Pivcevic- Husserl Versus Frege

Mind Association

Husserl versus FregeAuthor(s): E. PivcevicSource: Mind, New Series, Vol. 76, No. 302 (Apr., 1967), pp. 155-165Published by: Oxford University Press on behalf of the Mind AssociationStable URL: http://www.jstor.org/stable/2251767 .Accessed: 25/09/2013 01:14

Your use of the JSTOR archive indicates your acceptance of the Terms & Conditions of Use, available at .http://www.jstor.org/page/info/about/policies/terms.jsp

.

JSTOR is a not-for-profit service that helps scholars, researchers, and students discover, use, and build upon a wide range ofcontent in a trusted digital archive. We use information technology and tools to increase productivity and facilitate new formsof scholarship. For more information about JSTOR, please contact [email protected].

.

Oxford University Press and Mind Association are collaborating with JSTOR to digitize, preserve and extendaccess to Mind.

http://www.jstor.org

This content downloaded from 200.14.85.85 on Wed, 25 Sep 2013 01:14:04 AMAll use subject to JSTOR Terms and Conditions

VOL. LXXVI. No. 302] [April, 1967

M IND A QUARTERLY REVIEW

OF

PSYCHOLOGY AND PHILOSOPHY

I.-HUSSERL versus FREGE

BY E. PIVCEVIC

[IN his Phtlosophy of Arithmetic (Halle-Saale, 1891) Husserl made many critical references to Frege's theory of numbers. The extract translated below is from Chapter VII " Definitions of numbers based on the concept of equivalence " in which his objections are presented in a more or less systematic form. Later he withdrew his criticism of Frege's antipsychologism, but his basic critical attitude towards all theories based on the concept of equivalence did not change. He was in fact never prepared to accept Frege's viewpoint and, except for the attitude towards psychologism, their positions remained profoundly divided. There is also one other point that might be mentioned here. Husserl knew Frege's Begriffsschrift and, as his criticism shows, he concerned himself pretty closely with Frege's Foundation of Arithmetic. The latter book was also not unknown to other mathematicians at the time. It sounds therefore a little odd when Russell maintains that Frege's book " attracted almost no attention " and that " the definition of number which it contains remained practically unknown" until it was rediscovered by him in 1901 (Russell, Introduction to Mathematical Philosophy, Chapter II). The truth is that Frege's logicism did not appeal to mathematicians, and, on the Continent at least, it never had any influence comparable to that of formalism or intuitionism. The situation is hardly any different now, except that here too among mathematicians interest in logicism has waned.

In this translation I have left out irrelevant footnotes: only one, dealing directly with Frege's definition of number, has been translated.]

The Structure of Equivalence Theory In the preceding chapter, we have not without good reason

given much attention to clarifying the misunderstandings which ?w Basil Blackwell, 1967

6 155


156 E. PIVCEVIC: regularly crop up in connection with the definition of equinumerosity (Gleichzahligkeit) by means of one-one correspondence. These misunderstandings have indeed had unfortunate consequences, for they have led to a total mnisunderstanding of the concept of Number. It might perhaps be useful to consider the following train of thought which would enable us to present in as systematic a form as possible various views scattered here and there, disregarding for the time being the doctrines actually put forward.

The definitions of Equal (Gleichviel), More and Less, in the form in which they are here used as a basis, are independent of the concept of Number; what is required by them is only that one should take the sets to be compared, correlate their respective elements in one-to-one fashion and see if there are any elements left over. Therefore, without counting the sets and without even having to know what counting means, one is nevertheless in the position to reach a clear decision as to whether they are equinumerous (gleichzahlig) or not; one has only got to be care- ful not to attach to the word equinumerosity other meaning than that given to it by the definition. Therefore let us rather use the word equivalent instead of equinumerous, since the latter contains the concept of Number in its connotation, whereas the definition itself is not dependent upon this concept. If we now take an arbitrary concrete set S as a starting point, we can put all other, actual or imaginable sets, into correspondence with the set S and in this way separate all sets equivalent to S. It is in this sense that we shall speak of a class of sets K belonging to the set S. We can now add an arbitrary new element to S and form the respective class of equivalent sets; by adding yet another new element we can form a new class, and so on. The process, as one can see, continues in infinitum, for it is not possible to imagine a set to which we could not add new elements. We proceed in the same way in the opposite direction: we remove an element from S, no matter which, and form the respective class, then we remove yet another element and so on, till all elements of S have been exhausted. The classification of all thinkable sets achieved in this way, is the most rigorous that one can imagine. A set can never belong to two different classes at the same time. On the basis of the definition of equivalence given beforehand, any given set is assigned to a definite class, and to this class only. On the other hand, every class is completely determined by any of the sets belonging to it; each of its sets can therefore be used with equal justification as a foundation set for the construction of the class, and, consequently, regarded


HUSSERL versus FREGE 157 as its representative. It is also clear that the whole class could emerge from a single set as a result of all possible qualitative changes (and therefore not division) of the elements of the latter.

So far the collection of classes has remained an unordered collection. But we can easily find a principle of ordering; it is the same principle as the one which we have followed in the successive construction of classes. We take an arbitrary class K as a starting point. Let the set S be its representative. Suppose now that an element of S, no matter which, is removed from the set. The resulting set S' will represent the new class K' which we shall call the class immediately preceding the class K. It is easy to show that the class K' remains unchanged no matter which of the elements of S we choose to remove from the set, so that K' is univocally determined. If now we form a set S" by adding an arbitrary object to S, then the respective class K" will be called the class immediately following the class K. This class too is completely determined.

These definitions are clearly sufficient to enable us to order all classes in an unequivocal way in a single series in which each one of them has an entirely determined place.

The following simple consideration leads now to the concepts of numbers. Each particular class embraces all thinkable sets with the same cardinal number (Anzahl) ; to different classes there correspond different numbers. Our justification for attributing the same number to all sets belonging to a certain class can only be the existence of a property which all these sets have in common. And the property which they all share, and which distinguishes them from all other thinkable sets, consists in nothing else but in that they all belong to the same class, i.e. that they are mutually equivalent. In order to be able to express this property for a given set S we need a uniform symbolism which will mirror the classes in their natural relationships and their order of succession. A class can be unequivocally represented by any of its sets. Although it is inmaterial which one we choose, it is necessary to select one, in order to get a uniform system of symbolic repre- sentation for scientific use. We take the following sets which are formed by the repetition of the stroke 1, viz. by the repetition of the vocal complex one: 11, 111, 1111, . . ., or (in order to avoid confusion with certain complex numerical symbols in the Decadic system) 1+1, 1++ 1 +1, 1 + 1 + 1 + 1 . . ., as representatives of classes, and call then 2, 3, 4, . . . respectively. The sets thus formed out of strokes are the natural numbers, since they, by being the representatives of classes, represent at the same time the cardinal numbers as well.


158 E. PIVOEVIC:

The process of counting of a given set consists in looking for the natural number equivalent to it, whereby the given set is included in the respective class. We find the number corresponding to our set after a certain number of steps, by " register- ing" ("C mapping ", abbilden) each element of the set by means of a stroke; this results in a set of strokes equivalent to our set and this is its natural number. The numbers form an ordered series in accordance with the corresponding series of classes.

This might suffice as a characterization of a peculiar attempt to derive the concept of number from the concept of equinumerosity and to penetrate to the essence of the basic arithmetical concepts while at the same time avoiding all psychological analysis which, after all, might present some awkward problems.

Examples

In order to show that this theory is not a product of pure phantasy, I shall now, as an illustration, quote a passage from a recently published mathematical work to which I have referred earlier: Allgemeine Arithmetik by Stolz. After laying down the definitions of a collection (plurality, Vielheit) and the relations equal, greater and smaller, which are already known to us, Stolz gives the following explanation of the concept of Number, or, as he says, " natural number ".

"The common characteristic of all collections (Vielheiten, pluralities) which are equal to a given collection is expressed by a number-word. We compare collections with the sets of signs which are formed by the repetition of the stroke 1 (a one-sign, eine Emns, ein Einer): 11, 111, ... (These signs will not be introduced as the signs of the numbers eleven, one hundred and eleven until later). Anything that is capable of being repeatedly posited is called a denominate unit (benannte Einheit), a 1, the unit simpliciter. The natural number is a collection of units, i.e. of ones. Every other collection is called a denominate number (benannte Zahl). What we mean is that to every such collection there corresponds a natural number equal to it, and this number can be found so that we pick the units of the given collection, one by one, register them by using the stroke 1 and then set them aside. To the mutually equal collections there will correspond equal numbers; if a collection is larger, the corresponding number will be greater ".

" We can speak of equal natural numbers only in the sense that we can imagine such a number, like any concept, being posited as often as one likes."


HUSSERL versus FREGE 159

At first it might indeed seem that Stolz defines number as a mere set of strokes. But the first sentence says that the common characteristic of all collections which are equal to (or, as we would say, equivalent with) a certain collection is expressed by a number word. However, since there is not even the smallest mention of such a characteristic either before or after this, we must assume that the adjunct " which are equal to (equivalent with) a given collection " expresses this characteristic; which proves that his views coincide with the theory which we have expounded earlier.

Criticism We turn now to criticism. The fallacies of this extreme

relativistic' theory are very closely related to a misunderstanding of the essence of one-one correspondence and the role this relation plays in conveying the equality of two sets. The definition of equivalence, as we have established earlier, represents no more than a mere criterion for the existence of equality between two sets as to their cardinal number (Gleichheit der Anzahl), whereas here it is regarded as a nominal definition. But it is not true that the concepts " equivalence " and " equality as to cardinal number " (gleiche Anzahl) both have the same content; what is true, is only that their extensions are the same. If we identify equivalence with equality as to cardinal number, then it is of course natural to regard the equivalence as the source of the concept of Number itself, and to conclude that all equinumerous sets (i.e. equivalent sets, all belonging to one ' class') can after all have nothing else in common except the equinumerosity defined in the described way; consequently membership in a class would be the most important factor to be taken into account when considering the concept of a particular number. And attributing a number to a given set would mean nothing else but classifying it in the just given sense.

We cannot, of course, accept this line of reasoning. What the equivalent sets have in common is not merely the ' equinumerosity', or, more clearly, the equivalence, but the same Number in the true and proper sense of the word.

We say: Number in the true and proper sense of the word, because it can be easily shown that what we call numbers, in accordance with general usage in everyday life and in science, has nothing in common with what, according to this theory,

1 Husserl used the word " extrem-relativistische "; what he had in mind could perhaps be less ambiguously reproduced by " extreme relation- alistic ".


160 E. PIVCEVIC:

should be thus called. If numbers are defined as relational concepts based on equivalence, then every statement of number [concerning some given set] would only refer to the relations existing between this and other sets instead of aiming at the set as such. Attributing a number to a given set would mean classifying it among the mutually equivalent sets of a certain group; this, however, is not at all the meaning of a statement of number. Let us consider a concrete example. Do we give the number four to a group of walnuts in front of us only because this group belongs to a certain class of infinitely many collections which all can be put in one-one correspondence to each other ? Surely no one ever thinks of such things when stating a number, and there is hardly a practical motive which would stimulate our interest in this. What we are really interested in is the fact that we have a walnut and a walnut and a walnut and a walnut. However, we soon modify this impractical and clumsy idea (and how impractical and clumsy it would be in the case of larger sets!) by giving it a form which is more convenient both for thinking and speaking: we use the general form of a collection (allgemeine Mengenform), in our case, one and one and one and one, which has the name four. The undetermined 'one' becomes determined if we add the generic name (of the things counted) to the number word; this determination goes no further than our logical interest does : if, referring to a walnut, we say ' a walnut', our interest is concentrated on the walnut as an individual of a certain kind, not on this particular walnut with all its properties. It is the visible advantages which this already brings at the level of the most ordinary usage that stimulate our interest in extracting the general form of a collection, or Number. On the other hand, the equivalence-relations between a given set and other sets, to which the theory in question resorts in trying to explain the origin and the meaning of the concept of number, appear entirely useless and irrelevant.

The recourse to the sets of strokes 11, 111, . . ., which serve as standard representatives (in a sense, as Jtalons) of classes and with the help of which a set to be counted is included in the respective class, does not of course improve the position of this theory either. It is quite absurd to call these sets of strokes " natural numbers " and to regard the names two, three, etc., as the names of these sets ; it is no less absurd to identify the concept of a unit with such a stroke. It is quite clear that we do not attribute the number four to a group of walnuts, and the n-u,mber one to each particular walnut because they can be " mapped " by 1111 and 1 respectively.


HUSSERL versus FREGE 161 But then the question might be on what grounds are we entitled

to designate all objects that we count-and nothing is imaginable that cannot be counted-by means of such a stroke ? If this designation (Bezeichnung) is to have a true foundation, it must be based on a property which is common to all objects without distinction. There is however only one all-embracing concept: the concept of something. Consequently, the stroke 1 can designate every object only as a something, and Number is, accordingly, something and something etc. It might seem therefore that the smallest amount of reflection should already lead one from the erroneous to the right view; it might seem that it is already sufficient to put the above question, in order to arrive at the right answer. But the answer is too obvious and, at first sight, appears too trivial; and this is how some people, in an effort to avoid it, have come upon those remote and artificial theories which purport to construct the elementary arithmetical concepts out of their basic constitutive character- istics (Merkmale) given in definitions, but which change and misinterpret the meanings of these concepts to such an extent that they finally become completely extraneous conceptual constructions which are useless for practical purposes and equally useless to science.

Frege's attempt to solve the problem The appropriateness of these remarks is admirably illustrated,

among others, by the already often quoted ingenious book by Frege on the Foundations of Arithmetic which is entirely devoted to the analysis and definition of the concept of number. In fact, Frege raises the question how are we able to refer to all things by the name of one and devotes long explanations to this question. Occasionally he even approaches the right answer, but only to stray subsequently even farther from the truth. This is now the appropriate place to analyse Frege's remarkable attempt, because the view which he eventually adopts, is, in its essential features, closely related to the equivalence theory criticised above.

What Frege aims at, is not at all a psychological analysis of the concept of Number; he does not think that such an analysis could help to clarify the foundations of arithmetic. " Psychology should not imagine that it can contribute anything to the foundation of Arithmetic." And elsewhere too he is most resolute in his protests against the incursion of psychology into our field. One sees already in which direction he is moving. " Although ... mathematics must refuse all help from psychology,


162 E. PIVCEVIC: it cannot deny its relationship with logic." Frege's ideal is to lay the foundations of arithmetic on a set of formal definitions from which all theorems of this science could be deduced on a purely syllogistic basis.

It is surely not necessary to go into long discussions in order to show why I cannot share this view, considering that my whole analysis so far represents in itself a refutation of it. It is clear that only what is logically complex does lend itself to definition. As soon as we get to the basic, elementary concepts, no further definitions are possible. No one can define concepts like quality, intensity, place, time and other'similar concepts. The same is true of elementary relations and the concepts based on them. Equality, similarity, comparison (Steigerung), whole and part, plurality (Vielheit) and unit, etc., are concepts that cannot be given a formal logical definition at all. What one can do in these cases is only to point at the concrete phenomena from which, or through which, they are obtained by abstraction, and to explain how this is done; where necessary, one can, by means of various descriptions, rigorously circumscribe the relevant concepts and thus avoid confusion with other related concepts. What can reasonably be required from a linguistic explanation of such a concept (e.g. in the exposition of a science based on it) would then be this: this explanation must be such as to induce us to take the right kind of attitude, so that we can ourselves extract from inward or outward intuition those abstract features which are being referred to, viz. to reproduce in ourselves those mental processes which are necessary for the construction of the concept. All this will of course be both useful and necessary only when the name designating this concept, alone, is not sufficient for an understanding of its meaning, either because of some existing ambiguities, or because of some misinterpretations that had been occasioned by the concept. This is precisely the case with the concepts of numbers, and this is why we cannot really find anything objectionable if mathematicians preface their systems by " describing the way in which we arrive at the concepts of numbers ", instead of giving a logical definition of these concepts; only this description must be the correct one and it must fulfil its purpose.

Our analysis has shown beyond dispute that the concept of a plurality and the concept of a unit rest on certain basic, elementary mental data and, consequently belong to the concepts which are undefinable, in the stated sense. The concept of Number is, however, so closely connected with these concepts that it is hardly possible to talk here about a definition. The


HUSSERL versus FREGE 163

aim Frege set to himself must therefore be regarded as chimerical. Small wonder then that his analysis, in spite of all the acumen he displays, dissolves into sterile hyper-subtleties and finally ends without a positive result. It would take us too far off our course if we examined his reasoning in detail. Here it will be sufficient to take and examine some of his more important definitions. In order to make them intelligible, it must be said at once that, according to Frege, a statement of number contains a statement about a concept. The number does not apply either to a single object or to a set of objects; it applies to the concept under which the counted objects fall. When we say "Jupiter has four moons " we assign the number four to the concept " Moon of Jupiter ".

The main idea which Frege follows in his analysis coincides with that of the equivalence theory to which we have referred earlier, in the sense that he too tries to get to the concept of number through a definition of " equinumerosity ". The method he adopts represents, in his view, a special case of a general logical method which makes it possible to obtain a definition of what should be regarded as equal from an already familiar concept of equality. " Admittedly, this seems to be a very unusual type of definition, to which logicians have not yet paid enough attention; but that it is not altogether unheard of may be shown by a few examples. The judgement: 'the straight line a is parallel to the straight line b ', or, using symbols,

a /b can be interpreted as an equation. If we do this, we obtain the concept of direction, and say: 'the direction of the straight line a is the same as the direction of the line b.' Thus we replace the symbol ' / / ' by the more general ' = ', by dividing the particular content of the former between a and b. We carve up the content in a way different from the original way, and this yields us a new concept."'

Frege also gives another example: "From geometrical similarity is derived the concept of shape, so that, e.g. instead of

the two triangles are similar ' we say ' the two triangles have the same shape' or 'the shape of the one triangle is the same as the shape of the other '."

Parallelism and geometrical similarity represent in these examples " the familiar concepts of equality ". Let us now see

1 This and the following extracts from Frege's Foundations of Arith- metic are here reproduced with a few minor alterations in the translation by J. L. Austin.


164 E. PIVCEVIC:

how Frege uses these concepts in order to obtain the definition of what should be considered equal, i.e. the definition of the direction of a straight line and the shape of a triangle. What follows is the result of a longer explanation:

"If line a is parallel to line b, then the extension of the concept 'line parallel to line a 'is the same as the extension of the concept 'line parallel to line b '; and conversely, if the extensions of the two concepts just named are the same, then a is parallel to b. Let us try, therefore, the following type of definition:

The direction of line a is the extension of the concept ' parallel to line a ' The shape of triangle t is the extension of the concept ' similar to triangle t '."

It is easy now to see how these ideas and definitions can be made use of in a definition of the concept of number. Just as direction applies to straight lines and shape to triangles, so does the number apply to concepts. Consequently, what we have to do is to replace lines and triangles by concepts. More- over, the place of parallelism and similarity is now taken by the here applicable concept of equality: the " equinumerosity " of concepts. The concept F is said to be equinumerous with the concept G if the objects falling under one of them can be put into one-one correspondence to those falling under the other. In this way we get the following definition:

" The number which applies to the concept F is the extension of the concept ' equinumerous with the concept F' "

which, together with those given earlier, form the basis of a whole series of other definitions and subtle reflections that accompany them.

I fail to see how this method can represent an enrichment of logic. Its results are such that one only wonders in astonishment how anybody could accept it, even temporarily, as the correct one. In point of fact, what this method enables us to define, are not the contents of the concepts direction, shape, cardinal number, but their extensions. Thus, it produced, among others, the following definition: " The direction of line a is the extension of the concept 'parallel to line a '." Now, suely what we mean by the " extension of a concept" is the collection (collective, Inbegriff) of objects falling under this concept. The direction of line a would therefore be the collection of lines parallel to a. Similarly, we had: " The shape of triangle t is the extension of the concept 'similar to triangle t' ", i.e. the collection of all


HUSSERL versus FREGE 165 triangles similar to t. Finally, " the number which applies to the concept F " was also defined in a similar way, as the extension of the concept " equinumerous with the concept F "; which means that the concept of this number would be the totality of concepts that are equinumerous with F, and hence the totality of infinitely many " equivalent " sets. Further comment is surely superfluous. One sees, by the way, that all definitions become true statements if the concepts to be defined are replaced by their extensions; but clearly they become as well quite trivial and valueless statements.'

1 Frege himself seems to have sensed the dubious implications of this definition, for he remarks in a footnote referring to it: " I think that for ' extension of the concept' we could say simply 'concept '." Let us consider what this would mean. The expression 'the number of the moons of Jupiter' would in this case mean the same as 'equinumerous with the concept Moon of Jupiter ', or, to put it more clearly: equinumerous with the collection of moons of Jupiter. As one can see, we get again concepts of equal extension but not of the same content. The latter of the two concepts is identical with the concept ' any set from the equivalence class which is determined by the collection of moons of Jupiter '. All these sets come under the number four. That here, however, we have different concepts, is quite obvious. One can also see that Frege, as a consequence of the just mentioned modification, drifts towards the already refuted, and in general, even more straightforward, theory of equivalence.


Article Contentsp. 155p. 156p. 157p. 158p. 159p. 160p. 161p. 162p. 163p. 164p. 165

Issue Table of ContentsMind, New Series, Vol. 76, No. 302 (Apr., 1967), pp. 155-308Front MatterHusserl versus Frege [pp. 155 - 165]Propositions and Speech Acts [pp. 166 - 183]The Argument from Illusion in Aristotle's Metaphysics (, 1009-10) [pp. 184 - 197]Observation and Reality [pp. 198 - 207]On Avowing Reasons [pp. 208 - 216]Visual Experiences [pp. 217 - 227]Fact and Value [pp. 228 - 237]On Blaming [pp. 238 - 249]An Ethical Paradox [pp. 250 - 259]Discussion NotesProbability and the Theorem of Confirmation [pp. 260 - 263]The Space-Time World [pp. 264 - 269]Evidence for Creation [pp. 270 - 274]On Moore's Paradox [pp. 275 - 277]In Reply to Professor Stephan Krner, on Science and Moral Responsibility [pp. 278 - 281]In Defence of Euclid: A Reply to B. Meltzer [p. 282]Third Possibilities and the Law of the Excluded Middle [pp. 283 - 285]A Note on the "Is-Ought" Barrier [p. 286]

New Books [pp. 287 - 307]Errata: To Gdel via Babel [p. 307]Obituary: Alexander Bryan Johnson (1786-1867) [p. 307]Notes [p. 308]Back Matter

Documents

Pivcevic- Husserl Versus Frege