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DRAFT
ON KURT GÖDEL'S PHILOSOPHY
OF MATHEMATICS
by
Martin K. Solomon
Department of Computer Science and Engineering
Florida Atlantic University
Boca Raton, FL 33431
ABSTRACT
We characterize Gödel's
philosophy of mathematics, as presented in his published works, with
possible clarification and support provided by his posthumously published
drafts, as being formulated by Gödel as an optimistic neo-Kantian epistemology
superimposed on a Platonic metaphysics. We compare Gödel's philosophy
of mathematics to Steiner's "epistemological structuralism."
§1.
Introduction
We show that Gödel's philosophy
of mathematics, as presented in his published works, with possible clarification
and support provided by his posthumously published drafts, can be considered
as being formulated by Gödel as an optimistic neo-Kantian epistemology
(obtained from Kant's epistemology regarding the physical world in terms
of sensory appearances as distinct from things in themselves, not obtained
from Kant's epistemology of mathematics as being synthetic a priori
knowledge) superimposed on a platonic metaphysics. By Platonic
metaphysics, we of course mean that abstract objects have an objective
existence. By neo-Kantian we mean obtained from the Kantian epistemology
with one important modification, namely, removing the doctrine of the
unknowability of things in themselves.
Indeed, we will see in section
2.2 that Gödel thought that abstract things in themselves may be progressively
knowable. Furthermore, it is pointed out in section 2.4 that he
explicitly indicated that the knowability of physical things in themselves
is possible through the progressive advancement of modern science.
It is also pointed out in section 2.4 that Gödel didn't think that
Kant would consider such a modification to be as significant as might
some of Kant's followers.
In other words, in Gödel's well-known
analogy of mathematical intuition to sense perception (see the passage
from [16, p. 268] in section 2.1), he is clearly using (what he views
as) a Kantian model of the sensory world of experience, optimistically
modified. However, apparently missing from this Kantian model
when Gödel applies it to abstract intuitions is the subjective a priori
component (i.e., Gödel never mentions a mathematical intuition analog
of anything akin to the a priori intuitions of space and time that Kant
held for physical perception). Based on Gödel's letter to Greenberg
and the passage from [12, p. 241] given is section 2.1, it appears that
Gödel agreed with Kant in the existence of such a subjective a priori
component for physical perception. Another bit of interesting
information supplied by Gödel's letter to Greenberg is that Gödel
in [16, p. 268] was specifically discussing set theoretical intuition
as not necessarily providing "immediate Knowledge of the object
concerned," whereas Gödel feels that geometric intuition ("in
its purely mathematical aspect") does provide such immediate knowledge.
In section 2.4 we point out that
Gödel apparently believed that mathematical intuitions are more "direct"
than sense perceptions, presumably because the mathematical intuitions
are abstract impressions of abstract objects, whereas we are advancing
toward knowledge of physical things in themselves only by viewing the
sensory world through the abstract lenses of modern physics. Also,
in section 2.4, we conjecture that the seeming absence of the above
mentioned a priori subjective component from Gödel's view of mathematical
perception, as contrasted with his apparent agreement that such an a
priori component exists for sense perception, could have contributed
to his considering sense perception less direct than mathematical perception.
In section 3, we compare Gödel's
philosophy of mathematics to (what we may call) the epistemological
structuralist philosophy of mathematics that is briefly presented by
Mark Steiner in his book "Mathematical Knowledge" [24].
We observe that, although there are some clear differences between the
approaches of Gödel and Steiner, there are also some surprising similarities
between their approaches. Specifically, both approaches center
around the distinction between mathematical things in themselves and
our intuitions regarding these things, both approaches consider intuitions
to synthesize unities out of manifolds, both approaches distinguish
between different kinds of mathematical intuition, and both approaches
consider the content of mathematical statements to regard the relationship
between abstract objects. Thus, we will see that Gödel's philosophy
of mathematics has some elements in common with a certain kind of structuralist
philosophy of mathematics.
One important difference between
the two approaches is that Steiner is pessimistic (as is Kant with regard
to physical things in themselves) in that he arguably considers abstract
things in themselves to be unknowable. (Steiner states that "the
only things of value to know about abstract objects are such relationships"
[24, p. 134]; we argue in section 3.1 that "the only things of
value to know" in his statement can be equivalently replaced with
"the only things that can be known".) On the other hand,
Gödel, as mentioned previously, is optimistic in that he allows for
the convergence on knowledge of things in themselves. Therefore,
Steiner's philosophy of mathematics can also be considered to be a variety
of neo-Kantian Platonism "that is more Kantian" than Gödel's
variety of neo-Kantian Platonism.
In our conclusion, we conjecture
that Gödel can be considered a neo-Kantian Platonist, not only for
mathematics, but regarding the physical world as well.
§2.
Gödel's philosophy of mathematics
2.1. Abstract reality
and appearance
Even in Gödel's 1944 "Russell's
Mathematical Logic," along with postulating the existence of an
abstract reality, there is a hint of Gödel's distinction between abstract
reality and what our intuition may provide us with concerning that reality.
Classes and concepts may, however,
also be conceived as real objects, namely classes as "pluralities
of things" or as structures consisting of a plurality of things
[Gödel is referring to the membership trees of Mirimanoff; see [1]
for a good discussion of Mirimanoff's ideas] and concepts as the properties
and relations of things existing independently of our definitions and
constructions. [10, p. 128]
... the objects to be analyzed
(e.g., the classes or proposition) soon for the most part turned into
"logical fictions". Though perhaps this need not necessarily
mean (according to the sense in which Russell uses this term) that these
things do not exist, but only that we have no direct perception of them.
[10 p. 121]
From these two passages it is clear
that:
In the 1964 version of "What
is Cantor's Continuum Problem?" [16] Gödel elaborates in more
detail his ideas concerning the above distinction, and he further clarifies
his point of view in his letter to Marvin Jay Greenberg, which was sent
to provide material for Greenberg's book [19]. To start, in [16,
p. 259 n14], Gödel states that "set of x's" exists
as a thing in itself, even though at the present time we do not have
a clear grasp of the general concept of set (or "random sets,"
as Gödel puts it):
(footnote 14)
The operation "set of
x's" (where the variable x ranges over some given kind
of objects) cannot be defined satisfactorily (at least not in the present
state of knowledge), but can only be paraphrased by other expressions
involving again the concept of set, such as: "multitude" ("combination",
"part") is conceived of as something which exists in itself
no matter whether we can define it in a finite number of words (so that
random sets are not excluded).
Observe the hint of optimism in
this footnote, in which Gödel implies that the gap between the set
concept as a thing in itself and our intuitions concerning that concept,
may be narrowed in the future. We can see from other remarks of
Gödel in section 2.2, that this may be more than a cautious parenthetical,
but actually may reflect an important optimistic component of Gödel's
philosophy of mathematics.
Then, in the following intriguing
(and much-cited) passage from [16], Gödel gives his most direct presentation
of his epistemological ideas:
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 upon 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.
It should be noted that mathematical intuition need not be conceived of as a faculty giving an immediate knowledge of the objects concerned. Rather it seems that, as in the case of physical experience, we form our ideas also of those objects on the basis of something else which is immediately given. Only this something else here is not, or not primarily, the sensations. That something besides the sensations actually is immediately given follows (independently of mathematics) from the fact that even our ideas referring to physical objects contain constituents qualitatively different from sensations or mere combinations of sensations, e.g., the idea of object itself, whereas, on the other hand, by our thinking we cannot create only qualitatively new elements, but only reproduce and combine those that are given. Evidently the "given" underlying mathematics is closely related to the abstract elements contained in our empirical ideas.40 It by no means follows, however, that the data of this second kind, because they cannot be associated with actions of certain things upon our sense organs, are something purely subjective, as Kant asserted. Rather they, too, may represent an aspect of objective reality, but, as opposed to the sensations, their presence in us may be due to another kind of relationship between ourselves and reality. [16, p. 268]
40Note
that there is a close relationship between the concept of set explained
in footnote 14 [16, p. 259 n14] and the categories of pure understanding
in Kant's sense. Namely, the function of both is "synthesis",
i.e., the generating of unities out of manifolds (e.g., in Kant, of
the idea of one object out of its various aspects). [16, p. 268
n40]
Here Gödel identifies what mathematical
intuition provides to us (actually, according to his letter to Greenberg,
Gödel means specifically set theoretic intuition) as being something
that synthesizes a unity out of a manifold (data of the second kind).
Gödel refers to such "data of the second kind" as "abstract
impressions" in [25], as we shall see in section 2.2. Data
of the second kind in mathematics provides abstract impressions of abstract
objects (objects which themselves, by the above footnote 14, also synthesize
unities out of manifolds).
Also, rather surprisingly, as part
of his argument that data of the second kind is also involved in our
cognition of the physical world, Gödel characterizes thinking in a
manner that Potter calls "trivial" [20, p. 9] (is it also
mechanical?). Given Gödel's well-known view that mind is more
powerful than machine, if thinking is mechanical then the intuition
"input facility" is what gives the mind its power. We
will reexamine Gödel's "trivial" concept of mind in section
2.3, when we consider the implications of Gödel's conjecture concerning
the existence of an abstract sense organ.
Gödel's letter to Marvin Jay Greenberg
further clarifies the above passages from [16]. Greenberg mailed
Gödel, asking Gödel's permission to quote as follows from that article:
I don't see any reason why
we should have less confidence in this kind of perception, i.e., in
mathematical intuition, than is sense perception, which induces us to
build up physical theories and to expect that future sense perceptions
will agree with them and, moreover, to believe that a question not decidable
now has meaning and may be decided in the future. The set theoretical
paradoxes are hardly any more troublesome for mathematics than deceptions
of the senses are for physics. ... Evidently the "given" underlying
mathematics is closely related to the abstract elements contained in
our empirical ideas. It by no means follows, however, that the
data of this second kind [mathematical intuitions]1, because
they cannot be associated with actions of certain things upon our sense
organs, are something purely subjective, as Kant asserted. Rather,
they, too, may represent an aspect of objective reality. But as
opposed to the sensations, their presence in us may be due to another
kind of relationship between ourselves and reality. [19, p. 305]
Gödel responded to Greenberg as
follows:
Dear Professor Greenberg:
I have no objection
to the quotation mentioned in your letter of September 5, provided
you add the following:
Gödel in this
passage speaks (primarily) of set theoretical intuition.
As far as geometrical intuition is concerned the following, according
to Gödel, would have to be added: "Geometrical intuition, strictly
speaking, is not mathematical, but rather a priori physical, intuition.
In its purely mathematical aspect our Euclidean 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'."
This addition
is absolutely necessary in view of the fact that your book deals with
geometry, and that, moreover, in your quotation, you omit the first
sentence of the paragraph in question. See Benacerraf-Putnam,
Philosophy of Mathematics, Prentice-Hall, 1964, p. 271.1
Sincerely yours,
Kurt Gödel
1I also have to
request that you give this reference in full because you omit
important parts of my exposition, and, moreover, the passage you quote
does not occur in my original paper, but only in the supplement to the
second edition. [18, pp. 453-454]
From Gödel's letter we can see
that:
In the case of geometry, e.g.,
the fact that the physical bodies surrounding us move by the laws of
a non-Euclidean geometry does not exclude in the least that we should
have a Euclidean "form of sense perception", i.e., that we
should possess an a priori representation of Euclidean space and be
able to form images of outer objects only by projecting our sensations
on this representation of space, so that, even if we were born in some
strongly non-Euclidean world, we would nevertheless invariably imagine
space to be Euclidean, but material objects to change their size and
shape in a certain regular manner, when they move with respect to us
or we with respect to them.
2.2 Gödel's optimistic epistemology
for abstract objects
We have already noted in section
2.1 the optimistic tone in Gödel's letter to Greenberg [18] and a footnote
in the 1964 version of "What is Cantor's Continuum Problem?"
[16, p. 259 n14].
In his 1946 "Remarks Before
the Princeton Bicentennial Conference," [11] Gödel expressed optimism
concerning the possibility of discovering, in the future, a concept
of demonstrability (with a nonmechanical, but humanly generated axiom
set) that is complete for mathematics (i.e., set theory), and hence
absolute, just as Turing discovered the absolute concept of computability
[11, p. 151].
Also, Gödel expresses in [25,
pp. 324-325] (where he is summarizing from his Gibbs lecture) the view
that there do not exist number theoretical propositions that are undecidable
for the human mind (i.e., that are absolutely undecidable). From
the argument he gives there, Gödel also seems to be rejecting the existence
of any mathematical truths that are absolutely undecidable. In
"The Modern Development of the Foundations of Mathematics in the
Light of Philosophy," [15], which apparently is a draft of a lecture
that Gödel planned to deliver before the American Philosophical Society
but never delivered, he states:
It is not at all excluded by
the negative results mentioned earlier [his incompleteness theorems]
that nevertheless every clearly posed mathematical yes-or-no question
is solvable in this way [by the "intuitive grasping of even newer
axioms"]. [15, p. 385]
In [25, pp. 84-85] Gödel further
elaborates on his optimistic epistemology for abstract objects.
In particular, he describes how we can begin with an abstract impression
(called data of the second kind in [16]) of an abstract concept (in
itself) that is vague, and we can end up with the sharp concept that
faithfully represents the abstract concept in itself:
Gödel points out that the
precise notion of mechanical procedures is brought out clearly by Turing
machines ... The resulting definition of the concept of mechanical by
the sharp concept of 'performable by a Turing machine' is both correct
and unique. ... Gödel emphasizes that there is at least one highly
interesting concept which is made precise by the unqualified notion
of a Turing machine. Namely a formal system is nothing but a mechanical
procedure for producing theorems. ... In fact, the concept of formal
systems was not clear at all in 1931. Otherwise Gödel would have
then proved his incompleteness results in a more general form. ... 'If
we begin with a vague intuitive concept, how can we find a sharp concept
to correspond to it faithfully?' The answer Gödel gives is that
the sharp concept is there all along, only we did not perceive it clearly
at first. This is similar to our perception of an animal first
far away and then nearby. We had not perceived the sharp concept
of mechanical procedures sharply before Turing, who brought us the right
perspective. And then we do perceive clearly the sharp concept.
There are more similarities than differences between sense perceptions
and the perceptions of concepts. In fact, physical objects are
perceived more indirectly than concepts. The analog of perceiving
sense objects from different angles is the perception of different logically
equivalent concepts. If there is nothing sharp to begin with,
it is hard to understand how, in many cases, a vague concept can uniquely
determine a sharp one without even the slightest freedom of choice.
... Gödel conjectures that some physical organ is necessary to make
the handling of abstract impressions (as opposed to sense impressions)
possible, because we have some weakness in the handling of abstract
impressions which is remedied by viewing them in comparison with or
on the occasion of sense impressions. Such a sensory organ must
be closely related to the neural center for language.
2.3 Digression on Gödel's abstract
sense organ
Please observe that Gödel's thoughts
in the above passage concerning a "necessary" abstract impression
physical organ can be viewed as being motivated by a desire to strongly
counter the "Kantian assertion" stated in [16, p. 268]:
It by no means follows, however,
that the data of this second kind, because they cannot be associated
with actions of certain things upon our sense organs, are something
purely subjective, as Kant asserted.
Similarly, the existence of such
a physical organ could be used to counter Benacerraf's criticism of
Platonism that is based on the causal account of knowledge [2].
Finally, we might comment on the
apparent inconsistency of Gödel's belief in the existence of such an
abstract sense organ with his view that the mind is not mechanical (this
view is expressed, in particular, in [25, pp. 324-326] and [17, p. 306]).
We express this apparent inconsistency by synthesizing Gödel's views
in the following argument:
However, there are several possible
ways out of the apparent inconsistency of the above argument with Gödel's
non-mechanical view of mind:
What Turing disregards completely
is the fact that mind, in its use, is not static, but constantly
developing, i.e., that we understand abstract terms more and more
precisely as we go on using them, and that more and more abstract terms
enter the sphere of our understanding. ... Therefore, although at each
stage the number and precision of the abstract terms at our disposal
may be finite, both (and, therefore, also Turing's number of
distinguishable states of mind) may converge toward infinity
...
However, the version of this
passage that appears in [25, pp. 325-326], a somewhat later version
according to Wang [27, pp. 123-124], is not explicit about there being
two ways for producing an infinite number of mental states.
Even if the finite brain cannot
store an infinite amount of information, the spirit [mind]2
may be able to. The brain is a computing machine [situated in
the special manner of being]3 connected with a spirit.
If the brain is taken as physical and as a digital computer, from quantum
mechanics there are then only a finite number of states. Only
by connecting it to a spirit might it work some other way. [27, p. 127]
This infinite amount of memory would be needed to store the
infinite number of intuitions that "a mind of an unlimited
life span" would generate. And Gödel told Wang that "a mind
of an unlimited life span" is what Gödel meant by mind, and is
"close to the real situation" where "people constantly
introduce new axioms"
[27, p. 121].
However, although such infinite memory might distinguish the
mind from the brain, it would not distinguish the mind from a
Turing machine (which also has potentially infinite memory), unless
the mind could store an infinite amount of information, such
as a burst of an infinite
number of axioms, at once.
We close this section with passages
from two letters provided in Gödel's Collected Works in which Gödel
succinctly presents his view that the intuition of abstract objects
provides the mind the ability to surpass the machine.
Nothing has been changed lately
in my results or their philosophical consequences, but perhaps some
misconceptions of them have been dispelled or weakened. My theorems
only show that the mechanization of mathematics, i.e., the elimination
of the mind and of abstract entities, is impossible, if
one wants to have a satisfactory foundation and system of mathematics.
[9, p. 176, Letter to Leon Rappaport]
What has been proved is only
that the kind of reasoning necessary in mathematics cannot be completely
mechanized. Rather constantly renewed appeals to mathematical
intuition are necessary. The decision of my "undecidable"
proposition ... results from such an appeal. ... Whether every
arithmetical yes or no question can be decided with the help of some
chain of mathematical intuitions is not known. At any rate it
has not been proved that there are arithmetical questions undecidable
by the human mind. Rather what has been proved is only this: Either
there are such questions or the human mind is more than a machine.
In my opinion the second alternative is much more likely. [9, p. 162,
Letter to David F. Plummer]
Here is one formulation of the philosophical meaning of my result, which I have given once in answer to an inquiry:
The few immediately evident axioms from which all of contemporary mathematics can be derived do not suffice for answering all Diophantine yes or no questions of a certain well-defined simple kind.
Rather, for answering
all these questions, infinitely many new axioms are necessary, whose
truth can (if at all) be apprehended only by constantly renewed appeals
to a mathematical intuition, which is actualized in the course of the
development of mathematics. Such an intuition appears, e.g., in
the axioms of infinity of set theory. [8, p. 330, letter to George A.
Brutian]
2.4 Comparison with Gödel's
optimistic epistemology for physical reality
In Gödel's drafts of [13] he apparently
feels that Kant's "pessimistic" view that physical things
in themselves are unknowable should be modified so as to bring Kant's
epistemology into agreement with modern science:
A real contradiction between relativity theory and Kantian philosophy seems to me to exist only in one point, namely, as to Kant's opinion that natural science in the description it gives of the world must necessarily retain the forms of our sense perception and can do nothing else but set up relations between appearances within this frame.
This view of Kant has doubtless its source in his conviction of the unknowability (at least by theoretical reason) of the things in themselves, and at this point, it seems to me, Kant should be modified, if one wants to establish agreement between his doctrines and modern physics; i.e., it should be assumed that it is possible for scientific knowledge, at least partially and step by step, to go beyond the appearances and approach the world of things.
The abandoning
of that "natural" picture of the world which Kant calls the
world of "appearance" is exactly the main characteristic distinguishing
modern physics from Newtonian physics. Newtonian physics ... is
only a refinement, but not a correction, of this picture of the world;
modern physics however has an entirely different character. This
is seen most clearly from the distinction which has developed between
"laboratory language" and the theory, whereas Newtonian physics
can be completely expressed in a refined laboratory. [12, p. 244]
... one may find a description
in more detail of these steps or "levels of objectivation",
each of which is obtained from the preceding one by the elimination
of certain subjective elements. The "natural" world
picture, i.e., Kant's world of appearances itself, also must of course
be considered as one such level, in which a great many subjective elements
of the "world of sensations" are already eliminated.
Unfortunately whenever this fruitful viewpoint of a distinction
between subjective and objective elements in our knowledge (which is
so impressively suggested by Kant's comparison with the Copernican system)
appears in epistemology, there is at once a tendency to exaggerate it
into a boundless subjectivism, whereby its effect is annulled.
Kant's thesis of the unknowability of the things in themselves is one
example; and another one is the prejudice that the positivistic interpretation
of quantum mechanics, the only one known at present, must necessarily
be the final stage of the theory. [12, p. 240 n24]
Note how the last passage also
nicely characterizes Gödel's epistemology of mathematics where we distinguish
intuitions of abstract objects from the intuitions of those objects
in themselves, but consider our intuitions to converge on those objects
in themselves. The applicability of this passage to Gödel's epistemology
of mathematics is not surprising, since, as we have seen in the preceding
sections, that in Gödel's well-known analogy of mathematical intuition
to sensory perception is clearly based on (what he views as) the Kantian
model for the sensory world of experience that is based on the contrast
between that world of experience with the world of things in themselves,
after having optimistically modified this model.
In the following passage Gödel
expresses his belief that the unknowability of things in themselves
is more a tenet of Kant's followers than Kant himself:
Moreover, it is to be noted
that the possibility of a knowledge of things beyond the appearances
is by no means so strictly opposed to the views of Kant himself as it
is to those of many of his followers. For (1) Kant held the concept
of things in themselves to be meaningful and emphasized repeatedly that
their existence must be assumed, (2) the impossibility of a knowledge
concerning them, in Kant's view, is by no means a necessary consequence
of the nature of knowledge, and perhaps does not subsist even for human
knowledge in every respect. [12, p. 245]
We conclude this section by observing
that in several places Gödel expressed the view that abstract objects
are perceived more directly than physical objects. This is expressed
in the [25, p. 85] passage, which was given in section 2.2, as "In
fact physical objects are perceived more indirectly than concepts."
It is also expressed, somewhat more demurely, in the passage from [21,
p. 217], which is given in section 2.5, as "we perceive mathematical
objects and facts just as immediately as physical objects, or perhaps
more so." This is presumably the case because mathematical
(or conceptual) intuitions are abstract impressions that converge on
the abstract objects in themselves, whereas the advance toward knowledge
of physical things in themselves only by viewing the sensory world through
the abstract lenses of modern physics. Also, we conjecture that
the seeming absence of an a priori subjective component from Gödel's
view of mathematical intuition (at least he never mentions such a component),
as contrasted with his apparent agreement with Kant that such an a priori
component exists for sense perception (see section 2.1), could have
contributed to Gödel's judgement that sense perception is less direct
than mathematical intuition.
2.5 Intuition of abstract objects
and mathematical facts
Throughout Gödel's published philosophical
writings, it appears to us that he considers mathematical intuition
as providing both objects as well as facts (truths) to the mind, but
he is not very explicit about this. However, in [21, p. 217],
Gödel explicitly states this to be the case:
There exist experiences, namely
those of mathematical intuition, in which we perceive mathematical objects
and facts just as immediately as physical objects, or perhaps more so.
It is arbitrary to consider "this is red" an immediate datum,
but not so to consider modus ponens or complete induction (or
perhaps some simpler propositions from which the latter follows).
For the difference, as far as it is relevant here, consists solely in
the fact that in the first case a relationship between a concept and
a particular object is perceived, while in the second case it is a relationship
between concepts.
Note also this passage suggests
that the content of a mathematical truth is the relationship between
concepts that it expresses. This is also indicated in a different
passage from the same draft:
... the reasoning which
leads to the conclusion that no mathematical facts exist is nothing
but a petitio principii, i.e., "fact" from the beginning is
identified with "empirical fact", i.e. "fact in the world
of sense perception." Platonists should agree that mathematics
has no content of this kind. For its content, according to Platonism,
does not consist in facts perceptible with the senses, but in relations
between concepts or other ideal objects. [21, p. 184]
and is indicated again in the following
passage from [14, p. 320]:
Therefore a mathematical proposition,
although it does not say anything about space-time reality, still may
have a very sound objective content, insofar as it says something about
relations of concepts.
2.6 Gödel diagram
classes
synthesizes which are
unity out abstract
abstract things
in
them-selves
of manifold impressions
data
of second
kind
Humans
have (from vague
intuitions
to
clear)
which
involving
of arising out of includes plurality
and
(by structure
employing
abstract
concepts
sense
organ)
mathe-matical truths
have which involve
intuitions
relationships
of between mathematical
and other
(arising out
of)
concepts
We will summarize the preceding
observations on Gödel's philosophy of mathematics in the following
diagram. First we remark that in the diagram, we place the "mathematical
truth" node on a branch separate from the "data of the second
kind" node because for simplicity of presentation we consider these
to be separate kinds of intuitions, as opposed to treating the intuition
of a mathematical truth as being a special case of the intuition of
data of the second kind. Although this separation makes sense
to us, it may or may not coincide with the way Gödel felt. (In fact,
we will see in section 3.2 that Wang [28, pp. 226-227] appears to have
a contrary impression.)
2.7 A few remarks on the Gödelian
view of concepts
What Gödel means by "concept"
is not completely clear. In [10, p. 128], Gödel defines concepts
as the properties and relations of things. Charles Parsons in
his note to [10] given in [6, p. 108] observes that:
Gödel evidently means objects
signified in some way by predicates.
Wang remarks in [26, p. 189] that:
Even though G seems to speak
of mathematical and conceptual realism interchangeably, the obvious
connotations are different. He takes mathematics as the study
of (pure) sets and logic as a more inclusive domain that studies (pure)
concepts. This already suggests that conceptual realism is a stronger
position than mathematical realism, except perhaps in the metaphorical
sense of having been suggested by his mathematical experience but covering
much more. At any rate, G's conceptual realism goes far beyond
mathematics.
In fact, Wang links Gödel's "optimistic"
view of philosophy as an exact science to Gödel's conceptual realism.
Gödel believes that the development of philosophy into an exact science
is not only possible, but will take place within the next one hundred
years or even sooner [25, p. 85]. In [26, p. 192], Wang states:
Philosophy as an exact theory
may be viewed as a special application of G's conceptual realism.
It is to bring about the right perspective so as to see clearly the
basic metaphysical concepts.
According to this view, the diagram
given in section 2.5 can be taken to represent not only Gödel's philosophy
of mathematics, but also a more general aspect of his approach to philosophy.
§3.
Epistemological structuralism and neo-Kantian Platonism
We now discuss a structuralist
philosophy of mathematics that is briefly presented by Mark Steiner
in his book Mathematical Knowledge [24]. In particular,
we will argue that, although there are clear differences between Gödel
and Steiner's views, there are also some surprising similarities.
We shall phrase the comparison between Gödel's and Steiner's view by
calling them different kinds of epistemological structuralism.
The comparison with Steiner's views will reveal that Gödel's philosophy
of mathematics bears some elements in common with certain (at least
Steiner's) structuralist philosophies of mathematics that postulate
the existence of abstract objects. The comparison will also indicate
such a structuralist philosophy of mathematics can be considered a variety
of neo-Kantian Platonism that is "more Kantian" than Gödel's
variety of neo-Kantian Platonism.
3.1 Steiner's view
Steiner's view is given by the
following statement:
We might view with Benacerraf
(1965) the point of mathematics as the study of "structures,"
rather than individual mathematical objects. The point, however,
is epistemological rather than ontological - we accept mathematical
objects, contra Benacerraf, but we agree that the only things to know
about these objects of any value are their relationships with other
things. (This is the mark of abstract objects.) Intuition
becomes then the intuition of [abstract] structures rather than the
intuition either of truths or of individual objects. This
new point of view has the virtue of conforming to the way mathematicians
speak of intuition. One speaks of set-theoretic intuition, analytic
intuition, geometric intuition, topological intuition, and so forth.
Intuition in one branch of mathematics, furthermore, is alleged not
to go with intuition in another. [24, p. 134]
To clarify Steiner's passage, let
us first consider the view of Benacerraf that Steiner references.
Benacerraf argues in [3] that "there are no such things as numbers"
[3, p. 294]. However, in that paper he doesn't make the same claim
about sets or any other mathematical object. (However, Steiner
makes his "epistemological" point about all mathematical objects.)
Benacerraf makes his "ontological" argument as follows:
The pointlessness of trying to determine which objects the numbers are thus derives directly from the pointlessness of asking the question of any individual member. For arithmetical purposes the properties of numbers which do not stem from the relations they bear to one another in virtue of being arranged in a progression are of no consequence whatsoever. But it would be only these properties that would single out a number as this object or that.
Therefore, numbers
are not objects at all, because in giving the properties (that is, necessary
and sufficient) of numbers you merely characterize an abstract structure
- and the distinction lies in the fact that the "elements"
of the structure have no properties other than those relating them to
other "elements" of the same structure. If we identify
an abstract structure with a system of relations (in intension, of course,
or else with the set of all relations in extension isomorphic to a given
system of relations), we get arithmetic elaborating the properties of
the "less-than" relation, or of all systems of objects (that
is, concrete structures) exhibiting that abstract structure.
[3, p. 291]
Now back to Steiner. We see
that Steiner is making a sharp distinction between the mathematical
appearances, namely, relationships, which mathematical intuition provides
us with, and the mathematical objects which participate in those relationships.
We have intuition of the abstract structures that encapsulate those
relationships, but not of the objects themselves. In fact, we
claim that Steiner's assertion that "the only things to know about
these objects of any value are their relationships with other things
(i.e., their appearances) is equivalent to "the only things it
is possible to know about these objects are their relationships with
other things." For the things that are of value about these
objects are the things about these objects that are reflected in our
world of experience, which are the only things that it is possible for
us to know. Thus, Steiner's inclusion "of any value"
can be viewed as epistemological sour grapes. Recall that Benacerraf
identifies an abstract structure with either the set of all relational
systems in extension that are isomorphic to a given system of relations,
or with a system of relations in intension. Since Steiner considers
sets as being mathematical objects, and hence beyond intuition, he would
presumably opt for the latter choice. But with either choice,
the abstract structure synthesizes a unity out of a manifold.
We present the following diagram
as representing Steiner's view:
Humans
Humans
synthesizes unknowable participants
unity out
of in
knowable relationships
abstract objects in them-selves
manifold
abstract structure
have
data of second kind
intuitions which involves
of
relationships
arising out of
(between)
intensional
form
underlying
isomorphism
type
3.2 Comparison with Gödel's
view
Although, both Gödel and Steiner
have their intuitions synthesizing unities out of manifolds, Steiner's
manifolds appear to be specifically isomorphism types of relational
systems (with the unity being the underlying intensional form of such
an isomorphism type), whereas Gödel presumably has these, but also
many more kinds of manifolds in mind. Steiner has us intuiting
structures that encapsulate relationships between abstract objects,
but not abstract objects or mathematical truths, whereas Gödel seems
to have us intuiting mathematical truths, the contents of which consists
of relationships between mathematical objects (Steiner would obviously
agree with Gödel that mathematical truths consist of such content),
as well as having us intuit abstract impressions that converge on abstract
objects. When Gödel discusses the intuiting of structure in his
letter to Greenberg (that was given in section 2.1), he means structure
to be a mathematical object (such as, presumably, the extension notion
of structure that Benacerraf mentioned).
We remark that Wang, in his posthumously
published [28] expressed an impression that is contrary to the evidence
(given in section 2.4):
In fact, my impression is that
mathematical intuition for him is primarily our intuition that certain
propositions are true - such as modus ponens, mathematical induction,
4 is an even number, some of the axioms of set theory, and so on.
Only derivatively may we also speak of the perception of sets and concepts
as mathematical intuition.4 [28, pp. 226-227]
If this is indeed the case, then
given Gödel's view that the content of mathematical propositions consists
of relationships between abstract objects, Gödel's view would be a
lot closer to Steiner's than we think it is: Gödel would have us intuiting
only mathematical truths, whose content consists of relationships between
abstract objects, whereas Steiner has us intuiting only abstract structures,
which encapsulate relationships between abstract objects.
Although both Gödel and Steiner
distinguish between abstract appearance and abstract thing in itself,
with Steiner the gap is unbridgeable, there is a complete separation,
whereas with Gödel the appearance can converge on the object.
In this way, Steiner is more Kantian than Gödel (see the passage from
[7, pp. 257-258 n27] that was given in section 2.4, and our comment
on it).
Observe that both Gödel and Steiner
are in agreement that there are different kinds of mathematical intuition,
e.g., set theoretical intuition and geometric intuition (see Gödel's
letter to Greenberg that was given in section 2.1).
We conclude this section by formulating
the following definitions of epistemological structuralism, optimistic
epistemological structuralism, and pessimistic epistemological structuralism
as constituting a vehicle of comparison between Gödel's and Steiner's
views. We consider both Gödel and Steiner to be epistemological
structuralists, as defined here, with Gödel being optimistic, and Steiner
being pessimistic.
Epistemological structuralism
distinguishes between abstract things in themselves, which it postulates
to exist, and our intuitions concerning these things in themselves.
An important property of our intuitions is that they synthesize unities
out of manifolds.
Optimistic epistemological structuralism
is a form of epistemological structuralism where our intuitions of abstract
things in themselves vary as to their degree of sharpness, and sharp
intuitions can provide direct knowledge of these abstract things in
themselves.
Pessimistic epistemological
structuralism is a form of epistemological structuralism where it
is postulated that the gap between abstract things in themselves and
our intuitions concerning these things in themselves (we do not
have direct intuitions of these things in themselves) is unbridgeable
and consequently the abstract things in themselves are unknowable to
us (they are completely outside our world of experience). These
intuitions constitute the underlying form that synthesizes a unity out
of a manifold of concrete relational systems.
§4.
Conclusion
In this paper we observed that
Gödel appeared to have formulated his philosophy of mathematics as
a neo-Kantian epistemology superimposed on a Platonic metaphysics, i.e.,
by basing an epistemology for abstract objects, which are postulated
to exist, on (what Gödel considers to be) Kant's epistemology for the
physical world. Gödel modified Kant's epistemology for the physical
world in an "optimistic" way, so as to allow for knowledge
of physical things in themselves, and Gödel appeared to have his epistemology
for abstract objects inherit this optimistic feature.
Gödel appears to have taken perception
of abstract objects through intuition to be more direct than the data
that sense perception provides of the physical world. This may
reflect a greater trust that Gödel may have had in abstractions than
in time - a time that he argued in [13] to be illusory. Gödel seems
to have felt that Kant was misled into holding the unknowability of
physical things in themselves, because the science of Kant's day was
insufficiently abstract to progress beyond the subjective world of appearances
(see section 2.4). Thus, the science of Kant's day couldn't progress
beyond the shadows in the cave.
In fact, it can be conjectured that Gödel's non-materialism was much stronger than he explicitly expressed in [14] and [15] (as strong as the non-materialism of Parmenides and McTaggert, whom Gödel sympathetically cites in [13]). Possibly, one can take the sum of Gödel's published philosophical writings as providing a Platonic metaphysics (penetrable by abstract intuition), not just for mathematics, but for the physical world as well: the physical world is an imperfect reflection of a more perfect abstract world, which science enables us to penetrate that underlying abstract world as it becomes increasingly abstract. Perhaps the sum of Gödel's published philosophy presents him as a neo-Kantian Platonist, not just for mathematics, but in his general world view as well.
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