FOM: foundations as a search for certainty
Michael Detlefsen
Detlefsen.1 at nd.edu
Fri Oct 2 14:22:16 EDT 1998
This is a note concerning the recent line of discussion on the three 'isms'
or modern f.o.m.. I believe that they are best understood if their
connection with the discovery of non-Euclidean geometry is made clear. This
is also true of Hilbert's formalist ideas though the connection there is
more complex.
Mainly, what I want to say is that contrary to what the recent
contributions of Reuben and Steve (and certain others) suggest, certainty
was not the main issue with which the big 'isms' were concerned. They were,
I believe, working on far more subtle and interesting projects. I also
don't believe that certainty is (or at least should be) the chief concern
of foundations of mathematics today. I'll leave the argument for that to
another occasion, however, and concentrate on trying to straighten out some
serious misunderstandings of the above-mentioned foundational projects.
I start with some basic historical background.
In the CRITIQUE OF PURE REASON Kant presented a view of mathematical
knowledge which asserted a certain 'symmetry' between arithmetic and
geometry--to wit that they were both based on synthetic a priori intuition.
Not long after Kant elaborated his views, mathematicians expressed doubts
concerning them. Gauss, for example, clearly stated his doubts concerning
the a priori character of geometry in an 1817 letter to Olbers (cf. Gauss
1976, pp. 651-2). He restated the same view in an 1829 letter to Bessel
(cf. Werke, VIII, p. 200) and added that it had been his view for nearly 40
years. Here's what he said (my translation):
"My innermost conviction is that geometry has a completely different
position in our a priori knowledge than arithmetic. ... we must humbly
admit that, though number is purely a product of our intellect, space also
has a reality external to our intellect which prohibits us from being able
to give a complete specification of its laws a priori."
So, where Kant stated the epistemological symmetry of geometry and
arithmetic, Gauss hypothesized their epistemological ASYMMETRY. The
subsequent discovery of non-Euclidean geometries confirmed this hypothesis.
THIS ASYMMETRY BECAME THE CHIEF PHENOMENON TO BE DEALT WITH IN A
SATISFACTORY FOUNDATION (=PHILOSOPHY) OF MATHEMATICS. The management of
this issue of the symmetry or asymmetry between arithmetic and geometry
became the (or at least a) major preoccupation of the work of the logicists
Frege and Dedekind (though not Russell), the intuitionists Brouwer and Weyl
and the so-called formalist Hilbert.
In the main, two basic types of reactions to the asymmetry thesis emerged.
These corresponded to the two basic ways of accommodating it. One was to
retain a Kantian conception of arithmetic (as based on an a priori
intuition of time) and adopt a non-Kantian conception of geometry (i.e. a
conception which saw geometry as based on something other than a priori
intuition (of space)). The other was to take a non-Kantian view of
arithmetic (i.e. a view of arithmetic as being based on something other
than a priori intuition (of time)) while retaining a Kantian conception of
geometry (as based on a priori intuition of space). The former reaction was
essentially that of the intuitionists Brouwer and Weyl. The latter, on the
other hand, became the central idea motivating the logicism of Frege and
Dedekind. Hilbert's views are more complex. He maintained both the
epistemological symmetry of arithmetic and geometry and their fundamentally
a priori character. He rejected, however, Kant's proposed a priori
intuitions of space and time as their bases (calling them 'anthropomorphic
rubbish').
FREGE'S LOGICISM
Frege's logicism (which one must always be careful to distinguish from
Russell's very different logicism) took this asymmetry as the fundamental
feature of mathematics that an adequate philosophy of mathematics must
explain. It was not (as Steve, Ruben and some others have implied) intended
to put mathematics on a certain footing. Rather, it was intended to explain
what Frege and others regarded as an otherwise very puzzling
phenomenon--namely, the asymmetry of geometry and arithmetic.
In his 1873 doctoral dissertation, Frege emphasized that "the whole of
geometry rests, in the final analysis, on principles that derive their
validity from the character of our intuition". 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." (my translation)
The same basic point concerning the "unintuitedness" of the objects of
arithmetic is made in sec. 105 of the Grundlagen, where Frege remarks that
"...reason's proper study is itself. 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 reason and, as
its nearest kin, utterly transparent to it. And yet, for that very reason,
these objects are not subjective fantasies. There is nothing more
objective than the laws of arithmetic."
This is also the point made in the very memorable and picturesque secs. 13,
14 of the Grundlagen. where Frege broached the question of the relative
places occupied in our thinking by empirical, geometrical and arithmetical
laws. His claim 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
different kinds of laws. Denying a geometrical law, he says, stands to do
more extensive damage to a person's cognitive orientation than denying a
physical law. The reason why is that it would lead to a conflict between
what a person can conceive and what she can spatially intuit. In doing so
it would bring severe disorientation to a person's cognition. It would
force them, for example, to deduce things that they formerly had been able
just to "see". These deductions would be strange and unfamiliar. The
disorientation caused by this strangeness would not, however, result in a
global breakdown of rational thought. No, such global breakdown in one's
rational functioning would come about only as the result of denying an
arithmetic law. Such denial would not only keep one from seeing what he had
formerly been able to see, it would, according to Frege, prohibit his
engaging in deduction or reasoning of any sort. In his words, it would
bring about "complete confusion", so that "even to think at all would be no
longer possible" (loc. cit.). The laws of arithmetic must, he concluded,
"be connected very intimately with the laws of thought" (loc. cit.).
This, then, is how Frege the logicist proposed to manage the conjectured
asymmetry of geometry and arithmetic. To repeat, it was management of this
asymmetry and not concern for certainty that was at the bottom of his
logicism. He allowed that, as Kant had said, geometry is synthetic a
priori. The perceived asymmetry between geometry and arithmetic, however,
could only be accounted for by bringing arithmetic closer to the seat of
all rational thought. This is what his logicism was designed to do.
"... the axioms of geometry are are independent of one another and of the
primitive laws of logic, and consequently are synthetic. Can the same be
said of the fundamental propositions of the science of number? Here, we
have only to try denying any of them, and complete confusion ensues. Even
to think at all seems no longer possible. The basis of arithmetic lies
deeper, it seems, than that of any of the empirical sciences, and even than
that of geometry. The truths of arithmetic govern all that is numerable.
This is the widest domain of all; for to it belongs not only the actual,
not only the intuitable, but everything thinkable. Should not the laws of
number, then, be concerned very intimately with the laws of thought"
(Grundlagen, sec. 14)
QUESTION IT WOULD BE PROFITABLE TO DISCUSS 1: Is or should the asymmetry
between arithmetic and geometry that Gauss and nearly all other 19th
century foundational thinkers believed in still be treated as a fundamental
'datum' of the foundations of mathematics today?
BROUWER AND WEYL'S INTUITIONISM
Like Frege's logicism, the intuitionism of the early part of this century
was also dominated by (i) the idea that what the mind brings forth purely
of itself cannot be hidden from it and (ii) the belief that the existence
of non-Euclidean geometries reveals important epistemological differences
between geometry and arithmetic. The direct predecessors of the
intuitionists seem to have been Gauss and Kronecker, who interpreted the
discovery of non-Euclidean geometries differently than Frege. Where Frege
proposed a realist modification of the creation principle (the principle,
enunciated by Gauss and others, that what the mind creates or produces of
itself is better known to it than that which comes from without) that in
order to account for the apparent differences between arithmetic and
geometry brought to light by the discovery of non-Euclidean geometry ,
Gauss and Kronecker, and the intuitionists after them, interpreted the
difference between arithmetic and geometry in light of the creation
principle, of which they adopted an idealist reading.
Thus, instead of maintaining Kant's synthetic a priori conception of
geometry and trying to account for the difference between geometry and
arithmetic by establishing arithmetic as analytic, the early intuitionists
rejected Kant's synthetic a priori conception of geometry and proposed to
account for the differences between arithmetic and geometry by seeing the
former as a priori and the latter as a posteriori. As Gauss and Kronecker
emphasized, arithmetic is purely a product of the human intellect, whereas
geometry is determined by things outside the human intellect. Years later,
Weyl (cf. [1949], p. 22) would reiterate the same theme, remarking that
"the numbers are to a far greater measure than the objects and relations of
space a free product of the human mind and therefore transparent to the
mind".
Brouwer expressed similar ideas, identifying as the primary cause of the
demise of intuitionism since the time of Kant (cf. Brouwer [1912]) the
refutation of Kant's belief in an a priori intuition of space by the
discovery of non-Euclidean geometries. At the same time, however, he
advocated resolute adherence to an a priori intuition of time, and even
argued that from this intuition one could recoup a system of geometric
judgments via Descartes' "arithmetization" of geometry. He considered the
"primordial intuition of time" - which he described as the falling apart of
a life-moment into a part that is passing away and a part that is becoming
- as the "fundamental phenomenon of the human intellect" (loc. cit., p.
127). From this intuition one can pass, via a process of abstraction, to
the notion of "bare two-oneness", which Brouwer regarded as the basal
concept of all of mathematics. The further recognition by the intellect of
the possibility of indefinitely continuing this process then leads it
through the finite ordinals, to the smallest transfinite ordinal, and
finally to the intuition of the linear continuum (i.e. to that unified
plurality of elements which cannot be thought of as a mere collection of
units since the relations of interposition which join them is not exhausted
by mere interposition of new units). In this way, Brouwer believed (loc.
cit., pp. 131-2), first arithmetic and then geometry (albeit only analytic
geometry), via reduction of the former to the latter through Descartes'
calculus of coordinates, come to be qualified as synthetic a priori.
The early intuitionists thus retained a semblance of adherence to Kant's
belief in the synthetic a priority of arithmetical knowledge while denying
his belief in the a priority of our knowledge of the base characteristics
of visual space. They were also staunchly Kantian in their conception of
mathematical inference. Poincaré and Brouwer, in particular, devoted
considerable attention to this point. Indeed, Poincaré, who carried on a
well-known debate with Russell in the early years of this century , made
the role of logical inference in mathematical proof the centerpiece of his
critique of logicism. So, too, in effect, did Brouwer, though his critique
was aimed at the use of logical reasoning in classical mathematics
generally, and not just at the logicists' programmatic demand regarding the
logicization of proof.
The intuitionists were thus at odds with the logicists over the question of
the nature of mathematical reasoning. The heart of their disagreement,
moreover, was not a dispute concerning which logic is the right logic, but
rather a deeper difference regarding the role that any logical inference -
classical or non-classical - can play in mathematical reasoning. To put it
another way, they were divided over the Kantian question of whether
intuition has an indispensable role to play in mathematical REASONING or
INFERENCE. The intuitionists sided with Kant in holding that it does. The
logicists took the contrary view.
QUESTION IT WOULD BE PROFITABLE TO DISCUSS 2: Is the asserted non-logical
character of mathematical reasoning that Kant, Poincare and Brouwer
believed to be so important to a proper understanding of it something with
which we ought to be concernd in foundational work today. What is the
proper way to understand the role of purely logical inference in genuinely
mathematical reasoning?
HILBERT'S FORMALISM
In the third major "ism" of the early 20th century, Hilbert's so-called
formalism, we find another form of Kantianism, and one which contrasts with
the intuitionist position in at least three important respects. The first
concerns the conception of knowledge of existence claims that was adopted.
The second concerns the epistemic importance attached to spatial or
quasi-spatial intuition in the foundations of mathematics. The third
concerns the distinction between genuine judgments and regulative ideals
that figured so prominently in Kant's general critical epistemology. Rather
than focus on any of these, however, I'd like to try to clarify something
that seems to have confused the discussion on the FOM from time to time. It
concerns the sense in which Hilbert is rightly regarded as a 'formalist'.
Hilbert's formalism developed in two stages. In the first or geometrical
stage, it took the following form: a system of axioms does not express a
system of truths particular to a given subject-matter but rather a network
of logical relations among its basic concepts--a network that can (and,
ideally, will) be shared by the concepts pertaining to other
subject-matters as well. In the second or arithmetical stage, it went
beyond this to a view which saw axiomatic theories as expressing not
intertheoretically applicable frameworks of logical relations among
concepts, but schemes of purely symbolical relations among formal terms.
The geometrical formalism was the product of basic forces that shaped 19th
century geometry. One of these was the emergence of the duality phenomena
of projective geometry which showed that certain central and important
geometrical principles remain true when their key geometrical terms are
replaced by others. This encouraged the view that truth in geometry may not
have so much to do with subject-matter or intuitive content as with (some
type of) form.
[N.B. Generally speaking, a duality is a pair of theorems one of which can
be obtained from the other by a simple and uniform scheme of substitution
of terms. Among the simpler examples are the following pairs of
propositions:
(1) For every two distinct points, there is exactly one line which is
incident with both.
(1D) For every two distinct lines, there is exactly one point which is
incident with both.
(2) For every three points that are not incident with the same line, there
is exactly one plane with which they all are incident.
(2D) For every three planes that are not incident with the same line, there
is exactly one point with which they all are incident.
(1D) and (2D) result from (1) and (2), respectively, by interchanging
'point' and 'line'. They illustrate a general principle-the principle of
duality for the projective plane-that covers many more instances. Similarly
for (2) and (2D), which illustrate the principle of duality for projective
space.
In addition to these simple and basic dualities, there are many others,
some quite striking. These include: (i) a duality linking theorems of the
Riemannian geometry of the plane to theorems of the Euclidean geometry of
the sphere under interchange of the terms 'straight line' and 'great
circle', and (ii) a duality linking theorems concerning lines in Euclidean
three-dimensional geometry to theorems concerning points in Euclidean space
of four dimensions under replacement of 'line' with 'point'.
Hilbert, however, was more taken with a duality that had been observed to
exist between geometry-specifically, the axioms of linear congruence in
Euclidean geometry-and a wholly non-geometrical subject-namely, the laws
governing proportions of trait-couplings in mutations of certain varieties
of fruit flies (Hilbert 1922/23, pp. 84-6; 1930, p. 380). This duality
suggested to Hilbert the idea that certain non-logical 'forms' of thought
might be so basic or useful to our thinking that they would belong to even
its most widely disparate parts (intuitionally speaking). He therefore
described it as 'more wonderful' than anything imagined in even the
'boldest fantasy' (Hilbert 1930, p. 380). Others, too, (see Weyl 1944, p.
635; 1949, pp. 26-7) were moved by dualities between remote
subject-matters. End note.]
The importance of the dualities for understanding Hilbert's formalist view
is twofold. Firstly, they suggest that in an axiom system, the mathematical
terms do NOT function as CONSTANTS but as VARIABLES of some sort (i.e. as
terms whose meaning can be treated as 'variable'). This in turn suggests
that axiom systems are not to be seen as designed to capture a single,
favoured interpretation that we think of as 'the truth'. Rather, they are
to be seen as structuring a number of different subject-matters. They are
not designed to serve as theories of a particular subject-matter but as
theory-forms-'empty frames' (Pasch 1924, p. 11), 'logical moulds',
'hypothetico-deductive frameworks' (Weyl 1949, pp. 25-6)-which provide a
logical framework or scaffolding for the description of a variety of
different subject-matters (Hilbert 1899b, pp. 40-1). As Hilbert himself put
it in an 1899 letter to Frege
"...it is surely obvious that every theory is only a scaffolding or schema
of concepts together with their necessary relations to one another, and
that the basic elements can be thought of in any way one likes. If in
speaking of my points I think of some system of things, e.g. the system:
love, law, chimney sweep ... and then assume all my axioms as relations
between these things, then my propositions, e.g. Pythagoras' theorem, are
also valid for these things. In other words: any theory can always be
applied to infinitely many systems of basic elements. One only needs to
apply a reversible one-one transformation and lay it down that the axioms
shall be correspondingly the same for the transformed things. This
circumstance is in fact frequently made use of, e.g. in the principle of
duality, etc. ... All the statements of the theory of electricity are of
course also valid for any other system of things which is substituted for
the concepts magnetism, electricity ...provided only that the requisite
axioms are satisfied. ... [this] can never be a defect in a theory, but is
rather a tremendous advantage, and ... in any case unavoidable."
Another motive for formalism was the formal-logical conception of
inference that emerged in the 18th century in response to a long-recognized
need for increased rigour in geometrical proof. To avoid lapses in rigour,
it was proposed that the validity of inferences in a proof should be
certifiable without appeal to the meanings of the geometrical terms
occurring in them. It should depend only upon the logical forms of the
propositions involved. This would force all geometrically contentful items
to be declared in the axioms and, so, would rid geometrical proofs of
undeclared geometrical assumptions.
[N.B. Historically, this moment had its origins in the 17th century
discovery (e.g. by Lamy and Wallis) that various proofs in Euclid's
Elements relied upon assumptions not stated in his axioms, postulates and
definitions. In the 18th century, Lambert proposed a remedy for such
failures of rigour. He reasoned that since they occur when one makes tacit
use of geometrical intuition while conducting an inference in a proof, they
ought to be eradicable by forcing all inferences to be validated without
appeal to geometrical intuition (see Lambert 1786, p. 162). This latter, he
maintained, could be accomplished by requiring the validation of inferences
to proceed in abstraction from the intuitive meanings of the mathematical
terms involved and to be warrantable solely on the basis of their 'symbolic
characteristics' (durchaus symbolisch vortrage) (ibid.). Axioms would thus
be treated like 'algebraic equations' (algebraische Gleichungen, ibid.) in
which mathematical terms are manipulated according to the logical positions
laid down for them in the axioms and not according to their intuitive
meanings.
This, essentially, was the conception of rigour that Pasch and Hilbert put
into effect in their axiomatizations of geometry a century later (see Pasch
1882 and Hilbert 1899a). End note.]
This being done, the axioms would implicitly define the concept-terms
contained in them because all appeals to the intuitive meanings of those
terms would be replaceable by appeals to the axioms. Concept-terms
appearing in the axioms would thus function as mere markers of logical
position in a general framework or scaffolding of logical relations among
terms identified only as to logical type and not content.
QUESTION THAT MIGHT BE PROFITABLY DISCUSSED 3: Is the existence of the
dualities a fact of fundamental importance in understanding the nature of
mathematics? Should foundational research today be just as concerned with
it as Hilbert was? Can the phenomenon of the dualities be adequately
accounted for by anything other than a 'formalist' philosophy like
Hilbert's?
The formalism of Hilbert's arithmetical period extended this view by
emptying even the logical terms of contentual meaning. They were treated
purely as ideal elements, like the points at infinity in plane projective
geometry. Their purpose was to secure (at the formal level) a simple and
perspicuous logic for arithmetic reasoning--specifically, a logic
preserving the classical patterns of logical inference.
Hilbert believed, however, that the use of ideal elements was legitimate
only when certain conditions were satisfied. Specifically, he believed that
the use of ideal elements should not lead to inconsistencies with
contentual or intuitive thinking. He thus undertook to prove, in his
so-called 'program', the consistency of ideal arithmetic with its
contentual or finitary counterpart and to do so by purely finitary means.
The dualities of 19th century projective geometry also motivated Hilbert's
formalism in their reliance upon the use of so-called 'ideal' or 'imaginary
elements'. These are elements that (need) have no intuitional or perceptual
basis and whose sole justification is the simplifying or generalizing
effects they have on our thinking about a given subject. Their use is
well-illustrated by the duality between (1) and (1D) above. It depends upon
the use of 'points at infinity' to serve as the meets of pairs of parallel
lines. Without them, one does not get the simple, general duality between
(1) and (1D) noted above, but only the more restricted duality between
(1*) For any two points, there is at most one line that is incident with both
and
(1D*) For any two straight lines, there is at most one point that is
incident with both.
To move from this restricted duality to the more general duality between
(1) and (1D), one requires that for any two straight lines, there be at
least one point at which they are incident. For this, however, ideal
elements-points at infinity-are needed to serve as the meets of pairs of
parallel lines. Ideal elements are thus used to achieve simplicity and
generality in dualities.
Hilbert embraced the use of ideal elements in axiomatic theorizing. Indeed,
he essentially identified the axiomatic method with the method of ideal
elements (see Hilbert 1926, p. 383). Specifically, he took the conditions
for justified use of the ideal method to be the same as those for justified
use of the axiomatic method; namely, (i) that it be consistent with the
underlying contentual practice to which it is applied, and (ii) that it
bring simplicity or efficiency to the production of mathematical knowledge
(ibid., pp. 370, 372-3)).
In Hilbert's view, then, the justification of an axiomatic system depends
upon more than its mere consistency. It depends as well upon its promotion
of epistemic efficiency, which is what use of the method of ideal elements
has to offer. The geometric dualities illustrate this point well; for every
contentual proof, they yield not one but two theorems-one directly provided
by the contentual proof, the other from application of the substitution
scheme of the duality. One thus, roughly speaking, obtains two theorems for
the 'price' (i.e. the genuine proof) of one, a clear gain in efficiency.
The second, 'free' theorem will, of course, imply not only real
propositions but also ideal or imaginary propositions (e.g. the
intersections of parallel lines at infinity). Not everything that it covers
thus constitutes a gain in real knowledge. Nevertheless, a great part of
what it covers does, and it produces this with superior ease or efficiency.
Hilbert described the axiomatic or ideal method as the expression of an
important intellectual "freedom"--namely, the freedom to "create" and use
imaginary elements. This freedom can be taken as far as one wants to take
it provided that it does not clash with an associated body of contentual
practice and that it promoted simplicity or efficiency in the production of
real knowledge (see Hilbert 1900, pp. 439-40; 1905, pp. 135-6; 1926, p.
372, 379). Dedekind also spoke of the 'freedom' of the axiomatic method.
However, he understood it in a sense importantly different from
Hilbert's-namely, as a freedom that contrasts axiomatic thinking not with
contentual thinking, as in Hilbert, but with 'genetic' thinking. Dedekind
observed that genetically constructed objects (e.g. the finite cardinals
constructed as set-theoretic objects) inevitably take on features (e.g. the
infinity of the individual cardinals thus constructed) that are not
relevant to their mathematical functioning. In Dedekind's view, the
axiomatic method provides an alternative to this by giving us the freedom
simply to declare that a set of items has exactly the properties laid down
for them in a given set of axioms. Items introduced in this way are 'free
creation(s) of the human intellect' (see Dedekind 1888, sec. 73) which
retain only the relevant features of their genetically constructed
counterparts.
Axiomatic freedom in Hilbert, on the other hand, was a freedom to introduce
admittedly unreal, imaginary entities into our thinking when doing so
increases efficiency (over against purely contentual reasoning) in the
production of contentual knowledge (see Hilbert 1926, pp. 370-73, 379,
392). He thus assigned ideal elements a role similar to that which Kant
assigned the so-called 'ideas of reason' in his critical philosophy. Kant
saw ideas of reason as non-descriptive, regulative, conservative devices
for the efficient development of the judgments of the understanding (see
Critique of Pure Reason, 2nd ed, pp. 85-6, 362, 385, 536-7, 825-27). [N.B.
Kant even stated a kind of conservation thesis in which he stated the
conservativeness of his ideal methods (the so-called concepts or ideas of
reason) over literal or real methods: "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." This, I believe, is a striking earlier formulation of
the kind of conservation or soundness principles that Hilbert formulated.
End note.]
On Kant's view, then, we are free to use 'ideas of reason' so long as we do
not confuse them with objective, descriptive laws applying to an external
reality, which confusion leads to inconsistency. In similar fashion,
Hilbert saw ideal elements as psychologically efficient (perhaps even
indispensable) means of developing our real or contentual knowledge (see
Hilbert 1905, p. 135; 1926, pp. 383, 392). We are free to use them so long
as doing so does not lead to inconsistencies with contentual thinking (see
Hilbert 1905, p. 136; 1926, p. 383; 1928, p. 471).
This basic conception of the ideal or axiomatic method was a key ingredient
of Hilbert's thinking in both the foundations of geometry and the
foundations of arithmetic.
QUESTION THAT MIGHT BE PROFITABLY DISCUSSED 4: Is there an element of
freedom or creativity that is of fundamental importance in understanding
the nature of mathematics? Is this freedom or creativity well-captured,
expressed or at least illustrated by Hilbert's Kantian conception of ideal
elements and methods?
I've gone on too long and will end here. Hope what I've done here helps
some on this list to achieve a better understanding of the projects of the
big 'isms' of 19th and 20th century foundations and their possible
relationship(s) to foundational research today.
**************************
Michael Detlefsen
Department of Philosophy
University of Notre Dame
Notre Dame, Indiana 46556
U.S.A.
e-mail: Detlefsen.1 at nd.edu
FAX: 219-631-8609
Office phone: 219-631-7534
Home phone: 219-232-7273
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