22C:096
Computation, Information, and Description
Department of Computer Science
The University of Iowa
Lecture Notes
Last modified: 12 March 1997
Hilbert's Program
David Hilbert was arguably the most ingenious mathematician of this
century. He solved many difficult problems in particular branches of
mathematics, and he also was concerned with the foundations of
mathematics as a whole. The attempt to give all of mathematics a
secure foundation in set theory had foundered on the ``paradoxes'' of
set theory, which were actually presentations of an inherent
self-contradiction in the assumptions of set theory. Those
assumptions were so intuitively appealing that many great
mathematicians, particularly Gottlob Frege, accepted them as a secure
starting point from which to develop geometry, algebra, number theory,
real analysis, and all other branches of mathematics. The discovery of
a contradiction was rather scary to those who cared for the certainty
of mathematical reasoning.
Hilbert proposed a program to fix this problem: ``I should like to
eliminate once and for all the questions reagarding the foundations of
mathematics, in the form in which they are now posed, by turning every
mathematical proposition into a formula that can be concretely
exhibited and strictly derived, thus recasting mathematical
definitions and inferences in such a way that they are unshakable and
yet provide an adequate picture of the whole science.'' Specifically,
Hilbert's program has two parts:
- Provide a single formal system of computation capable of
generating all of the true assertions of mathematics.
- Prove mathematically that this system is consistent, that is,
that it contains no contradiction. This is essentially a proof of
correctness.
Strictly speaking, conistency does not guarantee that a system is
correct, just that it doesn't contradict itself. But, the
detail of Hilbert's program and his argument in favor of it shows that
he really intends to show correctness. Hilbert argues that only
certain simple assertions are judged to be true or false based on
their content. These simple assertions are essentially the equations
between integer expressions with no variables, which can be checked by
the arithmetic operations that we all learned in grade school. More
complex assertions, particularly those expressed with variables, and
asserting the existence of numbers with certain properties, or
asserting that some property holds of all numbers, are ``ideal''
propositions, and there is no fixed standard of truth for
them. Rather, a system for deriving ideal propositions is deemed
correct if the numerical equations that it produces are
true, and it is deemed useful if we find it practically helpful for
organizing our understanding and manipulation of numerical
equations. In the standard sorts of logical systems that Hilbert and
others use for mathematics, consistency implies this sort of
correctness, which is more properly called conservative
extension.
Hilbert leaves the question of usefullness out of his theory,
since it is a pragmatic question having to do with the applications of
mathematics, rather than a mathematical question per se. But, he
clearly believes that the most useful systems of mathematics may be
incorporated on his foundation.
Hilbert's program founded a loosely defined school in the philosophy
of mathematics, called formalism. Mathematicians of today
seem to acknowledge formalism as the basis for their work, but they
mostly seem not to understand it. Kurt Gödel showed that
Hilbert's program is impossible. But, the clear statement of the
program was an immense contribution to our understanding. As you read
Hilbert's key papers on this topic, try to separate out the conceptual
essentials from the particulars that were determined by mathematical
practice and convention at the time. For example, Hilbert's ideas have
nothing specifically to do with integer numerical mathematics
vs. non-numerical mathematics. At the time, and to this day, the
mathematics of numbers attracts a large share of attention. And, it is
well known that integer numbers can represent all strictly finite
mathematical objects. I will separate out the crucial from the
accidental as much as I can in the notes.
Aside from a general understanding of Hilbert's ideas, please try to
judge whether his papers support the following
Thesis: The first half of Hilbert's program, to provide a
single absolutely rigorous formal computational foundation for all of
mathematics, is essentially Descarte's program restricted to
mathematics, and expressed more precisely. Hilbert goes further, and
insists on a proof that the computational foundation is consistent.
Über das Unendliche
David Hilbert
(1925)
This is the text of a lecture by Hilbert in Münster on 4 June
1925 at a meeting of the Westphalian Mathematical Society in honor of
the memory of Weierstrass. I am using an English translation, ``On the
infinite,'' by Stefan Bauer-Mengelberg in From Frege to
Gödel: A Source Book in Mathematical Logic, 1879-1931,
edited by Jean van Heijenoort, Harvard University Press, 1967. I refer
to page number in the van Heijenoort book. I number paragraphs
starting with the first paragraph to begin on a given page. A partial
paragraph carried over from the previous page, I number as 0.
- Title. Hilbert wished to place all of mathematics on a
completely reliable computational foundation. He realized that all
finite objects are trivially computable. Computation is
essentially a way of presenting the infinite number of values of a
function in a precise way through a finite program. So, Hilbert
expressed his problem as one of rigorously explaining the
infinite.
- 369, par 1-2. The word ``arithmetic'' in this paper
means the theory of
real analysis. Today, it is often used to mean the theory
of the integers, or even the calculation of sums, differences,
products, and quotients on the integers. Real analysis was
particularly important because it supported the differential and
integral calculus and other key mathematical techniques used in
the physical sciences. It was well known that rationals may be
represented by pairs of integers, and that reals may be
represented by infinite sequences of rational
approximations. While Turing was concerned with the computability
of individual real numbers, Hilbert was more concerned with the
computable verification of correctness of proofs regarding the
real numbers.
- 370 par 1. ``Modes of inference employing the infinite
must be replaced generally by finite processes that have precisely
the same results.'' That is, intuitive references to the infinite
must be converted into objective computations.
- 370 par 2. Hilbert's (translated) words, but not quite
in order: ``The aim of my theory is to endow mathematical method
with definitive reliability.''
- 370-371. ``The definitive clarification of the
nature of the infinite has become necessary, not merely
for the special interests of the individual sciences, but rather
for the honor of the human understanding itself.'' This
has nothing to do with my thesis, but it's a marvellous example of
an emotional attitude toward mathematics.
- 371, par 2. Notice the word ``contentual.'' Hilbert
uses it a lot in this paper. I am pretty sure that the German word
is ``inhaltlich.'' This word is mentioned in Combinatory
Logic by Haskell B. Curry, Robert Feys, William Craig,
North-Holland (1958), but they don't specifically say that it
comes from Hilbert. I am not sure whether this is a commonly used
German word, or whether Hilbert cooked it up for his study. There
is certainly no direct translation into commonly used
English. Curry translated it as ``contensive.'' It is sometimes
rendered as ``intuitive,'' but that seems to me very misleading.
``Contentual'' here means referring to the content
of assertions or formulae. It is used as the opposite of
``formal,'' which means referring to the form of
assertions. Most people, even many mathematicians,
misunderstand the use of the word ``formal'' in logic, and think
that its opposite is ``relaxed,'' or ``unrigorous,'' rather than
``contentual.'' The proper understanding of formal
vs. contentual reasoning is crucial to our study, and to
the understanding of mathematical logic. Formal reasoning is
essentially computation. Contentual reasoning is essentially
inspiration. Sometimes they can be translated into one another,
but only formal judgements can be thoroughly objective and
reliable by themselves.
- 371 par 3, through 372 par 0. This is off of my thesis
again, but Hilbert states the incorrectness of Euclidean geometry
as a description of physical space (at least according to today's
physics) very nicely here. These paragraphs could be the end of a
long trace of the impact of geometry on European thought, starting
with Euclid and the ancient Greeks.
- 372 par 1. Hilbert contrasts ``specific numerical
equations,'' which might as well have been 2+2=4 instead of his
more complicated examples, with ``the solution of an arithmetic
problem'' given by a formula with variables, which represents the
``infinitely many propositions'' resulting from
substitutions for the variables. The language may mislead you
here. He disparages the importance of ``specific numerical
equations,'' since they ``can be verified by computation.'' The
point is that each such equation can be verified by a
single computation. Once we all look at that computation,
there is no real problem agreeing objectively that the equation is
true. On the other hand, formulas with variables stand for global
properties of the computations of certain programs on all possible
inputs. The crucial distinction is between verifying the
correctness of a single computation, and discovering a property of
all the infinitely many computations by a particular
program.
- 372 par 2. This is the first use of the word
``ideal,'' but the concept was used without the name in 370 par 4,
with the more familiar example of the imaginary numbers. Hilbert
claims that mathematics consists of a very small core of
contentual concepts, whose meanings are fixed in
advance. Assertions about the core must be absolutely true. Other
concepts are introduced to help us generate the true core
assertions in a systematic and understandable way. These
additional concepts are the ``ideal'' elements. Assertions about
ideal elements do not have fixed meanings. There is no question of
correctness regarding the ideal elements, only regarding the core
assertions that they help generate. I think that essentially all
mathematicians accept this sort of use of ideal elements. But
Hilbert classifies only the specific numerical equations as core
assertions, and leaves all other assertions in the world of
ideals. In particular, he treats all assertions of existence, or
of universal properties, as ideals. Many mathematicians,
particularly intuitionists, insist on a larger class of core assertions.
- 374 bottom of page. The thingumajigs shown here
with all the omegas are called the ordinals. They extend
the positive integers into the infinite, using them for ordering
rather than expressing size. A different system of infinites,
called the cardinals, is used for size. The ordinals are
not directly relevant to my thesis of the day. But, we will
discuss them briefly later. In an interesting and paradoxical way,
they represent the simplest known way to present complexity. So,
take a look at them as you go by.
- 375 par 4-5. Hilbert's program could have been an
entirely positive application of Descarte's longing for certainty
to the whole scope of mathematics, where it seemed likely that
certainty could be achieved once and for all. But, earlier
attempts at a fixed foundation for all of mathematics led to the
``paradoxes,'' that is iherent contradictions, of naive set
theory. This shook some of mathematicians' confidence that they
knew precisely what they were talking about. Hilbert is trying to
repair the damage, and create a security that will never again
suffer such an attack. There were also inconsistencies in the early
development of the differential calculus, associated with the
treatment of infinitesimals as if they were finite quantities. I
am less familiar with these problems, but Hilbert seems to regard
them more or less on a par with the inconsistencies of naive set
theory.
- 375 par 5. ``And where else would reliability and truth
be found if even mathematical thinking fails?'' This sentence can
be read as an emotional justification for pulling Descarte's
program back to mathematics, as a sort of intellectual strategic
retreat.
- 376 par 1. ``It is necessary to make inferences
everywhere as reliable as they are in ordinary elementary number
theory.'' Notice the rough analog between Descarte's desire to
extend the certainty of geometry to all knowledge, and Hilbert's
more modest desire to extend the certainty of number theory to all
mathematics. I think that the substitution of number theory for
geometry is not particularly significant, but just represents a
different emphasis of mathematical interest in different
centuries. It is ironic, though, that while the number theory of
Hilbert's day seems as secure today as he thought it, we now know
that it is incomplete, and has no secure completion.
- 376 par 3. Here Hilbert admits contentual judgements
into his secure mathematics, but only ``when applied to
real objects or events.'' He has already argued that real objects
are all finite. I am a bit puzzled by the exact scope of ``real
objects or events.'' I think that he does not mean all
physically real objects, but just the ones that constitute
mathematical assertions. Physical uncertainty appears to preclude
objective certainty regarding quantum events. Toward the end of
the paragraph, he requires that ``it must be possible to survey
these objects completely in all their parts, and the fact that
they occur, that they differ from one another, and that they
follow each other, or are concatenated, is immediately given
intuitively, together with the objects, as something that neither
can be reduced to anything else nor requires reduction.'' This
sound a lot like Turing's justification of the primitive steps of
the Turing machine, and it applies only to discrete symbols, not
to arbitrary physical objects. The requirement that basic objects
and their basic relations cannot ``be reduced to anything
else'' makes no sense to me. Every object can be described in
terms of smaller components. The crucial part is that we do not
need such a reduction to make objective judgements. The last two
sentences support my restriction to mathematical formulae, and not
all physical objects: ``This is the basic philosophical position
that I consider requisite of mathematics and, in general, for all
scientific thinking, understanding, and communication. And in
mathematics, in particular, what we consider is the concrete signs
themselves, whose shape, according to the conception we have
adopted, is immediately clear and recognizable.'' Notice that
Hilbert refers to ``thinking, understanding, and communication,''
and in particular to ``concrete signs,'' but not to other sorts of
physical objects, such as electrons. Hilbert doesn't use the word
``formal'' here, but the ``shape'' of signs, and the relations in
which they are placed (``they follow each other, or are
concatenated''), while they are contentual properties of the
signs, they are formal properties of the assertions expressed with
those signs.
- 377 par 0. ``It is possible to obtain in a purely
intuitive and finitary way, just like the truths of number theory,
those insights that guarantee the reliability of the mathematical
apparatus.'' I think that the ``truths of number theory'' here are
just the ``numerical equations'' mentioned earlier in the
paragraph. In the first sentence of the paragraph, he referred to
``ordinary finitary number theory,'' and I believe that ``number
theory'' in the later sentence is an abbreviation for ``ordinary
finite number theory,'' that is numerical equations, rather than
a reference to the theory of universal and existential assertions
about integers. First, Hilbert is asking for other forms of
mathematical assertion to be placed on a computational basis like
numerical equations. Second, he is particularly asking for a proof
of the consistency of mathematics to be done in this way.
- 377 par 3 through 378 par 3. A ``disjunction'' here is
a finite sequence of alternatives, joined by ``or.'' Assuming that
the individual alternatives are testable by computation (for
example, they can be numerical equations), a disjunction can be
tested for truth or falsehood by checking each of the
alternatives. But, the obvious direct way to test an assertion of
existence leads to an infinite computation. Because one may stop
the computation as soon as a true alternative arises, but one
never gets an answer when the existential assertion is false,
Hilbert says that existentials are ``from the finitist point of
view incapable of being negated.''
- 379 par 2. ``Let us remember that we are
mathematicians and as such have already often found ourselves
in a similar predicament.'' Another marvellous appeal to
emotion. I find it moving. Hilbert doesn't make it explicit here,
but he is touching on his faith that he and his colleagues can
meet all possible mathematical challenges. Is this hubris? Will
someone write a term paper analyzing Hilbert's program as a Greek
tragedy, according to Aristotle's definition?
- 380 par 1. The repetition of ``a+b=b+a'' in two
different fonts may puzzle you. In the first version, the germanic
a and b are actually being used as variables, to indicate the
contentual assertion that the infinitely many equations with
numbers substituted for a and b are all true. Hilbert allows us to
judge the truth of each individual numerical equation
contentually, but not the universal truth of infinitely many of
them. The second version, with the roman a and b, is understood
formally instead of contentually, and an ideal assertion rather
than a basic one from the core of mathematics.
- 381 par 0. Hilbert finally uses the f word:
``It is necessary to formalize the logical operations and also the
mathematical proofs themselves.'' That is, he will express
mathematical proofs as computations, that can be checked for
correctness objectively and reliably by considering their
formal symbolic structure, rather than the
content expressed by each assertion in the proof. If we
find an error in a proof, we do not detect the falsehood of an
assertion, we merely detect a manipulation of symbols that does
not follow the rules.
- 381 par 2 through 383 par 1. The details aren't
important to us. Hilbert is giving some of the particular
computation rules that he proposes for a universal foundation of
mathematics. The important thing to us is that they are
computation rules.
- 383 par 3-5. Now, Hilbert notes that an acceptable
formal system of mathematics, although it may use ideal
assertions freely and without a contentual standard of
correctness, must generate only correct numerical equations. He
calls this property ``consistency.'' There is some confusion here,
due to accepting certain conventions as if they were
incontrovertible. Strictly speaking, a logical system is
consistent as long as it does not prove an assertion and
also the negation of the same assertion. Because Hilbert's system
of numerical equations is complete (that is, every true
equation is verified), and because he admits inequalities along
with equalities, an erroneous equation must provide an
inconsistency. And, because the particular logical rules that
everybody uses allow every formula to be proved in an inconsistent
system, all possible errors can be converted to the form 1!=1. If
he were more careful here, Hilbert would mention that in general a
system of ideals need only be a conservative extension of
the core contentual theory. That is, it must not provide proofs of
core assertions unless they are contentually true. In the popular
systems of logic today, theories are consistent precisely when
they are conservative extensions of the numerical equations. But
Hilbert, after arguing that we may define the ideals any way we
like, should not have relied so implicitly on particular
properties of his particular ideals when he expressed the
requirements for a formal system.
- 383 par 4. ``A formalized proof, like a numeral, is a
concrete and surveyable object. It can be communicated from
beginning to end.'' That is, a proof can be judged objectively as
a computation, just like a numerical equation.
- 384 par 1. ``Mathematics in a certain sense develops
into a tribunal of arbitration, a supreme court that will decide
questions of principle---and on such a concrete basis that
universal agreement must be attainable and all assertions can be
verified.'' This pretty well nails my thesis, I think.
- 384 par 3. ``Every mathematical problem can be
solved.'' Here's the hubris again. To paraphrase, ``are we not
mathematicians?'' I couldn't resist this recasting of Hilbert's
phrase analogous to a quote from a piece of sensational
literature, even though the comparison is grossly unfair to even
the grossest of mathematicians. 1 week bragging rights to anyone
who knows the quote that I refer to.
Jason Boer pointed out a problematic sentence at the end of the
same paragraph. I think that I need to quote the whole paragraph.
As an example of the way in which fundamental questions can be
treated I would like to choose the thesis that every mathematical
problem can be solved. We are all convinced of that. After all,
one of the things that attract us most when we apply ourselves to
a mathematical problem is precisely that within us we always hear
the call: here is the problem, search for the solution; you can
find it by pur thought, for in mathematics there is no
ignorabimus. Now, to be sure, my proof theory cannot
specify a general method for solving every mathematical problem;
that does not exist. But the demonstration that the assumption of
the solvability of every mathematical problem is consistent falls
entirely within the scope of our theory.
In what precise sense can we not ``specify a general method for
solving every mathematical problem,'' although ``every
mathematical problem can be solved''? I am pretty sure that
Hilbert intended a single formal system sufficient for
all of mathematics; in 383 par 2 he says, ``... we are
able ... to construct the system of provable formulas, that is,
the science of mathematics.'' The definite article seems to
indicate a single system. The first note for the ``Grundlagen''
paper also supports the requirement of a single formal
system. What then?
- 384 par 3. ``The demonstration that the assumption of
the solvability of every mathematical problem is consistent falls
entirely within the scope of our theory.'' Ironically, Hilbert's
method were used to show the contrary.
- 384 par 4 through 392 par 1. This is just mathematics,
not related to our topic and you can skip it. It is Hilbert's
attempt to prove an important assertion in set theory, called the
``continuum hypothesis.'' Today, the continuum hypothesis is known
to be independent of the widely accepted facts about sets. One may
consistently assume it to be true, or to be false. There seems to
be no solid intuitive consensus that can settle it one way or the
other.
- 392 par 2. Hilbert probably overemphasizes his reliance
on contentual judgements regarding numerical equations, versus
Frege's and Dedekind's attempts to derive mathematics from logic
without content. I am not convinced that there is a real
distinction in the work, as opposed to a distinction merely in the
way that it is described. Hilbert's contentual judgements about
numerical equations work precisely because they can also be
treated as formal judgements: in some sense the content of
numerical equations is a sort of form. In terms of foundational
accomplishement, Frege's work led to a detected
contradiction in the mathematical content. Hilbert's appears to
have been all self consistent in the mathematics, but his
metamathematical proposals were incorrect.
I think that you need to read this paper a few times, accumulating
questions each time and trying to answer them on subsequent
passes. There are some subtle shifts in the types of concepts being
discussed. Perhaps you can avoid being rattled by these shifts if you
consider that they are a lot like the shifts between different levels
of abstraction in computing systems: machine, program, data, etc.
Pay particular attention to the content vs. form axis. The
subtlety here is that the form of a mathematical assertion becomes
part of the content of a metamathematical assertion. A mathematical
assertion with variables, such as a+b=b+a, represents an infinite
number of equations, and cannot be judged contentually as it
stands. Instead, the form of the assertion, which is just a certain
pattern of the signs ``a'', ``b'', ``+'', and ``='', is embedded into
a larger formal structure, called a proof. The assertion that the
proof is correct according to a set of computational rules can be
judged contentually, just as a specific numerical equation
(e.g. 2+2=4). The content of the assertion of correctness of the proof
has to do with the form of the assertion a+b=b+a.
Recall that the problem of the infinite arose for Hilbert when
mathematics went from assertions that can be judged each by a single
computation, to assertions about a program P that carries out
infinitely many computations on infinitely many arguments. Hilbert's
method involves taking P as data to a verifier program. The
verifier program tries to generate a proof of some universal property
of all the computations of P. When a verifier succeeds, we now
have a single computation, whose correctness can be judged
objectively. We take this single computation to be a demonstration of
some property of the infinitely many computations of good old
P. But, the usefulness of this verification now depends on
verifying the verifier. Hilbert thinks we can do that, too.
Hilbert did a marvellous job of advancing the precision with which we
understand what is at stake in mathematical logic. His conclusions
were essentially all wrong. Probably his immense success in other
areas of mathematics made him extrapolate the success of his program
in logic.
Die Grundlagen der Mathematik
David Hilbert
(1927)
This is the text of another lecture by Hilbert in Hamburg in July
1927. I am using an English translation, ``The foundations of
mathematics,'' by Stefan Bauer-Mengelberg and Dagfinn Follesdal from
From Frege to Gödel: A Source Book in Mathematical Logic,
1879-1931, edited by Jean van Heijenoort, Harvard University
Press, 1967. I refer to page number in the van Heijenoort book. I
number paragraphs starting with the first paragraph to begin on a
given page. A partial paragraph carried over from the previous page, I
number as 0.
- 464 par 1 of the paper. ``I should like to eliminate
once and for all the questions reagarding the foundations of
mathematics, in the form in which they are now posed, by turning
every mathematical proposition into a formula that can be
concretely exhibited and strictly derived, thus recasting
mathematical definitions and inferences in such a way that they
are unshakable and yet provide an adequate picture of the whole
science.'' That's the first half of Hilbert's program, quite
explicitly. I think that the second half is implicit in the word
``unshakable.''
- 464-465. This is verbatim repetition from ``On the
infinite.''
- 465 par 2 through 467 par 2. These are just the
particular computation rules for Hilbert's particular system. We
care only that they are computation rules.
- 467 par 3. ``In my theory contentual inference is
replaced by manipulation of signs according to rules; in this way
the axiomatic method attains that reliability and perfection that
it can and must reach if it is to become the basic instrument of
all theoretical research.''
- 467 par 4 through 469 par 6. More detailed computation
rules. Not important to us, but you might find the explicit use of
recursion soothing, since it looks more like programming than the
proof rules of the earlier segment.
- 469 par 7-8. Here is the contrast between contentual
judgements regarding numerical equations (``elementary number
theory'') treated as core assertions, and the ideal propositions
of ``number theory'' using variables.
- 470 par 2. Hilbert calls the two aforementioned sorts
of assertions ``real propositions'' vs. ``ideal
objects.''
- 470 par 4 through 471 par 1. The focus on the law of
the excluded middle is misleading here. This is the ``law'' that
every proposition is either true, or its negation is true. This
rule allows indirect proof. For example, if the assumption that
every number has the property P leads to a contradiction,
excluded middle allows the conclusion that there exists a number
without the property P. Some mathematicians, called
``constructivists'' or ``intuitionists'' refuse to accept such
indirect evidence for existence, and demand a direct construction
of the number in question. The controversy regarding excluded
middle is fascinating, but it has nothing to do with the viability
of Hilbert's program. Precisely the same logical problems arise
with or without excluded middle. This still hasn't sunk in to
everybody's thinking.
- 471 par 3. Hilbert requires that the ideals form a
conservative extension of the theory of the real
propositions: ``the relations that result for the old objects
whenever the ideal objects are eliminated are valid in the old
domain.'' He equates this with ``consistency'': ``no contradiction
is thereby brought about.'' Conservative extension is the real
requirement. For the conventional sorts of logic, including
Hilbert's, it is equivalent to consistency. But, it doesn't have
to be.
- 471 par 4-5. Here you see some of the properties of
conventional logic that tie conservative extension to
consistency. The only point of these paragraphs is to make it
obvious to people who never programmed in Perl that
consistency can be expressed in terms of a simple sort of symbolic
pattern matching.
- 479 par 2. ``Mathematics is a presuppositionless
science. To found it I do not need God, as does Kronecker, or the
assumption of a special faculty of our understanding attuned to
the principle of mathematical induction, as does Poincaré,
or the primal intuition of Brouwer, or, finally, as do Russell and
Whitehead, axioms of infinity, reducibility, or completeness,
which in fact are actual, contentual assumptions that cannot be
compensated for by consistency proofs.'' Well, the tone sounds
like hubris again, but this is actually a relatively modest claim,
and largely substantiated by events. Mathematics may arguably be
understood as the science of forms, which have an objective
quality independent of individual beliefs. What is missing here is
the claim that a single formal system may embrace all of
mathematics at once, and that we may prove that it contains no
error. Notice that Hilbert objects to the ``actual, contentual
assumptions'' of Russell and Whitehead. But, recall that Hilbert
claims to treat numerical equations contentually, and even seems
to regard that as a virtue. It is not the mere contentual quality
of Russell's and Whitehead's assumptions that Hilbert objects to,
but the fact that each particular assumptions is not verifiable by
a single computation or finite observation, and furthermore that
there is not even a proof that they are consistent with basic
numerical equations. Of course, Hilbert's proposed system never
gets its consistency proof either. Oh well.