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Dear All,
Advice please - if you have not already debated this at length.
I take the historians party line on logicism (in the spirit of Russell) to
be this: the reduction of mathematics to logic fails. Hilbert tried to
establish logic as a part of mathematics, but in the end Gvdel destroyed
Hilberts programme, although what remained became the orthodoxy among
philosophers of mathematics. A number of stories are mixed in with this one,
such as axiomatics and intuitionism.
At its simplest, such a story blurs essential distinctions, such as those
between first and second order logic and set theory, and syntax and
semantics. The question of what has been established, and what historical
effect this had, is blurred (at least in my eyes).
My question is: What are the best historical accounts of the logicist
debates in the foundations of mathematics, 1900-1940?
I am looking for something that attains reasonably high standards in these
respects:
the treatment is comprehensive, the analysis of claims is detailed and
precise, the level of historical argument is high.
My impression is that there is an older historical literature, which is a
mixture of many high-level technical papers and some books that give the
party line. Recently there have been a number of fine studies that go more
carefully over the ground (eg, on Peirce, Russell, Gvdel, and Hilbert). But
I know of no account that integrates these separate accounts, and I find
myself unable to tell myself a satisfactory story about these events. It is
very likely that I am missing some good work, quite probably by members of
this list. Whence my question.
With best wishes, and thanking you in advance for your help
Jeremy Gray
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I have some remarks to make as regarding this. First some small (?) points
that present some basic distinctions.
(1) It is important to separate the positions of differnt thinkers who have
gone by the title 'logicist'. Frege, for example, was a logicist only as
regards arithmetic. He believed that Kant was wrong only with regards to
arithmetic: he found arithmetic reasoning to be too ubiquitous to rational
thought to plauibly be regarded as synthetic a priori in character. To
account for its ubiquity, one had to see it as analytic (i.e. logical) in
nature. He agreed with Kant as regards the nature of geometry (viz. that it
was synthetic a priori in character, and, hence, not properly reducible to
logic. Russell avowed a more encompassing form logicism. It included not
only arithmetic but also geometry (and even parts of mechanics).
The basics of Frege's viewpoint were evident in his writings from the very
start. Thus, already in his 1873 doctoral dissertation, he emphasized that
"the whole of geometry rests, in the final analysis, on principles that
derive their validity from the character of our intuition" (cf. Frege
[1873], p. 3, my translation). And, in his 1874 Habilitationsschrift, he
expanded this observation to include his view of the relation between
geometry and arithmetic vis a vis their dependency on intuition.
"It is quite clear that there can be no intuition of so pervasive and
abstract a concept as that of magnitude (Größe). There is therefore a
noteworthy (bemerkenswerter) difference between geometry and arithmetic
concerning the way in which their basic laws are grounded. The elements of
all geometrical constructions are intuitions, and geometry refers to
intuition as the source of its axioms. Because the object of arithmetic is
not intuitable, it follows that its basic laws cannot be based on
intuition." (again, my translation)
The same basic point concerning the "unintuitedness" of the objects of
arithmetic is made in §105 of the _Grundlagen_, where Frege remarks that
"In arithmetic we are not concerned with objects which we come to know as
something alien from without through the medium of the senses, but with
objects given directly to our reason and, as its nearest kin, utterly
transparent to it."
The same basic contrast between geometry and arithmetic is drawn in §§13,
14 of the _Grundlagen_. There Frege broached the question of the relative
places occupied in our thinking by empirical, geometrical and arithmetical
laws. His conclusion is that arithmetic laws are deeper than geometrical
laws, and geometrical laws deeper than empirical laws. He arrives at this
conclusion by conducting a thought experiment in which he considers the
cognitive damage that one might expect to be done by denying each of the
various kinds of laws. Denying a geometrical law, he concludes, stands to
do more extensive damage to a person's cognitive orientation than denying a
physical law. For it would lead to a conflict between what the person can
conceive and what she can spatially intuit. It would bring severe
disorientation to a person's cognition. It would force her, for example, to
deduce things that she formerly had been able just to "see". And it would
even make the deductions strange and unfamiliar. It would not, however,
result in a global breakdown of her rational thinking. Such global
breakdown in rational functioning is rather the characteristic effect of
denial of an arithmetic law. Denying an arithmetic law, Frege maintained,
would not only keep one from seeing what he had formerly been able to see.
It would as well prohibit his engaging in deduction or reasoning of any
sort. In Frege's words, it would bring about "complete confusion", so that
"even to think at all would seem no longer possible" (loc. cit.).
Frege believed that this projected global breakdown in rational thought
upon denial of an arithmetic law required seeing arithmetic law, unlike
physical and geometrical law, as universal. It pertains not only to that
which is physically actual and that which is spatially intuitable, but,
indeed, all that which is numerable - and that, according to Frege, was the
widest range possible, extending to all that which is in any coherent way
thinkable or conceivable. The laws of arithmetic must, therefore, he
concluded, "be connected very intimately with the laws of thought" (loc.
cit.) - that is, with the laws of logic. Hence his logicism.
Russell's logicism was quite different. In the first place, it was not
motivated primarily by the discovery of non-Euclidean geometries with its
attendant belief in the epistemological asymmetry between geometry and
arithmetic. Nor was it based on belief in such things as logical objects,
and the associated division of cognition into faculties of sense, intuition
and reason. Nor, finally, as mentioned above, did it restrict itself to
arithmetic. It took as its starting points (i) a certain general definition
of mathematics, (ii) a methodological principle to pursue ever further
generalization in science, and (iii) a belief that pursuing this principle
in mathematics would eventually lead one to a most general science of all,
namely, logic. It was fueled in these pursuits by the then rapid and
impressive advances in symbolic logic.
In the opening paragraph of _The Principles of Mathematics_ (cf. p. 3),
Russell offered the following definition of pure mathematics: "Pure
mathematics is the class of all propositions of the form "p implies q", and
neither p nor q contains any constants except logical constants." He then
went on to describe his logicist project as that of showing "that whatever
has, in the past, been regarded as pure mathematics, is included in our
definition, and that whatever else is included possesses those marks by
which mathematics is commonly though vaguely distinguished from other
studies." (ibid.). Russell also maintained that, in addition to asserting
implications, propositions of pure mathematics are characterized by the
fact that they contain variables (cf. [1903], p. 5), indeed, variables of
wholly unrestricted range (cf. [1903], p. 7).
Russell planned to defend this last claim, which he acknowledged as being
highly counter-intuitive, by showing that even such apparently
variable-free statements as "1+1=2" can be seen to contain variables once
their true meaning and form is revealed. The discovery (or, better,
recovery) of the true meaning and form of such statements was made possible
by the vast enrichment of the basic stock of logical forms made available
through the work of Peirce, Schröder, Peano and Frege. Using this work,
Russell produced analyses of the deep forms of ordinary mathematical
statements. "1+1=2", for example, was analyzed as "If x is one and y is
one, and x differs from y, then x and y are two". Analyzed in this way,
Russell maintained, the apparently non-implicational, variable free "1+1=2"
is seen both to contain completely general variables and to express an
implication, just as his logicist theory predicted would be the case (cf.
[1903] , p. 6).
Of course, "if x is one and y is one, and x and y are different, then x and
y are two" does not express a genuine proposition at all since it contains
free variables. It expresses instead what might be called a
proposition-form or a proposition-schema. Russell called it a "type of
proposition", and went on to say that "mathematics is interested
exclusively in _types_ of propositions" (loc. cit., p. 7) rather than in
individual propositions per se. On this view, the business of mathematics
is to determine which propositions can be generalized (i.e. which constants
can be turned into variables), and then to carry this process of
generalization out to its maximum possible extent (cf. [1903], pp. 8, 9).
This maximum will have been reached when we have penetrated to a level of
propositions whose only constants are logical constants and whose only
undemonstrated propositions are the most basic truths whose only constants
are logical constants (cf. [1903], p. 8). The logical constants
themselves, as a class, were to be characterized only by enumeration.
Indeed, by their very nature they admitted only of this kind of
characterization, since any other kind of characterization would be forced
to make use of some element of the class to be defined.
At bottom, then, Russell's logicism was motivated by a view of mathematics
that saw it as the science of the most general formal truths; a science
whose only indefinables are those constants of rational thought (the
so-called logical constants) that have the widest and most ubiquitous usage
and whose only indemonstrables are those propositions which set out the
most basic properties of those indefinable terms (cf. [1903], p. 8). In
his view, this provided the only precise description of what philosophers
have had in mind in describing mathematics as an a priori science (cf.
[1903], p. 8). Mathematics is thus in the business of generalization. Its
aim is to identify those truths that remain true when their non-logical
constants are replaced by variables (cf. [1903], p. 7). This process of
generalization may require some analysis in order to find the genuine form
of the sentence to be generalized. But once that form is found, the
generalization process should ultimately lead to the realization that the
mathematical truth in question expresses a formal truth whose variables are
completely general and whose only constants are logical constants.
Ideally, proper method in mathematics requires pursuit of this process of
generalization to the ultimate degree. At that point, Russell believed, we
will find formal truths of maximum generality - truths of a generality so
great as to render them incapable of further generalization - truths, that
is, that are so general that they would become non-truths were any of their
constants to be replaced, even through conceptual analysis, by variables.
This, in Russell's opinion, was the only point at which the method of
mathematics (i.e. the pursuit of maximal formal generalization) can
properly and naturally be brought to a close. He also believed that it is
in this domain of formal truths of the utmost generality, and in this
domain alone, that we can rightly expect to meet what are properly regarded
as 'laws of logic'.
According to Russell, these laws are justified inductively from their
consequences.
"...in mathematics, except in the earliest parts, the propositions from
which a given proposition is deduced generally give the reason why we
believe the given proposition. But in dealing with the principles of
mathematics, this relation is reversed. Our propositions are too simple to
be easy, and thus their consequences are generally easier than they are.
Hence we tend to believe the premises because we can see that their
consequences are true, instead of believing the consequences because we
know the premises to be true. ... thus the method in investigating the
principles of mathematics is really an inductive method, and is
substantially the same as the method of discovering general laws in any
other science."
Thus, contrary to what Kant had maintained, the pursuit of greater
generality (or what Kant referred to as "unification") has a natural and
fairly inevitable stopping point; specifically, that level of judgments
having broad scope, entirely general variables and utterly ubiquitous
constants.
Frege and Russell, then, though they agreed in their rejection of Kantian
intuition as the basis of mathematical knowledge, nonetheless differed with
regard to their estimates of the proper scope of logicism and the nature
and origins of its basic laws. They also differed on the important
question of our knowledge of the infinities with which mathematics deals,
and how, exactly, that knowledge is related to our knowledge of concepts.
I'll save this for another time ... as I am already going on too long here.
For Frege, the chief historical development leading up to logicism was the
advent of non-Euclidean geometries, with their suggestion of a fundamental
epistemological asymmetry between arithmetic and geometry. The second major
factor contributing to the weakening of Kant's influence in 20th century
philosophy of mathematics was the dramatic development of logic during the
latter part of the 19th and the early part of the 20th centuries. This
included the introduction of algebraic methods by Boole and DeMorgan, the
improved treatment of relations by Peirce, Schroeder and Peano, the
replacement of the Aristotelian analysis of form based on the
subject-predicate relation with the more fecund analysis of form based on
Frege's general notion of a logical function, and the advances in
formalization brought about by the introduction (by Frege, Russell and
Whitehead, and Peano) of precisely defined and managed symbolic languages
and systems.
These developments took logic to a point well beyond what it was in the
time of Kant, and this caused some to judge that it was the relatively
under-developed state of logic in Kant's time that was primarily
responsible for his belief in the need for a synthetic basis for
mathematical judgment and inference. Russell, for one (cf. [1903],
[1907], [1919]), took such a position, arguing that though Kant's views may
have seemed reasonable given the sorry state of logic in his day, they
would never have been given a serious hearing had our knowledge of logic
been then what it is now. [N.B. But though Russell saw the enrichment of
the analysis of logical form brought about by the modern logic of relations
and the functional conception of the proposition as being of particular
importance to the correction of Kant's deficiencies, he also believed that
certain developments in mathematics proper were of great importance. Chief
among these were (i) the arithmetization of analysis by Weierstrass,
Dedekind and others, and (ii) the discovery by Peano of an axiomatization
of arithmetic. These led to what Russell regarded as a codification of
pure mathematics within a certain axiomatic system of arithmetic (viz.
2nd-order Peano Arithmetic), and so provided for its "logicization".
Russell reckoned the significance of these developments for Kant's
philosophy of mathematics to be as great as that of the discovery of
non-Euclidean geometries (cf. [1903], [1907]).]
This, in brief, is how the two main logicists differed. There were
differences with other logicists too (e.g. Dedekind and Peano), but I'll
save this for another time.
Now for some larger points ...
(2) You seem to count Hilbert as a logicist (perhaps because of Poincare's
miscategorizations of him as such in various of his writings?), but I think
this is pretty far from the truth. Indeed, he himself claimed that logicism
could not offer a satisfactory foundation for mathematics. Nor, as you also
seem to suggest, was Hilbert's 'Program' to show that logic is part of
mathematics. Hilbert opposed Kant's particular ideas in the philosophy of
mathematics (indeed, he characterized them as 'anthropomorphic rubbish' at
one point). He placed great emphasis, however, upon the chief structural
feature of Kant's general critical epistemology (as opposed to Kant's
particular epistemology for mathematics).
Of particular importance here is Kant's distinction between genuine
judgments and regulative ideals. Hilbert took this distinction as the
basic model for his division of classical mathematics into a real and an
ideal part. The real propositions and proofs were taken to be the genuine
judgments and evidence of which our knowledge is constituted. Ideal
propositions, on the other hand, though they served to stimulate and guide
the growth of our knowledge, were nonetheless not considered to be a part
of it. They did not describe things that are "present in the world" (cf.
[1925G(erman)], p. 190). Nor were they "admissible as a foundation of that
part of our thought having to do with the understanding (in unserem
verstandesmaeßigen Denken)" (cf. [1925G], p. 190). They corresponded
instead to ideas "if, following Kant's terminology, one understands as an
idea a concept of reason which transcends all experience and by means of
which the concrete is to be completed into a totality" (ibid.).
Hilbert's ideal sentences are therefore not to be likened to the indirectly
verifiable "theoretical sentences" of a realistically interpreted
scientific theory familiar to us from logical empiricist epistemology.
Rather they are to be interpreted instrumentalistically, as having the same
general regulative function as Kantian ideas of reason. The objects and
states of affairs described in the "theoretical sentences" of a
realistically interpreted science clearly do not "transcend all
experience". Kant's ideas of reason, on the other hand, do.
Hilbert's ideal propositions thus function as regulative devices. They do
not "prescribe any law for objects, and [do] not contain any general ground
of the possibility of knowing or of determining objects as such" (Kant
[1787], p. 362). Rather, they are "merely subjective law(s) for the
orderly management of the possessions of our understanding, that by
comparison of its concepts it may reduce them to the smallest number"
(ibid.).
Hilbert also followed Kant in maintaining that the use of ideal methods
should be _epistemically conservative_. They should, that is, be only more
efficient means of producing real judgments which could, nonetheless, in
principle (though less efficiently) be developed through the exclusive use
of real methods. As Kant put it:
"Although we must say of the transcendental concepts of reason that they
are only ideas, this is not by any means to be taken as signifying that
they are superfluous or void. For even if they cannot determine any
object, they may yet, in a fundamental and unobserved fashion, be of
service to the understanding as a canon for its extended and consistent
employment. The understanding does not thereby obtain more knowledge of
any object than it would have by its own concepts, but for the acquiring of
such knowledge it receives better and more extensive guidance."
Kant
[1787], p. 385
Similarly in Hilbert. Ideal methods, he said, play an "indispensable" and
"well-justified" role "in our _thinking_" (cf. [1925], p. 372, emphasis
Hilbert's). They should not, however, be permitted, to generate any real
result that does not agree with the dictates of real evidence itself (cf.
[1925], p. 376; [1927], p. 471) . Their role is rather that of enabling us
to retain in our reasoning those patterns of inference in terms of which we
most readily and efficiently conduct our inferential affairs (cf. [1925],
p. 379; [1927], p. 476).
These patterns are the patterns of classical logic. Thus, Hilbert's
introduction of the so-called ideal elements was ultimately for the sake of
preserving classical logic as the logic of our mathematical reasoning.
Introduction of ideal methods was made necessary by the fact that there
exist certain real propositions (referred to by Hilbert as _problematic_
real propositions) that do not abide by the principles of classical logic.
By this it is meant that when these propositions are manipulated by the
principles of classical logic, they produce conclusions that are not real
propositions. In order to obtain, then, a system that both contains the
real truths and also has classical logic as its logic, Hilbert believed it
necessary to add the _ideal_ propositions. He also believed this to be the
minimal modification of real mathematics necessary to restore it to its
epistemically optimal classical logical state (cf. Hilbert [1925], pp.
376-79; [1927], pp. 469-71).
However, in thus restoring mathematical reasoning to its classical logical
state, Hilbert observed that the logical operators were no longer being
conceived of and employed in a semantical or contentual way as expressions
for operations on meaningful propositions. Rather, they were being used in
a purely syntactical way as part of a larger computationo-algebraic device
for manipulating formulas. As he put it:
"...we have introduced the ideal propositions to ensure that the customary
laws of logic again hold one and all. But since the ideal propositions,
namely, the formulas, insofar as they do not express finitary assertions,
do not mean anything in themselves, the logical operations cannot be
applied to them in a contentual way, as they are to the finitary
propositions. Hence, it is necessary to formalize the logical operations
and also the mathematical proofs themselves; this requires a transcription
of the logical relations into formulas, so that to the mathematical signs
we must still adjoin some logical signs, say '&', 'v', '-->' and '~'."
Hilbert [1925], p. 381
We thus find here a final step of abstraction from meaning in Hilbert's
ideal mathematics - namely, abstraction from the meanings of the logical
constants. It was made necessary by the decision to preserve the
psychologically natural laws of classical logic as the laws of mathematical
reasoning; a decision which, in turn, was the result of trying to preserve
the most effective "canon" available to us for the development of our real
mathematical judgments. Ultimately, then, this "formalism" of Hilbert's,
with its radical abstraction from meaning, derived from his Kantian
conception of the distinction between the real and ideal propositions
according to which he saw the cognitive or epistemic value of the ideal
elements as residing in their utility as instruments for extending our real
judgments.
(3) You say that Goedel showed Hilbert's Program to fail. I disagree ...
and have written a book and several papers arguing the matter. However,
I'll not go into that now ... though I'd be willing to pursue it later ...
if you're interested. The crux of my argument is a detailed analysis of the
proofs of Goedel's theorems. I believe such analysis shows them not to
refute Hilbert's Program ... though this failure does not point to any
POSITIVE idea for executing Hilbert's Program. Thus, though I do not
believe that Goedel's theorems refute Hilbert's Program, I do not claim to
know a way of carrying out Hilbert's Program. I view this as a dilemma
since I think that something like Hilbert's Program (especially with its
distinction between real and ideal methods in mathematics) must be a part
of any successful philosophy of mathematics.
(4) Finally, you ask for sources for further reading ... reading that
attempts to 'integrate' things pertaining to logicism, Hilbert, etc. in an
historically responsible way. Though it's lacking proper modesty, may I
suggest my own (long) essay
PHILOSOPHY OF MATHEMATICS IN THE 20TH CENTURY, in vol. 9 of the Routledge
History of Philosophy (pp. 50-123)
as well as Howard Stein's fine essay
LOGOS, LOGIC AND LOGISTIKE: SOME PHILOSOPHICAL REMARKS ON THE 19TH CENTURY
TRANSFORMATION OF MATHEMATICS in Aspray and Kitcher (eds), History and
Philosophy of Modern Mathematics (U of Minnesota Press, 1988)
and his informative, well-written entry on logicism in vol. 5 (pp. 811-17)
of the new Routledge Encyclopedia of Philosophy?
Mic Detlefsen
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Michael Detlefsen
Department of Philosophy
University of Notre Dame
Notre Dame, Indiana 46556
U.S.A.
e-mail: Detlefsen.1@nd.edu
FAX: 219-631-8609
Office phone: 219-631-7534
Home phone: 219-232-7273
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