Gottlob Frege (review date 1894)
SOURCE: A review of Dr. E. Husserl's Philosophy of Arithmetic, in Husserl: Expositions and Appraisals, University of Notre Dame Press, 1977, pp. 314-24.
[The following excerpt is a translation (by E. W. Kluge) of Frege's 1894 critical review of Husserl's Philosophy of Arithmetic, which played a significant role in causing Husserl to refocus the direction of his thought.]
The author decides in the Introduction [of Philosophy of Arithmetic] that for the time being he will consider (only) cardinal numbers (cardinalia), and thereupon launches into a discussion of multiplicity, plurality, totality, aggregate, collection, set. He uses these words as if they were essentially synonymous; the concept of a cardinal number1 is supposed to be different from this. However, the logical relationship between multiplicity and number (p. 9) remains somewhat obscure. If one were to go by the words “The concept of number includes the same concrete phenomena as the concept of multiplicity, albeit only by way of the extensions of the concepts of its species, the numbers two, three, four, etc.,” one might infer that they had the same extension. On the other hand, multiplicity is supposed to be more indeterminate and more general than number. The matter would probably be clearer if a sharper distinction were drawn between falling under a concept and subordination. Now the first thing he attempts to do is to give an analysis of the concept of multiplicity. Determinate numbers, as well as the generic concept of number which presupposes them, are then supposed to emerge from it by means of determinations. Thus we are first led down from the general to the particular, and then up again.
Totalities are wholes whose parts are collectively connected. We must be conscious of these parts as noticed in and by themselves. The collective connection consists neither in the contents' being simultaneously in the awareness, nor in their arising in the awareness one after another. Not even space, as all-inclusive form, is the ground of the unification. The connection consists (p. 43) in the unifying act itself. “But neither is it the case that over and above the act there exists a relational content which is distinct from it and is its creative result.” Collective connection is a relation sui generis. Following J. St. Mill, the author then explains what is to be understood by “relation”: namely that state of consciousness or that phenomenon (these expressions are supposed to coincide in the extension of their reference) in which the related contents—the bases of the relation—are contained (p. 70). He then distinguishes between primary and mental relations. Here only the latter concern us more closely. “If a unitary mental act is directed towards several contents, then with respect to it the contents are connected or related to one another. If we perform such an act, it would of course be futile for us to look for a relation or connection in the presentational content which it contains (unless over and above this, there is also a primary relation). The contents here are united only by the act, and consequently this unification can be noticed only by a special reflection on it” (p. 73). The difference-relation, whereby two contents are related to one another by means of an evident negative judgment, is also of this kind (p. 74). Sameness, on the other hand, is (p. 77) a primary relation. (According to this, complete coincidence, too, would be a primary relation, while its negation—difference itself—would be a mental one. I here miss a statement of the difference between the difference-relation and collective connection, where in the opinion of the author the latter, too, is a mental relation because perceptually no unification is noticeable in its presentational content.) When one is speaking of “unrelated” contents, the contents are merely thought “together”, i.e. as a totality. “But by no means are they really unconnected, unrelated. On the contrary, they are connected by the mental act holding them together. It is only in the content of the latter that all noticeable unification is lacking” (p. 78). The conjunction ‘and’ fixes in a wholly appropriate manner the circumstance that given contents are connected in a collective manner (p. 81). “A presentation … falls under the concept of multiplicity insofar as it connects in a collective manner any contents which are noticed in and by themselves” (p. 82). (It appears that what is understood by “presentation” is an act.) “Multiplicity in general … is no more than something and something and something, etc.; or any one thing and any one thing and any one thing, etc.; or more briefly, one and one and one, etc.” (p. 85). When we remove the indeterminateness which lies in the “etc.,” we arrive at the numbers one and one; one, one and one; one, one, one and one; and so on. We can also arrive at these concepts directly, beginning with any concrete multiplicity whatever; for each one of them falls under one of these concepts, and under a determinate one at that (p. 87). To this end, we abstract from the particular constitution of the individual contents collected together in the multiplicity, retaining each one only insofar as it is a something or a one; and thus, with respect to the collective connection of the latter, we obtain the general form of multiplicity appropriate to the multiplicity under consideration, i.e. the appropriate number (p. 88). Along with this number-abstraction goes a complete removal of restrictions placed on the content (p. 100). We cannot explain the general concept of number otherwise than by pointing to the similarity which all number-concepts have to one another (p. 88).
Having thus given a brief presentation of the basic thoughts of the first part, I now want to give a general characterization of this mode of consideration. We here have an attempt to provide a naive conception of number with a scientific justification. I call any opinion naive if according to it a number-statement is not an assertion about a concept or the extension of a concept; for upon the slightest reflection about number, one is led with a certain necessity to such conceptions. Now strictly speaking, an opinion is naive only as long as the difficulties facing it are unknown—which does not quite apply in the case of our author. The most naive opinion is that according to which a number is something like a heap, a swarm in which the things are contained lock, stock and barrel. Next comes the conception of a number as a property of a heap, aggregate, or whatever else one might call it. Thereby one feels the need for cleansing the objects of their particularities. The present attempt belongs to those which undertake this cleansing in the psychological wash-tub. This offers the advantage that in it, things acquire a most peculiar suppleness, no longer have as hard a spatial impact on each other and lose many bothersome particularities and differences. The mixture of psychology and logic that is now so popular provides good suds for this purpose. First of all, everything becomes presentation. The references of words are presentations. In the case of the word “number,” for example, the aim is to exhibit the appropriate presentation and to describe its genesis and composition. Objects are presentations. Thus J. St. Mill, with the approval of the author, lets objects (whether physical or mental) enter into a state of consciousness and become constituents of this state (p. 70). But might not the moon, for example, be somewhat hard to digest for a state of consciousness? Since everything is now presentation, we can easily change the objects by now paying attention, now not. The latter is especially effective. We pay less attention to a property and it disappears. By thus letting one characteristic after another disappear, we obtain concepts that are increasingly more abstract. Therefore concepts, too, are presentations; only, they are less complete than objects; they still have those properties of objects which we have not abstracted. Inattention is an exceedingly effective logical power; whence, presumably, the absentmindedness of scholars. For example, let us suppose that in front of us there are sitting side by side a black and a white cat. We disregard their color: they become colorless but are still sitting side by side. We disregard their posture: they are no longer sitting, without, however, having assumed a different posture; but each one is still at its place. We disregard their location: they are without location, but still remain quite distinct. Thus from each one we have perhaps derived a general concept of a cat. Continued application of this process turns each object into a less and less substantial wraith. From each object we finally derive something which is completely without restrictions on its content; but the something derived from the one object nevertheless does differ from that derived from the other object, although it is not easy to say how. But wait! This last transition to a something does seem to be more difficult after all; at least the author talks (p. 86) about reflection on the mental act of presentation. But be that as it may, the result, at any rate, is the one just indicated. While in my opinion the bringing of an object under a concept is merely the recognition of a relation which previously already obtained, in the present case objects are essentially changed by this process, so that objects brought under the same concept become similar to one another. Perhaps the matter is to be understood thus, that for every object there arises a new presentation in which all determinations which do not occur in the concept are lacking. Hereby the difference between presentation and concept, between presenting and thinking, is blurred. Everything is shunted off into the subjective. But it is precisely because the boundary between the subjective and the objective is blurred, that conversely the subjective also acquires the appearance of the objective. For example, one talks of this or that presentation as if, separated from the presentor, it would let itself be observed in public. And yet, no-one has someone else's presentation but only his own, and no-one knows how far his presentation—e.g. that of red—agrees with that of someone else; for the peculiarity of the presentation which I associate with the word “red,” I cannot state (so as to be able to compare it). One would have to have the presentations of the one as well as that of the other combined in one and the same consciousness; and one would have to be sure that they had not changed in the transfer. With thoughts, it is quite different: one and the same thought can be grasped by many people. The components of a thought, and even more so the things themselves, must be distinguished from the presentations which in the soul accompany the grasping of a thought and which someone has about these things. In combining under the word “presentation” both what is subjective and what is objective, one blurs the boundary between the two in such a way that now a presentation in the proper sense of the word is treated like something objective, and now something objective is treated like a presentation. Thus in the case of our author, totality (set, multiplicity) appears now as a presentation (pp. 15, 17, 24, 82), now as something objective (pp. 10, 11, 235). But isn't it really a very harmless pleasantry to call, for example, the moon a presentation? It is—as long as one does not imagine that one can change it as one likes, or produce it by psychological means. But this is all too easily the result.
Given the psychologico-logical mode of thought just characterized, it is easy to understand how the author judges about definitions. An example from elementary geometry may illustrate this. There, one usually gives this definition: “A right angle is an angle which is equal to its adjacent angle.” The author would probably say to this, “The presentation of right-angledness is a simple one; hence it is a completely misguided undertaking to want to give a definition of it. In our presentation of right-angledness, there is nothing of the relation to another adjacent angle. True enough; the concepts ‘right angle’ and ‘angle which is equal to its adjacent angle’ have the same extension; but it is not true that they have the same content. Instead of the content, it is the extension of the concept that has been defined. If the definition were correct, then every assertion of right-angledness, instead of applying to the concretely present pair of lines as such, would always apply only to its relation to another pair of lines. All I can admit is (p. 114) that in this equality with the adjacent angle we have a necessary and sufficient condition for right-angledness.” The author judges in a similar way about the definition of equinumerosity by means of the concept of a univocal one-one correlation. “The simplest criterion for sameness of number is just that the same number results when counting the sets to be compared” (p. 115). Of course! The simplest way of testing whether or not something is a right angle is to use a protractor. The author forgets that this counting itself rests on a univocal one-one correlation, namely that between the numerals 1 to n and the objects of the set. Each of the two sets is to be counted. In this way, the situation is made more difficult than when we consider a relation which correlates the objects of the two sets with one another without numerals as intermediaries.
If words and combinations of words refer to presentations, then for any two of these only two cases are possible: either they designate the same presentation, or they designate different ones. In the first case, equating them by means of a definition is useless, “an obvious circle”; in the other, it is false. These are also the objections one of which the author raises regularly. Neither can a definition dissect the sense, for the dissected sense simply is not the original one. In the case of the word to be explained, either I already think clearly everything which I think in the case of the definiens—in which case we have the “obvious circle”—or the definiens has a more completely articulated sense—in which case I do not think the same thing in its case as I do in the case of the one to be explained: the definition is false. One would think that the definition would be unobjectionable at least in the case where the word to be explained does not yet have a sense, or where it is expressly asked that the sense be considered non-existent, so that the word acquires a sense only through this definition. But even in the latter case (p. 107), the author confutes the definition by reminding us of the distinctness of the presentations. Accordingly, in order to avoid all objections, one would probably have to create a new root-word and form a word out of it. A split here manifests itself between psychological logicians and mathematicians. The former are concerned with the sense of the words and with the presentations, which they do not distinguish from the sense; the latter, however, are concerned with the matter itself, with the reference of the words.2 The reproach that it is not the concept but its extension which is being defined, really applies to all the definitions of mathematics. So far as the mathematician is concerned, the definition of a conic section as the line of intersection of a plane with a cone is no more and no less correct than that as a plane whose equation is given in Cartesian coordinates of the second degree. Which of these two—or even of other—definitions is selected depends entirely on the pragmatics of the situation, although these expressions neither have the same sense nor evoke the same presentations. By this I do not mean that a concept and the extension of a concept are one and the same; rather, coincidence of extension is a necessary and sufficient condition for the fact that between the concepts there obtains that relation which corresponds to that of sameness in the case of objects.3 I here note that when I use the word “same” without further addition, I am using it in the sense of “not different,” “coinciding,” “identical.” Psychological logicians lack all understanding of sameness, just as they lack all understanding of definitions. This relation cannot help but remain completely puzzling to them; for if words always designated presentations, one could never say “A is the same as B.” For to be able to do that, one would already have to distinguish A from B, and then these would simply be different presentations. All the same, I do agree with the author in this, that Leibniz' explanation “Eadem sunt quorum unum potest substitui alteri salva veritate” does not deserve to be called a definition, although I hold this for different reasons. Since every definition is an equation, one cannot define equality itself. One could call Leibniz' explanation a principle which expresses the nature of the sameness-relation; and as such it is of fundamental importance. I am unable to acquire a taste for the author's explanation that (p. 108) “We simply say of any contents whatever that they are the same as one another, if there obtains sameness in the … characteristics which at that moment constitute the center of interest.”
Let us now go into details! According to the author, a number-statement refers to the totality (the set, multiplicity) of objects counted (p. 185). Such a totality finds its wholly appropriate expression in the conjunction “and.” Accordingly, one should expect that all number-statements have the form “A and B and C and … Q is n,” or at least that they could be brought into such a form. But what is it that we get exactly to know through the proposition “Berlin and Dresden and Munich are three” or—and this is supposed to be the same thing—through “Berlin and Dresden and Munich are something and something and something”? Who would want to go to the trouble of asking, merely to receive such an answer? It is not even supposed to be said by this that Berlin is distinct from Dresden, the latter from Munich,...
(The entire section is 7460 words.)