28/42 [No Music Version] Kurt Gödel: Modern Dvmt of the Foundations Of Math In Light Of Philosophy


Uploaded by GaryGeckDotCom on 01.08.2011

Transcript:
This is Gary Geck of Gary Geck.com. Kurt Godel has been called the greatest logician since
aristotle and a genius at odds with the zeitgeist. The following is a reading from Godel's 1961
lecture called "Modern Development of the Foundations Of Mathematics In Light Of Philosophy".
As was typical of Godel's very private philosophical work, the lecture was never delivered. I will
now read it in it's entirety. I will now begin:
I would like to attempt here to describe, in terms of philosophical concepts, the development
of foundational research in mathematics since around the turn of the century, and to fit
it into a general schema of possible philosophical world-views [Weltanschauungen]. For this,
it is necessary first of all to become clear about the schema itself. I believe that the
most fruitful principle for gaining an overall view of the possible world-views will be to
divide them up according to the degree and the manner of their affinity to or, respectively,
turning away from metaphysics (or religion). In this way we immediately obtain a division
into two groups: scepticism, materialism and positivism stand on one side, spiritualism,
idealism and theology on the other. We also at once see degrees of difference in this
sequence, in that scepticism stands even farther away from theology than does materialism,
while on the other hand idealism, e.g., in its pantheistic form, is a weakened form of
theology in the proper sense. The schema also proves fruitful, however,
for the analysis of philosophical doctrines admissible in special contexts, in that one
either arranges them in this manner or, in mixed cases, seeks out their materialistic
and spiritualistic elements. Thus one would, for example, say that apriorism belongs in
principle on the right and empiricism on the left side. On the other hand, however, there
are also such mixed forms as an empiristically grounded theology. Furthermore one sees also
that optimism belongs in principle toward the right and pessimism toward the left. For
scepticism is certainly a pessimism with regard to knowledge. Moreover, materialism is inclined
to regard the world as an unordered and therefore meaningless heap of atoms. In addition, death
appears to it to be final and complete annihilation, while, on the other hand, theology and idealism
see sense, purpose and reason in everything. On the other hand, Schopenhauer's pessimism
is a mixed form, namely a pessimistic idealism. Another example of a theory evidently on the
right is that of an objective right and objective aesthetic values, whereas the interpretation
of ethics and aesthetics on the basis of custom, upbringing, etc., belongs toward the left.
Now it is a familiar fact, even a platitude, that the development of philosophy since the
Renaissance has by and large gone from right to left - not in a straight line, but with
reverses, yet still, on the whole. Particularly in physics, this development has reached a
peak in our own time, in that, to a large extent, the possibility of knowledge of the
objectivisable states of affairs is denied, and it is asserted that we must be content
to predict results of observations. This is really the end of all theoretical science
in the usual sense (although this predicting can be completely sufficient for practical
purposes such as making television sets or atom bombs).
It would truly be a miracle if this (I would like to say rabid) development had not also
begun to make itself felt in the conception of mathematics. Actually, mathematics, by
its nature as an a priori science, always has, in and of itself, an inclination toward
the right, and, for this reason, has long withstood the spirit of the time [Zeitgeist]
that has ruled since the Renaissance; i.e., the empiricist theory of mathematics, such
as the one set forth by Mill, did not find much support. Indeed, mathematics has evolved
into ever higher abstractions, away from matter and to ever greater clarity in its foundations
(e.g., by giving an exact foundation of the infinitesimal calculus and the complex numbers)
- thus, away from scepticism. Finally, however, around the turn of the century,
its hour struck: in particular, it was the antinomies of set theory, contradictions that
allegedly appeared within mathematics, whose significance was exaggerated by sceptics and
empiricists and which were employed as a pretext for the leftward upheaval. I say "allegedly"
and "exaggerated" because, in the first place, these contradictions did not appear within
mathematics but near its outermost boundary toward philosophy, and secondly, they have
been resolved in a manner that is completely satisfactory and, for everyone who understands
the theory, nearly obvious. Such arguments are, however, of no use against the spirit
of the time, and so the result was that many or most mathematicians denied that mathematics,
as it had developed previously, represents a system of truths; rather, they acknowledged
this only for a part of mathematics (larger or smaller, according to their temperament)
and retained the rest at best in a hypothetical sense namely, one in which the theory properly
asserts only that from certain assumptions (not themselves to be justified), we can justifiably
draw certain conclusions. They thereby flattered themselves that everything essential had really
been retained. Since, after all, what interests the mathematician, in addition to drawing
consequences from these assumptions, is what can be carried out. In truth, however, mathematics
becomes in this way an empirical science. For if I somehow prove from the arbitrarily
postulated axioms that every natural number is the sum of four squares, it does not at
all follow with certainty that I will never find a counter-example to this theorem, for
my axioms could after all be inconsistent, and I can at most say that it follows with
a certain probability, because in spite of many deductions no contradiction has so far
been discovered. In addition, through this hypothetical conception of mathematics, many
questions lose the form "Does the proposition A hold or not?" For, from assumptions construed
as completely arbitrary, I can of course not expect that they have the peculiar property
of implying, in every case, exactly either A or ~A.
Although these nihilistic consequences are very well in accord with the spirit of the
time, here a reaction set in obviously not on the part of philosophy, but rather on that
of mathematics, which, by its nature, as I have already said, is very recalcitrant in
the face of the Zeitgeist. And thus came into being that curious hermaphroditic thing that
Hilbert's formalism represents, which sought to do justice both to the spirit of the time
and to the nature of mathematics. It consists in the following: on the one hand, in conformity
with the ideas prevailing in today's philosophy, it is acknowledged that the truth of the axioms
from which mathematics starts out cannot be justified or recognised in any way, and therefore
the drawing of consequences from them has meaning only in a hypothetical sense, whereby
this drawing of consequences itself (in order to satisfy even further the spirit of the
time) is construed as a mere game with symbols according to certain rules, likewise not supported
by insight. But, on the other hand, one clung to the belief,
corresponding to the earlier "rightward" philosophy of mathematics and to the mathematician's
instinct, that a proof for the correctness of such a proposition as the representability
of every number as a sum of four squares must provide a secure grounding for that proposition
- and furthermore, also that every precisely formulated yes-or-no question in mathematics
must have a clear-cut answer. I.e., one thus aims to prove, for inherently unfounded rules
of the game with symbols, as a property that attaches to them so to speak by accident,
that of two sentences A and ~A, exactly one can always be derived. That not both can be
derived constitutes consistency, and that one can always actually be derived means that
the mathematical question expressed by A can be unambiguously answered. Of course, if one
wishes to justify these two assertions with mathematical certainty, a certain part of
mathematics must be acknowledged as true in the sense of the old rightward philosophy.
But that is a part that is much less opposed to the spirit of the time than the high abstractions
of set theory. For it refers only to concrete and finite objects in space, namely the combinations
of symbols.
What I have said so far are really only obvious things, which I wanted to recall merely because
they are important for what follows. But the next step in the development is now this:
it turns out that it is impossible to rescue the old rightward aspects of mathematics in
such a manner as to be more or less in accord with the spirit of the time. Even if we restrict
ourselves to the theory of natural numbers, it is impossible to find a system of axioms
and formal rules from which, for every number-theoretic proposition A, either A or ~A would always
be derivable. And furthermore, for reasonably comprehensive axioms of mathematics, it is
impossible to carry out a proof of consistency merely by reflecting on the concrete combinations
of symbols, without introducing more abstract elements. The Hilbertian combination of materialism
and aspects of classical mathematics thus proves to be impossible.
Hence, only two possibilities remain open. One must either give up the old rightward
aspects of mathematics or attempt to uphold them in contradiction to the spirit of the
time. Obviously the first course is the only one that suits our time and is therefore also
the one usually adopted. One should, however, keep in mind that this is a purely negative
attitude. One simply gives up aspects whose fulfilment would in any case be very desirable
and which have much to recommend themselves: namely, on the one hand, to safeguard for
mathematics the certainty of its knowledge, and on the other, to uphold the belief that
for clear questions posed by reason, reason can also find clear answers. And as should
be noted, one gives up these aspects not because the mathematical results achieved compel one
to do so but because that is the only possible way, despite these results, to remain in agreement
with the prevailing philosophy. Now one can of course by no means close one's
eyes to the great advances which our time exhibits in many respects, and one can with
a certain justice assert that these advances are due just to this leftward spirit in philosophy
and world-view. But, on the other hand, if one considers the matter in proper historical
perspective, one must say that the fruitfulness of materialism is based in part only on the
excesses and the wrong direction of the preceding rightward philosophy. As far as the rightness
and wrongness, or, respectively, truth and falsity, of these two directions is concerned,
the correct attitude appears to me to be that the truth lies in the middle or consists of
a combination of the two conceptions. Now, in the case of mathematics, Hilbert had
of course attempted just such a combination, but one obviously too primitive and tending
too strongly in one direction. In any case there is no reason to trust blindly in the
spirit of the time, and it is therefore undoubtedly worth the effort at least once to try the
other of the alternatives mentioned above, which the results cited leave open - in the
hope of obtaining in this way a workable combination. Obviously, this means that the certainty of
mathematics is to be secured not by proving certain properties by a projection onto material
systems - namely, the manipulation of physical symbols but rather by cultivating (deepening)
knowledge of the abstract concepts themselves which lead to the setting up of these mechanical
systems, and further by seeking, according to the same procedures, to gain insights into
the solvability, and the actual methods for the solution, of all meaningful mathematical
problems. In what manner, however, is it possible to
extend our knowledge of these abstract concepts, i.e., to make these concepts themselves precise
and to gain comprehensive and secure insight into the fundamental relations that subsist
among them, i.e., into the axioms that hold for them? Obviously not, or in any case not
exclusively, by trying to give explicit definitions for concepts and proofs for axioms, since
for that one obviously needs other undefinable abstract concepts and axioms holding for them.
Otherwise one would have nothing from which one could define or prove. The procedure must
thus consist, at least to a large extent, in a clarification of meaning that does not
consist in giving definitions. Now in fact, there exists today the beginning
of a science which claims to possess a systematic method for such a clarification of meaning,
and that is the phenomenology founded by Husserl. Here clarification of meaning consists in
focusing more sharply on the concepts concerned by directing our attention in a certain way,
namely, onto our own acts in the use of these concepts, onto our powers in carrying out
our acts, etc. But one must keep clearly in mind that this phenomenology is not a science
in the same sense as the other sciences. Rather it is or in any case should be a procedure
or technique that should produce in us a new state of consciousness in which we describe
in detail the basic concepts we use in our thought, or grasp other basic concepts hitherto
unknown to us. I believe there is no reason at all to reject such a procedure at the outset
as hopeless. Empiricists, of course, have the least reason of all to do so, for that
would mean that their empiricism is, in truth, an apriorism with its sign reversed.
But not only is there no objective reason for the rejection of phenomenology, but on
the contrary one can present reasons in its favour. If one considers the development of
a child, one notices that it proceeds in two directions: it consists on the one hand in
experimenting with the objects of the external world and with its own sensory and motor organs,
on the other hand in coming to a better and better understanding of language, and that
means - as soon - as the child is beyond the most primitive designating of objects - of
the basic concepts on which it rests. With respect to the development in this second
direction, one can justifiably say that the child passes through states of consciousness
of various heights, e.g., one can say that a higher state of consciousness is attained
when the child first learns the use of words, and similarly at the moment when for the first
time it understands a logical inference. Now one may view the whole development of
empirical science as a systematic and conscious extension of what the child does when it develops
in the first direction. The success of this procedure is indeed astonishing and far greater
than one would expect a priori: after all, it leads to the entire technological development
of recent times. That makes it thus seem quite possible that a systematic and conscious advance
in the second direction will also far exceed the expectations one may have a priori.
In fact, one has examples where, even without the application of a systematic and conscious
procedure, but entirely by itself, a considerable further development takes place in the second
direction, one that transcends "common sense". Namely, it turns out that in the systematic
establishment of the axioms of mathematics, new axioms, which do not follow by formal
logic from those previously established, again and again become evident. It is not at all
excluded by the negative results mentioned earlier that nevertheless every clearly posed
mathematical yes-or-no question is solvable in this way. For it is just this becoming
evident of more and more new axioms on the basis of the meaning of the primitive notions
that a machine cannot imitate. I would like to point out that this intuitive
grasping of ever newer axioms that are logically independent from the earlier ones, which is
necessary for the solvability of all problems even within a very limited domain, agrees
in principle 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 that 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. I believe it to be a general
feature of many of Kant's assertions that literally understood they are false but in
a broader sense contain deep truths. In particular, the whole phenomenological method, as I sketched
it above, goes back in its central idea to Kant, and what Husserl did was merely that
he first formulated it more precisely, made it fully conscious and actually carried it
out for particular domains. Indeed, just from the
terminology used by Husserl, one sees how positively he himself values his relation
to Kant. I believe that precisely because in the last
analysis the Kantian philosophy rests on the idea of phenomenology, albeit in a not entirely
clear way, and has just thereby introduced into our thought something completely new,
and indeed characteristic of every genuine philosophy - it is precisely on that, I believe,
that the enormous influence which Kant has exercised over the entire subsequent development
of philosophy rests. Indeed, there is hardly any later direction that is not somehow related
to Kant's ideas. On the other hand, however, just because of the lack of clarity and the
literal incorrectness of many of Kant's formulations, quite divergent directions have developed
out of Kant's thought - none of which, however, really did justice to the core of Kant's thought.
This requirement seems to me to be met for the first time by phenomenology, which, entirely
as intended by Kant, avoids both the death-defying leaps of idealism into a new metaphysics as
well as the positivistic rejection of all metaphysics. But now, if the misunderstood
Kant has already led to so much that is interesting in philosophy, and also indirectly in science,
how much more can we expect it from Kant understood correctly?
[Image terms: Godel's Incompleteness theroems, Nicholas of Cusa, Willard Quine, Mysticism,
Mystical, Infinity, Georg Cantor, Ernst Mach, Albert Einstein, Luitzen Egbertus Jan Brouwer,
Karl Popper, Gilles Deleuze, Goethe, Fitche, Hegel, Plato, Platonic academy, Princeton
Institute for Advanced Study, time travel, documentary, etc.]