How did Frege and Hilbert differ concerning the interpretation of mathematical
theories?
0.Introduction
Section 1 summarises the view of Frege; Section 2 summarises the view of Hilbert;
Section 3 presents two of Frege‟s objections to the view of Hilbert; Section 4 gives
Hilbert‟s responses to these objections; Section 5 is the conclusion.
1.Frege
Frege hoped that the theories of mathematics could be given bases of self-evident
axioms. As far as possible, the aim was the reduction of mathematics to logic; the axioms
grounding mathematical theories should be logical principles, and the rules of inference
used to move from the axioms to the theory should be rules of logic. Frege did not
believe, however, that the reduction to logic was achievable for all theory; the basis of
axioms for geometry, for example, would be the self-evident truths of geometry, rather
than self-evident logical principles.
Although Frege allowed that some axioms be principles of logic, and others not, he
maintained that all axioms should be axioms in the classical sense: they should have a
truth-value, they should be true, and we should be able to take their truth for granted.
Further, the terms of the axioms should not, in general, be left undefined. He recognised
the necessity of leaving some terms undefined—for it is not possible to create a language
with every term defined, without at some stage giving a circular definition,—but he held
that the undefined terms, the primitive terms, should in some sense have their meanings
fixed. This point will be made clearer when the contrast to Hilbert‟s view is made below.
2.Hilbert
Hilbert shared with Frege the conviction that mathematical theories should be given
axiomatic bases. Further, he shared the conviction that mathematical reasoning must be
made transparent, and that there should be no reference to the vague notion of
mathematical intuition in an account of mathematical theories.1 To this end, both Frege
and Hilbert sought to formalise their axiomatic methods in a way that made every step
explicit and perspicuous.
However, whilst Hilbert shared with Frege the axiomatic approach, he believed that
Frege‟s attempt to build mathematics on self-evident foundations was misguided. The
source of this belief was a scepticism about the concept of self-evidence; can there really
be principles whose truth is absolutely unquestionable? Hilbert thought not. He was thus
less absorbed by the problem of guaranteeing the truth of mathematical theories, than by
ridding mathematics of contradiction. For Hilbert, the study of mathematics is the study
of formal structures, and the hallmark of rigour in mathematical reasoning is consistency;
reasoning free from contradiction. The aim of the axiomatic method, then, should be to
demonstrate the consistency of the axiom-system, and further the consistency of the
1
This is not strictly true; Frege did think that the faculty of intuition was needed to give us knowledge of
self-evident axioms (see Frege (1980:37)). But he did want to minimise reference to intuition.
collection of rules of inference that comprise the mathematician‟s toolkit. This
divergence in outlook led Hilbert to a view of the nature and function of axioms that was
quite different to the view of Frege.
Frege‟s axioms were to be chosen with two criteria in mind: first, they had to be selfevident; second, there had to be a strong prospect of getting from the axioms to the theory
in question. Hilbert, on the other hand, was only interested in the second of these criteria.
For a given mathematical concept (an example would be Euclidean geometry), the
approach was to look for an axiom-system, and a collection of rules of inference, that
would codify the assumptions of, notions of, and reasoning within, the given concept. As
already mentioned, Hilbert placed emphasis on consistency rather than on truth. Unlike
Frege‟s method, axioms were not required to be true, and were not even required to be
specific enough to have truth-values. Wilder provides an example of a Hilbert-style
axiom-system in [1]. He intends the system as an example of an axiom-system that might
arise after considering the concept of two-dimensional Euclidean geometry.
Axiom 1. Every line is a collection of points.
Axiom 2. There exist at least two points.
Axiom 3. If p and q are points, then there exists one and only one line containing p
and q.
Axiom 4. If L is a line, then there exists a point not on L.
Axiom 5. If L is a line, and p is a point not on L, then there exists one and only one
line containing p that is parallel to L.
Undefined terms: Point; line.2
This example provides an illustration of the contrast between Fregean axiom-systems and
Hilbert-style axiom-systems. At least one of the terms “point” and “line” appears in each
of the axioms. These two terms are identified as the undefined terms. When we outlined
Frege‟s method, we alluded to the Fregean belief that even the primitive (undefined)
terms must in some sense have their meanings fixed. (For how else could the axioms
containing primitive terms take on truth-values?) Hilbert, however, intended that the
undefined terms have their meanings left entirely indeterminate; if, in the above axiomsystem, we replace the string “point” with the string “x”, and the string “line” with the
string “y”, then the resulting system will be identical.
Since the meanings of the undefined terms are left indeterminate, we can assign any
meanings that we choose. (An example given by Wilder is to let “point” mean book and
to let “line” mean library.) Once the axioms take on meanings, they take on truth-values.
If we find an assignment of meanings that makes all the axioms true, then we call this
assignment an interpretation for the axiom-system. For Hilbert, the existence of an
interpretation for an axiom-system is sufficient for proving the consistency of the axiomsystem.
So, to review: Hilbert‟s goal was to show that mathematical theories could be founded
on consistent systems of axioms. Given a particular mathematical concept, axiomsystems were to be chosen by considering that concept, and selecting the axiom-system
2
Wilder (1961:10)
that seemed to best describe the concept‟s assumptions, notions, and methods of proof.
An axiom-system would intentionally contain undefined terms; terms with indeterminate
meanings. The consistency of the axiom-system would be proven by finding an
interpretation, and it was the consistency of the concept that it was the goal to establish.
3.Fregean consistency and Hilbertian consistency
Frege regarded Hilbert‟s project as mistaken. There are two reasons for this:
First, Frege considered mathematical consistency to be a relation that holds, not
between sentences, but between the meanings of the sentences; and in Frege‟s theory of
meaning, the meanings are the nonlinguistic propositions or thoughts expressed by the
sentences.3 Now, as we have seen, a Hilbert-style axiom-system intentionally contains
terms with indeterminate meaning, and consequently, a Hilbert-style axiom-system must
be comprised of axioms with indeterminate meaning. Thus, for Frege, Hilbert-style
axioms-systems could not be said to be mathematically consistent or otherwise. Whilst
Hilbertian proofs of the consistency of Hilbertian axiom-systems establish syntactic
consistency, they do not, and cannot, establish what we should be striving for:
mathematical consistency. Further, the syntactic consistency of a Hilbertian axiomsystem is not sufficient to prove the mathematical consistency of the axiom-system that
results when meanings are designated to the undefined terms. (Frege took for granted that
Hilbert fallaciously believed otherwise.) So, in Frege‟s view, Hilbert‟s method cannot
fulfil its intended role.
Second, as stated above, axioms in a Fregean axiom-system should be axioms in the
classical sense: they should have a truth-value, they should be true, and we should be able
to take their truth for granted. If all axioms are known to be true then it is manifest that
the system is consistent; consistency proofs, for Frege, are unnecessary.
4.Hilbert’s response
Hitherto, this essay has implicitly sided with Frege in his characterisation of Hilbertian
axioms as having indeterminate meaning. We have taken for granted that if a term, such
as “point” or “line”, does not have its meaning fixed by means of an explicit definition,
then it has no meaning at all, and this is in accordance with the view of Frege. Perhaps,
however, there is more to be said about meaning.
Recall that an interpretation of a Hilbertian axiom-system is an assignment of meanings
to the undefined terms that makes all of the axioms true. It is not hard to see that for a
given axiom-system we might find any number of interpretations; for some axiomsystems we might not be able to find any, for others a finite number, and for others an
infinite number. The point, however, is that the axiom-system necessarily delimits the
number of interpretations that we might find. Consequently, perhaps we can think of the
axiom-system as delimiting the variety of meanings that the undefined terms can take on.
Look again at the example of a Hilbert-style axiom-system given by Wilder. Each
3
Blanchette makes this point, p.322.
axiom asserts a logical relation between the undefined terms, “point” and “line”. 4 Thus,
the axiom-system as a whole asserts a set of logical relations between the undefined
terms. It is this set of logical relations that delimits the variety of meanings that the
undefined terms can take on. Hilbert‟s view is that this variety describes the notions
denoted by the undefined terms in just the way that we should want. In a letter to Frege
he writes,
..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 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.5
The upshot of this is that, in Hilbert‟s eyes, the axioms of Hilbertian axiom-systems are
contentful, and that when we prove consistency by finding an interpretation, we prove
something valuable. This is, I think, how Hilbert would have replied to the first of
Frege‟s two objections identified above.
Frege‟s second objection was that the axioms of an axiom-system, or at least of a
genuine axiom-system, are known to be true; and if the axioms are known to be true then
it is manifest that the system is consistent, so consistency proofs are unnecessary. The
following excerpt (which is taken from the same letter as the above excerpt) nicely
encapsulates how Hilbert replied to this objection.
You [Frege] write: „I call axioms propositions...From the truth of the axioms it
follows that they do not contradict one another.‟ I found it very interesting to read
this sentence in your letter, for as long as I have been thinking, writing and
lecturing on these things, I have been saying the exact reverse: if the arbitrarily
given axioms do not contradict one another with all their consequences, then they
are true and the things defined by the axioms exist. This is for me the criterion of
truth and existence.6
Providing Hilbert is correct in saying that consistency is “the criterion of truth and
existence” then Frege‟s second objection clearly dissolves. We shall briefly examine the
claim that consistency precedes truth in the conclusion.
Before we do so, it is worth making a supplementary remark concerning Hilbert‟s
project: Wilder explains that in the Foundations of Geometry Hilbert established
consistency by means of an interpretation in real number arithmetic. But to know the
resulting arithmetical axioms true requires that real number arithmetic contain no
inconsistencies, since consistency is the criterion of truth. If we are to avoid ceaselessly
shifting the proof of consistency from one domain of mathematics to another, then we
must at some stage seek a direct proof of consistency. Hilbert attempted this for
arithmetic. A proper evaluation of his project would include an evaluation of this attempt,
4
The relations are logical because terms such as collection, there exist, one, every, and not, are logical
terms.
but I shall not provide one in this essay.
5.Conclusion
In an evaluation of the differing positions held by Frege and Hilbert it seems that there
should be two central questions. The first is, Can Hilbertian axiom-systems serve as
definitions? the second is, Is consistency the criterion of truth or does Hilbert have truth
and consistency the wrong way round?
On the first I am inclined to agree with Hilbert; mathematical notions can be adequately
characterised by their logical relations with other mathematical notions, and I imagine
that most mathematicians would concur. Furthermore, if we try to do more to pin down
the meaning of notions then we risk losing the benefits of generality. Blanchette writes,
As the nineteenth-century revolution in geometrical methods and results had made
clear, the view of axioms as only partially interpreted, and hence capable of
characterising similarities across abstract structures, was an undeniably fruitful
one.7
At the root of Frege‟s contention that Hilbertian axiom-systems cannot serve as
definitions was his Platonism; we may choose to make “point” and “line” our undefined
terms but they still have meanings, their meanings are determinate, and we come to
understand the meanings through the faculty of intuition. I reject this account, however,
for the simple reason that I do not think that most mathematicians think of mathematical
notions in this way.
I am also inclined to side with Hilbert on the second question. Frege thinks that selfevident axioms can be found to support the mathematical edifice. Many philosophers of
the twentieth-century, however, have doubted that this is so. Quine, for instance, wrote
To say that mathematics in general has been reduced to logic hints at some new
firming up of mathematics at its foundations. This is misleading. Set theory is less
settled and more conjectural than the classical mathematical superstructure that can
be founded upon it.8
If there are no unrevisable statements, mathematical or otherwise, then self-evident
axioms become a pipe dream. And if this is so, then perhaps we can do no better than turn
our attention from truth to consistency; if the hallmark of rigour in mathematical
reasoning is consistency, and if self-evident truth is unattainable, then why not make
consistency primary and define truth in terms of it? An analogy can be made here with
coherence theories of truth; such theories seem to be motivated by a conviction that
certain truth cannot be got at, combined with the recognition that coherence plays a
fundamental role in deciding which of our beliefs are justified.
One final remark: It is not just Frege‟s logicist project that has received criticism.
5
Frege (1980:40)
Frege (1980:39)
7
Blanchette (1996:335)
8
Lakatos (1976:203)
6
Twentieth-century philosophers have also done their best to undermine Hilbert‟s
program. Gödel‟s results showed that it is impossible to develop mathematics in a
complete, consistent formal system. Nonetheless, we can still apply Hilbert‟s methods to
branches of mathematics in isolation. For the reasons given above, this seems to me to be
a worthwhile exercise.
Bibliography
Blanchette, P. „Frege and Hilbert on Consistency‟, The Journal of Philosophy, Vol. 93,
No.7 (Jul., 1996), pp. 317-336
Dummett, M. „Frege on the consistency of mathematical theories‟, in his Frege and
Other Philosophers (OUP, 1991), pp. 1-16
Frege, G. & Hilbert, D. „The Frege-Hilbert correspondence‟, in G. Frege, Philosophical
and Mathematical Correspondence (Oxford: Blackwell, 1980)
Lakatos, I. „A Renaissance of Empiricism in the Recent Philosophy of Mathematics‟, The
British Journal for the Philosophy of Science, Vol. 27, No.3 (Sep., 1976), pp. 201-223
Mac Lane, S. „Mathematical Models: A Sketch for the Philosophy of Mathematics‟, The
American Mathematical Monthly, Vol. 88, No. 7 (Aug., 1981), pp. 462-472
Wilder, R. L. Introduction to the Foundations of Mathematics (New York: Wiley, 1961)