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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
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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, ex- cept 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
trans- lated.]
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
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156 E. PIVCEVIC: regularly crop up in connection with the
definition of equi- numerosity (Gleichzahligkeit) by means of
one-one correspond- ence. 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 com- pletely 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
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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
representa- tives 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.
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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 corre-
sponding 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 equi-
numerosity and to penetrate to the essence of the basic arith-
metical 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."
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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, repre-
sents 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 ' equi-
numerosity', 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 ".
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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.
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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,
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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
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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.
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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
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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: equi- numerous 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 equival- ence 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.
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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