by Leo Corry
The Argument:
The belief in the existence of eternal mathematical truth has been part
of this science throughout history. Bourbaki, however, introduced an
interesting, and rather innovative twist to it, beginning in the mid-
1930s. This group of mathematicians advanced the view that mathematics
is a science dealing with _structures_, and that it attains its results
through a systematic application of the modern axiomatic method. Like
many other mathematicians, past and contemporary, Bourbaki understood
the historical development of mathematics as a series of necessary
stages inexorably leading to its current state-meaning by this, the
specific perspective that Bourbaki had adopted and were promoting. But
unlike anyone else, Bourbaki actively put forward the view that their
conception of mathematics was not only illuminating and useful to deal
with the _current_ concerns of mathematics, but in fact, that this was
the ultimate stage in the evolution of mathematics, bound to remain
unchanged by any future development of this science. In this way, they
were extending in an unprecedented way the domain of validity of the
belief in the eternal character of mathematical truths, from the body
to the images of mathematical knowledge as well.
Bourbaki were fond of presenting their insistence in the centrality
of the modern axiomatic method as a way to ensure the eternal character
of mathematical truth as an offshot of Hilbert's mathematical heritage.
A detailed examination of Hilbert's actual conception of the axiomatic
method, however, brings to the fore interesting differences between
it and Bourbaki's conception, thus underscoring the historically
conditioned character of certain, fundamental mathematical beliefs.
1. Introduction
Throughout history, no science has been more closely associated to the
idea of eternal truth than mathematics. This goes without saying. Still,
the way this idea has been conceived has not in itself been eternal and
invariable; rather, it has been the subject of a historical process of
development and change. Yesterday's conception of the eternal character
of mathematical truth is not identical with today's. Surprisingly,
perhaps, this idea has never been subjected to more serious scrutiny
and attack than nowadays.
In the present article I briefly discuss a short chapter in the long
story of the development of the idea of eternal truth in mathematics.
I will focus on two central figures that are among the most influential
contributors to shaping both the contents of twentieth-century mathematics
and our conception of it: Hilbert and Bourbaki. My main point of
interest will be their respective conceptions of the role of the modern
axiomatic method in mathematics and of its significance concerning the
eternal status of mathematical truth. Considering the differences between
the two will illustrate the subtle changes that the status of truth in
mathematics may undergo as part of its historical process of development.
It will also clarify the background against which current debates on
these questions are being held.
For the purposes of the present discussion, it is useful to introduce the
distinction between the 'body' and the 'images' of scientific knowledge.
The body of knowledge includes statements that are answers to questions
related to the subject matter of any given discipline. The images of
knowledge, on the other hand, include claims which express knowledge about
the discipline _qua_ discipline. The body of knowledge includes theories,
'facts', methods, open problems. The images of knowledge serve as guiding
principles, or selectors. They pose and resolve questions which arise from
the body of knowledge, but which are in general not part of, and cannot be
settled within, the body of knowledge itself. The images of knowledge
determine attitudes concerning issues such as the following: Which of the
open problems of the discipline most urgently demands attention? What is
to be considered a relevant experiment, or a relevant argument? What
procedures, individuals or institutions have authority to adjudicate
disagreements within the discipline? What is to be taken as the legitimate
methodology of the discipline? What is the most efficient and illuminating
technique that should be used to solve a certain kind of problem in the
discipline? What is the appropriate university curriculum for educating the
next generation of scientists in a given discipline? Thus the images of
knowledge cover both cognitive and normative views of scientists concerning
their own discipline.
The borderline between these two domains is somewhat blurred and it is
historically conditioned. Moreover, one should not perceive the difference
between the body and the images of knowledge in terms of two layers, one
more important, the other less so. Rather than differing in their
importance, these two domains differ in the range of the questions they
address: whereas the former answers questions dealing with the subject
matter of the discipline, the latter answers questions about the discipline
itself qua discipline. They appear as organically interconnected domains
in the actual history of the discipline. Their distinction is undertaken
for analytical purposes only, usually in hindsight.
Stated in these terms, the issue of the eternal character of mathematical
truth is closely connected to the images of mathematical knowledge, since
it deals with our conception of the kind of knowledge that mathematics
produces. In the following sections I will discuss the images of
mathematics that underlie the works of Hilbert and of Bourbaki and how
they are connected to the question that occupies us here. Finally, I will
also discuss very recent debates on the status of mathematical truth,
while attempting to place these debates in a proper historical perspective.
2. Hilbert
David Hilbert was among the most influential mathematicians of the beginning
of this century, if not _the_ most influential one.
The impact of his ideas may be traced up to the present day in fields
as distant from one other as number theory, algebraic invariants, geometry,
mathematical logic, linear integral equations, and physics. His name is
often associated with the application of the "modern axiomatic approach"
to diverse mathematical disciplines and, in the context of the foundations
of mathematics, he is usually mentioned as the founder of the formalist
school. The term "Hilbert Program", in particular, refers to the attempt
to provide a finitistic proof of the consistency of arithmetic, an attempt
that G"odel's works in the 1930s proved to be hopeless.
As any German professor educated in the specific intellectual environment
of the end of the nineteenth century, the debates of his philosopher
colleagues were not absolutely foreign to Hilbert, and, in fact, his own
conceptual world was heavily loaded with Kantian and neo-Kantian images.
A quotation of Kant in the frontispiece of his famous _Grundlagen der
Geometrie_ (1899) is but one, well-known, instance of this. The lecture
notes of his courses in G"ottingen contain many more similar examples.
There is also abundant evidence of his interest and involvement in the
careers of philosophers like Edmund Husserl and Leonard Nelson, and of
his hope for a fruitful interaction between them and the G"ottingen
mathematicians. Likewise, because of his direct involvement in the
foundational debates of the 1920s and the influence of his works in this
domain on the subsequent developments of many metamathematical disciplines,
his name has pervasively appeared in the context of twentieth-century
discussions about the philosophy of mathematics.
But in spite of all this, one has to exercise great care when referring
to Hilbert's philosophy of mathematics. Over his long years of activity,
Hilbert came to deal with many different aspects of mathematics and of
physics, facing the development of ever new theories and empirical
discoveries and amidst changing historical contexts. Hilbert was fond of
making sweeping statements about the nature of mathematical knowledge,
about the relationship between mathematics and science, and about logic
and mathematics. These statements are abundantly recorded in both published
and unpublished sources. Sometimes the views expressed in them changed
from time to time, if current scientific developments demanded so, or if
for any other reason Hilbert had changed his mind. And yet the authoritative
tone and the total conviction with which Hilbert proclaimed his opinions
remained forever the same. Concerning the foundations of physics, for
instance, he changed his position around 1913 from a total and absolute
support of the idea that all physical phenomena can be reduced to mechanical
interactions between rigid particles, to an equally total and absolute
defense of an electromagnetic reductionism. He produced important works in
fields like the kinetic theory of gases and the general theory of relativity,
while holding each of these views respectively. His stress on the importance
of each of these positions as a starting point for physical research is
consistently recorded in his lecture notes. Still, neither in his
publications nor in his lecture notes one finds a clue to the fact that his
present view was different to the one held before, nor a word of explanation
about the reasons that brought about this change of perspective.
As Hilbert was a "working mathematician", whose main professional interests
lie in solving problems, proving theorems, and building mathematical and
physical theories, one should not a-priori expect to find any kind of
systematic philosophical discussions in his writings. When these writings
do discuss philosophical issues at all, they often contain claims which are
not always supported by solid arguments and which sometimes contradict
earlier or later claims. In his interchanges with Gottlob Frege and Luitzen
J.E. Brouwer, Hilbert even showed a marked impatience with philosophical
discussions. Certainly, it would be misleading to speak about "_the_
philosophy of mathematics of Hilbert", without further qualifications.
Instead, it seems to me much more historically illuminating to speak of
Hilbert's images of mathematics, and to attempt to elucidate what was more
or less steady and permanent in them, on the one hand, and, on the other
hand, what changed over time and under what circumstances. In the present
section, I will discuss some of those images, focusing especially on those
aspects which are relevant to the use of the axiomatic method and to the
question of the provisory or eternal status of mathematical truth.
The first publication in which Hilbert thoroughly applied the modern
axiomatic method was his book _Grundlagen der Geometrie_ (1899). Among
Hilbert's sources of inspiration when dealing with the issues covered in
this book, the most important ones included the German tradition of work
on projective geometry (in particular Moritz Pasch's text of 1882), and,
perhaps to a somewhat lesser extent, the recent work of Heinrich Hertz on
the foundations of mechanics. The _Grundlagen_ is often read in retrospect
as an early manifestation of the so-called "formalistic" position, that
Hilbert elaborated and defended regarding the foundations of arithmetic
since the 1920s. Under this reading, Hilbert conceived geometry as a
deductive system in which theorems are derived from axioms according to
inferences rule prescribed in advance; the basic concepts of geometry, the
axioms and the theorems are - under this putative conception purely formal
constructs, having no direct, intuitive meaning whatsoever. This reading of
the _Grundlagen_, however, does not reflect faithfully Hilbert's own
conception.
His approach to geometry, at the turn of the century, had a meaningful,
empiricist hard-core, in which the empirical issues of geometry were never
lost of sight. In fact, the famous five groups of axioms are so conceived
as to express specific, separate ways, in which our intuition of space
manifests itself. Hilbert's essentially empiricist conception of geometry
is one of those aspects of his images of mathematics that remained unchanged
over the years. The following quotation, taken from the lecture notes of a
course taught in K"onigsberg in 1891, gives an idea of how he expressed his
early conceptions.
Geometry - Hilbert said - is the science that deals with
the properties of space. It differs essentially from pure
mathematical domains such as the theory of numbers, algebra,
or the theory of functions. The results of the latter are
obtained through pure thinking ... The situation is completely
different in the case of geometry. I can never penetrate
the properties of space by pure reflection, much as I can
never recognize the basic laws of mechanics, the law of
gravitation or any other physical law in this way. Space is
not a product of my reflections. Rather, it is given to me
through the senses.
The borderline between those disciplines whose truths can be obtained
through pure thinking and those that arise from the senses sometimes
shifted in Hilbert's thinking: arithmetic, for instance, is found in his
writings in both sides of this borderline at different times. But geometry
invariably appears in Hilbert's writing as an _empirical_ science (Hilbert
sometimes even says 'experimental'), similar in essence to mechanics,
optics, etc. The kind of differences that Hilbert used to stress between
the latter and geometry concerned their historical stage of development,
rather than their essence. As he wrote in 1894:
Among the appearances or facts of experience manifest
to us in the observation of nature, there is a peculiar
type, namely, those facts concerning the outer shape of
things. Geometry deals with these facts ... Geometry is
a science whose essentials are developed to such a degree,
that all its facts can already be logically deduced from
earlier ones. Much different is the case with the theory
of electricity or with optics, in which still many new
facts are being discovered. Nevertheless, with regards to
its origins, geometry is a natural science.
In other words, eventually in the future, when other physical sciences will
attain the same degree of historical development than the one geometry has
already attained, then there will be no appreciable differences between them,
and the kind of axiomatic analysis that one applies now to geometry will be
equally useful for studying other physical sciences.
But what is then the meaning of applying a process of axiomatization to
geometry, one may ask, if this science is in essence an empirical one? The
aim of the axiomatic analysis that Hilbert presented in the _Grundlagen_ -
unlike that of the formalistic conception of axiomatization - was to
elucidate the logical structure of a given discipline, so that it will
become clear what theorems follow from what assumptions, which assumptions
are independent of which, and what assumptions are needed in order to
derive the whole body of knowledge in that discipline, as we know it at a
given stage of its development.
In fact, Hilbert's excitement about axiomatization was sparkled by his
discovery that a classical technical problem in geometry could be now
overcome, namely, that one does not need infinitesimals in order to
reconstruct plane geometry, whereas in space geometry one actually does.
This way of conceiving the role of the axiomatic analysis helps reading
Hilbert's early works on axiomatization from a perspective which is
basically different from the traditionally accepted one. The question of
the consistency of the various kinds of geometries, for instance, which
from the point of view of Hilbert's later metamathematical research and
the developments that followed it, might be considered to be the most
important one undertaken in the _Grundlagen_, was not even explicitly
mentioned in the introduction to that book. Hilbert discussed the
consistency of the axioms in barely two pages of it, and from the
contents of these pages it is not immediately obvious why he addressed
this question at all. In 1899 Hilbert did not seem to have envisaged the
possibility that the body of theorems traditionally associated with Euclidean
geometry might contain contradictions, since this was a _natural_ science
whose subject matter is the properties of physical space. Hilbert seems
rather to have been echoing here an idea originally formulated in Hertz's
book, according to which the axiomatic analysis of physical theories will
help clearing away possible contradictions brought about over time by the
gradual addition of new hypotheses to a specific scientific theory (Hertz
1894 [1956], 10). Although this was not likely to be the case for the
well-established discipline of geometry, it might still happen that the
particular way in which the _axioms_ had been formulated in order to
account for the theorems of this science led to statements that contradict
each other. The recent development of non-Euclidean geometries made this
possibility only more patent. Thus, Hilbert believed that in the framework
of his system of axioms for geometry he could also easily show that no
such contradictory statements would appear.
Hilbert established through the _Grundlagen_ the relative consistency of
geometry _vis-a\-vis_ arithmetic, i.e., he proved that any contradiction
existing in Euclidean geometry must manifest itself in the arithmetical
system of real numbers. He did this by defining a hierarchy of fields of
algebraic numbers. But in the first edition of the _Grundlagen_, Hilbert
contented himself with constructing a model that satisfied all the axioms,
using only a proper sub-field, rather than the whole field of real numbers
(Hilbert 1899, 21). It was only in the second edition of the _Grundlagen_,
published in 1903, that he added an additional axiom, the so-called "axiom
of completeness" (_Vollst"andigkeitsaxiom_); the latter was meant to ensure
that, although infinitely many incomplete models satisfy all the other
axioms, there is only one complete model that satisfies this last axiom
as well, namely, the usual Cartesian geometry, obtained when the whole
field of real numbers is used in the model (Hilbert 1903, 22-24).
The question of the consistency of geometry was thus reduced to that of
the consistency of arithmetic, but the further necessary step of proving
the latter was not even mentioned in the _Grundlagen_. It is likely that at
this early stage, Hilbert did not yet consider that such a proof could
involve a difficulty of principle. Soon, however, he would assign an
increasingly high priority to it as an important open problem of mathematics.
Thus, among the famous list of twenty-three problems proposed by Hilbert in
Paris in 1900, the second one concerns the proof of the "compatibility of
arithmetical axioms." In formulating this problem, Hilbert articulated his
views on the relations between axiomatic systems and mathematical truth,
and he thus wrote:
When we are engaged in investigating the foundations
of a science, we must set up a system of axioms which
contains an exact and complete description of the
relations subsisting between the elementary ideas of
the science. The axioms so set up are at the same time
the definitions of those elementary ideas, and no
statement within the realm of the science whose foundation
we are testing is held to be correct unless it can be
derived from those axioms by means of a finite number
of logical steps. (Hilbert 1902, 447.)
Views such as this one were at the basis of the well-known debate that
arose between Hilbert and Frege immediately after the publication of the
_Grundlagen_. The latter strongly disputed Hilbert's novel idea, according
to which logical consistency implied mathematical existence and truth;
for Frege, the axioms were necessarily consistent _because_ they were true.
For Hilbert, on the other hand, the freedom implied by the possibility
of creating new mathematical worlds based on consistent axiomatic systems
was enormously appealing, if not for anything else, for the potential
support it seemed to lend to a wholehearted adoption of Georg Cantor's
conceptions of the infinite. Moreover, this view endorsed the legitimacy
of proofs of existence by contradiction, and thus, _a-posteriori_, one of
Hilbert's early mathematical breakthroughs, namely, his proof of the
finite-basis theorem in the theory of algebraic invariants, which had
initially encountered with serious dissent by mathematicians of older
generations.
Still, it would be misleading to believe that the mathematical freedom
pursued by Hilbert implied a conception of mathematics as a discipline
dealing with arbitrarily formulated axiomatic systems devoid of any
intuitive, direct meaning. The analysis that Hilbert applied to the
axioms of geometry in the _Grundlagen_ was based on demanding four
properties that need to be met by that system of axioms: completeness,
consistency, independence, and simplicity. It is true that _in principle_,
there should be no reason why a similar analysis could not be applied to
any other axiomatic system, and in particular, to an arbitrarily given
system of postulates that establishes mutual abstract relations among
undefined elements arbitrarily chosen in advance and having no concrete
mathematical meaning. But _in fact_, Hilbert's own conception of axiomatics
did not convey or encourage the formulation of abstract axiomatic systems
as such: his work was instead directly motivated by the need for better
understanding of existing mathematical and scientific theories. In Hilbert's
view, the definition of systems of abstract axioms and the kind of axiomatic
analysis described above was meant to be carried out, retrospectively, for
'concrete', _well-established and elaborated_ mathematical entities. In
this context, one should notice that in the years immediately following the
publication of the _Grundlagen_, several mathematicians, especially in the
USA, undertook an analysis of the systems of abstract postulates for algebraic
concepts such as groups, fields, Boolean algebras, etc., based on the
application of techniques and conceptions similar to those developed by
Hilbert in his study of the foundations of geometry. These kinds of systems
provided an archetype on which Bourbaki eventually modeled the basic
definitions of the mathematical structures that constitute in his view the
heart of the various mathematical disciplines. Thus, this is one of the
points at which Bourbaki saw his work as a direct continuation of Hilbert's
intellectual legacy. However, we have no direct evidence that Hilbert showed
any interest in the work of the American postulationalist, or in similar
undertakings, and in fact there are many reasons to believe that such works
implied a direction of research that Hilbert did not contemplate when putting
forward his axiomatic program. It seems safe to assert that Hilbert even
thought of this direction of research as mathematically ill-conceived.
Hilbert's actual conception of the essence of the axiomatic method is lucidly
condensed in the following passage, taken from a 1905 course devoted to
exposing the principles of the method and its actual application to diverse
mathematical and scientific domains:
The edifice of science is not raised like a dwelling,
in which the foundations are first firmly laid and
only then one proceeds to construct and to enlarge the
rooms. Science prefers to secure as soon as possible
comfortable spaces to wander around and only subsequently,
when signs appear here and there that the loose
foundations are not able to sustain the expansion of
the rooms, it sets to support and fortify them. This
is not a weakness, but rather the right and healthy
path of development.
After the publication of the _Grundlagen_, Hilbert continued to work on the
foundation of geometry for the next two-three years, but soon he switched
to the next domain of inquiry in which his interests focused over the next
period of time: the theory of integral equations. Some of his collaborators
in G"ottingen, however, continued to explore the application of the
axiomatic method to many domains. Thus, for instance, Ernst Zermelo studied
in detail the axiomatic foundation of set-theory, while Hermann Minkowski
discussed the application of the axiomatic analysis to the latest
developments in the electrodynamics of moving bodies. Hilbert followed all
these developments closely, and to a certain extent, actively participated
in them.
Over time, the issue of consistency became increasingly central to the
axiomatic analysis as conceived by Hilbert, especially given the increasing
centrality of this question to the foundations of arithmetic. Eventually,
the requirements of completeness and simplicity of axiomatic systems were
paid no more attention, and only independence and consistency of the axioms
mattered. Simultaneously, the connection between the axiomatic analysis and
the foundational aspects of mathematics attained more prominence in Hilbert's
thought. Hilbert continued to relate to the axioms of a given theory as
historically determined and subject to change, but at the same time he also
developed a differentiation between at least two kinds of axioms. This idea
was exposed in a now famous lecture held in 1917 in Z"urich, where Hilbert
explained the essentials of the axiomatic method as he then conceived it.
Hilbert opened his Z"urich lecture by presenting again the idea that every
elaborated scientific and mathematical theory can be reorganized in such a
way that its whole body of propositions can be derived from a very limited
number of them - the axioms of that theory. Hilbert mentioned many different
kinds of examples of this situation, among them: the parallelogram law as a
basic axiom of statics, the law of entropy as a basis for thermodynamics,
Kirchhoff's laws of emission and absorption for the theory of radiation,
Gauss's error law as the basic axiom of the calculus of probabilities, the
theorem establishing the existence of roots as basis for the theory of
polynomial equations, and - especially interesting for the present
discussion - the Riemann conjecture, concerning the purely real character
and the frequency of the roots of the function V($s$), as the "foundational
law" of the theory of prime numbers (I will return to this example below).
All these examples, by the way, had already been mentioned by Hilbert in
many earlier occasions, and he had shown in a more or less detailed fashion
how the derivation of the whole discipline can in fact be realized. The
axiomatic derivation of the theory of radiation from the Kirchhoff's law,
for instance, constituted an original, and important, contribution of
Hilbert, whose publication attracted much attention (though not always a
favorable one).
All these examples, Hilbert explained, illustrate _provisory_ solutions to
foundational questions concerning each of the mentioned theories. Very
often in science, however, the need arises to clarify, whether these axioms
can themselves be expressed in terms of more basic propositions belonging
to a deeper layer. It has been the case, that "proofs" have been advanced
of the validity of some of the axioms of the first kind mentioned above:
the linearity of the equations of the plane, the laws of arithmetic, the
parallelogram law for force-addition, the law of entropy and the theorem
of the existence of roots of an algebraic equation. Hilbert discussed this
situation in the following terms:
[The] critical test for these "proofs" is manifest in
the fact, that they are not themselves proofs, but
that at bottom they enable the reduction to deeper-lying
propositions which from now on have to be considered as
new axioms, instead of the original axioms that we
intended to prove. Thus emerge what are properly called
today _axioms of geometry_, of arithmetic, of statics,
of mechanics, of the theory of radiation, or of
thermodynamics... The operation of the axiomatic method,
as it has been described here, is thus tantamount to a
deepening of the foundations of the individual scientific
disciplines, very similar to that which eventually becomes
necessary while an edifice is enlarged and built higher,
and we then want to avail for its safety. (Hilbert 1918,
148. Italics in the original)
Hilbert thus stuck to the edifice metaphor as an explanation of the role
of the axiomatic method in science, but, at the same time, he laid some
stress on the more basic role that certain axioms play from the point of
view of foundations. The differentiation suggested here by Hilbert did not
explicitly appear in many other places among his writings. It seems to me,
however, that the ambiguous attitude inherent in this passage implicitly
comes to the fore in many opportunities, giving rise to diverging
interpretations of Hilbert's views according to which of the two aspects,
the empirical or the formal, is more strongly stressed.
How are these issues related to the question of eternal truths in
mathematics? In the first place, it has already been made clear that
Hilbert's interest in the axiomatic method was closely connected with his
awareness to the constant changes that scientific theories undergo in the
course of their historical development. This applies to physical as well
to mathematical disciplines. One of the aims of the axiomatic analysis of
theories was for Hilbert, the possibility of analyzing whether the adoption
of new hypothesis into existing theories would lead to contradiction with
the existing body of knowledge, a situation that in his view had been very
frequent in the history of science. Hilbert thought that the axiomatic
analysis of theories could help minimizing the appearance of difficulties
in the logical structure of theories, but certainly not avoid them
completely. Still, the question arises how Hilbert thought that open
questions at the level of the images of knowledge should be settled, and
whether the axiomatic method would play any role in this, as Bourbaki was
later to believe. The answer to this question is that Hilbert was somewhat
ambiguous towards it.
Aware of the power of reflexive mathematical reasoning, Hilbert obviously
thought that some meta-questions about mathematics can be solved _within_
the body of mathematical knowledge, definitely endorsing the answers by
means of standard mathematical proofs. The formalistic program for the
foundations of mathematics, in which he was involved in the 1920s, was
based precisely in transforming the very idea of a mathematical proof
into an entity susceptible of mathematical study in itself. Occasionally,
Hilbert also suggested that additional meta-questions could also be solved
with the help of axiomatic analysis. In his 1917 lecture on axiomatic
thinking, Hilbert explained that a solid foundation for the whole of
mathematics would be attained if logic could be properly axiomatized in
terms of a consistent system of abstract postulates. But he also
mentioned additional issues, as being closely related to the latter task:
On closer reflection - Hilbert wrote - we soon
recognize that the question of consistency is not an
isolated one concerning the integer numbers and the
theory of sets alone, but rather that it is part of
a larger domain of very difficult, epistemological,
questions of a specific mathematical hue: in order
to characterize this domain of questions briefly, I
mention the problem of the _solvability in principle
of every mathematical question_, the problem of the
retrospective _controllability_ of the results of
any mathematical investigation, then the question of
a _criterion for the simplicity_ of a mathematical
proof, the question of the relation between _contents
and formalism_ in mathematics and in logic, and finally,
the problem of the _decidability_ of a mathematical
question in a finite number of steps. (Hilbert 1918,
153. Italics in the original)
As a matter of historical fact, Hilbert did not himself deal with all
these problems in the way he formulated them here. Only the last of these
problems, the _Entscheidungsproblem_, subsequently became the basis of an
actual, fruitful research program. But the specific point I want to make
here is that in instances like this one, Hilbert did discuss the
possibility of solving metamathematical issues inside the body of
mathematical knowledge, and more specifically, with the help of the
axiomatic method. Hilbert, however, was aware of the limitations of this
approach to solving such issues, and on many occasions he also stressed
the contextual, historical or social, factors that affected the actual
answers to questions such as the relative importance of mathematical
theories, or the appropriate way to organize mathematical knowledge. In
the opening lecture of a course on the foundations of physics, taught in
G"ottingen in 1917, Hilbert expressed very clearly this position, while
discussing the interrelation of physics and geometry in the aftermath of
the development of general relativity. In a passage that brings to the
fore once again his empiricist view of geometry at a relatively later
stage of his career, he said:
In the past, physics adopted the conclusions of geometry
without further ado. This was justified insofar as not only
the rough, but also the finest physical facts confirmed
those conclusions. This was also the case when Gauss measured
the sum of angles in a triangle and found that it equals two
right ones. That is no longer the case for the new physics.
_Modern physics must draw geometry into the realm of its
investigations_. This is logical and natural: every science
grows like a tree, of which not only the branches continually
expand, but also the roots penetrate deeper.
Some decades ago one could observe a similar development in
mathematics. A theorem was considered according to
Weierstrass to have been proved if it could be reduced to
relations among integer numbers, whose laws were assumed
to be given. Any further dealings with the latter were laid
aside and entrusted to the philosophers. Kronecker said
once: 'The good Lord created the integer numbers.' These
were at that time a touch-me-not (_noli me tangere_) of
mathematics. That was the case until the logical foundations
of this science began to stagger. The integer numbers turned
then into one of the most fruitful research domains of
mathematics, and especially of set theory (Dedekind). The
mathematician was thus compelled to become a philosopher,
for otherwise he ceased to be a mathematician.
The same happens now: _the physicist must become a geometer_,
for otherwise he runs the risk of ceasing to be a physics
and vice versa. The separation of the sciences into professions
and faculties is an anthropological one, and it is thus foreign
to reality as such. For a natural phenomenon does not ask about
itself whether it is the business of a physicists or of a
mathematician. On these grounds we should not be allowed to
simply accept the axioms of geometry. The latter may be the
expression of certain facts of experience that further
experiments would contradict.
The 1920s are the years of Hilbert's more intense involvement with the
questions of foundations. But even in this period there is plenty of
evidence that his basic views on the place of uncertainty in mathematics
did not change in any essential way. In 1919-20, Hilbert gave a series
of popular lectures in G"ottingen under the general title of "Nature and
Mathematical Knowledge."
In these lectures Hilbert sharply criticized accepted views of mathematics
and physics. He explicitly discarded the view that mathematics can be
reduced to a formal game played with meaningless symbols according to rules
established in advance, while stressing the role of intuition and experience
as a source of mathematics. He also discussed the place of conjectural
thinking, and the fallibility of mathematical reasoning. Of particular
relevance for the present discussion is the connection that Hilbert
established between the axiomatic method and conjectural thinking in
mathematics.
It is clear that any axiomatically developed theory has an hypothetical
character, in the sense that the conclusions of the theory are valid whenever
the validity of the axioms is assumed. For instance, in every mathematical
situation in which the conditions of the elementary axioms of the theory of
rings are satisfied, the theory provides certain theorems of unique
factorization that are valid in that situation. This is the way in which
Bourbaki, as we will see, presented the essence of the axiomatic method,
and clearly this description applies to a large extent to Hilbert's view
as well. But in these lectures of 1919-20, we also find a broader conception
of the application of the method as Hilbert conceived it: the possibility
of incorporating into the body of mathematical knowledge theories that are
based on unproved, tough perhaps plausible, theorems of significant content.
Hilbert had in mind, in this case, the particular example of the already
mentioned Riemann conjecture. From the point of view of the calculus of
probability, this conjecture certainly appears, _a-priori_, as a rather
implausible one, since it demands that the zeros of a certain function
will all lie on a very delimited region of space. Still, what we know
about mathematics, and in particular, our knowledge of the fruitful
results that seem to follow from this conjecture lead us to assign it a
high plausibility of being true. Moreover, Hilbert saw it as legitimate
to build a full mathematical theory based on the assumption of the validity
of this conjecture, but only insofar as a correct application of the
axiomatic method will help us keeping track of the limitations of such a
theory. Hilbert repeated in these lectures many of the ideas exposed in
1917 in Z"urich, and in particular the idea of two different layers of
axioms, bearing a different foundational character. Echoing this
distinction, Hilbert explained the possible use that can be made of
conjectures such as Riemann's. Thus Hilbert wrote:
In discussing the method of mathematics, I have
already stressed that when building a particular
theory, it is a fully justified procedure to assume
still unproved, but plausible, theorems (as axioms),
provided one is clear about the incomplete character
of this way of laying the foundations of the theory.
(Hilbert 1919-20 [1992], 78)
In 1922-23 Hilbert gave another series of popular lectures in G"ottingen.
Among other topics he discussed the place of error in the history of
mathematics. Hilbert plainly declared that errors had played a significant
role in the development of this science. The passage where he explains
his view presents an image of how progress is attained in mathematics,
which, again, is totally opposed to the one Bourbaki tried so hard to put
forward more than two decades later. I quote Hilbert in some extension:
Every time that a new, fruitful method is invented in
order to solve a problem, in order to expand our knowledge,
or in order to conquer new provinces of science, there are,
on the one hand, critical researchers who distrust the
novelty, and on the other hand, the courageous ones, who
before all others deplete the inexhaustible and productive
source, swiftly achieve innovation and soon even gain
overweight of it, so that they can silence the objections
of the critics. This is the period of the swift advancement
of science. Often the best pioneers are those who dare to
advance deeper and are the first to arrive to unsafe territory.
Signs of the latter are unclearness and uncertainty in
the results obtained, to the point that even visible
contradictions and countersenses - the so-called paradoxes -
arise. At this moment reappear on the stage the critical
tendencies, that until now have stood aside. They take
possession of the paradoxes, uncover real mistakes and thus
attempt to incriminate the whole method and to reject it.
The danger exists that all the progress achieved will be
lost. The main task in such a situation is to hold this
criticism back (_einzud"ammen_) and to look after a
reformulation of the foundations of the method, so that it
remains safe from all its false applications and, at the
same time, that the ordinary results of the established
portions of mathematical knowledge can be incorporated into
it. (Hilbert 1922-23, 38-39)
In the past, Hilbert had himself played the role of a courageous pioneer
in the framework of his early work on algebraic invariants. In 1888 he
proved the existence of a finite basis for every system of invariants
using new methods that were, at first, harshly criticized by more
conservative mathematicians. In 1922, he was clearly referring to his
current concern with the foundations of mathematics. From very early
on Hilbert had defended the new conception of the infinite implied by
the work of Cantor on sets, and his formalist program was conceived as
way of countering the criticism of those who thought that the acceptance
of the actual infinite in mathematics was damaging, as the appearance
of paradoxes suggested. But in any case, Hilbert's open-minded attitude
towards alternative and innovative views in mathematics, highly
contrasts with that of Bourbaki, as will be described below. And again,
Hilbert did not believe that the axiomatic conception of mathematical
theories would totally safeguard against error.
Hilbert's conception of the axiomatic method and of its role in science,
then, contained various, somewhat diverging, elements. Small wonder,
then, that different mathematicians at different times derived different
ideas from it. Clearly, Bourbaki's conceptions are elaboration of some
of these elements, but leave aside others. But even among Hilbert's
students and colleagues, and still during his lifetime, one can observe
how these various elements are differently stressed.
On the occasion of Hilbert's sixtieth birthday, the journal _Die
Naturwissenschaften_ dedicated one of its issues to celebrate the
achievements of the master. Several of his students were commissioned
with articles summarizing Hilbert's contributions in different fields.
Max Born, who as a young student in G"ottingen attended many of Hilbert's
courses, and later on as a colleague continued to participate in his
seminars, wrote about Hilbert's physics. Born was perhaps the physicist
that expressed a more sustained enthusiasm for Hilbert's physics. He
seems also to have truly appreciated the exact nature of Hilbert's
program for axiomatizing physical theories and the potential contribution
that the realization of that program could yield. His description of
the essence of this program stressed its empiricist underpinnings,
and at the same time attempted to explain why, in general, physicists
tended not to appreciate it. Curiously, he directly addressed the
issue of the relationship between the modern axiomatic method and
eternal mathematical truth. Born put it in the following words:
The physicist set outs to explore how things are
in nature; experiment and theory are thus for him
only a means to attain an aim. Conscious of the
infinite complexities of the phenomena with which
he is confronted in every experiment, he resists
the idea of considering a theory as something
definitive. He therefore abhors the word "Axiom",
which in its usual usage evokes the idea of
definitive truth. The physicist is thus acting in
accordance with his healthy instinct, that
dogmatism is the worst enemy of natural science.
The mathematician, on the contrary, has no business
with factual phenomena, but rather with logic
interrelations. In Hilbert's language the axiomatic
treatment of a discipline implies in no sense a
definitive formulation of specific axioms as eternal
truths, but rather the following methodological
demand: specify the assumptions at the beginning of
your deliberation, stop for a moment and investigate
whether or not these assumptions are partly superfluous
or contradict each other. (Born 1922, 591)
In Born's view, then, Hilbert's axiomatic approach is not applied in
order to attain eternal truth, as Bourbaki's views later implied, but
rather in order to enable a clearer understanding of the nature of our
provisory conceptions, and in order to provide the means to correct
errors that might arise in them. But in the same issue of that journal
dedicated to Hilbert, a somewhat different (although not contradictory)
assessment of his work appears, in an article by Paul Bernays on
"Hilbert's Significance for the Philosophy of Mathematics." Bernays was
at that time Hilbert's closest assistant, and together they dedicated
most of their current efforts to foundational questions, and, in
particular, to lay down the basis for the realization of the so-called
"Hilbert Program." Clearly, when presenting Hilbert's ideas, Bernays
stressed mainly those connected with his foundational concerns. Bernays
explained the essence of Hilbert's axiomatic conception, and claimed
that the main task of his analysis was the proof of consistency of
the theories involved. This was certainly true for Hilbert's current
concerns, but, as I claimed above, it had been much less the case for
his early investigations of the foundations of geometry and of physics.
as a constant point of major interest for Hilbert. After explaining
the motivation for Hilbert's current interest in investigating the
nature of mathematical proofs, Bernays explained the philosophical
meaning of the master's entire endeavor, in the following terms:
While clarifying the workings of mathematical logic,
Hilbert transformed the meaning of this method [of
the logical calculus] in a way very similar, to that
earlier applied for the axiomatic method. Very much
like he had once striped off the visualizable
(_Anschauulich_) contents out of the basic relations
and of the axioms of geometry, so he detached now the
mental contents of deductions from the proofs of
arithmetic and analysis, which he had made the
subject-matter of his investigations. He did so by
taking as his immediate object of consideration the
systems of formulae through which those proofs are
represented in the logical calculus, cut off from any
logical-contentual interpretation. In this way he
could substitute the methods of proofs used in analysis
with purely formal transactions, which are performed on
determinate signs according to fixed rules.
By means of this approach, in which the separation of
what is specifically mathematical from anything that
has to do with contents reaches its peak, the Hilbertian
conception of the essence of mathematics and of the
axiomatic method attains for the first time its true
realization. For we recognize from now on, that the
sphere of the mathematical-abstract into which the
mathematical way of thinking translates all what is
theoretically conceivable, is not the sphere of what has
logical content, but rather it is the domain of the pure
formalism. Mathematics thus appears as the general
theory of formal systems (_Formalismen_), and, since we
are able to conceive it that way, also the universal
meaning of this science becomes clear at once.
(Bernays 1922, 98)
As we will see below, Bernays' characterization of Hilbert's axiomatic
method is very close to the one accorded to him by Bourbaki, and to the
one put forward in Bourbaki's mathematics. This is also the one that has
come to be more closely associated with Hilbert's name. But as we have
seen, this is only one aspect of Hilbert's much more complex conception,
and an excessive stress on it runs the risk of leading to misinterpretation:
this is particularly the case when it comes to the link between the
axiomatic method and eternal truth in mathematics.
[end of part 1/3]