowledge that a statement is true is intuitive knowledge if it involves the intuition of those objects which make that statement true (p. 224).(4)

As we shall see, Page questions whether the necessary condition is satisfied in some relevant cases.

Another underlying idea of a conception of intuitive knowledge is that intuition should be in some way sufficient for the knowledge in question. By looking at the analogous perceptual case we can see that this is going to give rise to difficulties. Propositional knowledge requires something more than perceptual capacity or even actual perception; to know that a rabbit is there it is not enough to see a rabbit (which usually means simply seeing something that is in fact a rabbit), but something further of a nature that is naturally called conceptual: one must recognize what one sees as a rabbit (and as being there, whatever counts as that). I shall try, in responding to Page’s claims, to bring out how far I understand intuition in my sense as being sufficient for intuitive knowledge.

I

Before I reply to Page’s criticism, however, I shall mention another prior difficulty for my conception of intuition (and that of Hilbert which was its model) suggested by some published or forthcoming discussions.(5)

The difficulty touches most directly the “modal nominalist” construal of types in my first systematic presentation (1971, [section] III) of my conception of intuition of such objects. But let us consider, to simplify things, a purely nominalist approach to the stroke language.(6) The nominalist approach has certain difficulties. It presupposes that there is a relation of concrete (let us suppose physical) objects which we can call “being of the same type”. But this relation will have to be an equivalence relation. In practice, however, it is to a certain degree vague what counts as a token of a given type. But then it is not clear that such a relation can be defined that will be transitive. For example it might be vague with regard to strings of strokes what the distance between them might be. Consider the examples

////

/// /

/// /

/// /

/// /.

At what point does the inscription cease to be of the same type as the initial one? They all satisfy Page’s criterion of consisting of four strokes. So one might say at no point. But that is not at all plausible; no one would say, for example, that the first of the above strings continued by “/” 1000 miles to the right constitute a single string of five strokes.

We naturally think of the strings as generated inductively, beginning with a simple stroke and successively attaching an additional stroke on the right of a given string.(7) The above example illustrates the potential vagueness of the relation |y results from x by attaching a stroke to the right of x’. Vagueness could equally infect the notion of being a stroke (as a predicate of inscriptions). This would be clear if one relies on the usual ostensive explanation, given by writing or printing a sample.(8) Regarding the transitivity of the same-type relation, we can define it inductively parallel to the proposed inductive definition of string, and prove (using induction according to the definition) that it is transitive. But we should distrust this proof for well-known reasons, since it relies on application of induction to vague predicates.

The nominalist may not be troubled by these matters; he will point out that one might characterize in an exact way, with the help of physics, just what inscriptions count as strokes and what the operation of adding a stroke on the right is. However that may be, this sort of reply is not open to me. For it is hard to see how its use could square with the idea that consciousness of types (on this construal, consciousness of tokens together with an equivalence relation of “being the same type”) is essentially perceptual. Such an exact concept of a stroke inscription will make a finer discrimination between strokes and non-strokes than perception can make. For example, a stroke might be required to be exactly 0.3 cm high, but one that is 0.30000000001 cm high would not be distinguishable from it.(9)

In later writing I have backed away from the modal nominalist understanding of types.(10) Without it, one has to understand intuition of types and other forms in a way that does not yield a reduction of the language concerning them to language in which reference to objects of this kind is absent. But the ubiquity in our everyday experience of linguistic expressions and other “quasi-concrete” abstract objects such as sense-qualities and shapes should lead us to question why a reduction should be necessary or desirable. Behind the technical difficulties arising from vagueness is the rather basic consideration that one may identify such a form, as it were in the flux of experience, without identifying an object of which it is the form. A spoken expression is a type of which the tokens are presumably events of some kind, but to identify just what event a given token is presumably implies locating it in space and time (and possibly in a causal nexus), although the expression itself does not have a location. I hold that neither the identification of the token nor that of the type is a necessary condition of the other.

If I read a sentence, and take in what sentence it is, of course I do see a certain physical inscription. In that sense my perception of the sentence is founded on perception of the inscription. But I can identify what sentence it is without identifying what inscription it is, and certain kinds of illusion about the latter would not disturb the correctness of my perception of the former. And it is not an evidently necessary condition of such perception that there be either a criterion stated in terms of physical characteristics of inscriptions for an inscription to be an inscription of a particular sentence, or a criterion stated somehow in terms of sensible appearances for a given perception to be of the sentence. We do not demand such a criterion in the case of everyday objects.

Thus as a first answer to this difficulty, the phenomenological claim that our identification and reidentification of expression types is perceptual is not undermined by the fact that we cannot give a sharp equivalence relation of more “basic” objects that serves as a criterion of identity. There is not such a relation for physical objects either.

However, we have so far only perception, not mathematical intuition. What makes intuition mathematical intuition is that it gives objects that instantiate concepts that have a sharp, precise character. At least for statements in the mathematical vocabulary, there is no vagueness in their application to strings of strokes. There may indeed be vagueness as to whether what is before us is or is not a token of a given string, but not about the question whether one string, say, consists of two more strokes than another. When I spoke above of perception of types, was I in a position to rule out vagueness or puzzle cases about their identity, as there might be both in the case of everyday objects? Pure intuition as Kant understood it was evidently supposed somehow to get us across the divide between the fuzzy phenomenal world (including the world of everyday objects) and the sharp, precise realm of the mathematical, in terms of which mathematical conceptions of the physical world are developed.(11)

Let us consider a somewhat different case from that of types, where we can see very clearly where a problem about vagueness will arise: the idea, developed in some detail in early writings of Husserl, of intuition of finite sets. According to this a certain way of taking a surveyable plurality of objects of perception, say the eggs in a carton,(12) is intuition of a set whose elements are the eggs in the carton. Eggs belong to what I have called the phenomenal world, but on the face of it sets of eggs are mathematical objects. Given such a set a, one would expect that it would be quite determinate, for a given x, whether or not x[epsilon]a. But suppose x is one of the eggs, and let x’ be an egg that we encounter some time later. If x’ = x, then of course x'[epsilon]a. But suppose (as it seems that we cannot rule out) that x’ is an egg such that our criteria of identity for eggs are not able to settle whether it is x or not, although they determine that it is not one of the other eggs in a (perhaps they were consumed before we encountered x’).(13) Then it seems that it is after all indeterminate whether x'[epsilon]a. Cantor thought of sets as consisting of “well-distinguished objects”; and the practice of set theory ever since has been to assume that problems of individuation of the basic elements will not arise. This is a case of “idealization”: the application of a theory of sets of eggs, even small finite sets, might break down if it is made to cases where the eggs are not “well-distinguished”, that is if there are questions of the identity of “two” potential elements of a set that cannot be resolved.(14) For example, there might be an unresolvable question about the cardinality of the set. In the present case, however, it seems clear that one can see that there are three eggs in the carton.(15) Let x and x’ be as before. Then if x = x’, x’ does not add another egg, and if x'[is not equal to]x, then our hypotheses imply that x’ is not in the carton.(16)

It seems from these considerations that the idea of perception or intuition of sets is ambiguous. If we perceive the set of eggs in the carton in the above-described situation, then we do not get from that the knowledge that it consists of well-distinguished objects. Thus on this view we perceive the set a; perhaps when we have encountered x’ we can perceive or intuit the set a’= {x’, x’, y, z}. But of a’ it seems to be undetermined whether it has three or four elements; likewise whether or not a = a’. The conclusion to draw is that individual sets of concrete objects can inherit from their elements any troubles about the individuation of the latter.

It could be a matter of dispute whether the concept of set that allows vagueness of this kind is really the mathematical one. Set theory would not lose its applicability to everyday situations by ruling out such sets, since they will in any case be rare, and one will often get correct results by ignoring such vagueness; thus in the present case one would get the right cardinality for the set of eggs in the carton.

What now is the impact, if any, of the vagueness we have called attention to on concepts of expression-types and a mathematics that deals with such types? One can see that a similar vagueness will arise on an attractive conception of strings, namely that they are sequences of symbols. Some have wanted to think of strings in this way while conceiving the notion of sequence in some abstract, possibly set-theoretic way, while thinking of the individual symbols as sets of their tokens or in some other way where the concrete tokens play a constitutive role.(17) Such an approach will leave the identity of strings no more precise than that of individual symbol-types; for example given the two-symbol strings st and uv, st = uv if and only if s = u and t = v. But then any indeterminacy left by our criteria of identity for individual symbol types will infect the identity of strings.

If we are not antecedently committed to this construal, the case of types is different from those of sets and sequences in that it is possible to shunt off any threat of vagueness in their individuation onto the relation to their tokens or possibly the individuation of the tokens. Thus it may seem to be vague whether the type instantiated by

(a) ////

is the same as the type instantiated by

(b) /// /.

But that conclusion is forced on us only by the unwarranted idea that “the type” instantiated by each of these inscriptions must be uniquely determined. If we allow ourselves to talk of the type instantiated by (a), presumably because we can re-identify it and relate it to other types we have admitted, and then find it vague whether (b) is of that type, then our difficulty concerns a relation between a type, the type of (a), and the token (b). That relation can be vague without the relations between types becoming vague. The vagueness may generate ambiguity as to what type “the type instantiated by (b)” designates (if we do not take it to fail of reference because of failure of uniqueness). But if we admit that it does designate a type, then the identity statement

the type instantiated by (a) = the type instantiated by (b)

cannot be vague, although it may be ambiguous.(18)

Considering strings of strokes as types, does someone who has the concept of such a string and (we will suppose) intuits a string when he perceives (a) also intuit a string when he perceives (b)? The answer is not altogether clear. Let us suppose that he approaches the situation with the concept of a string of the language that concerns us, and not with the concept of any other kind of string. Then, it seems, the only string he could intuit would be (a); that is, he would intuit a string only if (a) and (b) are of the same type. Let us assume he will read (b) as consisting of four tokens of the unique symbol of our alphabet. So then the question is whether he will recognize the attachment of the last symbol as an instance of the concatenation relation. But obviously he may or may not. We must then ask whether he would be right in making one or the other choice. But this is just where the vagueness comes in: if (b) is really a borderline case of being of the same type as (a), then our agent’s situation must be a borderline case of intuition of the type that we are assuming to be instantiated by (a). This is only true, however, if we assume that his intuition must be founded on his actual perception. Perception of (b) might prompt imagination that would found intuition of the type, which our agent would then presumably recognize to be only imperfectly instantiated by the perceived token (b).(19)

II

Without going further we can begin to examine the question whether the propositions (PA1′)-(PA5′) resulting from interpreting the axioms of arithmetic with respect to strings of strokes are intuitively known. We can leave aside the induction principle (PA5′), of which I have denied that it is intuitively known.(20) (PA3′), which is the principal object of Page’s criticism, raises further issues, and I will defer it to [section] III. About the others I have not said much in past writings, and Page is therefore justified in asking why they should be held to be intuitively known.

One might begin with the observation that surely the concept of a string of strokes has a certain ostensive character; part of explaining what they are, and convincing ourselves that they exist, is exhibiting them. I understand this as intuition of the kind I undertook to describe. Even if one rejects this, there is a role of ordinary perception (which in my usage also counts as intuition). It follows that any sentence containing singular reference to such strings could not be known without intuition and in that sense will “involve” intuition.

Page concedes this role of intuition in application to (PA1′). That he seems to question that the necessary condition is satisfied in the two remaining cases (PA2′) and (PA4′) may rest on the not explicitly formulated expectation that there should be some further role of intuition in these cases. With regard to (PA2′), “/ is not the successor of any string”, he gives an argument containing the step, “One string can succeed another only if it consists of at least two strokes” (p. 226). This will lead to the conclusion only with the help of the obvious statement “/ does not consist of at least two strokes”. Page would need to argue that that can be known without intuition. I don’t see how he would argue this. It may be helpful to step away from the matter of abstract intuition and consider the same proposition about the physical inscription. That seems to me a very clear case of perceptual knowledge.

I also fail to see the ground of Page’s doubt about (PA4′), which might best be reformulated as “If strings x and y have the same successor, then x = y.” Page himself uses the language of “seeing” in this case:

We can see that the stroke string inscriptions which result from adding one stroke to each inscription or from taking one stroke away from each inscription are also tokens of the same stroke string type (p. 227).

I have to speculate about what troubles Page; it may be merely that it has not been explained clearly enough how this knowledge is intuitive.

I can imagine its being argued that (PA2′) and perhaps other elementary truths of this kind are analytic and that for that reason knowledge of them does not rest on intuition. In the case of the concept of string of strokes, where intuition is needed even to understand the concept, even if one accepted the claim that these statements are analytic in some sense it would not follow that they were independent of intuition. If that is the ground of Page’s doubt, he must be making the demand for some additional role of intuition. I haven’t claimed to meet that demand, but let us look into the matter anyway.

This analyticity might well be claimed for (PA4′), since what is the “successor” of a string x but something obtained by attaching one more stroke to x? This “attaching” takes place in the space outside that occupied by x, so that if x differs from y, then there is no way in which one could, by adding a stroke to each of x and y, obtain identical results. But if someone argues that this obtains by virtue of the concepts involved, or by virtue of the meaning of such terms as “stroke”, “string”, and “attach”, although I am skeptical about the general idea I would not insist in the present context that this cannot be so. On that the question whether the demand for an additional role of intuition is satisfied may turn. So I will leave the question open.

What is clear is that the successor operation is exhibited clearly enough in intuition so that we cannot deny that (PA 4′) is intuitive on the sort of grounds on which we would deny it for the principle of induction and presumably for those applications of induction or recursion that involve abstract concepts, however we draw the limits of the intuitive.

The same will also be true of the related statement that if two strings are identical, they have identical successors. That will be a truth of logic if we allow ourselves a function symbol for “successor”, but there is surely a prior question whether we are entitled to that. One could be misled by the fact that the particular intuitive structure we are concerned with was singled out with the natural numbers in mind. If we consider a symbolism with just two symbols, say | – ‘ in addition to “/”, and again the successor of a string is the result of adding one more symbol on the right, then successor is no longer unique; for example the string “// – /” will have the two successors “// – //” and “// – / -“. One might put the matter by saying that although the uniqueness of predecessors is a general feature of strings, the uniqueness of successors depends on the fact that our alphabet consists of a single symbol.

It does follow that if we recognize an extension of a given string as belonging to the “language” at all, there is only one possibility as to what the new symbol can be. Moreover there is similarly only one concatenation relation, even in the case where the alphabet has more than one symbol.(21)

The reader may legitimately feel that more should be said about equality of strings. The idea seems natural that in our simple case equality is “generated” by the two principles that two strings each consisting of a single symbol are equal, and equal strings have equal successors. Applied directly to types, however, these principles are trivial and by themselves give rise only to self-identities. This ceases to be so, however, when they are combined with recursion equations for additional operations.

In spite of the general inadequacy of the nominalist view, two corresponding principles about the notion of being of the same type are informative and describe a canonical procedure for determining sameness of type, modulo the problem of vagueness discussed above. They can be stated as follows:

(1) If a and b are each single stroke-tokens, then a and b are of the same

type.

(2) If a and b are of the same type, and c results from a by attaching a stroke

token on the right, and d results from b by attaching a stroke token on

the right, then c and d are of the same type.

III

Page’s objections to the claim that PA3′ is known intuitively are the heart of the matter, because without PA3′ the theory of strings would not model the infinity of the natural numbers. Intuitive knowledge of PA3′ gives intuition a central role in our knowledge of infinity.

PA 3′ amounts to the statement that every string can be extended. That can be verified by an imaginative thought experiment, as described in 1980, pp. 156-7. One should not, however, overestimate what this accomplishes. It tells us that we can always add an additional stroke. But the idea that there are infinitely many strings involves two further elements. The first is that the operation of adding another string can be indefinitely iterated. The second is that the string obtained by adding another stroke is a genuinely new string in the sense of being of a different type. Both of these involve recursion or induction, as we shall see below. The thought experiment is not uniquely determined in its details. That seems to me not a difficulty; it rather confirms that the conclusion can be arrived at in different ways. As Page observes, however, I point out that imagining an arbitrary string involves either imagining it vaguely, so that one is not imagining a very specific string (or imagining anything very definite about its length), or choosing a specific string as a paradigm, in which case the specific internal structure must be seen to be irrelevant.

Let us consider the first alternative. Page objects that if one imagines an inscription in this way, and sees that it can be extended by an additional stroke, one will not be sure that one has an inscription of length one greater, because it could be that in the process some other stroke “dropped out” (p. 228). It seems to me that part of what is imagined is that what is first imagined is unchanged. Moreover, since we have to do with spatial form, the causal considerations that would lead one to fear that a stroke might “disappear” are irrelevant.

Page expresses concern that, if one imagines strings without imagining them to be of a specific length, the maxim “no object without identity” is violated, since one does not have the information to determine whether they are the same string. This seems to me to be an unreasonable demand to make on imagination in general. Quite apart from the present context, surely I can imagine a and b, and also imagine either that a = b or that a [is not equal to] b, without necessarily imagining a and b in such detail that their identity or non-identity is plain from the other content of the imagining. For example, an adult man might imagine his mother as she now is and then try to imagine how she might have looked when younger, say at the time of his birth. It is simply part of what he is trying to imagine that it is her that he is imagining; it doesn’t follow that his imaginative picture will somehow determine it to be her rather than someone else.

In the present context, I imagine a string, in such a way that I am not imagining a particular string of a definite length, that is, imagining one string as opposed to another. In seeing that it can be extended, I continue to imagine this string. That is just part of the normal working of imagination. That I can imagine this, Page rightly says, is a fact about the structure of the way space occurs in perception, its figure-ground structure. Thus if I imagine a string, the space for it to be extended is always “there”. This is not to say that any such imagining shows that physical space is infinite. The conception of a physical inscription that is somehow “on the edge of space,” so that the space for it to be extended by another real physical stroke-token is lacking, is a very strange one. But I don’t claim that we know intuitively that it is impossible; I don’t even insist that it is impossible. What Page calls “imaginable space” is closer to what I had in mind.

On this interpretation Page writes: Here our knowing that any stroke string inscription can be extended depends on our knowing that our actual imaginative capacities are infinite. But if we assume that our actual imaginative capacities are infinite then we have again begged the question of how we know any stroke string inscription can be extended. Worse, to assume that our actual imaginative capacities are infinite is to assume something which is surely false (p. 230).

I fail to see how the thought experiments I have described depend on prior knowledge that “our actual imaginative capacities are infinite.” As I remarked above, we have infinity only when the operation of adding an additional stroke is indefinitely iterated. This idea can be formulated in different ways, but any of them will involve induction and recursion. Page has not made any case for the claim that these ideas are involved in my simple thought experiments. It is still a nice question whether and in what sense it is intuitively known that there are infinitely many strings.(22)

Page says it might be that “the relation of figure to ground for strings greater than [10.sup.100] strokes is different from what we can imagine” (p. 230). I’m not sure what he could mean by that unless he is claiming that if one has before one a string of more than [10.sup.100] strokes, that would cause the figure-ground structure of perception to be different. Even if that is relevant to the question whether it is physically possible for any stroke-string inscription to be extended, it is not relevant to the question at issue, which rather concerns mathematical possibility.

It should be clear that what we are concerned with is the figure-ground structure of perception as it is. This gives the possibility that what is now ground should become figure, yielding again the same figure-ground structure. The situation is essentially the same if one considers the matter from the point of view of temporal rather than spatial experience. Page criticizes my statement that if we think of a string as constructed step by step, it is “then evident that at any stage one can take another step” (1980, p. 156). What is being claimed concerns the structure of my experience of time as it now is, in which the present proceeds to an immediate future, which becomes present and embodies the same structure. The point is that one expects a future in which the immediate past is retained, and that this is possible.

Of course in both these cases the “expectation” concerns the world as well as one’s own experience: what one perceives as figure is perceived as in a surrounding space; that it is in such a space is part of one’s perceptual belief. Likewise, one’s present perception is of something in a time that continues.”

Once one has seen that every string can be extended, it is still another question whether the string resulting by adding another symbol is a different string from the original one. For this it must be of different type, and it is not obvious why this must be the case. For example, why can there not be a one-one correspondence of the strokes in the new string with those in the previous string? Thinking of the matter intuitively, it seems that in order to see this we have to appeal to the step-by-step construction of strings. The thought-experiment to show that every string can be extended is compatible with allowing strings that are not constructed from a one-symbol string in finitely many steps. And in fact, induction is needed to prove Sx [is not equal to] x in elementary arithmetic.(24)

The facts I have appealed to no doubt contribute to our having the idea of infinite space and time. But they would yield them only as mathematical abstractions; they do not imply that physical space or time is infinite. They belong to the “manifest image” of the world, and are subject to reinterpretation once mathematics and science have been built up, and we can entertain such ideas as that space or even time might, at scales much greater than that of perception, double back on themselves.

(2) In the preceding essay, which here is cited merely by page numbers. Citations merely by date and page number are of my own writings, unless the context clearly indicates another author. (3) It could be called simply intuition (and indeed is, in effect, in Parsons 1980, p. 155), but that would in practice give rise to confusion, on two counts: intuition of objects and intuition as a propositional attitude (intuition of and intuition that respectively) can come to be confused with one another, and concerning intuition that the question arises whether or not the term is to be used in a way that implies knowledge. Most use of the term “intuition” in contemporary philosophy is for the weaker sense of intuition that, i.e. so as not to imply knowledge. I argue elsewhere that this is true even for the contemporary philosopher of mathematics who claims most for mathematical intuition, Kurt Godel. But in this case as in others, one must recall that intuition can be a source of knowledge without being ipso facto knowledge. (4) This might seem to state a sufficient condition, but in a footnote Page qualifies the remark so that it is not affirmed as such. (5) Wetzel 1989 is an extended criticism of the view that expressions are more elementary and unproblematic objects than numbers. Her criticism is primarily directed at a skepticism or reductionism about numbers that I do not share, but since I have argued that there is intuition of some expressions and denied (in Parsons 1993) that there is intuition of numbers, I am still in some way in the camp that she is attacking. She appeals to the complexity of the type-token relation in natural language and even in artificial symbolisms, and she could have argued that my own expositions of intuition of expression types do not do justice to this complexity. She has suggested this in conversation, and a similar claim is made in Tymoczko 1993. With respect to natural language, accounts of this relation that discern factors unsuspected in my expositions are Bromberger 1989 and Bromberger and Halle 1992. (6) Modality is introduced as a way of providing for infinity, but in the present context we can ignore that problem. Evidently if difficulties arise in this simple case, they will affect nominalist or modal nominalist syntax quite generally. (7) That is how the Ziffern that are the objects of intuitive, finistist arithmetic are explained in Hilbert and Bernays 1934, p. 21. (8) Concerning the official explanation of Hilbert and Bernays, the question arises whether anything that we would recognize as a numeral “1” counts as an initial Ziffer for them, or only something whose size and shape are the same (or sufficiently close to) that of “1” as actually printed in their text. (9) Of course handwriting and even printing fail to make such fine discriminations.

For written and spoken natural language, Wetzel makes the stronger claim that two tokens of different types can be more similar to one another than either is to more “distant” tokens of its own type (1989, p. 187). This is probably correct. In an artificial symbolism such as primarily concerns me, one can avoid this consequence by suitable design of the symbolism. I don’t think it shows even for natural language that identification of types cannot be perceptual, but it does tell against a nominalist or modal nominalist understanding of expression-types of natural language where the relation “same type” is understood in terms of physical or visual or auditory similarity. A deeper consideration of what understanding of the relation would be satisfactory (as in Bromberger 1989) introduces ideas (such as speakers’ intentions) that hardly yield a nominalist construal of talk about types. (10) See particularly 1986, pp. 215-216. (11) Just for this reason, the term “phenomenal world” is not used exactly in a Kantian sense. (12) The example is from Maddy 1990, p. 58. Maddy defends the idea of perception of sets. She holds, for reasons not relevant to the present discussion, that a case like that in the text is one of perception, in a sense distinguishable from intuition.

I should say that although I use the Husserlian idea of intuition of finite sets for purposes of illustration and comparison, I do not wish to be committed to it. I hope to discuss it elsewhere. (13) If we take the letters x and x’ to be true variables, then it appears we are admitting x and x’ as “vague objects” such as were famously rejected by Gareth Evans (see Evans 1978). Clearly, however, we can admit vague identity statements without admitting vague objects. To avoid the commitment to vague objects, we should understand our letters as schematic letters for singular terms.

Since the question whether vague objects can be admitted is controversial, the formulation that allows them is still of interest. The discussion in the text would then argue that if x and x’ are vague objects, such that it is not determinately true or false that x = x’, then sets containing them are also vague objects; in particular it is not determinately true or false that {x, y, z} = {x’, y, z}. The situation is analogous to the rigidity of membership when set theory is combined with modal logic. (On the latter see for example Parsons 1983, pp. 298-308.)

The idea that sets must consist of well-distinguished objects may be interpreted simply to exclude vague objects as elements of sets, or it may exclude from the language of set theory singular terms that can enter into vague identity statements. In the text I am taking it in the latter, stronger sense. (14) This does not mean that we would have to be able to know whether x = x’. (15) For the sake of argument, I am assuming with Maddy that seeing that there are three eggs in the carton involves perceiving or intuiting the set of eggs in the carton. This is not in fact my view; see Parsons 1993, note 22. (16) One might argue that we cannot see at a time t, before we encounter x’, that there are three eggs in the carton, on the grounds that to know that there are three eggs in the carton we have to know that x’ is in the carton at t only if x’= x, and we can’t know that at t because, by hypothesis, we are totally unaware of x’. But we don’t have to know it; it is implied by |there are three eggs in the carton’ only together with certain facts about x’. (17) This should be distinguished from a more purely structuralist way of construing strings as sequences, in which the alphabet is just some arbitrary finite set of objects, about which nothing is assumed except the number of its elements. Such an approach is perfectly appropriate to the mathematical study of formal languages, but it simply does not engage the question how the relations of symbol- and string-types are related to those of concrete objects or are manifested in perception. (18) Even if we do not reject vague objects on general grounds (see note 13), types are not vague objects, and singular terms designating them can be indeterminate as to their reference only when they involve non-mathematical relations such as relations to tokens. (19) If we ask, in the above set case, whether someone intuits {x’, y, z}, we again have a borderline case, but one that rests on a borderline case of an identity statement concerning what is intuited. And there is no opening, in the case where he has x, y, and z before him, for intuition of {x’, y, z} founded on imagination. (20) This does not mean that there are not serious questions concerning instances of induction, particularly those used in finitist arithmetic. This issue, and related ones concerning recursion, are important issues concerning intuitive knowledge in elementary mathematics that I am not discussing here. More should be said than is said in Parsons 1986. (21) This latter would not obtain in a nonlinear symbolism. (22) This question is clearly bound up with the issues alluded to in note 19. (23) One might ask then about the experience of someone on the point of death. Is the temporal structure of his experience somehow different? It does not follow; he may continue up to the end to have the instinctive expectation of continuation; only that expectation won’t be fulfilled. I don’t think such an experience is relevant to the present question: the failure of fulfilment is the failure of a possibility to become actual; what concerns us is the possibility. It has, however, been speculated that a person at the point of death has an experience as of the “end of time”. Although I don’t think we have a real conception of such an experience, I think it too is irrelevant; what is relevant is the temporal structure of experience as it now is. (24) That it is unprovable in Robinson arithmetic Q is shown in Tarski, Mostowski, and Robinson 1953, p. 55. (Thanks to George Boolos for this reference.) If one is interested only in the independence of Sx [is not equal to] x, one can do with only one non-standard element. On the other hand a model with two will show all at once the unprovability in Q of a number of elementary arithmetic statements. See Boolos and Jeffrey 1974, exercise 14.2.

REFERENCES

Boolos, George S., and Richard C. Jeffrey, 1974. Computability and Logic. Cambridge University Press. (2d ed. 1981, 3rd ed. 1989.) Bromberger, Sylvain, 1989. “Types and tokens in linguistics”. In Alexander George (ed.), Reflections on Chomsky, pp. 58-89. Cambridge, Mass.: MIT Press. Reprinted in Bromberger 1992, pp. 170-208. _____ 1992: On What We Know We Don’t Know. Chicago: University of Chicago Press, and Stanford: Center for the Study of Language and Information. Bromberger, Sylvain, and Morris Halle, 1992. “The ontology of phonology”, in Bromberger 1992, pp. 209-228. Evans, Gareth, 1978. “Can there be vague objects?”. Reprinted in his Collected Papers, pp. 176-177. Oxford: Clarendon Press, 1985. George, Alexander (ed.), 1993. Mathematics and Mind. Oxford: Clarendon Press, forthcoming. Hilbert, David, and Paul Bernays, 1934. Grundlagen der Mathematik I. Berlin: Springer. (2d ed. 1968.) Maddy, Penelope, 1990. Realism in Mathematics. Oxford: Clarendon Press. Parsons, Charles, 1971. “Ontology and mathematics”. Reprinted in Parsons 1983, pp. 37-60. _____ 1980. “Mathematical intuition”. Proceedings of the Aristotelian Society N. S. 80 (1979-80), pp. 145-168. _____ 1983. Mathematics in Philosophy: Selected Essays. Ithaca, N. Y.: Cornell University Press. _____ 1986. “Intuition in constructive mathematics”, in Jeremy Butterfield (ed.), Language, Mind, and Logic, pp. 211-229. Cambridge: Cambridge University Press. _____ 1993. “Intuition and number”, in George 1993. Tarski, Alfred, Andrzej Mostowski, and Raphael M. Robinson, 1953. Undecidable Theories. Amsterdam: North-Holland. Tymoczko, Thomas, 199?. “Comment on Parsons” 1993, in George 1993. Wetzel, Linda, 1989. “Expressions vs. numbers”. Philosophical Topics 17, pp. 173-196.

COPYRIGHT 1993 Oxford University Press

COPYRIGHT 2004 Gale Group