The core of Godel's logical and mathematical achievements is comprised
of a remarkable series of demonstrations concerning formal underivability, beginning with his completeness theorem which shows that any statement formally underivable from first-order axioms can in principle be
shown to be so by a model. He soon discovered the underivability in formal number theory of a statement formalizing the assertion of its consistency, as well as a host of other sentences formalizing, their own underivability, qualifying them as prima facie candidates for 'intuitive truths'.
In a paper "Is Mathematics Syntax of Language ?" written...
This appeared to contrast with the parallel postulate, the discovery of
whose underivability from Euclid's other axioms had led most mathematicians to regard the question of its truth as meaningless within mathematics. Then there was Godel's proof of the consistency of Cantor's continuum hypothesis with the axioms of set theory, showing the impossibility of deriving its denial from them. But neither this nor Cohen's later
proof of its own underivability from them would stop Godel from looking for new 'set theoretical intuitions' that might settle the matter, which
he will distinguish from the 'apriori physical intuition' he sees in geometrical intuition. Finally, perhaps the most spectacular of all his discoveries,
Godel showed the impossibility of deriving the asymmetry of the relation
of temporal precedence from the field equations of general relativity,
leaving us unable to prove from the laws of nature the first and foremost
property our ' intuitive notion of time'. Godel saw in this result grounds
for claiming the essential compatability of Kant's philosophy with relativity theory, and indeed as a proof of his thesis of the ideality of time, one
which we shall examine at length after first comparing his approach to
intuition with that of Hilbert and Kant.
All but the last of these discoveries were solutions or contributions to
problems posed by Hilbert who along with Kant found in the discovery of
impossibility proofs the strongest evidence for the solvability of all mathematical problems. In opposition to the growing opinion that the development and arithmetization of analysis had discredited if not eliminated the
appeal to intuition in mathematics, Hilbert formulated an account of the
origin and growth of mathematic theories as a continual interplay of physical idealizations, intuitions, and formal thought. In analyzing the role of
intuition in this broadly dialectical development Hilbert appeals to
Kantian intuition of which he undertakes in the context of geometry, a
logical analysis. Godel's attempt to derive Kant's ideality of time from
relativity theory obliges him to consider the role of Kantian intuition in
this context in a more psychologistic manner than did Hilbert. He also
argues that in trying to prove the consistency of formal theories Hilbert
unduly restricted Kant's Anschauung to finite discrete configurations of
symbols, and that for the development of mathematical theories one will
need appeal to much stronger set theoretical intuitions. Indeed, since
Godel also based his philosophy on what Wang called a "dialectic
between our idealizations, thoughts and intuitions",
As we learn from his conversations with Godel reported...
it is instructive to
view his articulation of this philosophy as a development of the dialectic
first laid out as the background of Hilbert's 23 mathematical problems
and later refined in his metamathematical problems about formalisms in
Hilbert's philosophy of mathematics in 1900 took the form of (i)
a theory of the sources of mathematical problems, (ii) an account of the
rigor required for their solutions, and (iii) his rationale for believing that
all such well-posed problems were solvable. He found the origin of mathematical problems "in the world of external phenomena" as eventually
described in geometry, mechanics, astronomy, and physics. But, says
Hilbert, "in further development of a branch of mathematics : the human
mind, encouraged by the success of its solutions, becomes conscious of
its independence. It evolves from itself alone, often without appreciable
influence from without, by means of logical combination, generalization,
specialization...new and fruitful problems, and then appears itself as the
"Mathematical Problems", Bulletin of the American Mathematical...
Then, "while the creative power of pure reason is at
work, the outer world comes into play again, forcing upon us new questions from actual experience, opening up new branches of mathematics... And it seems to me that the numerous and surprising analogies and
that apparently prearranged harmony which the mathematician so often
perceives in the questions, methods, and ideas of the various branches of
his science, have their origin in this ever recurring interplay of thought
Ibid, p. 440.
In 1923 Hilbert envisages the development of the mathematical sciences as comprised of a 'constant interplay' between the formal derivation of theorems of a formalism and the intuitive proof of its
consistency upon the addition of ever new axioms.
"Logical Foundations of Mathematics", in From Kant...
As for rigor Hilbert required that "it shall be possible to establish the
correctness of the solution by means of a finite number of steps based on
a finite number of hypotheses which are implied in the statement of the
problem and which must always be exactly formulated."
"Mathematical Problems", p. 441.
was more radical in 1900 than it now seems, for these 'hypotheses' are
just the axioms that Hilbert implied must be found for any branch of
mathematics, contrary to the prevailing view that the most important such
branches were suspended in mathematical thought free of any axioms or
hypotheses. He also opposed a related view "advocated by eminent men"
that rigor resided only in arithmetic and analysis, a view which, taken
seriously, "would soon lead to ignoring all concepts arising from geometry, mechanics, and physics, to a stoppage of the flow of new material
from the outside world, and finally... to the rejection of the idea of the
continuum and of the irrational number."
Ibid, p.442. It is interesting that by 1920 Hilbert...
Thus Weierstrass' arithmetization of analysis used dialectic e-8 rigor to define irrational numbers independently of any limits presupposing such numbers, but he insisted that
the arithmetic to which he had reduced analysis neither had nor needed
any axioms, making it all the more difficult for him to discuss
Kronecker's claim that irrational numbers did not exist ! Nor was
Weierstrass alone : Cantor not only agreed that it was impossible to formulate any axioms for arithmetic, but took Newton's 'Hypotheses non fingo'
as the motto of his set theory, denying that it based on anything but his
definition of a set as 'any collection into a whole of definite objects of my
thought or intuition'.
G. Cantor' Gesammelte Abhondlungen. mathematischen...
Only 'metageometers' steeped in the tradition of
axiomatic geometry, said Cantor, could imagine that arithmetic even
admitted axioms. By means of analytic geometry even this discipline
could be reduced to analysis. Hilbert's Festschrift, however, which he
called 'a logical analysis of spatial intuition,' had just reversed this
process by constructing various kinds of fields out of segments by means
of geometrical propositions. He alluded to this in his lecture on mathematical problems by saying that the rigorous use geometrical figures, which
he calls "memonic symbols of space intuition", requires "a rigorous
axiomatic investigation of their conceptual content".
"Mathematical Problems", p 443. With Pascal's theorem,...
But the same was
true, he insisted, of the arithmetic symbols which are nothing but 'written
diagrams' governed by suitable axioms no less than the 'graphic formulas' of geometry.
As hard as some of his problems were, Hilbert argued his audience
that all well-posed mathematical problems were solvable in principle,
reminding them that "Occasionally it happens that we seek the solution
under insufficient hypotheses or in an incorrect sense and for this reason
we do not succeed. The problem then arises : to show the impossibility of
the solution under the given hypotheses, or in the sense contemplated.. . In
later mathematics, the question as to the impossibility of certain solutions
plays a prominent part, and we perceive in this way the old and difficult
problems, such as the proof of the axiom of parallels, the squaring of the
circle, or the solutions of equations of the fifth degree by radicals have
finally found full satisfactory and rigorous solutions, although in another
sense than originally intended. It is probably this important fact along
with other philosophical reasons that gives rise to the conviction (which
every mathematician shares, but not one has yet proved), that every definite mathematical problem must necessarily be susceptible of an exact
settlement, either in the form of an actual answer to the question asked, or
by a proof of the impossibility of its solution."
Ibid, p. 444
For example Hilbert
could only call his Festschrift a 'logical analysis of spatial intuition' by
virtue of its non-derivability and embedding theorems enabling him to
establish the role of the space axioms as well as Desargues' theorem as
their logical surrogate in the plane. Accordingly, "its ground rule was to
discuss every question that arises in such a way as to find out whether it
can be answered in a specified way with some limited means."
D. Hilbert : Foundations of Geometry, 1971. p. 106
reason the drive for knowledge cannot advance without establishing
impossibility theorems, and the reason Hilbert and his contemporaries
took their proofs as the most convincing evidence for the solvability thesis was that in such cases one had found a solution despite groping for
centuries in the wrong direction, after which one still had to carefully
analyze and formulate in appropriate language the 'limited means' permitted for their solution.
Kant's belief in the decidability of mathematics seems...
Godel shared Hilbert's belief in the solvability thesis which he took as
a premise, along with his incompleteness theorem, of an argument to
show that the human mind can prove more theorems than any machine.
Godel, however, based his belief in the thesis on the accumulation of positive results in number theory and diophantine equations, whereas Hilbert
would have found the negative solution of his 10th problem on deciding
diophantine equations by 'a finite process' an even stronger confirmation
of it for requiring a successful analysis of such 'finite processes'. Godel's
incompleteness theorem was another such solution to one of Hilbert's
1928 list of metamathematical problems. He also solved important cases
of Hilbert's Entscheidungsproblem, but his most important contribution
to this problem turned out to be all those undecidable sentences found in
his first theorem. Turing's negative solution to this problem depends on
what Godel regarded as his 'correct and unique' analysis of 'the intuitive
notion of a mechanical process.' Turing realized, however, that it was precisely all those nameless undecidable sentences unearthed by Godel that
prevented one from refuting his analysis by any kind of diagonal argument that was explicit enough to be expressed in a formal system.
As A. Watson : "Mathematics and its Foundations", Mind...
Hilbert proposed to meet Brouwer's objection to the application of
classical logic to an infinite domain by proving the consistency of formal
number theory by reasoning based on 'the intuition of the finite.' Godel
demonstrated in 1931 that the consistency of such a system could not be
proved in P if in fact-P was consistent, but he stressed that the result "does
not contradict Hilbert's formalistic standpoint. For this viewpoint presupposes only the existence of a consistency proof in which nothing but finitary means of proof is used, and it is conceivable that there exist finitary
proofs that cannot be expressed in the formalism of P."
claimed his consistency proof for number theory using transfinite induction up to e(), was finitary, arguing that in the progression through the ordinals beyond to to e0 "nothing new ever really occurs".
The Collected Papers of Gerhard Gentzen, 1969, p. ...
In his analysis of
Gentzen's proof in 1938a Godel granted one could give "an intuitive picture of this number", even a "very intuitive construction procedure" for it
"by the countable iteration of the transition from a to 2a.
G3, pp 105,107.
e0 is only
"immediately given" once this transition is : "Nonetheless, one will not
deny a high degree of intuitiveness to the inference by induction on e0
thus defined, as in general to the procedure of defining an ordinal by
induction on ordinals (though this is an impredicative procedure)". While
expressing a high regard for Hilbert's goal of proving consistency "in a
purely intuitive and finitary way" Godel later concluded that "What
Hilbert means by 'Anschauung' is substantially Kant's space-time intuition confined, however, to configurations of a finite number of discrete
objects.. .it is Hilbert's insistence on concrete knowledge that makes finitary so surprisingly weak and excludes many things that are just as incontrovertibly evident to everybody as finitary number theory, e.g. the general principle of primitive recursive definition, because it contains the
abstract concept of function. There is nothing in the term 'finitary' which
would suggest a restriction to concrete knowledge. Only Hilbert's special
interpretation of it introduces this restriction."
G2, p. 272. See note 49. For the emergence of this...
Gentzen's principles of
induction and accessibility of ordinals, Godel now said, "create the deceptive impression of being based on a concrete intuition of certain infinite
procedures, such a 'counting beyond co'or 'running through' the ordinals
smaller than an ordinal a. We do have such an intuition but it does not
reach very far in the series of ordinals, certainly no further than
Ibid, p. 272
Godel was convinced that Gentzen's induction "cannot be
made immediately evident, as is possible...in the case of co2."
Ibid, p. 273.
admitted that "due to a lack of a precise definition of either concrete or
abstract evidence there exists, today, no rigorous proof of the insufficient
(even for the consistency proof of number theory) of finitary mathematics."
Ibid, p. 273
Godel was convinced that we need abstract knowledge in addition
to concrete intuition, but not necessarily as a matter of principle, conceding rather that "Whether the necessity of abstract concepts for the proof
of induction from a certain point on in the series of constructive ordinals
is due solely to the impossibility of grasping intuitively the complicated
(through only finitely complicated) combinatorial relation involved or
arise for some essential reason, cannot be decided off hand."
Ibid, p. 273-4. In particular, Godel was not convinced...
Wang has emphasized that Godel's own sense of intuition represents
'a vast extension of Kant's conception of Anschauung' which Hilbert had
adapted to the demands of metamathematical investigations, as a brief
review of the extent of Godel's intuitions is set theory will show. It will
also lead us to his analysis of Kant' 'space-time intuition', specifically, of
the euclidean space intuition and the intuitive concept of time on which
he believes Kant rested his account of intuition.
Godel stressed that the question of the truth of a sentence of a mathematical theory need not lose its meaning when its independence from the
axioms is established, in set-theory than in geometry : "the difference is
only that in geometry the meaning usually adopted today refers to physics
rather than to mathematical intuition, and that therefore a decision falls
outside the range of mathematics."
Ibid, p. 267. In contrast to Cantor's adoption of 'hypotheses...
The objects of set theory do not
belong to the physical world, according to Godel, having only 'a loose
indirect connection' with it due to the minor role played by set theory in
the physical sciences : "but despite their remoteness from sense experience, we do have something like a perception also of the objects of set
theory, as is seen from the fact that the axioms force themselves on us as
being true. I don't see any reason why we should have less confidence in
this kind of perception, i.e. in mathematical intuition, than in sense perception which induces us to build up physical theories and ....to believe
that a question not decided now may be decide in the future.... That new
mathematical intuitions leading to a decision of such problems as
Cantor's continuum hypothesis are perfectly possible."
Ibid, p. 268.
Indeed, Godel believed this hypothesis would "turn out to be wrong"
by virtue of implying the existence of certain 'thin' subsets of the real line
that he regarded as "paradoxical consequences" violating his set theoretical intuitions. He granted that point set theory had also reached some
"highly unexpected and implausible" results without using Cantor's
hypothesis, but "still the situation is different there, insofar as in those
instances (such as, e.g. Peano's curves) the appearance to the contrary can
in general be explained by the lack of agreement between our intuitive
geometrical concepts and the set theoretical ones occurring in the theorems."
Ibid, p. 264.
But the balance of expert opinion has weighed against Godel
here, in seeing neither the implausibility of his 'paradoxical consequences' of Cantor's hypothesis nor any essential difference between
them and cases like the Peano curve.
See p. 165 of G. Moore's introductory note in G2 for...
However new axioms might force themselves on us as true, the intuition that underwrites them takes time to develop : as time passes "more
and more abstract terms enter the sphere of our understanding".
critique of what he called the "philosophical error" of Turing's argument
that "mental procedures cannot go beyond mechanical procedures",
Godel argued that the discovery of "systematic methods of actualizing
this development" would show that "although at each stage the number
and precision of abstract terms at our disposal may be finite, both (and,
therefore, also Turing's number of distinguishable states of mind) may
converge toward infinity in the course of the development of the procedure."
Ibid, p. 306. Godel says that "something like this...
Godel's claim that Turing disregarded that "mind, in its use, is not
static, but constantly developing " presupposes that the collection of states
of mind as well as its fund of abstract concepts is really changing in time
as it proves more and more theorems from axioms intuited in newly distinguished epistemic states. But unlike Brouwer, who grounded mathematics on a Kantian intuition of time, Godel does not incorporate this
temporal dimension of mathematical activity into the epistemology of
mathematics, but rather tries to demonstrate the unreality of time implicit in Kant's thesis of the ideality of time, which indeed was the 'conclusion' Kant drew from the fact that time was a pure intuition. But since his
final proof of this fact in the B-deduction of the categories depends via its
linear representation on the ideality of space, we first need to look at how
Godel connected his own intuition to Kantian categories as well as his
view of Kant' intuition of space.
Godel grants that "mathematical intuition need not be conceived as a
faculty giving immediate knowledge of the objects concerned. Rather it
seems that as in the case of physical experience we form our ides of those
objects on the basis of something else which is immediately given. Only
this something else is not...the sensations". Godel thinks that this something else is indicated by our need for a 'concept of an object', even to
perceive physical objects and suggests that the given underlying mathematics is "closely related to the abstract elements in our empirical
Ibid, p. 268.
He insists that being uncaused by anything impinging on our
sense organs does not make this 'something else subjective as it was for
Kant, though he does see "a close relationship between the concept of a
set...and the categories of pure understanding in Kant's sense."
Ibid, p. 268.
mere existence of this mathematical intuition is enough to give meaning
to the continuum hypothesis despite its independence from our current
axioms, unlike the case, he thinks, of Euclid's parallel postulate.
Thus it happened that when Greenberg asked Godel's permission to
quote his famous passage we have been dissecting in his book on geometry, Godel granted it only on the condition that he point out to the reader that this passage was dealing with 'set theoretical as opposed to geometrical intuition', and then add the explanation that according to Godel :
"Geometrical intuition, strictly speaking, is not mathematical, but rather
apriori physical intuition. In its purely mathematical aspect our Euclidian
space intuition is perfectly correct, namely, it represents correctly a certain structure existing in the realm of mathematical objects. Even physically it is correct 'in the small'".
G4, p. 454-5.
This 'certain structure' correctly represented by our Euclidean space intuition is just the arithmetical model constructed by Hilbert to prove the consistency of the axioms of Euclidean
geometry. Godel's remark echoes Hilbert's claim to Frege that 'the more
precise propositions' of Euclid's geometry do not hold in nature but only
in an algebraic structure. Neither Hilbert nor Godel would say that these
axioms of Euclid are true per se but only that they hold in a model. Frege
had replied that Euclidean geometry was obviously consistent since all its
axioms were true - period. While Godel's claim that 'our Euclidean space
intuition only holds exactly in a model may seem to lean to Hilbert's side
in this matter, his belief that out space intuition is Euclidean after all
shows some sympathy Frege's attitude.
This raises a question Greenberg, in thanking Godel for his permission, could not resist putting to him : "Are you here equating "geometrical
intuition with 'Euclidean space intuition'? If so, perhaps I should emphasize that my book will be titled Euclidean and Non-Euclidean
Geometries. What about the intuition of J. Bolyai and Lobachevsky had
for Hyperbolic Geometry before it was shown that there were Euclidean
models ? What about Riemann's intuition ? It seems to me that your
remarks which I quoted do apply to Non-Euclidean Geometry, not just to
set theory (as you qualified them in your letter to me). Am I mistaken ?"
Ibid, p. 455.
Greenberg knew that Bolyai and Lobachevsky believed in the truth of
their axioms as much as Godel in the axioms of set theory, and that his
explanation of the truth of Euclid's axioms applied equally to theirs : they
hold in a structure existing in the mathematical universe. Were not their
intuitions of this structure as legitimate as Godel's of his preferred structure of sets ? Was the hyperbolic pathology any more vexing to space intuition that the paradoxes to 'set theoretical intuition'?
Godel was somewhat taken aback by Greenberg's question but was
prepared to admit that our intuitions can be altered by training and practice. He drafted a reply in which he wrote that : "I am not sufficiently well
acquainted with the original papers of Lobachevsky, Bolyai, and Riemann
to know whether they claimed to have developed a non-Euclidean space
intuition. Note that for developing a theory from given axioms an intuition of its objects is quite unnecessary. If you can quote to me statements
by them or bring reports to this effect I'll be very much interested. It is not
impossible that out of the Euclidean space intuition (which we all have) a
non-Euclidean space intuition could be developed by combining its elements differently by sufficient practice...even though in order to do that
one would have to solve, e.g. the problem of imagining two lines assymtotically approximating each other and being at the same time everywhere
concave toward each other (which exists trivially in hyperbolic geometry)".
G4, p. 455. Godel never sent this letter.
These remarks show that Godel had in fact equated geometrical
intuition with Euclidean space intuition, and that Greenberg's question
had prompted him to reconsider a line of thought he had explored over
may years concerning the compatibility of what the took to be Kant's
view that we have "an innate intuition of Euclidean space" with relativity theory.
In 1946/9-C1 Godel claimed that Kant may be right in the sense that
"we should have a Euclidean 'form of sense perception', i.e. that we
should possess an apriori representation of Euclidean space and be able to
form images of outer objects only by projecting our sensations on the representation of space, so that, even if we were born in some strongly non-Euclidean world, we would nevertheless invariably imagine space to be
G3. p. 255, fn. 20. By "referring to reality" Godel...
Godel defined geometrical concepts satisfying Euclid's
axioms in a non-euclidean world, showing that "an innate Euclidean geometrical intuition which refers to reality and is apriori valid is logically
possible and compatible with the existence of non-Euclidean geometry
and with relativity theory."
Ibid. p. 255, fn. 20.
However, wrote Godel, "I do not think the
question whether we have an innate intuition of Euclidean space... has yet
been decided; nor the related question whether we are able (in our world)
to learn to imagine a non-euclidean space."
If Godel's notion of imagination here is no less psychologistic that that of Descartes, according to
whom we cannot imagine a chiliagon - since we cannot "intuit by our
powers of discernment" all of its 1000 sides as distinctly present to our
mind - the same may thus be true of his notion of geometrical intuition as
In 1961 Godel returned to the status of Kant's views of geometry,
intuition, and the solvability of mathematical problems in the light of
modern developments in the axiomatization of geometry as well as his
own discovery of the incompleteability of arithmetic. Godel argues that
"the intuitive grasping of ever new axioms" necessary for the solvability
of all problems even in the limited domain of arithmetic "agrees with the
Kantian conception of mathematics. The relevant utterances by Kant are,
it is true, incorrect if taken literally, since Kant asserts that in the derivation of geometrical theorems we always need new geometrical intuitions,
and therefore a purely logical derivation from a finite number of axioms
is impossible. That is demonstrably false. However, if in this proposition
we replace the term 'geometrical' by 'mathematical' or 'set theoretical',
then it becomes a demonstrably true proposition."
Ibid, p. 385.
Although Kant was
wrong to (allegedly) assume the impossibility of finitely axiomatizing
geometry (something Godel presumably took Hilbert to have shown in his
Festschrift), his views of mathematics "in a broader sense contain deep
Specifically Godel grants that "It is not at all excluded by the
negative results mentioned earlier that nevertheless every clearly posed
yes-or-no question is solvable in this way."
Ibid.p. 385. See note 12.
However, Godel's claim that
'the relevant utterances by Kant' implied his belief that geometrical
proofs would require 'ever new axioms' and 'always need new geometrical intuitions' is seriously in error, if, as seems plausible, he meant the
famous passage at A717 where Kant explains how the geometer demonstrates Euclid's angle sum theorem by drawing an auxiliary parallel
whereupon his proof is 'guided throughout by intuition'.
G3, p. 367.
wrote this passage he was unaware of there being any 'problem of parallels' but when on being apprised of it shortly after he explored the matter
extensively, he came to the conclusion that the key fact (equivalent to the
axiom of parallels) needed to prove this theorem could itself only be
proved "philosophically from concepts". Indeed, any reference to
"Anshauung" is conspicuous by its absence from Kant's reflections on the
See Kants Gesammette Schriften, Vol. 14, pp. 22-51.1...
This circumstance undercuts any way of finding in Kant a
conscious commitment to any innate Euclidean space intuition, but we can
still look for a less psychologistic account of Kantian space intuition than
Godel's that respects its strengths while better respecting Kant's texts.
We have seen that according to Godel, Hilbert's 'Anschauung' was
basically 'Kant's space-time intuition confined to configurations of a
finite number of discrete objects'. Godel had in mind here Hilbert's
famous concession to Kant that as a precondition of the application of logical inference to infinite domains "something must already be given to our
faculty of representation, certain extralogical concrete objects that are
intuitively present as immediate experience prior to all thought. If logical
inference is to be reliable, 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, or are concatenated, is immediately given intuitively, together with the objects, as something that neither can be reduced to anything
else nor requires reduction."
"On the Infinite", in From Frege to Godel, edited by...
The role of the infinite, said Hilbert, is
"merely that of an idea - if, in accordance with Kant's words, we understand by an idea a concept of reason that transcends all experience and
through which the concrete is completed so as to form a totality."
also reaffirmed his belief he shared with Kant in the solvability of all
mathematical problems, though with an important qualification : "Now, to
be sure, my proof theory cannot specify a general method for solving
every mathematical problem; that does not exist."
Ibid, p. 384.
Was Hilbert expressing disbelief in a positive solution to his Entscheidungsproblem, and if so,
why ? The seeds for such disbelief were sown in the Festschrift for whose
'logical analysis of spatial intuition' Hilbert had chosen as a motto Kant's
famous conclusion of the Dialectic that "all human cognition begins with
intuitions, goes forth from there to concepts, and ends with ideas."
Hilbert separated his axioms into five groups, those of incidence
(plane and space), order, congruence, parallelism, and continuity, "each of
which express certain basis facts of our intuition which belong together"
Lectures, p. 437.
But from his earliest lectures of 1891 on geometry the incidence axioms
were the Grundgestze der Anschauung, the simplest and most direct
expression of our space intuition being the space axioms about planes,
whose most immediate consequence he calls the Raumsatz. viz
Lectures, p. 31. The theorem states that two triangles...
This theorem can be used to prove the plane case
(D) of Desargues by projection; but Hilbert constructed an ingenious
model to show that (D) cannot be proved from the axioms of a plane incidence, order, parallels, and continuity alone without using those of congruence, which he calls Bewegungsaxiome since they formalize the concept at the motion of rigid bodies.
Lectures, xi- 104-5. p. 166.
On the other hand Hilbert shows that
(D) is all one needs to prove that the space axioms hold of the 'space
points, lines and planes' one can define in the plane. Together with the
non-derivability of (D) in the plane, this embedding shows, as Hilbert puts
it, that "Desargues' theorem is the only one which the plane gains for itself
from space and we could say that everything is provable with Desargues
in the plane that is provable in space generally."
Lectures, p. 240. This result is part of Hilbert's...
Hilbert also shows that
(D) is sufficient to prove all the algebraic laws of a field for his
Streckenrechnung except the commutativity of its multiplication, which
he shows to be equivalent to Pascal's theorem (P). The key result here is
(H):(P) cannot be proved from the axioms of incidence, order, and parallels even if (D) is assumed,
which Hilbert established by constructing a rather complicated non-Archimedian model, showing that in the absence of the concept of a rigid
motion one cannot prove (P) from (D) in the plane without continuity
axioms requiring appeal to some idea of infinite series or sets.
Lectures, pp. 506-10.
As for his
Kant motto, Hilbert could have said fairly straightforwardly that his logical analysis shows that if we begin our cognition of space with the simplest intuitions of planes, they yield (D) immediately in space : but to
prove it in the plane it is necessary to introduce a concept of congruence
'foreign' to (D), while to prove (P) in order to do analytic geometry without this concept it is necessary to introduce continuity axioms involving
an idea of infinity.
Recall that the key space axiom that a unique plane...
As for the Entscheidungsproblem, Hilbert and Ackerman did motivate
its importance in 1928 precisely be reference to (H): they construct a first
order formula which they show would be valid were (H) false, so that a
positive solution to this problem would have enabled Hilbert to have discovered his non-Archimean model establishing (H) by a purely mechanical calculation !
D. Hilbert and W Ackermann : Grundziige der theoretischen...
I dare say he would have been loathe to believe this,
given the difficulty he had personally experienced just sorting out the logical relations between (D) and (P), let alone concocting models to demonstrate the impossibility of deriving the full congruence axiom for triangles
from that restricted to triangles of the same orientation without continuity. It seems plausible that, when Hilbert claimed in 1926 that his proof
theory 'cannot specify a general method for solving every mathematical
problem, that does not exist', he was thinking at least of the unlikelihood
that all the effort and ingenuity he had expended in constructing models
to establish necessary conditions of geometrical theorems could have
been spared by routine calculation.
J. von Neumann : "Zur Hilbertschen Beweistheorie",...
No wonder it was called the 'leading problem of mathematical logic'. Nevertheless, its negative solution by
Church and Turing surely comprised the very sort of impossibility result
that had persuaded Hilbert, Kant and others of the solvability of all mathematical problems. It is unclear whether Godel found this quite as persuasive, or what he thought of Hilbert's modest logical analysis of space intuition as opposed to his own psychologistic and physicalist analysis of
Kant's innate intuition of Euclidean space. In any case, Kant's theory of
time as the form of inner intuition is the real lynchpin of his philosophy.
This brings us to Godel's most remarkable discovery, namely, that one
cannot derive the asymmetry of the relation of temporal precedence from
the postulates and equations of general relativity. In fact, Godel tells us
that he discovered his rotating solutions that establish this result "while
working on the relationship between Kant and relativity theory and, more
particularly, on the similarity which subsists between Kant and relativistic physics insofar as in both theories the objective existence of a time in
the Newtonian sense is denied."
G3, p. 274.
In view of its principle of the relativity
of simultaneity, Godel thought that the special theory of relativity already
precludes any objective meaning for judgments of the simultaneity and
temporal precedence of events A and B "insofar as another observer, with
the same claim to correctness, can assert that A and B are not simultaneous (or that B happened before A)."
G2, p. 202
This relativity of simultaneity,
Godel believed, would of itself have far reaching consequences for our
"intuitive idea" of time, comprising "an unequivocal proof for the view of
those philosophers who, like Parmenides, Kant, and the modern idealists,
deny the objectivity of change and consider change as an illusion or an
appearance due to our special mode of perception."
But did Kant deny
the objective reality of Newtonian time ? In the Dissertation he defended
his thesis that "time is not something objective and real" by arguing that
neither Newton's not Leibniz' account of its objective reality were adequate. Newton's idea of time as "a continuous flux within existence was
a most absurd fabrication", while Leibniz' account "completely neglects
simultaneity, the most important corollary of time."
Theoretical Philosophy 1755-1770, translated and edited...
But not Newton's,
as Kant explains : "For simultaneous things are joined together at some
moment of time, just as successive things are joined together by different
moments. Accordingly, though time has only one dimension, yet the ubiquity of time (to speak with Newton), in virtue of which all things which
can be thought sensitively are at some time, adds a further dimension to
the magnitude of actual things, in so far as they hang, so to speak, from
the same point of time. For, if you were to represent time by a straight line
extended to infinity, and simultaneous things of any point of time by lines
drawn perpendicular to it, the surface thus generated would represent the
phenomenal world in respect of both substance and accidents."
Kant wholly embraced the very property of Newtonian time that is rejected by relativity theory : absolute simultaneity. Still, Godel finds in the
Critique evidence of the compatibility of Kant's ideality of time with relativity which we examine below.
Meanwhile, Godel's argument against the objective reality of change
from special relativity runs as follows : "Change becomes possible only
through the lapse of time. The existence of an objective lapse of time,
however, means (or, at least, is equivalent to the fact) that reality consists
of an infinity of layers of 'now' which come into existence successively.
But if simultaneity is something relative in the sense just explained, reality cannot be split up into such layers is an objectively determined way.
Each observer has his own set of 'nows' and none of these various systems of layers can claim the prerogative of representing the objective
lapse of time."
G2, pp. 202-3
But as Godel notes, the equivalence of all inertial frames
on which this argument depends, holds only in the abstract empty space-time frame of special relativity. The introduction of matter produces a
space-time curvature governed by Einstein's equations for general relativity which destroys this equivalence of observers in a way which allows
one, in all solutions of these equations known hitherto, to piece together
the local times distinguished observers into a cosmological time whereupon, as Godel admits, "it becomes possible to consider this time as the
'true' one which lapses objectively."
Ibid, p. 204.
In fact, Jeans had already concluded from this construction that, as Godel puts it, "there is no reason to
abandon the intuitive idea of an absolute tie lapsing objectively."
Godel discovered, however, an entirely new kind of rotating solution
which turns out to describe a world in which such a global time function
cannot be defined. Godel's model is temporally orientable in the sense
that pairs of light cones can be partitioned into two classes, future ones
and past ones. One can then define a relation P< Q of temporal precedence which holds just in case there is a smooth future-directed timelike
line from P to Q, representing the history of a body traveling from P to Q
while moving always into the future. However, this relation is not asymmetrical in Godel's world, for as he puts it, "if P and Q are any two points
on a world-line of matter, and P preceeds Q on this line, there exists a
time-like line connecting P and Q on which Q preceeds P; i.e. it is theoretically possible in these worlds to travel into the past, or otherwise influence the past."
Godel concludes from this that not only does the argument quoted above for 'the non-objectivity of change' hold in worlds represented by his solutions, but that "it is possible in these worlds to travel
arbitrarily far into the future or the past and back again exactly as it is possible in other worlds to travel to distant parts of space."
But how can
Godel infer from the unreality of time and change in his rotating worlds
their unreality in the world we live in ?
Godel grants that the static nature of his model is inconsistent with the
observed red-shift which is taken to indicate that we live in an expanding
universe. But he points out that he alas also found other rotating models
which are expanding and that "in such universes an absolute time might
also fail to exist, and it is not impossible that our world is a universe of
But in the end Godel rests his case for the non-objectivity of
time and change on the sheer possibility of worlds with no distinguished
absolute time and (in which) therefore, no objective lapse of time can
exist : "For, if someone asserts that this absolute time is lapsing, he accepts
as a consequence that whether or not an objective lapse of time exists, i.e.
whether or not a time in the ordinary sense of the word exists, depends on
the particular way in which matter and motion are arranged in the world.
This is not a straightforward contradiction; nevertheless, a philosophical
view leading to such consequences can hardly be considered satisfactory."
Ibid, p. 206-7.
The very existence of solutions showing the compatability with the
laws of nature of worlds in which there is no objective change implies that
time and change exist only contingently in worlds in which it can be
defined. By Godel's lights a satisfactory theory of objective time must
show that it necessarily exists.
How then do these arguments fit in with the rest of Godel's epistemology of mathematics, and how suitable are they for supporting Kant's thesis of the ideality of time ? How did he reconcile his conclusion that time
and change are 'an illusion or appearance due to our special mode of perception' with his vision of the development in time of the mathematical
mind grasping successively ever new axioms as true by virtue of its ever
enlarging repertoire of internal states ? I doubt that Godel thought he was
succumbing to an illusion when he argued that the mathematical intuition
of sets allows one 'to believe that a question about them not decided now
may be decided in the future', and that we may develop 'new mathematical intuitions leading to a decision of such problems' about them that now
elude us. Nor presumably did he believe that the relativity of simultaneity should be relevant to the structure of the time in which such intuitions
unfold. But Godel did claim that relativity favors those philosophers who,
like Kant, consider change an illusion or 'and appearance due to our special mode of perception', and this does sound like Kant. Godel even
claims that Kant "expresses this view" in the Critique when he says that
"those affections which we represent to ourselves as changes, in beings
with other forms of cognition, would give rise to a perception in which the
idea to time, and therefore also of change, would not occur at all"(A37);
a formulation, says Godel, that "agrees so well with the situation subsisting in relativity theory that one is almost tempted to add : such as, e.g. a
perception of the inclination relative to each other of the world lines of
matter in Minkowki space."
This however is a description better left
alone, for it seems that when understood properly, Kant's claim just quoted is not something that could possibly be elucidated, much less confirmed by a physical theory.
The passage quoted by Godel occurs in Kant's Elucidation of his thesis of the transcendental ideality of time written specifically to answer the
objections made by Lambert and Mendelssohn to Kant's original formulation of this thesis in the Dissertation. Lambert argued that the thesis was
incompatible with the reality of change, since "All changes are bound to
time and are inconceivable without time. If changes are real, then time is
real, whatever it may be. If time is unreal, then no change can be real. I
think, though, that even an idealist must grant at least that changes really
exist ad occur in his representations, e.g. their beginning and ending."
Kant : Correspondence, translated and edited by A....
Implicit in Lambert's formulation of the change argument, and made
explicit by Mendelssohn, is the point that 'even the idealist' must grant
that he observes a succession of real changes in his self, and hence knows
immediately an object undergoing real changes in time. Kant tried to meet
this argument in the Critique, by factoring consciousness into its active
and passive components, called 'apperception' and 'inner sense' respectively, and then showing that time is 'nothing but the form of inner sense'
which as such, Kant argues, is a 'pure intuition', and that even the idealist can only apprehend his self as it appears to him in inner sense, not as
it really is in itself. This 'paradox of inner sense', as Kant calls it, follows
from the ideality argument that, since time is only a pure intuition comprising the form of inner sense, it cannot be a property of things in themselves that would "thus remain if one abstracted from all subjective conditions of the intuition of them" (A32); for otherwise, he argued, "time
could not be cognized and intuited a priori through synthetic propositions"(A33)
Here Kant means the 'axioms of time' referred to at...
The transcendental ideality of time then claims that time "is
nothing at all if one abstracts from the subjective conditions of sensible
intuition" (A36), and it is this crucial abstraction thesis that Kant tries to
elucidate in the passage at A37 quoted repeatedly by Godel in his efforts
to find the 'relativity of time' in Kant.
What Kant says is that "If I or another being could intuit myself without this condition of sensibility, then these very determinations, which we
now represent to ourselves as alterations, would yield us a cognition in
which the representation of time and thus also of alteration would not
occur at all."(A37)
Kant's text reads : "Wenn aber ich selbst, oder ein...
Here then is the sense in which time lacks 'absolute
reality' for Kant : 'another being' could intuit my inner state as changeless.
But to carry the burden of the abstraction thesis and hence of ideality, this
cannot be just any old 'being with another form of cognition' as Godel
reads Kant's elucidation, but only one like God with an intellectual intuition of my unchanging self as it is in itself.
Godel comes at the crucial Elucidation passage from...
Nothing less will do. Kant's
ideality of time, by which he tries to make the causality of nature compatible with free will and morality, thus requires much more of another being
that what Godel's argument requires of it, namely, just another moving
observer whose judgments about the temporal relations of two physical
events A and B observed by both of us could justifiably conflict with my
own judgments about them. The fact that such an observer could, because
of a chronometry based on the constancy of the speed of light, see A
before B as well as I could see B before A is no reason to suppose that he
would intuit no change or any succession at all in my inner states. What
else could he assume than that a representation of B occurred in my inner
states before one of A ?
Godel further elaborated that : "For a being which had no sensibility of
all (i.e. not contact through sensations with reality), no time would exist.
But this is exactly what Kant means, who frequently explains the things
in themselves as the things such as pure understanding would perceive
them. (This of course does not apply to human understanding which is not
pure insofar as it apprehends concepts only with the help of sensual
images and symbols and therefore depends on sensibility and on time").
This does apply to the noumenal self or 'intelligible character' of human
beings which 'does not stand under any condition of time', though Godel
is correct in saying that things in themselves are things as intuited by a
pure understanding. As Kant said : "the concept of a noumenon, i.e. a thing
that is not thought of as an object of the senses but rather as a thing in
itself (solely through a pure understanding), is not at all contradictory; for
one cannot assert of sensibility that it is the only possible kind of intuition". (A254/B310). There is then at least this important similarity
between Kant's case for the ideality of time and Godel's argument from
general relativity : just as Godel argued against objective time and change
that it would be consistent with the laws of nature to imagine a world in
which no such time existed, Kant argued that there is no contradiction in
supposing a pure understanding who intuited our inner state intellectually as changeless. Of course, Godel actually produced a model demonstrating this consistency, something Kant could hardly been expected to do.
Since God is by Kant's lights the only being who could indubitably be
said to intuit thing intellectually, it should seem that his ideality of time
thesis depends as much on God as does Berkeley's comparable thesis
about space. In any case, it does not appear at all to have been understood
by Kant in a way that lends itself to support or clarification from relativity theory in any way envisaged by Godel.
Unlike Kant, whose motive for the ideality thesis for time is clearly to
achieve compatibility between free timeless moral agents and the laws of
time bound phenomena, it is more difficult to discern the philosophical
motive Godel's idealism with respect to time. Reichenbach once suggested that "the fear of death has greatly influenced the logical analysis which
philosophers have give of the problem of time. The belief that they had
discovered paradoxes in the flow of time is called 'projection' in modern
Hans Reichenbach : The Direction of Time, 1971, p....
Whether such an influence was at work in
Godel's arguments I could not say. In any case, they are extraordinarily
challenging whatever their source.