At 11:10 PM 1/11/99 -0500, Michael Detlefsen wrote:
[GRAY:]
>Jeremy Gray wrote:
>
>+++++++++++++++++++
>
>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
>
>++++++++++++++++++++++
>
[DETLEFSEN:]
>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 fact that the *Principia Mathematica* of Russell and Whitehead never
got to a projected volume 3 on the foundations of geometry, together with
the fact that Whitehead later (I think it was later) wrote a Cambridge
series little book on differential geometry shows something about Russell's
attempts to include geometry under the scope of his logicism. Am I right?
I suggest that a similar fate befell Bourbaki, whose series, as I recall,
never got to differential geometry.
>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)
>
Is it fair to say using biological terms (but not too rigorously) that
Frege thought that a concept (notion, idea, neural structure) of
"magnitude" is inherent in humans at conception, or inevitably develops in
them at some early age, barring unusual abnormalities? Is the use of the
slippery word "intuition" or "Anschauung" here meant to refer to a
component of sense perception or other non-inevitable experience in the
formation of a concept of "magnitude"?
If so, are the basic laws of arithmetic as exemplified in some set of
axioms (e.g., those of Peano) put forth as a description of inherent
operations of the human body, especially the brain?
>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."
>
Ah ha! This seems to confirm to some degree my suggestions above. So, in
Kant's terminology, I suppose, "arithmetic" as practiced by humans is
analytic, or leads to analytic judgments. Is it fair to say then that
these are given by our biological structures, and that "result of built-in
structures of normally functioning human nervous systems (or at least, a
great many human nervous systems) is a reasonable translation of "given
directly to our reason"?
>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.).
>
Does "deeper" here ("tiefer"?) refer to the fact that arithmetic judgments
are the result of a built-in feature of humans unaffected, and perhaps
unaffectable, by sense experience? On another point, does Frege have in
mind here problems that arise when one tries to square what one senses "out
there" with what one knows about the implications of axioms for a
non-euclidean geometry, as obtained by rationally tinkering with a set of
axioms which presume to express how normal brains function (e.g., Euclid,
Hilbert, etc.)?
>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.
>
So would it be fair to say that Frege regarded "logic" as a built-in,
inevitable feature of human brains, alongside "arithmetic"? Would he say
something similar about non-aristotelian logics that he seems to have said
about non-euclidean geometries, as I suggested above?
>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.
Here it appears that sense, intuition and reason are separate. Does this
suggest that intuition is some kind of combination of sense and reason,
maybe even with something else thrown in?
>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).
>
What is the status of the term "one" in "x is one"? Does it mean something
like "x is not made of parts"? What function(s) of human brains is (are)
involved in deciding that that "x is one" is true or false (or something
else, e.g. probable, etc.)?
>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.
>
Is it fair to translate "if x is one and y is one, and x and y are
different, then x and y are two" as "if x has not parts and y has no parts,
and x and y are different, then x and y considered as a unit has two
parts"? Is the use of the word "two" here by way of a definition of "two"?
If not, there would appear to be some sort of circularity involved, no?
>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.
>
Mathematicians often like to generalize, but don't they usually mean
something extending a proposition which applies to some structures to apply
to a larger class of structures? How, if at all, is this kind of
generalization related to the kind of generalization which results in
propositions with variables in them? Is this latter kind of generalization
meant to describe how the process of the former kind of generalization
actually works in humans?
>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'.
>
So, what happened to geometry?
>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.
>
How are ubiquitous constants arrived at? Is the concept of a "straight
line" a ubiquitous constant? No, I guess not, since of course, "straight
lines" in a euclidean geometry are different from "straight lines" in
elliptic or hyperbolic geometries. Are the only ubiquitous constants in
mathematics ordained by Russell to be those of logic, like conjunction,
disjunction, equality, non-equality, etc.? If so, how do the senses and
intuition interact with these "constants of reason"? Where do "circles"
come from? Of course, we could claim that when "mathematicians" deal with
circles, they are doing "applied" rather than "pure" mathematics, or
perhaps doing "physics". But I would claim there are very few
mathematicians in the world, in the usual sense of "mathematicians" (and
equivalents or near-equivalents in other languages), who don't consider
geometry as a part of mathematics. It appears that some logicians think or
thought so, too, e.g. Frege. Maybe we should talk about "arithmeticians"
as distinguished from "mathematicians"?
>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.
>
Rejection of Kantian intuition as the basis of all mathematical knowledge,
or just arithmetical knowledge? I thought you said above that Frege
believed "intuition" was involved in geometrical knowledge, which I take to
be included in mathematical knowledge?
>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]).]
>
But Bertie, where's the beef? I mean, where's the geometry? I somehow
thought that Kant could be interpreted as saying that we humans *perceive*
in a euclidean manner (which may or may not be true), but not that
non-euclidean geometries developed by a combination of reason and our
perceptions (analogy, etc.) were out of the question?
>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.).
>
Well, I suppose Hilbert in referring to "ideal ideas" (if I may call them
this) is here speaking of the sort of thing I was speaking of above, which
I called (loosely) built-in concepts (or neural patterns, or whatever).
What in Kant did Hilbert refer to as "anthromorphic rubbish"? Some alleged
view of Kant that there could be one and only one study worthy of the name
"geometry", namely euclidean geometry? As I indicated above, I thought
Kant was only saying that there is one and only one kind of geometry which
we can arrive at through a combination of sense perception, and something
or the other to do with our Vernunft (reason) and/or Versta"ndnis
(understanding), I forget now what.
>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.
>
So ideal sentences according to Hilbert are those we can't help having --
if we're "normal" enough? We need to use them as tools in mathematicssince
we have no others which aren't, so to speak, tainted by our experiences in
our worlds? So the foundations of mathematics, according to Hilbert,
should be put forth in uninterpreted axioms, using something like logical
constants in their formulation, but, to avoid being like Russell, we
shouldn't think of the non-constants, the uninterpreted symbols in the
axioms, as variables? Or what?
>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.).
>
Ah, subjective! (But "merely"? Come now!) Or, as I've said above,
built-in? Somehow, the distinction between Russell's logicism and
Hilbert's formalism is becoming blurred in my mind!
>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
>
>
How do ideas other than transcendental ones "determine an object"? Does
this mean that our built-in neural processes act on what they are applied
to, i.e. our experiences?
>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).
>
Ah! So we have to be careful that we haven't described or axiomatized our
neural processes in some reprehensible way? What sort of thing is the
"real evidence" referred to here? Something to do with consistency or
coherence or non-contradiction?
>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).
>
In Hilbert's view, what constituted the departure of "real" mathematics
from its "optimal classical logical state"? Intuitionism a la Brouwer?
Something to do with set theory? Or what?
>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:
>
Does this mean even the logical operators, which I take it is what Russell
considered as logical "constants" are uninterpreted in the way "points" and
"lines" are in the axioms of geometry (according to Hilbert)?
>"...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.
>
Well, I see I've anticipated you. So a major difference between Russell's
so-called logicism and Hilbert's so-called formalism lies in the status of
the logical "constants" (Russell) or "signs" (Hilbert)? Which reminds me,
didn't Hilbert say somewhere something to the effect that the basis of
mathematics, in the end, lies in the intuition (I suppose "Anschauung") of
mathematicians? Was it that "popular" book he wrote with Cohn-Vossen?
This would help explain how he could formulate *completely* uninterpreted
axioms -- that is what he hoped to be doing, isn't it?
>(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.
>
Please do elaborate on why Goedel's theorems don't point to a failure of
some major part of Hilbert's program, and more especially why Hilbert's
program *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?
>
>>From snow-packed, sub-zero South Bend,
>
>Mic Detlefsen
>
>
>
>**************************
>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
>**************************
>
>
>
Best wishes from Harrisonburg VA where the weather is somewhat better
Gordon Fisher gfisher@shentel.net