Frege''s Reduction
History and Philosophy of Logic
Vol 15 (1994), 85-103
Patricia A. Blanchette
Department of Philosophy, University of Notre Dame
Notre Dame, IN 46556
2
1
Frege''s logicism, the thesis that "the laws of arithmetic are analytic''''
1
is
standardly taken to be an important epistemological thesis. The traditional
view of Frege''s work is that his "reduction'''' of arithmetic to logic was intended
to provide the cornerstone of an argument that the truths of arithmetic are
knowable a priori and independently of anything which Kant would have
labelled “intuition”. The truth of Fregean logicism would, on this view, have
repudiated the explanations of arithmetical knowledge offered by Kant and by
Mill. It would have provided an explanation of arithmetical knowledge which
was acceptable from a generally empiricist perspective, and which preserved
the intuition that arithmetical truths are necessary and knowable a priori.
As against this received view, Paul Benacerraf has recently argued in
"Frege: The Last Logicist"
2
that Frege''s project was not an epistemological one,
and was in particular not an attempt to counter Kant''s view of the nature of
arithmetical knowledge. Though Frege explicitly claims to be engaged in
demonstrating the analytic, a priori nature of arithmetical truth, Benacerraf
claims that Frege has so re-construed the notions of analyticity and a priori
truth that the entire project is, from the very beginning, a non-
epistemological one. As Benacerraf puts it, Frege''s "attempt to establish the
analyticity of arithmetic [is] not to be construed as an attempt to enter an
ongoing philosophical debate between Kant and the empiricists, and indeed ...
his very construal of the question took it out of that arena.”
Benacerraf''s claim is an alarming one. First of all, Frege''s project clearly
looks like an epistemological one, and one intended to provide an alternative to
Kant''s view of the nature of arithmetical knowledge. In the conclusion of the
Grundlagen, Frege sums up that work as follows: "I hope I may claim in the
present work to have made it probable that the laws of arithmetic are analytic
judgments and consequently a priori,''''
3
and that in showing this result, "we
achieved an improvement on the view of Kant.''''
4
The early Grundlagen
discussion of the "distinctions between a priori and a posteriori, analytic and
synthetic'''' are accompanied by the note: "I do not, of course, mean to assign a
new sense to these terms, but only to state accurately what earlier writers,
Kant in particular, have meant by them.''''
5
Secondly, the incompatibility of Frege''s project with epistemological goals
is due largely, on Benacerraf''s reading, to the fact that Frege allows multiple
reductions of the numbers to logical objects. But if the inference here is
warranted, then it is difficult to see how any reductionist project, whether
2
logicist or not, can ever be of epistemological significance. For no standard
reduction of the numbers can be claimed to be uniquely correct. Thus if
Benacerraf is right, it seems that we must not look to arithmetical reductions
in the attempt to understand the nature of arithmetical knowledge.
It is the purpose of this paper to defend the epistemologically-loaded
reading of Frege''s project. Frege''s view of the purpose and method of
mathematical reductions was strikingly different from those conceptions
which have supplanted his. I hope to make it clear that the view of Frege''s
project as having been irrelevant to epistemological concerns is due to a
confusion of Frege''s reduction with a reduction of a more modern variety. As
a corollary, I take it that Frege''s own reductionist project, in general outline,
is one which must be taken seriously by those who are concerned with the
nature of arithmetical knowledge.
1. The Project
Analyticity, for Frege, is first and foremost a proof-theoretic concept: an
analytic truth is anything provable using only definitions and laws of logic.
6
As Frege views proofs,
The aim of proof is ... not merely to place the truth of a
proposition beyond all doubt, but also to afford us insight into the
dependence of truths upon one another.
7
The relations of dependence revealed by proofs is the key to what Frege
regards as the epistemological status of the proposition proven. In particular,
to have shown upon which propositions a given truth depends is to have
demonstrated that knowledge of those propositions suffices to ground
knowledge of the truth in question. As Frege puts it in 1897,
...I was looking for the fundamental principles or axioms upon
which the whole of mathematics rests. Only after this question is
answered can it be hoped to trace successfully the springs of
knowledge upon which this science thrives.
8
A crucial feature of Frege’s conception of analyticity is that the concept
applies never to sentences, but only to the nonlinguistic items expressed by
sentences.
9
This view is owed to the close connection in Frege''s system
between analyticity, provability, and dependence, along with the view that
dependence is a relation between nonlinguistic items. Regarding dependence,
Frege holds that
3
We have to distinguish between the external, audible, or visible
which is supposed to express a thought, and the thought itself. It
seems to me that the usage prevalent in logic, according to which
only the former is called a sentence, is preferable. Accordingly,
we simply cannot say that one sentence is independent of other
sentences; for after all no one wants to predicate this
independence of what is audible or visible.
10
Thus:
What we prove is not a sentence, but a thought. And it is neither
here nor there which language is used in giving the proof.
11
The claim that the truths of arithmetic are analytic is a claim that each of a
particular collection of nonlinguistic propositions is provable from, and
hence dependent solely upon, laws of logic and whatever is expressed by the
relevant definitions.
Two things go, for Frege, by the name “definition.” The first is simply a
stipulation that a newly-introduced piece of notation is to abbreviate a longer,
already-meaningful series of marks. Only definitions of this kind occur in
Frege’s formal proofs, and all of these introduce conservative extensions of
the formal system and hence are eliminable.
12
Such definitions express no
extra-logical content. Thus a proof using laws of logic and definitions will
demonstrate that the proven proposition depends for its truth solely on
principles of logic. As Frege puts it, the result will be that “every proposition
of arithmetic [is] a law of logic, albeit a derivative one.”
13
The second thing which Frege calls “definition” is the much less clearly-
characterized conceptual analysis which precedes stipulative definition. Here
Frege starts with an ordinary arithmetical concept and “reduces” it to simpler
or more general concepts. The question of the definability in this sense of the
concept of cardinal number is, Frege claims, “the point which [the
Grundlagen] is meant to settle.”
14
In general, the reduction of a concept to
concepts expressible in primitive Begriffsschrift notation is followed in the
formal work by a stipulative definition, one which introduces a new term as
shorthand for the purported analysans.
15
The abbreviation is then used in
proofs, instead of its less rigorously-defined ordinary counterpart. We return
to the central issue of the relationship between the ordinary and defined
terms in §2. The point to be stressed here is that (as Frege himself stresses) the
definitions which occur within the proofs are always stipulative and
eliminable.
4
Despite their dispensability within proofs, stipulative definitions play a
crucial role in the overall project. A sentence containing a defined term
expresses just what its definitional transcription expresses, and hence the
proposition expressed by a sentence containing defined terms is determined to
a large extent by the definitions of those terms. Had Frege''s definitions been
different from what they are, the propositions expressed by sentences
containing defined terms would have been correspondingly different.
Where a proof is a series of propositions, the last of which is the
proposition proven, let us use the word “derivation” for a series of sentences
which together express a proof. As long as each sentence in a derivation
either (a) expresses a truth of logic, (b) stipulates a notational convention, or
(c) expresses a proposition which follows by logical principles from previous
lines, then a derivation will demonstrate that the proposition expressed by its
last line is analytic. Hence the importance of Frege’s stipulative definitions:
which propositions will have been shown to be analytic depends to a large
extent on the definitions of those defined terms occurring in the derived
sentences.
2. Definitions, Propositions and Multiple Reductions
If Frege’s proofs are to be, strictly speaking, proofs of the truths of
arithmetic, his definitions must, together with the other semantic features of
the language, ensure that the derived sentences express arithmetical claims.
Assuming that we express arithmetical truths in our ordinary arithmetical
discourse, Frege’s proofs will succeed in this way only if his derived sentences
express the same propositions as do sentences of ordinary arithmetic. Taking a
cue from Frege’s talk of analysis and reduction, we might fairly presume that
sentences containing defined terms are intended to express claims about
“fully-analysed” versions of arithmetical objects and concepts, and that the
success of the analyses underwrites the identity (or sufficient similarity)
between proven proposition and ordinary arithmetical truth. If this is the
appropriate description of the project, then the constraints on Frege’s
analyses and definitions are clear: The definition of an arithmetical term in
the formal language must assign a meaning sufficiently close to that of the
ordinary arithmetical term that sentences incorporating the defined term
express the same propositions as (or ones sufficiently similar to) those
expressed by sentences incorporating the ordinary term.
5
The question, now, is whether Frege''s definitions do in anything like this
sense "preserve the meanings" of the sentences of ordinary arithmetic. A
central part of Benacerraf’s argument is that, unlike the clearly
epistemologically-motivated logicism of the logical positivists, Frege’s logicism
does not involve the kind of meaning-preservation required for
epistemological significance.
We should note, first, that the importance of some kind of semantic-value
preservation is clear to Frege, and underlies various of his complaints against
competing foundationalist projects. In particular, Frege''s response to Mill
includes the objection that "Mill understands the symbol + in such a way that it
will serve to express the relation between the parts of a physical body or of a
heap and the whole body or heap; but such is not the sense of that symbol;"
and that "In order to be able to call arithmetical truths laws of nature, Mill
attributes to them a sense which they do not bear."
16
In response to Hankel''s
attempt to define addition in terms of our intuitions of magnitude, Frege
objects:
The definition can perhaps be constructed, but it will not do as a
substitute for the original propositions; for in seeking to apply it
the question would always arise: Are Numbers magnitudes, and is
what we ordinarily call addition of Numbers addition in the sense
of this definition?
17
The response to Newton contains a similar worry. Even if Newton''s definitions
can be given in a non-circular way, Frege claims, "Even so, we should still
remain in doubt as to how the number defined geometrically in this way is
related to the number of ordinary life..."
18
Given this concern with the preservation of ordinary meaning, one might
expect from Frege some clear criterion of sameness of meaning, or of
sameness of proposition expressed. But there seems to be no such workable
criterion. The closest Frege comes to a clear conception of proposition-identity
is the post-1890’s relation of identity of sense, or of thought expressed. But the
criterion of sense-identity most readily reconstructed from Frege’s writings of
this period is one on which two sentences express the same sense only if they
are fairly obvious synonyms. And such a criterion is clearly not what Frege
had in mind: no derived sentence in Frege’s system is anything like an easily-
recognizable synonym of its ordinary counterpart. Nor should it be, of course;
if we are to have informative reductions of any kind, we cannot require that
6
reduced and reducing sentences be in any immediate and obvious sense the
same in meaning.
19
An alternative criterion to which one might naturally hold Frege''s
definitions is that the defined singular terms and predicative expressions must
refer to the same things as do their ordinary counterparts.
20
The idea here
would be that one first determines which objects and relations are referred to
by the parts of ordinary arithmetical sentences, and then assigns these objects
and relations as the referents of the corresponding terms in Frege''s formulae.
The goal of the definitions would be, on this picture, to "preserve the
reference" of the arithmetical terms.
But such atomistic reference-preservation is, as Benacerraf points out,
clearly not what Frege had in mind. Frege''s definition of natural number
turns on the definition of "number which belongs to the concept ..."; that
definition is as follows:
"The Number which belongs to the concept F is the extension of
the concept ''equinumerous with the concept F''."
21
This definition is used to provide the definitions of each of the natural
numbers; each number is on this account the number which belongs to some
first-level concept. Thus each number is the extension of a particular second-
level concept.
22
Immediately having given the above definition, Frege
remarks in a footnote to which Benacerraf draws attention that
I believe that for "extension of the concept'''' we could write
simply "concept.''''
A similar claim occurs in the concluding pages of the Grundlagen. Here Frege
says: "I attach no decisive importance to bringing in the extensions of
concepts at all.''''
23
The thrust of these passages is that though he has in fact defined the
numbers as extensions, the purposes of the project could as easily have been
met by defining them as concepts.
24
As Benacerraf puts it, "Frege ... allows
that different definitions, providing different referents ... might have done as
well;'''' consequently: "The moral is inescapable. Not even reference needs to
be preserved.''''
25
One might hope to avoid Benacerraf''s conclusion by pointing out that the
passages in question are inconsistent with much of the rest of the Grundlagen,
and are repudiated by the time of the Grundgesetze. Frege argues at length in
7
the Grundlagen that the numbers are objects, and that they cannot be
concepts. Further, one of the demands of logicism is that the existence of the
natural numbers be established by purely logical means.
26
Thus if the
numbers are to be objects, they must be the kind of objects which are
guaranteed to exist via the principles of Frege''s logic. Since the only such
objects are the extensions of concepts, Frege''s claim to "attach no decisive
importance" to the use of extensions shows that he was perhaps not entirely
clear about the demands of the project at this point.
Frege seems quickly to realize the indispensability of extensions. For the
problematic Grundlagen claim is never repeated after 1885,
27
and we find in
the Grundgesetze the claim about extensions (courses of value) that "we just
cannot get on without them."
28
Even this repudiation, however, does not
clarify matters entirely with respect to the issue of multiple reducibility. For
though it is clear that, as Frege claims, the numbers must be defined as
extensions, it is not at all clear that there is a unique sequence of extensions
which must serve as the referents of the formal numerals.
In fact, it appears that Frege himself proposes two distinct reductions of
the numbers to extensions. In the Grundlagen, the numerals refer to the
extensions of second-level concepts, while in the Grundgesetze they refer to
the extensions of first-level concepts.
29
This would seem to clinch the case in
favor of Frege''s tolerance of multiple reductions.
30
But here again there are
difficulties. For there are no clear criteria by means of which to conclude that
the relevant first-level and second-level extensions are in fact distinct. First,
though every well-formed identity-sentence of the Grundgesetze receives a
truth-value, no sentence which identifies the extension of a second-level
concept with that of a first-level concept occurs in the Grundgesetze notation.
For there are no singular terms in this notation for extensions of second-level
concepts.
31
Secondly, there is no natural way to extend the Grundlagen''s or
Grundgesetze''s comprehension principles so as to deal with such "cross-level"
identities.
32
Were Frege to extend the Grundgesetze notation to include names
for the extensions of second-level functions, he would need to stipulate which
of the new identity-sentences were true and which false.
33
But in the absence
of such a stipulation, there is simply no fact of the matter about whether the
extensions in question are the same or distinct.
34
Despite the fact that the indeterminacy here raises alternatives to reading
Frege as admitting multiple reductions, it does nothing to restore the view of
8
Frege''s definitions as essentially reference-preserving. For this
indeterminacy illustrates a general feature of Frege''s reduction: that the
answers to a wide range of identity-questions concerning the referents of the
numerals are simply irrelevant to the ability of these objects to play the
appropriate role within the reduction. In the case of the Grundlagen, the only
answerable identity-questions concerning the referents of the numerals are
those which question the identity of these objects with extensions of second-
level concepts. In the Grundgesetze, the only numerical identity-questions
which receive answers are those concerning the identity of a numeral''s
referent with an extension of a first-level concept. Consider now the
referents of the numerals of ordinary arithmetic. Unless it is already clear
that these objects are, or that they are not, first- (second-) level extensions,
then the question of their identity with the Grundlagen''s or Grundgesetze''s
numerals simply receives no answer. Whatever Frege''s definitions do, then,
they do not pick out a sequence of objects independently defensible as "the
numbers" and assign them to the Fregean numerals.
3. Mathematics, Epistemology and Preservation of Structure
The decidedly non-linguistic flavor of Frege’s reduction underscores the
central claim of Benacerraf’s paper: that Frege’s logicism was of a very
different kind from that of his twentieth-century followers. And as
Benacerraf points out, Frege’s acceptance of apparently distinct reductions
reinforces this point: the positivist aim of preserving meaning term-for-term
is no part of Frege’s goal.
But the further lesson which Benacerraf draws from this difference is one
which, I shall argue, we ought to resist. On Benacerraf’s view, the distinction
between Frege and the logical positivists marks a distinction between a
primarily mathematical concern with foundations and a primarily
philosophical concern:
[A] concern [with the foundations of arithmetic] might be
interpreted in two different ways, corresponding to the interests
of a philosopher and to those of a mathematician. Typically, the
philosopher takes a body of knowledge as given and concerns
himself with epistemological and metaphysical questions that
arise in accounting for that body of knowledge, fitting it into a
general account of knowledge and the world. That is Kant''s
stance. He studies the nature of mathematical knowledge in the
context of an investigation of knowledge as a whole. And that
9
was the positivist’s stance, though they reached quite different
conclusions.
But a mathematician''s interest in what might be called
"foundations'''' is importantly different. Qua mathematician, he is
concerned with substantive questions about the truth of the
propositions in question ...
35
As Benacerraf remarks, this distinction cannot be drawn sharply, and “the
interests of the two groups are not disjoint.” But Frege does have primarily, as
Benacerraf sees it, the mathematician’s concern; his proofs were intended to
demonstrate the truth of propositions which stood in need of such
demonstration, and thereby to provide a needed justification for our
acceptance of the truths of arithmetic. They were not, on Benacerraf’s
reading, intended to contribute to the kind of philosophical project in which
Kant was engaged.
Before deciding whether the failure of Frege''s definitions to preserve
reference should incline us toward such a reading, it is worth noting that
Frege himself seems to draw substantially the distinction drawn above by
Benacerraf, but places himself on the "philosophical" side of the divide:
My purpose necessitates many departures from what is customary
in mathematics. ... Generally people are satisfied if every step in
the proof is evidently correct, and this is permissible if one
merely wishes to be persuaded that the proposition to be proved is
true. But if it is a matter of gaining an insight into the nature of
this ''being evident'', this procedure does not suffice; we must put
down all of the intermediate steps, that the full light of
consciousness may fall upon them. Mathematicians generally
are indeed only concerned with the content of a proposition and
with the fact that it is to be proved. What is new in this book is
not the content of the proposition, but the way in which the
proof is carried out and the foundations on which it rests. That
this essentially different viewpoint calls for a different method
of treatment should not surprise us.
36
We should note, secondly, that Frege''s acceptance of multiple reductions
and the consequent distinction between his and positivist logicism does not by
itself offer any clear support to a "purely mathematical" reading of the
Fregean project. If the failure of Frege''s definitions to preserve reference
entails that the final sentences of his derivations do not express the truths of
arithmetic, then his proofs will not provide an immediate demonstration of the
analyticity or of the truth of these propositions.
10
But this is not to say that epistemological and mathematical significance
need in general go hand-in-hand. Had Frege’s reduction been an attempt to
provide e.g. a model for the true sentences of arithmetic, then his project
would arguably have been of the kind outlined by Benacerraf. The
availability of distinct w-sequences makes the existence of distinct models for
the arithmetic of the natural numbers immediate. And the provision of a
model would suffice to establish the truth of arithmetical claims while
contributing little of value to the epistemological debate. For any sentence
which expresses a truth about an w-sequence (or about members thereof) and
which is provable solely from the fact that the sequence is an w-sequence, will
continue to express a truth when interpreted as being about the natural
numbers. But the analyticity of such a proposition will in no way guarantee
the analyticity of the corresponding arithmetical truth. For this further
conclusion demands the additional argument that the fact that the natural
numbers form an w-sequence is itself a matter of logic - and not, e.g., a matter
of geometry. The mere existence of w-sequences can tell us nothing about
whether, say, Mill is right about the nature of arithmetical knowledge.
But it is clear that Frege’s extensions and courses of value are intended to
provide more than just a model. Consider, first, Frege''s rejection of attempts to
found arithmetic on geometrical or formalist constructions. Two of the central
objections here are that such foundations would mis-construe the meanings of
arithmetical claims, and that they would fail to explain the applicability of
arithmetic to non-spatial questions.
37
These objections make no sense from
the point of view of a project whose goal is simply to assign to the numerals
members of a collection which exhibits the right structure. Additionally, the
rejected projects provide perfectly good models: there are any number of ways
of constructing w-sequences out of geometrical or syntactic objects.
38
Further, Frege explicitly rejects the model-theoretic approach in his
objections to Hilbert''s "reduction" of geometry to arithmetic. On Frege''s view,
the use of partially-interpreted sentences in the axiomatization of geometry
entails that the axioms are not in fact about geometry at all.
39
Were Frege''s
reduction of arithmetic to logic simply the provision of a model for partially-
interpreted arithmetical sentences, precisely the same objection would apply:
the resulting interpretation in terms of extensions would bear only a
superficial resemblance to arithmetic. We should note also in this connection
that Frege does not see in Hilbert''s work any reason to modify the view that
11
geometry is synthetic. If Frege''s reductions were answerable only to the
criteria met by Hilbert''s, then there would be no sense to be made of Frege''s
view that the foundations of arithmetic and geometry are fundamentally
different.
To return to the central issue, then. The final sentences of Frege''s
derivations are not in any standard sense synonymous with the sentences of
ordinary arithmetic. This is clear even without the phenomenon of multiple
reducibility, and is sufficiently clear that presumably no such correlation as
obvious synonymy was intended by Frege to hold between these sentences. On
the other hand, the fact that the referents of Frege''s formal numerals can be
mapped in an order-preserving way onto the referents of the ordinary
numerals is, while necessary, not sufficient for the success of the project as
Frege conceives it. The connection between the sentences of ordinary
arithmetic and the various formal counterparts which are acceptable from a
Fregean point of view must lie somewhere between these two extremes.
4. Another Response to Benacerraf
An account of Frege''s program which attempts to reconcile the
acknowledgement of multiple reducibility with the epistemological project has
recently been offered by Joan Weiner.
40
On Weiner''s account, the reason that
Frege''s assignment of referents to the formal numerals was not required to
preserve the reference of the ordinary numerals is that these numerals have,
prior to Frege''s work, no reference. Frege''s goal, on this reading, is to provide
the numerals (and other parts of the language) with reference, as part of the
project of providing a replacement for ordinary arithmetic. The replacement
is to be an improvement on its ordinary counterpart in terms of precision and
rigor. The point of Frege''s definitions would be the provision of reference for
the arithmetical terms in such a way that all and only the previously-accepted
arithmetical sentences (or their formal counterparts) end up expressing
truths.
If Frege''s goal were indeed that of providing the numerals with reference
for the first time, then there would certainly be several different ways of
doing this, consistent with the demands of the overall project. But note that
such a project must abandon the idea that the things we know (prior to Frege''s
work) are both the truths of arithmetic and the things expressed by ordinary
arithmetical sentences. For if the ordinary numerals lack reference, then the
12
sentences of ordinary arithmetic lack truth-value. The claims they express
are not truths of any kind, let alone truths of arithmetic. If we do (prior to
Frege''s work) know any truths of arithmetic, then the things we know are not
expressed by the sentences we use to communicate. This conclusion cannot be
Frege''s, in light of his view that the things "grasped" in judgment and thought
are precisely the semantic values of our utterances. On the other hand, if the
things we know are expressed by the sentences we use, then on this account
we know, prior to Frege''s work, no truths of arithmetic. And in this case, it is
difficult to see what the epistemological import of Frege''s work could have
been. For he will have demonstrated of some interesting collection of
propositions that they are all analytic, but these will have been propositions
which, prior to his work, nobody had ever known.
A large number of issues are raised by this interpretation of Frege, most of
which must wait until another time. The central issue here is that if we are to
view Frege''s project as an attempt to explain the epistemological status of (or
even to demonstrate the truth of) claims which are actually known by those
who do ordinary arithmetic, then Weiner''s reading is not the appropriate one
to give. We cannot explain Frege''s acceptance of multiple reductions on the
grounds that there are no antecedent referents to be "preserved" via these
reductions.
41
5. Frege’s Analyses
Despite his own emphasis on the provision of analyses, Frege himself
gives no clear account of the nature of analysis, or of the conditions which
must be met by an adequate analysis.
42
But we should recall that the crucial
relationship is that between the proven propositions and the truths of
arithmetic; if Frege’s analyses and definitions are such that the analyticity of
the former propositions guarantees the analyticity of the latter, then the
definitions will have been successful. Note that a proposition is analytic in
Frege’s sense just in case all of its logical equivalents are as well. This section
is an examination of Frege’s analysis of the finite cardinals, from which, I
shall argue, a picture of Frege’s procedure emerges according to which: (a)
the definitions are intended to provide provable propositions whose
analyticity fairly clearly entails that of the truths of arithmetic; and (b) the
existence of multiple reductions is to be expected, due to the fact that obvious
logical equivalents are as good as one another for Frege’s purposes.
13
That feature of number with which Frege is primarily concerned is that
numbers are used to count things. In particular, Frege''s analysis of number
centers on an analysis of statements of the form "there are n F''s." With
respect to such statements, the claim which Frege later regards as "the most
fundamental of [his] results" is that "a statement of number expresses an
assertion about a concept." The Grundgesetze''s account of arithmetic, we are
told, "rests upon this" result.
43
What Frege means here is best illustrated by his
own examples:
If I say "Venus has 0 moons", ... what happens is that a property is
assigned to the concept "moon of Venus", namely that of
including nothing under it. If I say "the King''s carriage is
drawn by four horses", then I assign the number four to the
concept "horse that draws the King''s carriage."
44
The argument for this view comes entirely from considerations about ordinary
arithmetical discourse, and occupies the bulk of sections 18 through 52 of
Grundlagen. Frege''s claim here would appear to be that, in some ordinary
sense of analysis, claims of the form "there are n F''s" are adequately analyzed
as claims which "assign a number" to a concept.
The account of what it is to assign a number to a concept forms the basis of
the analysis of number in general. In particular, to say what it is to "assign"
zero to a concept, or to say that zero is the number which belongs to a concept,
will be to give an account of the number zero. To say what it is to assign the
successor of a number to a concept will be to give an account of the relation of
successor. This, then, is intended to suffice for the account of the natural
numbers, as these are to be defined in terms of zero and successor.
As a preliminary but ultimately unsatisfactory account of such
assignments, Frege suggests that to assign zero to a concept F is to say,
intuitively, that there are no F''s, and that to assign n+1 to a concept F is to say
that for some object a falling under F, there are n F''s other than a. More
precisely, Frege''s suggestion is that
(a) zero is the number which belongs to the concept F
and
(b) n+1 is the number which belongs to the concept F
be analysed as, respectively,
(A) The proposition that a does not fall under F is true
universally, whatever a may be,
and
14
(B) There is an object a falling under F and such that the number
n belongs to the concept "falling under F, but not a."
These preliminary definitions, says Frege, "suggest themselves so
spontaneously in the light of our previous results, that we shall have to go into
the reasons why they cannot be reckoned satisfactory."
45
The
spontaneousness with which the definitions suggest themselves is not difficult
to feel, if the suggestion is that the sentences (A) and (B), as normally
understood, make essentially the same claims as do "there are no F''s" and
"there are n+1 F''s". There is of course room for argument here. But what is
crucial is that (A) and (B) do offer, arguably, analyses of the claims expressed
by their ordinary counterparts.
Despite their naturalness, Frege of course rejects (A) and (B). Their
unsatisfactoriness is due to the fact that they give no account of the uses of
numerical terms in contexts other than those of the form "the number of F''s".
In particular, they do not allow us to make sense of ordinary identity-
statements involving numerals. But this is not to say that Frege rejects the
view that (A) and (B) do provide analyses of the ordinary claims. Far from it.
For the definitions with which Frege is eventually satisfied, both in the
Grundlagen and the Grundgesetze, assign to (a) and (b) claims which are
logically equivalent with (A) and (B).
46
The final Grundlagen account of (a) rests on analyses of the relation
"number which belongs to the concept," and of the number zero. The
definitions corresponding to these analyses are as follows:
(D1) "The number which belongs to the concept F" is shorthand
for "the extension of the concept equinumerous with the concept
F."
(D2) "0" is shorthand for "the number which belongs to the
concept not self-identical."
The concept of equinumerosity involved in (D1) is itself cashed out in terms of
one-one mappings.
Using these definitions, (a) is shorthand for a cumbersome sentence
which is provably equivalent, using Frege''s laws of logic, with (A). The
situation is similar with successor. Using the Grundlagen''s analysis of
successor, (b) is logically equivalent with (B). The analysis of the rest of the
natural numbers is then straightforward: for every first-level concept F and
numeral n, Frege''s "n is the number which belongs to the concept F" is
logically equivalent with the analysis one would have obtained using the
15
preliminary (A) and (B). Each such claim will in fact be logically equivalent
with our own familiar first-order rendition of "there are n F''s."
If Frege is right that the natural numbers are to be treated as essentially
measures of cardinality,
47
then this last result is crucial: it shows that to
whatever extent ordinary claims about the numbers can be analyzed as claims
about the cardinality of concepts, they are logically equivalent with claims
about the referents of Frege''s numerals. Much more is needed in the way of
analysis before it is clear that for each arithmetical claim there is such an
equivalent. In particular, in order to be able to give logical equivalents of
such purely arithmetical claims as the Peano axioms, Frege needs to give an
account of how claims about addition and multiplication are to be cashed out in
terms of the cardinality of concepts. But the groundwork has been laid; the
accounts of equinumerosity, of zero, of successor, and of natural number
suffice to generate logical equivalents of a variety of arithmetical claims
involving reference to and quantification over the natural numbers. Consider
for example Frege''s version of "every natural number has a successor."
Unpacking definitions, the Fregean claim is logically equivalent with the
claim that for every finite first-level concept F there is a concept G under
which one more object falls.
48
If one accepts the Fregean account on which
natural numbers are essentially numbers of concepts, this latter claim
provides an adequate analysis of the ordinary claim that every natural
number has a successor. The derivation of the Fregean sentence would have
demonstrated the analyticity of the ordinary claim.
Do Frege''s definitions "preserve meaning"? Probably not, for to say that
the number of F''s is zero is presumably not to make the same claim as that
made by (a), where we take (a) to be shorthand for its definitional
transcription. But in this simple case at least, they preserve what is crucial: If
we wanted to know, for a particular F, whether the fact that there are zero F''s
depends on purely logical principles, we would have to look no further than
(a).
So far, it is a straightforward matter to make sense of the multiple
"reducibility" of the numbers. Replacing definition (D1) above with
(D1'') "The number which belongs to the concept F" is shorthand
for "the extension of the concept under which fall all and only
the extensions of concepts equinumerous with F"
16
gives us (a natural-language version of) the Grundgesetze''s account.
49
Whether the object referred to by "the number which belongs to the concept
F" is the same in the Grundgesetze as it is in the Grundlagen is no longer of
any significance. The Grundgesetze''s version of (a), using (D1''), is logically
equivalent with the Grundlagen''s version, and hence with (A). Similarly, the
Grundgesetze''s version of (b) is, like its predecessor, logically equivalent with
(B). If the goal is to come up with identity-sentences which are logically
equivalent with the proposed analyses, then there are infinitely many ways to
do this. It is in this case no longer surprising that the change from the
Grundlagen version to the technically nicer Grundgesetze version was
something which Frege saw no need to defend.
It is now clear why not just any w-sequence would do for Frege''s purposes.
Supplied with any such sequence
0
, a
1
, ...>, one can correlate concepts with
members of the sequence in such a way that all and only the concepts under
which nothing falls are correlated with a
0
, and a concept F is correlated with
a
n+1
iff there''s an object a falling under F and such that the concept "falling
under F but not identical with a" is correlated with a
n
. It will then be the case
that for each concept F and natural number n, F is correlated with a
n
iff there
are exactly n F''s. But this is as close as such an arbitrary reduction will get to a
Fregean reduction. The claim that a
n
is correlated with F will in general be
only materially, and not logically, equivalent with the ordinary claim that
there are n F''s. In particular, if the existence of the sequence
0
, a
1, ...
> is
grounded in principles of e.g. geometry or set theory, then no appeal to laws
of logic will enable one to derive the "reduced" from the ordinary claim.
50
Herein lies a crucial distinction between Frege''s reduction and modern
reductions of arithmetic to set theory. A standard reduction to set theory
clarifies the relationships between properties shared by all isomorphic
structures. By showing that certain truths about a collection of objects and
relations follow solely from structural features of that collection, the
reduction to set theory will help explain why arithmetic, if it is about such a
structured collection, must contain these truths. So too will a reduction of
arithmetic to geometry. But neither of these reductions will help to clarify
what Frege calls the "ultimate grounds" of the truths of arithmetic. For there
is no reason to suppose that the claims of ordinary arithmetic are true for the
same reasons as are their set-theoretic or geometrical surrogates. In
particular, no reduction of arithmetic to set theory can help to say whether
17
the structural features of the arithmetical subject-matter are due to empirical
facts (as Mill would have it), facts about the structure of intuition (as Kant
would have it), or something else altogether. For none of these alternatives
has the least bearing on the "reducibility" of arithmetic to set theory. If Frege
had been right, however, and if his "logic" had not succumbed to Russell''s
paradox, the situation would have been quite different with his own reduction.
For Frege was in a position to argue that the claims of ordinary arithmetic
were logically equivalent with their Fregean surrogates, and hence that the
grounds of the latter sufficed to ground the former.
6. Frege and the Tradition
Both Frege and Kant view proofs as establishing the grounds of the
proven proposition.
51
And both take these grounds to be epistemologically
significant. Assuming the correctness of Frege’s view of the natural numbers
as essentially measures of cardinality, and of equi-cardinality as a matter of
the existence of 1-1 mappings, Frege’s proofs would have shown arithmetical
truths to be grounded in principles which he counted as purely logical.
Setting aside for a moment the implications of his error in this last regard, we
can ask whether the kind of analyticity which would have been conferred
upon the truths of arithmetic via these proofs would have sufficed to show
them analytic in Kant’s eyes.
There are some clear differences between the two accounts of analyticity,
perhaps the most striking of which concern Frege’s rejection of the
importance of the subject-predicate nature of propositions, and his conception
of analytic truths as capable of carrying existential import. Further, analytic
judgments are never for Kant ampliative, while they are clearly so for Frege.
52
But there is a striking parallel, one which arguably justifies Frege’s claim
to be using a modernized version of Kant’s own notion.
53
For Kant, the notion
of contradiction is central; an analytic truth is often characterized by him as
one whose negation is self-contradictory. That the principle of contradiction
suffices to ground analytic judgments is tied to the fact that, for Kant, the
truth of an analytic proposition follows just from an analysis of the concepts
involved in that proposition. Synthetic truths on the other hand, for Kant,
“cannot possibly spring from the principle of analysis, namely, the law of
contradiction, alone. They require a quite different principle from which
they may be deduced...”
54
Here Frege is in full agreement. The laws of logic
18
are, for Frege, laws which can only be denied on pain of self-contradiction.
55
An analytic truth is one for which an analysis of constituent concepts,
according to the rules of logic, suffices to demonstrate its truth. Synthetic
truths, on the other hand, are those which cannot be proven “without making
use of truths which are not of a general logical nature.” If we assume a shared
understanding of “contradiction” here, Frege and Kant are directly at odds
over arithmetic. For Kant, no amount of analysis of arithmetical concepts will
ever show that an arithmetical truth is true on pain of self-contradiction. For
Frege, this is precisely what his analyses and proofs demonstrate.
Thus no Kantian can agree with Frege’s analyses and with the view that
Frege’s laws of logic become self-contradictory when denied. Similarly, no
Millian can accept the analyses and the weaker view that Frege’s laws are
knowable a priori. The fact that Frege’s logicism poses no threat to these
programs is owing simply to the fact that his laws turned out to satisfy neither
description; the lack of a clear confrontation is due to Russell’s paradox, not to
the irrelevance of Frege’s program to the epistemological debate.
7. The Failure of the Project
Russell''s paradox shows that Frege''s comprehension principle, which he
had taken to be a "fundamental law of logic,"
56
is simply false. This principle
is central to the analyses outlined above: without the law of comprehension,
Frege''s method of arriving at logical equivalents of the truths of arithmetic is
entirely undermined. Thus Russell''s paradox does not just destroy the validity
of Frege''s proofs; it also shows that the derived sentences fail to express truths
whose analyticity would guarantee that of the truths of arithmetic. As Frege
puts it, "My efforts to become clear about what is meant by number have
resulted in failure."
57
The failure of the comprehension principle spells the failure of the
logicist reduction as Frege conceives it. For there is little reason to believe, in
the absence of such a principle, that the existence of an appropriate sequence
of objects can be regarded as a matter of logic. Thus there is little hope that,
for instance, claims about the equinumerosity of concepts can be regarded as
logically equivalent with claims about the identity of objects. If the
appropriate analysis of arithmetical claims is one which treats the numbers as
objects, then arithmetical truth is not grounded solely in logical truth. This is
the conclusion Frege himself comes to by about 1925, if we are to take his late
19
diary entries and unpublished manuscripts seriously. The conclusion is again
an epistemological one for Frege: Because arithmetic is about objects, and
because the "logical source of knowledge" is insufficient to "yield us any
objects," arithmetical knowledge turns out not to be simply a species of logical
knowledge.
58
If we accept Frege''s realism about the numbers, then we must agree with
his conclusion that there is more to arithmetic than pure logic. Whether we
should agree in this way is, I think, still an open question. But one thing
which should be clear is that the Fregean conception of reduction is one
which has an epistemological payoff. If, as Frege speculated late in life,
arithmetic is reducible in the Fregean manner to geometry, then the source of
arithmetical knowledge is to be found in our knowledge of the principles of
geometry. For such a reduction would be a demonstration that, properly
analyzed, the truths of arithmetic are themselves truths about geometrical
objects and relations. If on the other hand there is, contra Frege, an analysis
of arithmetic which makes no reference to numerical objects, then the door is
open once again to a Fregean reduction of arithmetic to logic.
59
And such a
reduction would be evidence that the truths of arithmetic are, in something
very like Kant''s sense, analytic.
60
1
Grundlagen 99. Abbreviations used in this paper for Frege''s works are:
Bs (Begriffsschrift): "Begriffsschrift, a formula language, ..." in From Frege
to Godel, van Heijenoort (ed), Harvard 1967, pp 5-82.
CP: Collected Papers, McGuinness (ed); Blackwell 1984.
FG: On the Foundations of Geometry and Formal Theories of Arithmetic,
Kluge (ed), Yale 1981.
Gg (Grundgesetze): The Basic Laws of Arithmetic, Furth (ed), California 1964.
Gl (Grundlagen): The Foundations of Arithmetic, Austin (ed), Blackwell 1980.
PMC: Philosophical and Mathematical Correspondence, McGuinness (ed)
Chicago 1980.
PW: Posthumous Writings, Hermes et al (eds), Chicago 1979.
2
Midwest Studies in Philosophy VI, French, Uehling, & Wettstein (eds);
Minnesota, 1981, pp 17-36. Hereafter, "FLL".
3
Gl 99.
4
Gl 118-9.
5
Gl 3.
6
Gl 4
7
Gl 2
8
“On Mr. Peano''s Conceptual Notation and My Own”, CP 235. See also Gg 3:
“Because there are no gaps in the chains of inference, every ‘axiom’, every
‘assumption’, ‘hypothesis,’ or whatever you wish to call it, upon which a proof
20
is based is brought to light, and in this way we gain a basis upon which to
judge the epistemological nature of the law that is proved.”
9
I.e., to contents of possible judgment, or simply contents in the pre-1890''s
writings (see e.g. Bs pp 11-13); to senses of sentences, or thoughts in the later
writings. See footnote 11.
10
FG 101. Here Kluge has translated ``Satz'''' as ``proposition''''; I take it that the
context makes it clear that in this case ``sentence'''' is preferable.
11
PW 206. See also FG 101: "When one uses the phrase "prove a proposition'''' in
mathematics, then by the word "proposition'''' we clearly mean not a sequence
of words or a group of signs, but a thought; something of which one can say
that it is true." The view of nonlinguistic items as the bearers of
analyticity/provability dates back to the Begriffsschrift, in which the
nonlinguistic judgment is derived, as opposed to its representation “in signs.”
(Bs 28). In the Grundlagen, Frege reports that he has already, in the
Begriffsschrift, “given a proof of a proposition, which might at first sight be
taken for synthetic, which I shall here formulate as follows: ...” (Gl 103). The
proposition proven is neither the Begriffsschrift sentence (formula 133) nor
the Grundlagen sentence which follows this remark; it is the nonlinguistic
counterpart which these purportedly share. The thing which has been
demonstrated to be analytic, that thing "which might at first sight have been
taken for synthetic,'''' is the nonlinguistic item.
12
This is again a view which remains constant from Frege’s early work
through his latest writings. The first definition of the Begriffsschrift is
accompanied with the explanation that “we can do without the notation
introduced by this proposition and hence without the proposition itself as its
definition; nothing follows from the proposition that could not also be
inferred without it. Our sole purpose in introducing such definitions is to
bring about an extrinsic simplification by stipulating an abbreviation” (Bs
§24). In the Grundlagen, a definition “only lays down the meaning of a
symbol” (Gl 78), and in the Grundgesetze: “The definitions ... merely introduce
abbreviated notations (names), which could be dispensed with were it not that
lengthiness would then make for insuperable external difficulties” (Gg 2). See
also Frege’s 1899 letter to HIlbert (PMC 36), and FG 23, FG 24. Note that the
newly-introduced sign will have the same meaning (same content and, later,
same sense and reference) as does the definiens (see e.g. Gg §33, PW 211). This
is to be distinguished from the issue of meaning-preservation raised in the
next section.
Benacerraf (FLL 28) points out the apparent tension between this view
and the Grundlagen view that definitions must be “fruitful” (Gl 81). I take it
that the tension is no more than apparent, as the “definitions” which Frege
takes to be fruitful are the conceptual analyses (see next paragraph) which
precede stipulative definitions and do not occur within proofs, while Frege
consistently observes in his formal work the stipulative and eliminable
character of definitions which do appear in proofs. For further discussion of
fruitfulness and stipulative definitions, see Joan Weiner: "The Philosopher
Behind the Last Logicist" in Frege: Tradition and Influence, Wright (ed),
Blackwell 1984, and Frege in Perspective, Cornell 1990, pp 89-92, and M.
Ruffino: “Context Principle, Fruitfulness of Logic and the Cognitive Value of
Arithmetic”, History and Philosophy of Logic 12 (1991) 185-194. My own
understanding of the fruitfulness requirement has benefitted from an
unpublished paper by Jamie Tappenden (though he would not agree with the
above).
21
13
Gl 99. Frege does not seem to share his successors’ concern that logical
truths involve only “logical” concepts. It would appear that anything
provable using only the self-evident fundamental laws of logic will count as a
(derivative) truth of logic. The purpose of the conceptual analysis (of, e.g. the
concept of number) is not to eliminate “non-logical” concepts, but to present
the logical structure to which one must appeal in giving the proof. See e.g. Gl
4. In any case, concepts which we would count as non-logical will occur only
inessentially in any truths which Frege counts as logical.
14
Gl 5
15
It is of course irrelevant whether the “new” term is typographically
distinguishable from its ordinary counterpart.
16
Gl 13.
17
Gl 18
18
Gl 26
19
The importance for the epistemological project of a semantic connection
between the ordinary and formal discourse is noted by Gregory Currie (“Frege,
Sense and Mathematical Knowledge”, Australasian Journal of Philosophy 60
(1982) 5-19). On Currie’s view, the definitions are intended to preserve sense,
as a means to preserving reference. But as Currie notes, this requires that we
attribute to Frege at least two distinct notions of sense; we are left with no
clear criteria of identity for the relevant one. Regarding reference-
preservation, see below. A second criterion of proposition-identity which
occurs occasionally in Frege (see e.g. letter to Husserl of 9 December 1906) is
that two sentences express the same proposition iff they are logically
equivalent. But note that (a) this criterion would imply that if logicism is true,
there is only one truth of arithmetic; and (b) the criterion cannot be applied,
as the relation of logical equivalence applies in the first instance, for Frege, to
propositions.
20
Frege seems to suggest such a criterion in his 1894 Review of Husserl’s
Philosophy of Arithmetic in response to Husserl''s objection that his
definitions do not preserve (something like) sense. On the strangeness of this
suggestion, see Dummett, Frege Philosophy of Mathematics, Harvard Press 1991
(Hereafter, “FPM”) p 142.
21
Gl 79-80.
22
That is, each numeral is, via these definitions, shorthand for a singular term
which refers to the extension of a concept.
23
Gl 117.
24
Frege often holds that a concept-word occurring in subject-position refers
not to a concept, but to an associated object. (See e.g. "On Concept and Object",
CP 184) If this position is applied to these passages, then one can read Frege''s
proposed substitution of "concept" for "extension of the concept" as a
substitution of one term referring to an object for another term referring to
an object. As Michael Resnik notes ("Frege''s Theory of Incomplete Entities";
Philosophy of Science 1965, pp 329-341; esp. p. 333n), Frege later suggests such
a reading (op cit p 48). Resnik takes the suggestion to imply, further, that the
associated object is the extension of the concept. This stronger claim is also
argued for by Burge ("Frege on Extensions of Concepts", Philosophical Review
XCIII, 1 (Jan 1984)) and Cocchiarella (Logical Studies in Early Analytic
Philosophy, Ohio 1987, pp 76 ff). On this reading, Frege''s proposed change in
the Grundlagen footnote would be merely terminological.
This reading seems doubtful to me, for two reasons. (1) The
identification is sufficiently simple, and important, that had Frege made it, it is
22
reasonable to suppose that he would have been explicit about it. But he
nowhere claims that the "associated objects" are extensions. (2) Following the
proposal, Frege notes that it would be open to the objections that (a) it
"contradicts my earlier statement that the individual numbers are objects",
and (b) "concepts can have identical extensions without themselves
coinciding." The first objection would be badly misguided had Frege taken the
change to be merely terminological, and it is strange that he does not
immediately explain this. The second objection would be beside the point;
again, Frege''s silence on such an easily-remedied matter is at least odd.
Note also that the proposed reading of the Grundlagen footnote will not
help with Frege''s claim that he attaches no "decisive importance" to bringing
in extensions, since on this construal, the alternative definition would have
"brought in" extensions. In any case, there is nothing in Frege''s texts which
necessitates the proposed reading; this is what is important for Benacerraf''s
point.
25
FLL 30.
26
Note e.g. that the claim that every number has a successor must be provable
via the laws of logic.
27
An endorsement of this claim seems to appear in Frege''s 1885 reply to
Cantor''s review of the Grundlagen. See CP 122.
28
Gg 6.
29
Michael Resnik suggests (op cit) that Frege takes the second occurrence of
"the concept F" in the Grundlagen''s definition to refer to the extension of the
concept F, so that this definition in fact assigns the extension of a first-level
concept to each numeral. In this case, as Resnik suggests, the Grundlagen and
Grundgesetze definitions would be "essentially the same." But note that on this
reading we must take each phrase of the form "the concept ..." in the
Grundlagen''s definition of equinumeracy as referring to an extension, so that
equinumerosity is properly a relation between extensions. But this conflicts
with the precise rendering of this definition in the Grundgesetze, §§38-40, in
which "the concept F" and "the concept G" are replaced by "-x?G" and "-z?D",
which are clearly concept-terms. The Grundgesetze''s version of the number
of F''s is the extension of that first-level concept under which fall all and only
the extensions of concepts equinumerous with F, so that even though the
number is the extension of a first-level concept, equinumerosity is still a
relation between concepts. See footnote 49.
30
Or to show that he changed his mind. But as Frege nowhere notes the change
in definition, and nowhere claims that the new definition does a better job (or
even an equally good job) at securing for his numerals the referents of the
ordinary numerals, this seems unlikely.
31
This way of putting it is not quite accurate, since of course if the extensions
of second-level concepts are extensions of first-level concepts, then they have
names in the Grundgesetze. More precisely: the Grundgesetze has no term-
forming operator which applies to the name of a second-level concept and
gives its course of values.
32
For the comprehension principles tell us that the extensions of two concepts
are the same iff the same things fall under those concepts. But if the concepts
in question are of different levels, then the question of the identity of the
"things" falling under the concepts is, on Frege''s own principles, non-
sensical. Specifically: Where F is the relevant first-level concept and F the
relevant second-level concept, the purported comprehension principle would
23
imply: ext(F) = ext(F) iff (x)(Fx iff F(x)). If the range of x includes objects,
then "F(x)" is ill-formed; if it includes concepts, then "F(x)" is ill-formed.
33
See Gg §10.
34
Further, there is as far as I can see no reason that the stipulations in
question ought to be made one way rather than another. (The relevant
question would seem to be which such stipulations would preserve consistency
- but of course this question makes little sense, in light of the inconsistency of
the system as a whole.)
35
FLL 23
36
Gg 4-5.
37
See Gl 18-21, 101-102.
38
The "modelling" interpretation of Frege is not helped by including the
sentences of applied arithmetic in the collection to be satisfied by the model.
For the assignment of reference to numerals in such a way as to model applied
sentences involving finite cardinalities requires only that there be a function
f from first-level concepts to the referents such that for each concept F, f(F) =
the referent of the numeral n iff there are n things falling under F. And the
existence of such a function requires only that the referents do in fact form
an w-sequence.
39
See "On the Foundations of Geometry" (1906), in FG pp 49 - 112.
40
op. cit.
41
Weiner''s positive argument for the view that ordinary numerals lack
bedeutung is, I think, problematic in its insistence that Frege''s requirement of
sharp delimitation and total definition of concepts is applicable to ordinary
language. These requirements are better seen as constraints on the formal
language; they are in particular required for rigor in that the failure of these
constraints will result in syntactically well-formed names which lack
reference. This will make it impossible to give purely syntactic rules of
inference which are literally truth-preserving. Another reason to doubt that
the ordinary numerals lack bedeutung is that if they do, then it is difficult to
understand those arguments of Frege''s which turn on the fact that the
numerals are singular terms. The term "Odysseus" is a singular term in the
sense in which Weiner would construe the term "two", but Frege would not
take the occurrence of the former as evidence of any kind of realism. In
short: Frege''s remarks about the functioning of number-words in ordinary
language only carry the weight they need if the sentences which embed them
have truth-values.
42
This is pointed out by Dummett (at e.g. FPM 30-31, 143). On Dummett’s
account, Frege’s definitions are (or ought to be) intended to “come as close as
possible to capturing the existing sense.” Dummett cashes out this criterion in
terms of a number of conditions (see esp. FPM 152-3) which if met ensure that
the system of definitions “comprises everything that must be implicitly
known by anyone who understands all those expressions” (FPM 154). But
there is more to be said. What advantage do definitions in terms of extensions
have over definitions in terms e.g. of geometrical constructions by way of
“comprising everything that must be implicitly known” about the numbers?
The following is in large part an attempt to answer this further question. If
the account in this section is correct, then it seems that one need not attribute
to Frege a conception quite as “holist” as does Dummett (FPM 32, 154). But
further pursuit of this issue must be postponed.
43
The claim occurs in Gl 58-61, 67. The quoted remark occurs at Gg 5.
44
Gl 59
24
45
Gl 67
46
Logically equivalent, that is, on Frege’s problematic view of logic. The
inconsistent comprehension principle is essential here.
47
It is interesting to note that in his last writings, Frege seems to hold that his
earlier view of finite cardinals as fundamental was mistaken. See "Numbers
and Arithmetic" (1924/25); PW 275-7.
48
That is: for every finite concept F, there is a concept G and an object a falling
under G such that the number which belongs to the concept "falling under G
but not identical with a" is the same as the number which belongs to the
concept F.
49
The Grundgesetze''s version of the number which belongs to the concept F,
i.e., eF(e) is, by definition of and using modern notation for the quantifier
and conjunction:
(1) a($q)(eF(e) ? (a ? > q) & a ? (eF(e) ? > q)) (§ 40).
I.e., it is the extension of a concept under which an object a falls iff:
(2) ($q)(eF(e) ? (a ? > q) & a ? (eF(e) ? > q)).
Recall that for a function F (or, as Frege would write it, F(z)), eF(e) is that
function''s course of values (cov). Where F is a concept, this cov is the
extension of F. The symbol "?" is defined (§ 34) in such a way that a?u = G(a),
where G is "the u-function", i.e., the function whose cov is u. (Note that, in
particular, if u is the extension of a concept G, then a?u = T iff a falls under G.)
Where f is the q-relation (i.e., q=aef(ea)), eF(e) ? (a ? > q) above expands
(using modern terminology for the material conditional) to:
(e)(d)(f(e,d) … (c)(f(e,c) … d=c)) & (d)(F(d) … ($c)(f(d,c) & c?a))
(§§ 34, 36-39).
Similarly, a ? (eF(e) ? > q) expands to:
(e)(d)(f(d,e) … (c)(f(c,e) … d=c)) & (d)(d?a … ($c)(f(c,d) & F(c))).
Here, "c?a" will refer to y(c), where y is the function such that a=ey(e).
Thus (2) says that:
(3) ($q)(the q-relation maps the a-concept one-one onto F),
where by "the q-relation" we mean the relation whose cov is q, and by "the a-
concept", that concept whose extension is a.
So (1) names the extension of the concept under which fall all and only the
extensions of those concepts which can be mapped 1-1 onto F.
50
Unless, contra the pre-1920''s Frege, the ordinary claim is at bottom
geometrical.
51
That is, they share this view with respect to direct proofs. The view that a
demonstration of grounds is the primary purpose of proofs explains Kant’s and
Frege’s shared mistrust of indirect proofs, and Frege’s view that premises and
axioms must always be true. For a discussion of the legacy of the rejection of
proofs by contradiction, see Paolo Mancosu’s forthcoming Philosophy of
Mathematics and Mathematical Practice in the Seventeenth Century, Oxford
University Press.
52
I suspect that this prima facie difference is not as striking as it at first
appears, as Frege does not seem to share Kant’s notion of “ampliative”. But I
leave this question for another time.
53
That Frege’s analytic/synthetic distinction is essentially Kant’s has been
argued by Philip Kitcher in “Frege’s Epistemology” (Philosophical Review 88
(1979) 235-262). The opposite has been argued by Newton Garver in
“Analyticity and Grammar” (Kant Studies Today, L.W. Beck (ed), Open Court: La
Salle 1969).
25
54
Prolegomena to Any Future Metaphysics, Beck translation (Liberal Arts Press
1950), p. 15
55
See e.g. Gl 20-21, in which the fact that the truths of geometry can be denied
without self-contradiction is a mark of their synthetic nature. See also Gg 15.
I have recently seen a similar emphasis on contradiction in Kant and Frege in
an unpublished paper by Jamie Tappenden.
56
"Function and Concept", CP 42.
57
Diary entry of March 28, 1924; PW p. 263.
58
PW 278-279; manuscript of 1924/25.
59
For such an analysis, see Harold Hodes: “Logicism and the Ontological
Commitments of Arithmetic”, Philosophical Studies 41 (1982) 161-178.
60
Many of the ideas in this paper have been presented in colloquia to a
number of departments of philosophy; thanks to many members of these
audiences for helpful discussions and criticism. I would particularly like to
thank David Schmidtz, Michael Resnik, John Etchemendy, Paolo Mancosu,
William Demopoulos, Andrew Irvine and Gottfried Gabriel for helpful
discussion and/or comments on earlier versions.
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