19. Mathematical existence and consistency
(Note: Page references to the paper and to others we have translated are to the pdf files of the
translations on the project web site. At the typesetting stage, these references should be changed
to references to pages in our volume.)
During the years after the Second World War, Bernays wrote a number of
essays on the philosophy of mathematics. Two-thirds of his collection
Abhandlungen consists of these writings. However, no individual one is as
extended or as wide-ranging as "Die Philosophie der Mathematik und die
Hilbertsche Beweistheorie" [1930b] or "Sur le platonisme dans les
mathématiques" [1935d]. The present essay, published in a 1950 in a Festschrift
for his Zürich colleague Ferdinand Gonseth,1 is nonetheless the longest and
probably most significant of his post-war philosophical essays.
Gonseth was like Bernays a mathematician by origin. However, his
extensive writings are largely in philosophy of mathematics and science.
Concerning him and his long and close relation to Bernays, the reader is referred
to Gerhard Heinzmann's introduction to the preceding essay [1946a]. The
present paper makes only occasional reference to Gonseth's views. However, it
deals in a general way with mathematical ontology, that is what can be said
about mathematical existence and the "nature" or lack of it of mathematical
objects. Although one could discern such issues in earlier foundational debates,
few writers focused on them explicitly. Gonseth's idea of logic as a physique de
l'objet quelconque (physics of arbitrary objects) is an exception, and Bernays surely
found it suggestive, although he does not discuss the details of Gonseth's writing
1 The editors of this volume are anonymous, but it is extremely likely that they were
Gonseth's two fellow principal editors of Dialectica, Gaston Bachelard and Bernays.
2
on the subject. But it may have been a reason why Bernays chose a theme in
mathematical ontology for a work honoring Gonseth.
However, it is an idea older than Gonseth's work that Bernays seeks to
make sense of in this paper. The thesis that "existence, in the mathematical sense,
means nothing but consistency" is found in Poincaré's writings and was in some
way a motivating idea of Hilbert's program. Bernays sees the thesis as opposed
to a traditional form of platonism according to which mathematical entities have
"ideal being" independent of being thought of and of being "determinations" of
something real. He criticizes this thesis as doing no methodological work.
Although he does not immediately address the vexed question what is meant by
'independent', it appears that he gives it quite a strong interpretation. This is
indicated by the fact that so much else of what he says is compatible with
platonism characterized in roughly this way. To begin with he rejects the
program of eliminating reference to what he calls theoretical entities, which in
the context we might call abstract entities.2
Bernays goes on to sketch four interpretations of existence statements
about theoretical entities, none of which he takes to imply independent existence.
Of these (b) and (c) are certainly compatible with independence of thought in the
sense of philosophical realism, and (d) might be interpreted that way. In
denying that (d) introduces independent existence of theoretical entities, he
comments:
2 That would be enough to make him count as a platonist as that term is used in much
contemporary discussion, where it contrasts with 'nominalist'. In [Bernays, 1935d] the contrast is
rather with constructivism. That is not much at issue in the present paper, where he anyway
stays away from the label 'platonism'.
3
The existence statement is kept within the particular conceptual context, and no
philosophical (ontological) question of modality which goes beyond this context is
entered into (p. 6).
That suggests that he has in mind something formally analogous to Carnap's
distinction of internal and external questions, and an affirmation of "independent
existence" would be a metaphysician's affirmative answer to an external question
of existence.3
In this general discussion sense (c) of existence for theoretical entities is
one of the key concepts of the paper. It is existence relative to a structure (in a
broad sense), what he calls relative existence (bezogene Existenz). But his route to
the development of this idea goes by way of a more direct discussion of the
equation of mathematical existence and consistency. Applied in general to
existence statements individually, of individual objects or of objects satisfying
some condition, the thesis is criticized on generally familiar grounds (pp. 6-9).
Bernays suggests that its appeal may have rested on a too simple understanding
of consistency.
Bernays proceeds to consider the case of existence axioms of an axiomatic
theory. Here there is an obvious objection to equating existence with
consistency, that consistency is a property of the system of axioms as a whole (p.
10). But since the axiom system may be regarded as a description of a structure,
the existence claims of the axioms can be understood as statements of relative
When writing the present essay, Bernays is unlikely to have known [Carnap, 1950], the
locus classicus of the distinction of internal and external questions; however, the idea is certainly
foreshadowed in earlier writings of Carnap. In his essay [1961c] on Carnap, Bernays does not
comment on the distinction. However, in [1957e], also largely about Carnap, he comments that it
is in general incorrect to pose the question "Is there such and such?" in an "absolute sense" and
that here the criticism by logical empiricism is justified. As in the present essay, he maintains that
questions of existence are sensible only in a specific conceptual context. That suggests that he
viewed Carnap's distinction sympathetically.
3
4
existence. Thus Bernays comes to the point, much less familiar at the time than it
is now, that typical existence statements in mathematics are in the context of a
structure. (His example is Euclidean geometry.)
The philosophical question is in his view only shifted by this observation,
since the question then arises of the existence of the structure. Here, there is
some justification for equating existence with consistency, as in the case of nonEuclidean geometry (p. 11). But typically consistency is made out in such cases
by the construction of a model, which leads to a potential regress of relative
existences. This is the point where Hilbert shortly after 1900 saw the necessity of
a syntactic consistency proof as a way to end such a regress.
At this point Bernays does not turn so quickly to proof theory. He writes:
We finally reach the point at which we make reference to a theoretical framework
(ideeller Rahmen). It is a thought-system that involves a kind of methodological
attitude; in the final analysis, the mathematical existence posits relate to this
thought-system (p. 12).
He says that mathematical experience has tested the consistency of this
framework to such an extent that there is "de facto no doubt about it", and that is
what is needed for the existence posits made within the framework, which are
presumably of relative existence.
Bernays is not as explicit as one would wish as to what this framework
might be. He probably thinks there are different options even within classical
mathematics; a little later he speaks of "indeterminacies" in demarcating a
framework (p. 14). The somewhat obscure remarks about the number system (p.
13) are clearly meant to be illustrative. His point is perhaps that it is not so clear
as appears at first sight what positing the number system consists in. Although
5
in the first instance it is a matter of the existence of objects, conceptual
understanding and with it logic are essentially involved.
Bernays' thesis that mathematical practice operates within a framework
that is not uniquely determined is another point of contact between his thought
and that of Carnap. For Carnap, a framework is essentially linguistic and
specifiable precisely with the methods of mathematical logic, a formal system at
least in a generalized sense, since the characterization may involve infinitary
rules. Bernays' description of the framework as a "thought-system" implies that
he thinks of it in a more informal and perhaps mentalistic way. However, he
clearly thinks that a framework can be delineated more or less precisely (cf. p. 15)
and that axiomatization and formalization are means of attaining greater
precision.
The importance of the question how he precisely thinks of the concept of
framework is somewhat diminished by a point on which he places some
emphasis, that a precisely delineated framework for mathematics intends "a
certain domain of mathematical reality" that is at least to a certain degree
independent of the "particular configuration" of the framework (p. 15). What he
has specifically in mind is the fact that different axiomatizations and
constructions (e.g. of the real numbers) can be equivalent even if they differ in
their ontology. Bernays seems to recognize that the concept of relative existence
loses some of its sharpness by its being viewed in this way (p. 17). He adds the
observation that mathematical reality is not fully exhausted by a delimited
framework, what Gödel called the inexhuastibility of mathematics.
Bernays' understanding of typical mathematical existence statements in
terms of relative existence makes his view a structuralist view in a general sense.
However, he does not pursue an idea central to later structuralist views of
6
mathematical objects, which are based on the idea that such objects have no more
in the way of a "nature" than is given by the basic relations of a structure to
which they belong.4
Bernays mentions two difficulties for his perspective on mathematical
existence, that it might reinstate the "ideal existence" that he has rejected, and
that it might be a form of relativism. He considers what he has already said
about the independence of mathematical reality of the particulars of a framework
as a sufficient answer to the second objection. About the first, he makes the
observation that in natural science the central assertions of existence are of the
"factually real", and other talk of existence, for example of laws, "appears as mere
improper existence" (p. 17). There is no such contrast in mathematics. "It is not a
question of being (Dasein) but of relational, structural connections and the
emergence (being induced) of theoretical entities from other such entities (p. 17).
To me that leaves the rejected sense of independent existence somewhat elusive.
I don't believe that Bernays would have taken the Carnapian line that, to the
extent that they make sense at all, answers to external questions are simply
pragmatic recommendations of one type of framework over another.
Along the way Bernays has addressed an epistemological issue, observing
that if one tries to specify a framework for mathematics in which "one relies
solely on what is absolutely trivial or absolutely evident" (p. 14), there will be no
agreement. He then proposes, with Gonseth, to reject the demand for a
foundation that would be absolutely a priori and to regard mathematical ideas as
instead growing "out of a kind of intellectual experimentation", where in
particular the existence of competing theories should be expected. The attitude
4 Thus I would no longer affirm what I wrote in [1990], p. 303, that the paper gives a
"clear general statement" of such a view.
7
toward the a priori that Bernays expresses here and attributes to Gonseth is in
agreement with the holistic epistemology sketched at about the same time in
[Quine, 1951] and of course developed in detail in the latter's subsequent
writings. However, Bernays questions the idea of the a priori already in [1937k],
which shows no evidence of Gonseth's influence. Evidently he thought that the
traditional notion of a priori knowledge did not capture what is distinctive about
the justification of mathematical statements. In [1961c], pp. 10-11, again
following Gonseth but here at least superficially differing from Quine, Bernays
proposes to replace the notion of the a priori by the more modest one of the
antecedent (vorgängig, Gonseth préalable), which comprises the concepts and
propositions that are presupposed, possibly instinctively, in a given inquiry.5
The last pages of the paper consist largely of remarks about
constructivism and Hilbert's program that are apart from his main argument. He
views a concern with construction as complementary to the "attention toward the
objectual" (p. 19) that he has thus far assumed in something like the way the
theoretical language of physics is complemented by that of the experimentalist.
As he has before, he rejects a formalistic interpretation of Hilbert's program and
emphasizes the idealization involved in the methods of proof theory and
intuitionism.
This does not mean that he endorsed Quine's epistemology of mathematics, although he
apparently thought there was something right about Quine's citicism of the analytic-synthetic
distinction as deployed by Carnap. However, in [1969], p. 4, he writes that the distinction
captures that "between mathematical facts and facts about natural reality" and that "mathematical
statements are justified in a different sense from statements in physics". (Bernays discusses
Quine's criticism in [1957e], pp. 237-39, but the discussion is not illuminating about these issues.)
Unlike the members of the Vienna Circle and many others of his time, Bernays does not seem to
have thought that the notion of a priori knowledge was made more helpful and less problematic
by the thesis that all a priori knowledge is analytic. He seems to have thought that if something is
known a priori it must have a kind of definitive certainty. Then perhaps he had too much
experience of lack of such certainty in mathematics.
5
8
This paper is surely one of the most important contributions to
mathematical ontology of its time, not as sharply formulated as the
contemporary writings of Carnap and Quine but more sensitive to actual
mathematics. However, it seems to have fallen dead-born from the press. Even
to this day it has excited little comment in the literature.6 I don't think this is
accounted for by the fact that it was written in German; in the 1950s a reading
knowledge of German was standard equipment for workers in the foundations
of mathematics, and German writers have not done more than others to draw
attention to the paper. However, it has a certain dense quality that makes its
ideas not easy to assimilate. Other reasons can be cited, for example the rather
obscure Festschrift in which it appeared, and Bernays' own habit of not referring
to his earlier philosophical essays in his later ones.7 Furthermore, questions of
ontology in relation to logic and mathematics tended in the 1950s to be framed in
terms derived from Quine or Carnap, so that Bernays' taking as his point of
departure an earlier foundational issue probably did not find an echo.
Charles Parsons
However, [Bernays, 1953d] is a summary of the paper in French, and it was belatedly
reviewed in the Journal of Symbolic Logic; see [van Heijenoort, 1957]. Still, I think its existence did
not register with me until 1972, when Gert Müller gave me copies of a number of papers of
Bernays, this one included. In contrast, [1935d] seemed to me a quite well-known paper when I
undertook in 1962 to translate it into English.
6
7 A paucity of references to other philosophical writings is a general characteristic of
Bernays' style. In particular, in spite of what seems to have been his high regard for Gonseth, his
fairly frequent mentions of ideas of Gonseth are almost never accompanied by references to
specific writings. That hardly served the dissemination of Gonseth's philosophy.
9
Additional references
Bernays, Paul, 1937k. Grundsätzliche Betrachtungen zur Erkenntnistheorie.
Abhandlungen der Fries'schen Schule, neue Folge, 6, 279-290.
-----------------, 1953d. Existence et non-contradiction en mathématiques. Revue
philosophique de la France et de l'étranger 143 (1953), 85-87. With a note by G.
Bouligand.
Carnap, Rudolf, 1950. Empiricism, semantics, and ontology. Revue internationale
de philosophie 4, 20-40. Reprinted in Meaning and Necessity, 2d ed.
(University of Chicago Press, 1956).
Quine, W. V., 1951. Two dogmas of empiricism. Philosophical Review 60, 20-43.
Reprinted in From a Logical Point of View (Cambridge, Mass., Harvard
University Press, 1953).
Parsons, Charles, 1990. The structuralist view of mathematical objects. Synthese
84, 303-346.
van Heijenoort, Jean (John), 1957. Review of [Bernays, 1950b]. The Journal of
Symbolic Logic 22, 210-211.