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.]

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.]