FOM: Hilbert and solvability, etc.

Michael Detlefsen Detlefsen.1 at
Mon Jun 21 09:38:38 EDT 1999

Neil Tennant, Jacques Dubucs and others have recently been discussing the
matter of Hilbert's early understanding of notions of solvability and
decidability. I have a few comments to make on this exchange.

(1): A careful reading of the 1900 'Mathematical Problems' sheds at least a
little light. In speaking of the solvability of every mathematical problem,
H says (my translation):

"Occasionally it is the case that we seek the solution under insufficient
hypotheses or in an incorrect sense, and for this reason do not succeed.
The problem then arises: to show the impossibility of the solution under
the given hypotheses, or in the sense entertained. Such proofs of
impossibility were produced by the ancients ... In more recent mathematicsm
the question as to the impossibility of certain solutions plays a dominant
role, and we ascertain in this way that old and difficult problems ... have
at last received fully satisfactory and rigorous solutions, although not in
the sense originally intended. Along with other philosophical reasons, it
is in all likelihood this important fact that induces the conviction (which
every mathematician shares, but which no one has as yet supported by a
proof) that every definite mathematical problem must necessarily be
susceptible of a precise resolution, either in the form of an actual answer
to the question asked, or by the proof of the impossibility of its solution
and therewith the necessary failure of all attempts."

Here, I believe, we find fairly convincing indication that H did NOT think
of the solvability of every mathematical problem (what he sometimes
referred to as "the axiom of solvability") in terms of a universal decision
method for all mathematical problems.

We also, I believe, see the influence of Kant. For, after the passage just
noted, H goes on to write ... in striking echo of Kant ...

"Is this axiom of the solvability of every problem a characteristic that is
peculiar to mathematical thought alone, or is there possibly a general law
inherent in the nature of the mind that all questions it asks must be

H goes on to answer the question in the affirmative. But the thing I want
to call attention to is the parallel with the opening paragraph(s) of the
preface to the first edition of the Critique of Pure Reason. No one who has
read that can fail to think of it when reading the above passages (and
other accompanying remarks) in the 'Problems' address. For your
convenience, I quote the paragraph here:

"Human reason has this peculiar fate that in one species of its knowledge
it is burdened by questions which, as prescribed by the very nature of
reason itself, it is not able to ignore, but which, as transcending all its
powers, it is also not able to answer."

H's project was, I believe, to show a way out of the dilemma posed by Kant
... and to do so in a way which was itself faithful to the general
structure of critical epistemology. Since I have argued this at length
elsewhere, I will not repeat the argument again here. H is reacting to Kant
(and some other thinkers of certain neo-Kantian tendencies) but also
accepting Kant. He urges that there is in mathematics "no ignorabimus', but
he does so on the strength of his advocacy (in Kantian fashion) of
'transcendental' solutions (i.e. solutions establishing the unsolvability
of the problem under the methods according to which a solution had been

(2) There is another ground for H's advocacy of "no ignorabimus" as well,
and that is the influence (clear in the 1904 Heidelberg adress and the 1905
lectures 'Logische Prinzipien des mathematischen Denkens') that Dedekind
had on his conception of the axiomatic method. (This was also present in at
least embryonic form in the 'Problems' address in the several places where
H emphasized the 'creative freedom' of the mathematician.) In the 1904
address to the Heidelberg congress, H sets out to develop the 'fundamental
idea' of what he refers to as 'a method I would call axiomatic' (Hilbert
1904, 131). He sets out three basic laws that he takes to govern this
method. The first of them is something he refers to as the creative
principle. He states it as follows:
"Once arrived at a certain stage in the development of the theory, I may
say that a further proposition is true as soon as we recognize that no
contradiction results if it is added to the propositions previously found
true ..."
							Hilbert 1904, 135

H then elaborates this a bit further by saying that "the creative principle
..., in its freest use, justifies us in forming ever new notions, with the
sole restriction that we avoid contradiction" (l.c., 136).

In this conception of axioms as conditions which themselves implicitly
defining the properties of the 'objects' being axiomatically described, H
was following Dedekind, who related his axiomatic description of the whole
numbers to the ideal of free creation in the following way:

"If in the consideration of a simply infinite system N set in order by a
transformation f we entirely neglect the special character of the elements;
simply retaining their distinguishability and taking into account only the
relations to one another in which they are placed by the order-setting
transformation f, then are these elements called natural numbers or ordinal
numbers or simply numbers, and the base element 1 is called the base-number
of the number-series N. With reference to this freeing the elements from
every other content (abstraction) we are justified in calling the numbers a
free creation of the human mind. The relations or laws which are derived
entirely from the conditions a, b, g, d ... form the first object of the
science of numbers or arithmetic."
							WsuwsdZ, 68
It was thus by a free exercise of a certain power of abstraction, that
Dedekind took himself to have arrived at the first axiomatization of
arithmetic. His is no mere 'genetic' logicist attempt to define the numbers
in terms of more basic entities. Indeed, he expressly repudiates this
suggestion and says that the numbers are to be freed from all content save
that which is laid down for them in their defining 'axioms' a, b, g, d. For
that reason, I believe it proper to regard him as having linked the free
creation of concepts to the axiomatic method in such a way as to suggest
the former as the (or at least a) typifying characteristic of the latter.
Further support for this view comes from his well-known letter to Weber of
1888. There he warns against confusing his proposal with one which merely
seeks to define the numbers as classes. He wrote:

"There is much to say about such a class (e.g. that it is a system of
infinitely many elements, namely, all similar systems), a weight one would
not gladly hang about the neck of the number itself. But doesn't everyone
gladly soon forget that the number four is a system of infinitely many
things? (That the number 4 is the child of the number 3 and the mother of
the number 5, however, will remain in everyone's consciousness.)"
							Dedekind 1932, 490

To relate this to the proper understanding of H's conception of
solvability/decidability, I would note this: There is (and would have been
for H) no reason to think of a set of axioms as having the power to decide
every question frameable in its language. Nor would this even have been
desirable since not all questions frameable are automatically interesting.
Indeed, that's the general type of thing from which the 'freedom' of our
power to abstract is supposed to free us. In his early (pre-1917) writings,
then, H was not, I believe, at all of the view that even a 'local' axiom
system ought ideall to furnish answers to all questions frameable in its
language ... except in the sense described in (1) above, according to which
a proof of undecidability counts as an 'answer'.

By the way, I would like to register my reservations regarding Wilfried's
suggestion that H was pursuing a logicist project around the time of the
1917 essay and that he was, in that, heavily influenced by Russell. I think
there's little in the way of evidence to support such a view. He expressly
rejected logicism in either its Fregean, its Dedekindean or its Cantorian
forms in the 1904 essay. This rejection appeared again, in expanded form,
in his work in the 20s. The chief influence of Russell was, I believe, in
his (Russell's) having provided the vehicle of formalization which was
needed to give the 'third' option of H's 'kein ignorabimus' view (i.e. the
option of 'unsolvable under such-and-such conditions') a precise form. I've
looked at the lecture notes from this period and read parts of the material
of H's student Behmann that are supposed to 'evidence' the connection with
Russell but see nothing in any of this that gives much credence to the idea
that H was influenced by the substantive logicist views of Russell during
this period.

Michael Detlefsen
Department of Philosophy
University of Notre Dame
Notre Dame, Indiana  46556
e-mail:  Detlefsen.1 at
FAX:  219-631-8609
Office phones: 219-631-7534
Home phone: 219-273-2744

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