a) The Platonic Dogma

Past and present philosophers seem to agree that, whatever else may be true of numbers, numbers cannot be perceived. Steiner, for example, says that “No one today, however, upholds hard-core intuition—the direct intuition of mathematical objects, the first type of mathematical intuition mentioned above. No one, with the possible exception of Goedel on one reading of the passage quoted above, claims to having perception of individual numbers” (M. Steiner, Mathematical Knowledge, p. 131). If Steiner is correct, then there are at most two people who believe that we can perceive numbers, namely, Goedel and Grossmann. I must confess that I find the company congenial. However, I am not sure that Goedel really shares my view, for he does not speak univocally about perception in regard to numbers. The fact remains that the overwhelming majority of contemporary philosophers consider my view to be doubtful at best, absurd at worst. What accounts for this unanimity? What is responsible for this conviction?

It seems that the common view is based on a certain type of argument to the effect that we cannot perceive numbers because we cannot causally interact with them. I shall call this “the argument from causal interaction.” Before we take a close look at it, let us locate it in the philosophical landscape.

Let us go backward, from the conclusion to the premises:

Why can we not perceive abstract things?

The argument thus rests, firstly, on a certain view about causality, according to which abstract things cannot causally interact; and, secondly, on a certain view about perception, according to which perception is a matter of causal interaction between perceiver and perceived object. It must be emphasized that these two somewhat dubious views are combined for one and only one purpose, namely, to make premise (2) of the argument plausible. They are exhibited briefly, so to speak, in order to defend (2), what I called earlier “the Platonic dogma,” which otherwise would be nothing but an unfounded thesis. Of course, there is an easier way to defend (2): One could maintain that there are no abstract entities. But this would stand things on their heads, for we are here interested in an epistemological argument against abstract entities, and within this context it would be circular to assume that there are no such things. No, the reasoning goes the other way: If we cannot perceive abstract entities, one may reasonably doubt that there are such things.

I shall try to sever the connection between (3) and (4), on the one hand, and (1) and (2), on the other. In regard to (4), we obviously cannot here discuss the pros and cons of all of the theories of causality that have been proposed. In a nutshell, the question is: Does the causal nexus, whatever its nature may be, hold between concrete things (individuals) only? I do not think so. It seems to me that the most plausible theory of causality holds that causal relations obtain among states of affairs. But be that as it may, my purpose is to call your attention to the fact that the argument from causal interaction rests on a specific and, may I say, not too plausible view about the terms of the causal nexus. This reminder alone may suffice to shake your confidence in the argument.

But even if this specific view about causality were acceptable, a connection has to be established between it and perception. What does causality have to do with perception? Well, it is a piece of common-sense that our perceptions of green apples and multi-colored rainbows is caused by apples and rainbows. This belief is quite true. It may even help to explain, as some philosophers have claimed, the difference between true perception, on the one hand, and perception which, though veridical, is only true by accident, on the other (see, for example, H. P. Grice, “The Causal Theory of Perception”). But it does not show that we cannot perceive abstract things. Consider a simple example: I see that the pencils on my desk are yellow. Since this is not a hallucination, we believe that there are some pencils on my desk and that these pencils cause me to see that the pencils on my desk are yellow. This does not show, as I said, that I cannot see abstract things. Quite to the contrary! It seems to me to be certain that in this situation I see the color yellow. And since I believe that this color is an abstract entity, I conclude that I see in this situation an abstract entity.

This last argument reveals my strategy. I am convinced by it that (2) is false. Therefore, if (3) and (4) imply (2), I must conclude that either (3) is false, or (4) is false, or both are false. I propose to attack the argument by attacking (2) only, because I think it is easy to prove that (2) is false. That (2) is false follows immediately from the following two true propositions:

No philosophical argument could possibly convince me that (a) is false. (b), however, is of a different sort. Arguments against (b) I would not just dismiss as being silly. How could I? What is at stake in regard to (b) is nothing less than the issue of the existence of universals. Fortunately, I need not go into this controversy, for two things are true in any case. Firstly, one cannot very well argue that there are no abstract entities because (2) is true, and then use this conclusion in order to argue that (2) must be true. Secondly, (2) must be false, it seems to me, if colors are abstract entities.

This second truth shows the weakness in our opponent’s position. The question is: Can we or can we not perceive abstract things? Well, can we not perceive colors, and are colors not abstract things? If our opponent answers that we do not perceive colors, nothing more, I think, can be done to advance the dialectic of the two opposing views. Most likely, though, he will deny that colors are abstract things. All right then, what about shapes? We are told that they are not abstract either. And we get the same reply in regard to pitches. Are there then any abstract things at all, we ask, and if the answer is negative, as it most likely is, then the argument shifts from epistemology to ontology. The question is now: Are there abstract entities? It is clear that our opponent’s answer to the epistemological question is based on his answer to the ontological one, and not the other way around. If numbers, in case they existed, were abstract things, and if there are, as a matter of fact, no abstract things, then it is obvious that we cannot perceive them, and the case is closed.

What else can our opponent do but rely on the ontological issue? But if he does, then the power of the color example becomes apparent. Surely, colors are abstract things; they are not located in space and time like the things which are colored. And is it not obvious that we see colors? Who could deny it? Who could think otherwise?

The case for the perception of numbers is as strong as the case for the perception of colors. Let us assume that you see, not that there are yellow pencils on the desk, but that there are two pencils on the desk before you. In this situation, I maintain, you see the number two just as in the earlier situation you saw the color yellow. And just as you may say that the yellow pencils on the desk before you caused you to have that perception, so may you say that the two pencils on the desk before you caused you to see that there are two pencils on the desk. If there had been three pencils on the desk instead, you would have seen that there are three pencils on the desk, just as you would have seen that there are blue pencils on the desk, had there been blue instead of yellow pencils on the desk before you. Under normal conditions, what there is in front of your nose determines, in the simple-minded way we are employing, what you see.

b) The Causal Theory of Perception

Steiner, trying to come to grips with the so-called causal theory of perception, considers the following version of if: (C) One cannot see an F, unless the F participates in an event that causes one to have some perceptual experience (Steiner, p. 118). And he goes on to argue that it is not obvious that (C) is true and, hence, that perception of abstract objects is impossible. I would like to argue along an entirely different line. First of all, it should be noted that Grice’s version of the causal theory, as distilled in (C), is designed to distinguish between accidentally true perception and genuine veridical perception. Secondly, I do not think that the truth of (C) implies that one cannot perceive abstract things. Let us apply (C) to our pencil example: One cannot see a yellow pencil, unless the yellow pencil participates in an event that causes one to have some perceptual experience. At this point, we must recall one of our most basic assumptions—one of our “dogmas,” if you wish—namely, that perception is propositional. This means that one cannot see a yellow pencil unless one sees some such thing as that this is a yellow pencil or that there is a yellow pencil on the desk. From our point of view, therefore, (C) becomes:

I think that this version of (C) is true. But it does not exclude the perception of abstract entities, for example, of colors. For the case of color, we have, corresponding to (C):

Remembering our thesis that perception is propositional, we get:

I think that (D’) is just as true as (C’). Finally, we can adapt (C) to the perception of numbers:

(C), as I pointed out earlier, was formulated by Grice in order to avoid certain problems about perceptions that happen to be accidentally true. Assume that you are in bed, fast asleep, and dreaming that you are in your office. In your dream, you see that there are two yellow pencils on your desk. Assume also that there are two yellow pencils on your desk and that you saw yesterday, while you were sitting in your office, that there are two yellow pencils on your desk. Are you now, in your dream, seeing those yellow pencils on your desk? Phenomenally, what you see seems to be the same state of affairs that you saw yesterday, namely, the state of affairs that there are two yellow pencils on this desk before me now. Yet, you do not really see the pencils on your desk. How could you? You are not even in your office. You are merely dreaming that you are in your office. Grice’s (C) is designed to distinguish between these two different seeings. In the first case, genuine seeing, the pencils on the desk actually cause you to see them; in the dreaming situation, they do not: They are miles away, your eyes are closed, etc. Similarly, in our version of Grice’s explanation, the fact that there are two yellow pencils on the desk plays a causal role in the veridical case but does not play a role in the dreaming situation.

But does Grice’s explanation work, not just for individual things like pencils, but also for abstract things like colors and numbers? One may argue that while you cannot see the pencils on your desk when you are asleep with your eyes closed, you can see the color yellow in your dream (see Steiner, p. 120). After all, while the pencils are miles away, in your office, the color yellow, since it is an abstract thing, is not located anywhere and, therefore, can never be “in your vicinity.” One may therefore argue that (C) breaks down for abstract things: No causal interaction needs to take place; all that is required is that a phenomenal perception of the abstract entity occurs. But it would be a mistake, it seems to me, to conclude that you can literally and veridically see the color yellow with your eyes closed, just because it is not located in one place like individual things are. The case for properties is the same as the case for individual things: You can see neither the pencils on your desk nor their color when you are home in bed, fast asleep. But, of course, you may dream that you see the yellow pencils on your desk.

I have argued that we can see that there are two pencils on the desk just as we can see that there are yellow pencils on the desk. I have also argued that to see that there are two pencils on the desk is to see the number two, just as to see that there are yellow pencils on the desk is to see the color yellow. And finally I argued that the fact that there are two pencils on the desk must play a causal role in one’s seeing that there are two pencils on the desk, just as the fact that there are yellow pencils on the desk must play a causal role, in the genuine case, in one’s seeing that there are yellow pencils on the desk.

Let me put it more generally. Firstly, I hold that to perceive a pencil, a color, or a number is to perceive a certain fact; for example the fact that there are two yellow pencils on my desk. Secondly, I maintain that if one perceives, in the literal sense, a certain fact, then that fact must “play a causal role in one’s perceiving it.” Thirdly, I believe that such a perceived fact may contain, as constituents, both concrete and abstract things. The fact that this is a pencil, for example, consists not only of the individual pencil, but also of the property of being a pencil. The fact that there are two pencils on the desk contains, among other things, the number two.

c) Sets “Concreticized”

Philosophical fashions wash like waves over the philosophical seashore without being resisted. The argument from causal interaction is such a wave. When looked at more carefully, it turns out to be nothing more than the Platonic dogma in disguise. This dogma is mistaken. But dogmas do not yield to arguments. Nor are they well defended. Jubien, for example, takes for granted that one cannot perceive abstract things and concludes that, therefore, they cannot “be given by ostension”: “Now apparently the method of ostension does not offer much hope given a deliberately platonist posture. For part of that position is that mathematical entities are not sensible” (M. Jubien, “Ontology and Mathematical Truth,” p. 135). Jubien’s remark calls to our attention yet another terminological matter.

What Jubien seems to have in mind when he mentions a “platonist posture” is ontological Platonism, that is, the view that there are abstract entities. But this Platonism must be distinguished from the Platonism mentioned in the second of his sentences, namely, the view that abstract entities are not sensible (cannot be perceived). In Plato, of course, these two views go hand in hand. But they need not be combined. My philosophy is a case in point: while I am a Platonist in the ontological sense, I reject what I have called the Platonic dogma, namely, the view that abstract things cannot be perceived. In order to avoid confusion, I shall call myself a realist (in regard to abstract entities), and also an empiricist (in regard to how we know those abstract things).

I wish to comment on another remark of Jubien’s: “Although it seems to me possible that we have mathematical intuition in the sense of a clear, distinct, and perhaps immediate apprehension of certain mathematical verities, such an intuition does not presuppose any intuition of objects of the sort under consideration” (Jubien, p. 136). This view, I wish to point out, differs radically from the one I have outlined and advocated. I cannot be sure what mathematical verities Jubien has in mind, but the ones that come to my mind could not be apprehended without an intuition (perception) of mathematical entities. When I perceive that there are two apples on the table, I perceive (ipso facto) the number two. When I perceive that two apples plus two apples are four apples, I perceive (ipso facto) the numbers two and four. I also perceive, and this perhaps needs to be emphasized, the sum relation between those numbers. In short, these very simple “mathematical truths” can only be apprehended if one apprehends the mathematical things involved.

Among these mathematical things, as we must constantly remind ourselves, our opponents usually list sets, even though it is perfectly obvious that sets are no more “mathematical” than, say, relations are. There are all kinds of relations: There are spatial and temporal relations; there are relations among people; there are relations among color hues; and there are also arithmetic relations, for example, the sum relation. But the fact that there are arithmetic relations does not make the category of relations an arithmetic (“mathematical”) one. Similarly for sets: There are all kinds of sets: There are sets of people, sets of color hues, and sets of numbers. The fact that there are sets of numbers, however, does not make the category of set a “mathematical” category, or sets in general mathematical entities. It is true, though, that sets, sets of people as well as sets of numbers, are commonly taken to be abstract things: They are not located in space and/or time. And this creates a problem for those philosophers who (a) accept some form of the argument from causal interaction, and (b) think of arithmetic as set theory. They must argue that this common view is mistaken, that sets are really concrete things.

Maddy is an example (see P. Maddy, “Perception and Mathematical Intuition”). She starts out by explaining the standard claim that sets, if they are abstract, cannot be “baptized by ostension,” in distinction, for example, from gold. In the case of gold, so the familiar story goes, the Baptist looks at some samples of gold and announces that these, and all things like them, are gold. Maddy then asks why one should not be able to do the same thing with a set of books by announcing that “all the books on this shelf, taken together, regardless of order, form a set,” and that this group and all other groups of things are sets? (Maddy, p. 167). Why not, indeed? As Maddy sees it, the objection to “this picture of set theoretic reference is that, while the gold dubber causally interacts with some samples, the set dubber interacts only with the members of some samples” (Maddy, p. 167). She goes on to argue that a realist may claim in response to this objection that the gold dubber has only interacted with a fleeting aspect of the pieces of gold, and that the same may be true for the set dubber. I am not sure that I understand what she has in mind. But another reply seems to me to be obvious. If it is granted that one can dub a property like gold, why should one be unable to dub a property like set? The property of being gold is just as much an abstract entity as the property of being a set. In order to dub gold, the dubber must interact with some lumps that have the property of being gold. He must perceive that certain lumps before him have this property. And this means that he must perceive the property gold. Similarly, in order to dub the property of being a set, the dubber must causally interact with some things that are sets. She must perceive that some things are sets. And this means that she must perceive the property of being a set. If one can causally interact with the property of being gold, why should one not be able to causally interact with the property of being a set?

There is a difference between the two cases, though, and I suspect that it is this difference which is thought to make a difference. Lumps of gold are concrete things, while sets are abstract things. In other words, while in the first case, the things that have the (abstract) property are concrete, in the second case, they are themselves abstract. Lumps of gold are located in space and time, while sets are not. One can literally point at lumps of gold, but one cannot literally point at examples of sets. I suspect that the view under discussion rests on the assumption that one can dub lumps of gold, but not sets, because the former but not the latter can be perceived by the dubber; and the former but not the latter can be perceived by the dubber, because the former but not the latter can causally interact with the dubber.

The example of the gold dubber suffers from an important ambiguity. It is not clear what it is that is being dubbed in the situation: Is it the lumps of gold or is it the property? In the background lurks the common-name doctrine and its nominalistic motivation. Is ‘gold’ a (common) name of the individual lumps, or is it a (proper) name of the property? If the former, then the dubber can be said to dub the individual lumps; if the latter, then she is dubbing the property by pointing at the lumps which have the property. Since I think that the common-name doctrine is false, I can only assume that the dubber is naming the property rather than the individual pieces of gold. I once heard a joke that makes my case against the common-name doctrine. A drunk, having been thrown out of a bar, lands on all fours on a piece of grass in front of the bar. He notices a grasshopper a few inches in front of his nose. “Do you know, little fellow,” he addresses the grasshopper, “that they have named a cocktail after you?” The grasshopper looks at him very perplexed and says: “Oh, Irving?”

Can we dub properties of abstract things? Well, we can only name such properties if we can perceive that certain things have these properties. And we can only perceive that certain things have these properties if we can perceive these certain things. All depends, therefore, on the old question of whether or not we can perceive abstract things. And this question leads us back to the argument from causal interaction. That we cannot baptize sets (really: the property of being a set) follows from this argument. But the argument is not sound.

If it were sound, to look at it from another angle, then it would also follow that we cannot dub colors (the property of being a color). We can construct a case for colors that exactly parallels the case for sets. Assume that there are a number of differently colored and differently shaped pieces of cardboard on the table before you. The pieces are individual things, perceivable according to the Platonic dogma. But their colors and shapes are abstract things, in the same category with sets. The properties of being a color and being a shape are properties of these abstract things. If the argument from causal interaction were sound, then it would follow that you could not dub the colors (or the shapes). You could not perceive the colors and, hence, could not perceive that they are colors, just as you allegedly cannot perceive the sets and, hence, cannot perceive that they are sets. But surely, we are all familiar with the property of being a color (or with the property of being a shape), because we have seen that certain properties are colors, while others are, for example, shapes. And this shows, as far as I can see, that the argument from causal interaction as used in the dubbing case is not sound.

Maddy, on the other hand, accepts the argument. But she, too, wants to hold that we can perceive (some) sets. Thus she is forced to maintain that some sets are concrete things, located in space and/or time. From our point of view, she turns abstract things into concrete ones, because she is under the spell of the Platonic dogma. Her argument revolves around perceiving three eggs in an egg carton: You see that there are three eggs left in an egg carton. According to Maddy, you “acquire the perceptual belief that there is a set of eggs before [you], that it is three-membered, and that it has various two-membered sub-sets” (Maddy, p. 179). This talk about “acquiring a perceptual belief’ is a piece of jargon of which certain philosophers happen to be fond at the moment. We can safely replace it by straightforward talk about perception (or seeing, in our case). Maddy mentions the belief about the two-membered sub-sets because in her example, you are looking for two eggs for a recipe that calls for two eggs. I do not think that in this situation you see what Maddy thinks you see. When you see that there are three eggs left in the carton, you do not see a set of eggs, or any other set. You see precisely what you do see, namely, that there are three eggs left in the carton. I would argue, as we know by now, that you see, therefore, among other things, the number three and the property of being an egg. But you do not see a set. Not that you cannot see a set. But you typically do not see sets when you look for two eggs in an egg carton.

But let us go to the heart of the matter: Maddy maintains that the set of three eggs which you allegedly see “is located in the egg carton, that is, exactly where the physical aggregate made up of the eggs is located” (Maddy, p. 179). Shades of John Stuart Mill!

At this place, Maddy seems to distinguish between the physical aggregate and the set, but at other times she seems to identify the two. I hope that I do not misinterpret her if I assume from now on that the physical aggregate is not identical with the set. The dialectic, then, looks like this. Maddy holds that there is a spatial aggregate of eggs which is located within the egg carton. There also exists a set of eggs, in addition to the aggregate. This set is a concrete thing, just like the aggregate. It is located precisely where the aggregate is. Just as you can see the aggregate of eggs, so you can see the set. And you see it exactly where the aggregate is. Assume that one of the three eggs is located, not in the carton, but in the door of the refrigerator. In this case, the aggregate is different, the set is the same, and the set is now located differently, but again “where the aggregate is.” If all three eggs are in the carton, then the set is in the carton; and if one of the eggs is in the door, then the set is not in the carton, but somehow spread out from the carton to the door.

I must admit that I do not know how to show that the set of eggs is not in the carton, as Maddy claims. Nevertheless, I cannot help feeling that it is a trick to locate the set wherever the aggregate is. It seems to me obvious that we do not see that the set is in the carton. To answer the question: Where is the set of three eggs? we are instructed to look where the eggs are that are members of the set. Since these eggs are located in space, they stand in certain spatial relations to each other. As a consequence, they form a spatial configuration, the aggregate mentioned earlier. This aggregate consists of the three eggs in their mutual relations. (I shall leave out temporal relations.) This aggregate is also a concrete thing; it, too, is located in space (and time). Thus we see with our eyes, looking into the carton, that the eggs are arranged in a certain way, forming a certain aggregate, and that this aggregate is in the carton. Having located the aggregate with our eyes, we are told that we have located the set of eggs: It is precisely where the aggregate is. According to our conception of sets, the set is not located anywhere. In particular, it is not where the aggregate is. What is true is, rather, that the members of this set form an aggregate, and that this aggregate is located in the carton. The trick works for all sets consisting of individual things, because all of these things form spatio-temporal aggregates.

A similar trick works for properties. The property gold, in my view, is an abstract entity; it is not spatio-temporal. According to another view, this property is universal in that it can belong to many different individual things, but it is also concrete (see D. Armstrong, Nominalism and Realism, Cambridge: Cambridge University Press, 1978). It is located in space (and time). Where is the property? Well, we are told, it is wherever there are lumps of gold. Thus this property is at many different places at once. In this case, too, we locate the property not directly, but by means of the lumps which have the property. If I want to know where the color olive green is, I am instructed to look for things which are olive green, to locate them, and then to conclude that the color is where the colored individuals are. In this case, too, the color is located by means of two facts: the fact that it is exemplified by certain things, and the fact that these things are located in space. In other words, the trick is accomplished because the property stands in a certain relation (exemplification) to things which are truly located in space.

My suspicion that trickery is at work is reinforced by the fact that the trick no longer works when the relation between the individual and the entity in question, the entity to be located, is not a one-one relation. Assume that point B lies between points A and C. Where is the relation of between? Obviously, it is not where any one of the points is. Nor is it located somewhere between points A and B. The trick no longer works, since the relation is “coordinated,” not just to one individual thing, like a property is, but to three concrete things. But we have learned from Maddy’s view how to overcome this obstacle: We must find an entity which (a) is located in space, and (b) is “associated” with the relation. We can then locate the relation wherever that entity is. Nor is it hard to find such an entity. The configuration (aggregate) of the three points will do. The relation between, we could claim, is located precisely where the aggregate is located which consists of the things that stand in this relation. As a result of this piece of legerdemain, we now have the set of the three points as well as the relation of being to the left of located in the same place, namely, where the configuration of the three points is. What convinces me that neither the set nor the relation is really located anywhere is the fact that their alleged location can only be specified in terms of the location of the three points. If they really were concrete things, then one would be able to determine their spatio-temporal positions directly, as we are clearly able to do in regard to the points A, B, and C.

d) Goedel on Mathematical Intuition

Of all the recent views about the nature of mathematical knowledge, Goedel’s view comes closest to the one I am defending. But this proximity is due, not to a close resemblance between our two views, but rather to the distance that separates our two views from the rest of the field. (Maddy’s position is an exception.) However, it is not possible to compare Goedel’s view with ours in detail, for Goedel merely hints at his view. Nevertheless, it may be instructive to have a look at his view and to talk briefly about his commentators.

I am under the impression that two views actually appear in Goedel’s papers. In the paper on Russell’s mathematical logic, for example, we find the following remark:

It seems to me that the assumption of such objects [classes and concepts] is quite as legitimate as the assumption of physical bodies and there is quite as much reason to believe in their existence. They are in the same sense necessary to obtain a satisfactory system of mathematics as physical bodies are necessary for a satisfactory theory of our sense perceptions and in both cases it is impossible to interpret the propositions one wants to assert about these entities as propositions about the “data”, i. e., in the latter case the actually occurring sense perceptions.

(K. Goedel, “Russell’s Mathematical Logic,” p. 220)

This passage, it seems to me, can be interpreted in two different ways. Interpreted in a way that is most congenial to my own view, Goedel claims that we need be no more suspicious of sets and properties than we are of perceptual objects; and that it is just as impossible to interpret statements about sets and properties as being really about other kinds of things, as it is to interpret statements about perceptual objects as being really about our sensations. Needless to say, this interpretation agrees fully with my sentiments. Properties, sets, and numbers, in my view, are just as much part of the furniture of the world as apples and electrons. Nor is it in the least plausible to claim that statements about sets are about anything but sets; statements about numbers, about anything but numbers; and statements about apples, about anything but apples. We must steadfastly reject the fashionable view which tries to snatch realism from the jaws of idealism by “constructing” numbers, sets, and perceptual objects out of other things. The loci classici of this view are Russell’s Our Knowledge of the External World and Carnap’s Der logische Aufbauder Welt.

The second interpretation, perhaps the more natural one, takes Goedel to be saying that just as perceptual objects must be postulated in order to have a satisfactory theory of perception, so mathematical entities have to be postulated in order to arrive at a satisfactory set theory (and arithmetic). What is implied by Goedel’s view is that we are acquainted neither with perceptual objects nor with mathematical things, but that we have to postulate their existence in order to get adequate theories. Realism in regard to perceptual objects and in regard to mathematical entities is thus secured by postulation. This kind of realism is particularly popular among philosophers of science. The Gordian knot of the idealism-realism controversy, which has so vexed modern philosophy since Descartes, is cut with the sword of postulation. Whatever cannot be defended against the persuasive arguments of the idealist or skeptic is simply postulated. At times, one even finds astonishment that we bother to worry about such controversies, when a simple act of postulation will do the trick. Who does not remember the claim that such disputes are nothing but “Scheinprobleme”?

But we are not intimidated by our scientifically-minded colleagues. What is a “satisfactory” theory, we demand to know. One obvious answer will not do. It may be replied that, quite obviously, a satisfactory arithmetic, for example, must, among other things, deal with the arithmetic entities there are. In other words, it must be about numbers. And a satisfactory theory of perception must be about the perceptual things there are, that is, about perceptual objects. With this answer in mind, the view under discussion reduces to the truism that a theory of number must be about numbers, a theory of sets must be about sets, and a theory of perception must be about perceptual objects. Of course they must. But it is now assumed that we already know that there are numbers, sets, and perceptual objects, so that there may be theories about them, and the question is, precisely how we know this.

But talk about a “satisfactory” theory may take on a different meaning. What comes to mind is Russell’s delightful example, which we discussed earlier, of the cat that walks from one corner of his room to the other. First, he sees it in one corner, then in the other, but he does not see it cross the room. Russell asks why we assume that there exists a cat that crosses the room, even if we do not see it, rather than assume that there are only the sense-impressions which Russell experiences when he sees the cat, first in one corner, then in the other. Why do we believe, in other words, that there are perceptual objects in addition to our sense-impressions? Russell’s answer is: “Since this belief does not lead to any difficulties, but on the contrary tends to simplify and systematize our account of our experiences, there seems no good reason for rejecting it” (B. Russell, The Problems of Philosophy, p. 24). What Russell claims here is that a theory of our experiences is simpler and more systematic if we assume that there exist perceptual objects in addition to sense-impressions, and that this fact is our justification for believing in perceptual objects. But this is surely mistaken. We believe that there are apples and tigers, not because these beliefs fit a theory, but simply because we see apples and tigers. We believe that there are perceptual objects because we perceive them. As for the claim that this belief somehow simplifies and systematizes, the dialectic inevitably dissolves into pragmatic ambiguity, and we do not have the space here to pursue even one of its many sides. But it is clear that the only good reason for postulating perceptual objects or numbers, unless one is acquainted with them, is that one believes on other grounds that such things exist. According to our view, however, we are acquainted with perceptual objects and numbers, and, therefore, there is no need to postulate them.

Perhaps the analogy is supposed to be drawn, not between numbers and perceptual objects, but between numbers and physical objects; not between numbers and apples, but between numbers and electrons. Compare, for example, the following quite typical comment by George Berry:

How then do we find out about this realm of extra-mental nonparticular, unobservable entities? Our knowledge of them, like our knowledge of the extra-mental, unobservable objects of the physical sciences, is indirect, being tied to perceived things by a fragile web of theory. In both cases—physics and logic—our hypotheses about the unperceived are tested by their success in accounting for the character of the perceived. Misreading this similarity, one might easily conclude that a faculty of non-sensory perception, call it ‘intuition’, is necessary to play a part in logic parallel to the role of sensation in physics.

(G. Berry, “Logic with Platonism,” p. 261)

I think that Berry’s diagnosis is false. Those who appeal to some kind of “intuition” of logical or mathematical things, are not guided by the parallel which he describes. Rather, they want to draw a parallel between the perception of perceptual objects and the intuition of logical and mathematical things. It is at any rate true that some philosophers argue that numbers are like electrons: Neither numbers nor electrons can be perceived, yet we know that they exist, and we know this by inference. I admit that this is roughly true for electrons. They cannot be perceived, but their effects can, and from these effects, we can infer their existence by means of certain laws. But this case of inference is quite different from Russell’s case of postulation. In the case of electrons, we maintain that they exist, period; not merely that the assumption of their existence simplifies or systematizes. We know that electrons exist in the same way in which Robinson Crusoe knew that Friday existed, because he saw his footprints in the sand. There is a fundamental difference between the way in which we know that there are perceptual objects and the way in which we know that there are physical objects. We know perceptual objects, not by inference from our sensations, as I have time and again emphasized, but because we perceive them. On the other hand, we know electrons, not because we perceive them, but by inference from what we perceive. Some philosophers, taking as their model our knowledge of electrons, mistakenly believe that our knowledge of apples is of the same sort. The question before us is this: Is our knowledge of numbers (and sets) like our knowledge of apples or like our knowledge of electrons? I am arguing that it is like our knowledge of apples: We perceive some numbers (and sets) just as we perceive apples.

Goedel, I said at the beginning, seems to hold, at different places, two different views. We have briefly discussed one of these two views, the one expressed in his article on Russell. We shall now turn to the other view. It is contained in the classic paper on the continuum hypothesis:

But, despite their remoteness from sense experience, we do have something like a perception also of the objects of set theory, as is seen from the fact that the axioms force themselves upon us as being true. I don’t see any reason why we should have less confidence in this kind of perception, i. e., in mathematical intuition, than in sense perception. . . .

It should be noted that mathematical intuition need not be conceived of as a faculty giving an immediate knowledge of the objects concerned. Rather it seems that, as in the case of physical experience, we form our ideas also of those objects on the basis of something else which is immediately given. Only this something else here is not, or not primarily, the sensations. That something besides the sensations actually is immediately given follows (independently of mathematics) from the fact that even our ideas referring to physical objects contain constituents qualitatively different from sensations or mere combinations of sensations, e. g., the idea of object itself, . . . Evidently the “given” underlying mathematics is closely related to the abstract elements contained in our empirical ideas. It by no means follows, however, that the data of this second kind, because they cannot be associated with actions of certain things upon our sense organs, are something purely subjective, as Kant asserted. Rather they, too, may represent an aspect of objective reality, but, as opposed to the sensation, their presence in us may be due to another kind of relationship between ourselves and reality.

(K. Goedel, “What is Cantor’s Continuum Problem?”, pp. 271-72)

I think that this long excerpt contains a number of philosophical themes. It rings a series of philosophical bells. But not all of the tones harmonize with each other. The first paragraph starts with what seems to be the straightforward assertion that, in addition to ordinary perception, there is another kind of perception, “mathematical intuition,” of mathematical things. Goedel seems to be saying, in Husserl’s terms, that in addition to ordinary intuition, there is eidetic intuition, and that the latter is just as direct, just as immediate, as the former. The latter is a “vision,” too, but a vision of what is abstract rather than concrete. Goedel defends the existence of this kind of intuition by pointing out that the axioms of set theory force themselves upon us as being true. The same, of course, could be said about the axioms of arithmetic. I have argued, in opposition to Goedel’s (and Husserl’s) view, that there is no second kind of intuition. There is no special kind of mental act which acquaints us with abstract or mathematical things. Perception alone, since it is propositional, acquaints us with both individual things as well as abstract entities.

As to Goedel’s remark about the axioms, I believe that it applies to our view as well as to his. I have repeatedly claimed that just as we know that midnight blue is darker than lemon yellow by means of perception, so that this truth “forces itself upon us,” so we know by perception that two things plus two things are four things, or that the set of fruit before us is the union set of a set of apples and a set of bananas, so that these truths, too, “force themselves upon us.” Things are not quite that simple, when we turn to axioms, that is, to general facts. I can learn by means of perception that the sum of 3 and 2 is the same as the sum of 2 and 3, but I cannot learn by means of perception that this holds for all numbers. Similarly, I can learn by means of perception that there exists the union set consisting of the set of apples and the set of bananas, but I cannot learn in the same way that there exists a union set for any two sets. The truth of these axioms forces itself upon us in the same way in which all inductive truth forces itself upon us, namely, through the persuasion of its instances. Paraphrasing Goedel, we could say: Only because there is a perception (mathematical intuition) of the instances of these laws, are we forced, to whatever degree we are forced, to accept the laws.

Our understanding of Goedel’s remark differs sharply from Chihara’s (see C. Chihara, Ontology and the Vicious-Circle Principle, pp. 78-79). Chihara draws a parallel between the way in which, according to Goedel, the axioms of set theory force themselves upon us and the way in which conceptual truths about God, conceived of as the perfect being, may force themselves upon us, even if we are atheists. We do not see this parallel. In addition to the fact that hypothetical statements about God (God is a necessary being) are not laws, there is the difference that perception is lacking in the case of God but not in the case of set theory. An atheist may agree that if there is God, he is omniscient, but he would insist, in the same breath, that there is no perceptual evidence for God’s existence in the first place. For sets, however, Goedel could and would insist that he is acquainted with sets and certain relations among them. The axioms of set theory force themselves upon him, not because of his concept of a set, divorced from all acquaintance, but precisely because he is acquainted with sets.

Returning to the rest of our quotation from Goedel, he considers next the possibility that mathematical intuition is after all not immediate, but mediated like the perception of perceptual objects. I take it that he adopts here, perhaps only tentatively, the position which we criticized a few paragraphs earlier, namely, the view that just as our knowledge of perceptual objects is a matter of inference from sensations, so our knowledge of mathematical things may be a matter of inference. If this interpretation is correct, then what I said earlier applies to Goedel’s possibility.

But Goedel adds a Kantian theme to his considerations. He seems to be arguing that just as we are aware of perceptual objects by being directly acquainted with sensations, so we are indirectly acquainted with sets by being directly acquainted with something else. This something else, though, does not consist in sensations or, at least, does not primarily consist in sensations. And then he states that even in perception, something in addition to sensation must play a role. This is the Kantian twist which I just mentioned. As an example of the “something else” in perception, Goedel mentions the idea of object. Without this idea, he may have reasoned, we could not possibly fashion out of the buzzing, booming confusion of our sensations the notion of a a perceptual object. What is given to the mind, in addition to raw sensations, are certain abstract elements, such as the property of being an object, and mathematical intuition is somehow based on such abstract elements. Goedel avoids the idealistic consequences of this view by pointing out that those abstract elements may not be purely subjective, as Kant concluded, but could be representing aspects of reality. (There occurs at this point also a hint of the infamous argument from causal interaction.) All of this is rather vague, so that we may feel free to interpret Goedel’s statements in the spirit of our point of view. If we do, then we must purge them of talk about sensations in favor of talk about perception. Perception, we have repeatedly emphasized, acquaints us with abstract things, not just with individuals. The property of being an individual (perceptual) object, for example, is presented to us in perception, in addition to the individual things themselves. In short, what according to Goedel occurs on the level of sensation, occurs, according to our view, on the level of perception.

In the last sentence quoted from Goedel, he surmises that there may be a relationship other than causal interaction between us and reality. We know what this relationship is: It is the intentional nexus between a mental act and the state of affairs which is its object. When you see that there are two apples on the table in front of you, your mental act of seeing stands in a unique, noncausal, relationship to a certain fact. And this relationship obtains, in the case of perception, always between a mind (mental act), on the one hand, and an abstract thing, namely, a state of affairs, on the other. It is far from true that only concrete things can be perceived; it is rather the case that concrete things can only be perceived through the medium of abstract entities. Only as constituents of states of affairs do concrete things appear before the mind in perception.

This consideration may shed some light on the popularity among some philosophers of the causal theory of perception. The causal theory of perception is merely a part of a causal theory of knowledge. And the causal theory of knowledge is developed in opposition to a theory based on intentionality. The causal theory replaces the intentional nexus between mind and world with a causal connection. I suppose that there are many reasons for this rejection of intentionality. There exists an unfounded suspicion of the mental, an explicit or implicit materialism. Physics or physiology is to be preferred to “folk psychology”; philosophers without minds, to philosophers with minds. There is also a misplaced loyalty to science. Intentionality is unscientific, causality is not. Intentionality smacks of magic, causality does not. Finally, there is an unjustified aversion to abstract entities. Causality, one may mistakenly think, involves nothing but individual things in space and/or time. Allowing myself to stray from the straight and narrow path of detailed discussion, I venture the following sweeping diagnosis: Most recent views about the nature of mathematical knowledge are based on nominalism and physicalism, the twin ailments of analytic philosophy.