@make(Article)
@style(spacing = 2)
@style(size=12)
@device(postscript)
@PageHeading(left = "George Xixis", right = @value"page")
@begin(titlepage)
@begin(titlebox)
@MajorHeading(Epistemology, Nature and the A priori)
@end(titlebox)
by
George Xixis
December 12, 1986
24.211 Paper 2
@end(titlepage)
Epistemology is not a discipline that relishes in standing alone. the
theory of Knowledge is very concerned with the foundations and
development of the sciences. Mathematics in particular seems to be the
best modality in which to illuminate the realtion between epistemolgy and
the natural world.
Mathematical studies according to Quine (1969) divide into two types
conceptual and doctrinal. The conceptual studies are concerned with the
meaning, clarification and definition of concepts. The doctrinal studies
deal with the establishment of laws. Ideally the definitions in their final
reductions would exist as "clear and distinct" ideas . All theorems would
be generated from a set of self-evident truths. Although, it has been found
that the reducibility/simplification of mathematics is in no sense
complete (it can not be distilled down to logical principles) it still serves
as a reasonable illumination of epistemological concepts.
The conceptual side of epistemology has seen the greatest advances
from Descarte's time. There have been a number of schools of thought on
how to approach the conceptual side of epistemology. The first, more
classsical view, sees the universe of discourse as a real independant
construct. We relate to it by perceiving some type of information about a
pre-existing universe. Hume's handling of the conceptual, identifies bodies
by their sense impressions . The collection of sense impressions we
receive from a particular object x is what constitutes our knowledge of
that body x.
Here is where we run into our first bit of trouble. Hume's analysis of
this condition demands a significant disconnection between the perception
of the object and the actual concept of the object in question. Consider the
following case: In front of you there is a grey wall, you consequently
perceive a grey wall. Now paint the wall blue. Has anything besides the
color of the wall changed? Has that which makes the wall in front of you
(call it wall w) the one and only wall w? No, you in fact still perceive wall
w. You may argue that the object w has in fact changed. It is now, no
longer w, a grey wall, it is some object y a blue wall. Not so. Not so.
Hume's thesis states that what we obtain is sense data. Sense data is by
its nature an internalization. Now, imagine this grey wall in front of you
is now blue. Close your eyes and imagine the grey wall slowly turning blue.
It is still the same wall. A property of the wall not the wall itself has
changed.
There appear to be some properties that are associated directly with
objects in the real world, and are consequently inherent in determining
their being. Others, such as color, position and temporal location seem to
be associated more with our relation to the particular object. Seemingly
motivated by the Cartesian search for certainty in an otherwise uncertain
universe as Quine puts it "To endure the truths of nature with the full
authority of immediate experince was as forlorn a hope as hoping to endow
the truths of mathematics with the potential obviousness of elementary
logic."
Quine has also made it clear that the language contains in it a number
of truths that can be considered analytic but, are not true because of their
logical form. His classic exaple "no bachelor is married" is true no matter
what universe the proposition is considered in , yet, it can not be
considered true soley on the basis of logic.
Carnap in his radical reconstructionist theory deals with analyticity
in the light of an "artificial observational language that can be
constructed by laying down rules". In this new view, unlike Hume, the
meanings of the words are not associated directl;y with and soley with the
physical world as sense data impressions. The rules do not necessarily
enumerate the full meanings of words but, instead they deal with meaning
relations between specific words.
A typical construction would appear as follows: (looselm Carnap)
(S1) The term "animal" reprents all andonly those properties (1).......
(2)........... (3).............. (4)................ (5).............
(S2) The term "mammal" represents all and only those properties (1).........
(2)............(3)..............(4)...................(5)............... (as in
S1 above ) plus the
properties (6).......... (7)............ (8)..........
(S3) The term "dog" represents all and only those properties (1).............
(2).............(3)...............(4)................(5).............. as in S1
above, (6).............
(7)..............(8)..................as in S2 above plus the properties
(9).........
(10).............(11)..................
In such a way many of the descriptive terms in the language which
appear irreducible otherwise can be fully specified in the form of a
realtional matrix.
There are a number of problems with this relational matrix that may
not be immediately apparent. Firstly, there seems to be a problem with the
length of the list. It is obvious that in a very short time the number of
properties listed would far outstrip our mental capacity. Carnap states
that it is therefore sufficient that these analytic-postulates (or
A-postulates) can be controlled by specifying the meaning realtions that
hold among the language's descriptive terms for example:
(a1) All mammals are animals
(a2) All dogs are mammals.
It is not clear though, that the problem has been solved In these cases
statements of the form "all chairs are not animals, all tables are not
animals, etc.. would seem to still be necessary to fully elaborate the
concepts involved.
There also seems to be a "circularity" bottleneck. If all the analytic
terms of the language are described by this construction and are internal
to this construct then where do the logical/mathematical operators by
which we manipulate the analytic postulates reside? Above we have the
statements a1 and a2 from them we can deduce the statement (a3) All
dogs are animals. Somewhere we must have the concept of deductive
inference. What I claim is that the concept must appear outside the set of
A-postulates in order to avoid a circularity problem.
Quine suggests some departures from conventional theory in his
views on the analytic and tthe realtion of language to thought. As we have
seen earlier depending on only sense impressions as the direct source of
information without some more developed structure leads us to problems.
What is the nature of these sense impressions? How are they used? Quine's
view is that what we manipulate are not sense impressions but
observation sentences. These observational sentences are not reports of
sense impressions nor "statements of an elmentary sor about the external
world." They are sentences that when one learns language are strongly
associated with particular sensory stimuli.
There are problems with the previous definition though. Most
statements depend on a wealth of stored Knowledge that must be
co-referenced to determine the validity of a particular statement. Thus
Quine redefines the observation sentence such that all verdicts on the
truth or falsity of that statement depend on no exterior knowledge besides
that which goes into understanding the sentence (vocabulary).
This illustrates a question that Quine raises in a great number of his
writings. How are we to tellwhat goes into understanding a sentence and
what goes into extending that knowledge. This brings us to a distinction
between two types of knowledge the analytic and the synthetic. In his
"Two dogmas of empiricism" Quine argues taht all attempts to redefine the
distinction between the analytic and the synthetic fail due to a cicularity
problem. What Putnam has realized is that Quine speaks about two froms
of the analytic. The first so-called linguistic notion is similar to Kant's .
A sentence is analytic if it can be obtained from a truth of logic by putting
synonyms for synonyms. The invalidation of this version of the analytic
rests on the fact that using this exchange system and the rules of logic
the statement " Necessarily all and only bachelors are unmarried men" has
as its only property that it is true. The system does not in any way tell us
whether the realtion between bachelor and and unmarried man depends
upon the meaning of the word or coincidence. What has to be equated is the
analyticity of the statements as well as the meaning. Hence the
circularity problem.
The second definition of the analytic is the notion, according to
Putnam, taht "analytic truth is one that is confirmed no matter what" This
is also one of the conventional definitions of the a priori. Quine believes
that the sum total of our "facts" is , " a man made fabric which impinges
on experience only along the edges". Truth values that have been previously
assigned to particular statements become significantly redistributed from
time to time. When these "truths" change the realtions that they once held
to each other also change. Since the entire field of our knowledge is
therefore indeterminant we have a choice in re evaluating any particular
statement , the only condition being the mainatince of the equilibrium of
the system as a whole.
Consider the case of classical versus relativistic physics. Until the
turn of the century classical physics was thought to be the a priori "truth"
of the laws of the universe. In fact theory had reached such a high degree
of development that scientists of the time were heard to remark "There is
nothing more to know. We understand the universe, now let's go fishing." ,
or words to that effect. With the advent of relativistic physics we
realized that our perceptions of the universe were flawed. Yet, to this day
in our daily interactions with physical laws we still base our decisions on
classical principles. What we do use is an approximation to the "truth". The
system we had previously constructed was self-consistent and uilitarian.
It gave us reproduceable results which we chose to consider as true.
Quine's point is that no statement is immune from revision. One might
ask though isn't this a case of "You were just wrong". No, the concept
wrong implies that an incorrect judgement. The reasoning itself was
correct given this particular system. Carnap elaborates and makes the
discussion more concrete. He uses the case of quantum logic as an
exxample . He takes it to be the "only realistic interpretation" of quantum
mechanics as follows: " It is the only realistic interpretation of the
present theory... If I am right in the foregoing then anyone who concedes
that the present theory could be true should concede that there is a strong
'case' for the possibility of a quantum logical universe." In other words,
sone statements can only be overthrown by a rival theory, but since the
existance of such an object is always possible the existance of an
unreviseable statement is impossible.
It is interesting to see the parallel drawn between Carnap's
interpretation of Quine and Wiottgenstein's view on certainty.
Wittgenstein advances the thesis that with sentences or statements are
associated warrants for the sentence's assertion or denial. His
interpretation of 'I am certain that p' psychologically implies p. Can this
explain how we come to believe ceratin facts about the universe? As our
warrants for a model of a particular universe change we will change our
"current" view of the model to fit the warrants. If though our warrants
sufficiently support both the old and new models we can then in some
sense accept both.
The final analysis appears to be that even though there are few if any
a priori truths there do exists analytic truths. These are the truths are
such by logic , or language. Analytic truths are not unrevisable though,
they are only unrevisable until we chang ethe langiage or the logic.
Now, let us examine in more detail the existance of the a priori. Does
it exist? WE seems to be on the horns of a dilemma when considering this
problem. We have previously stated that there is no a priori truth. If we
are correct then is this not an a priori truth? It appears so. There also
seem to be other parts of our knowledge that lead us to belive in the
existance of a priori knowledge. Putnam's example 'When I open this box
you will see the sheet of paper is red and the sheet of paper is not red' is a
statement that has no meaning. We know a priori that it can not be the
case that this is true. (Assuming, a normal piece of paper etc.). This argues
against the theory that what is really happening when we develop the
illusion of the apriori is that we substitute the contextual sense of the a
priori for the absolute (see Quine re: Euclidean Geometry).
There may or may not be other forms of knowledge that are a priori.
The best candidates logic and meathematics have been thought of as a
priori for a long time. We have argued that these concepts are so
entrenched in our language and thinking that are in effect the bounds by
which we define a rational argument. In both forms of reasoning,
inductive and deductive, we use the fact that there exists a transitive
connective that satisfies modus ponens. This is necessary and central to
the nature of reasoning. Aristotle himself comments on how that given
someone who says that he disbelieves a law of logic and begins to argue
with us we could easily convince him that his own arguments assume the
fundamental structure of the logic he is attempting to refute. How then
can we define rationality in light of the total rejection of the a priori. It
may be that rationality is not as necessary as we believe. It is apparently
impossible at this point to show (with a good argument) that any specific
statement about the real universe is such that it would be irrational for
one ever to give it up given sufficient evidence. If somehow some creature
was able to show you the negation of the law of the excluded middle (A
sheet of paper that was both completely red and completely green)
wouldn't your conceptualizaon of the universe change?
How does this possible non-existance of the a priori effect
mathematics then? It has long been proposed that mathematics is the
only discipline in which, given sufficient ability and the axioms of
set theory, all of the field could eventually be derived. We come
to a difficulty at this point. We can not prove that mathematics or
logic are proveable by convention. To apply their principles in
particular cases requires us to reason from generalities, but such
reasoning is alas, the province of logic itself.
Quine (1938) addresses this in his "Truth by Convention". Take
a language L, for example, with a particular syntax. Describe then a
special class of truths of logic in L (true in L). One could then say
that logic was defineable by convention in the language L. There are a
number ways in which this description could be accomplished. The set
could be described as a "page" in the language on which are listed
(enumerated) all the logical truths. Since this does not make use of
the concept of "general statements" it is of little use. The second
method consists of an enumearation of a set of principles (each
principle's equivalent list would have been infinite). The logical
truths in L would be all and only those truths proveable as such by
means of the set of principles. The problem here arises that to
understand this "convention" you have to know what it is to be a
logical truth (ie. that statement S is proveable from premises
P1,...Pn) To adopt the convention you need an understanding of how it
is to be applied. It is not a simple enumeration any longer. It is
possible that this list of a length sufficient that we could on learn
about logical truth by reading the elements of the list. What we see
is that an understanding of logic is needed to adopt the logical
conventions, hence logic can not be true by convention. By a simple
extensionif logical can not be true by convention neither can
mathematics.
Mathematics therefore need not be true by convention unless
the radical Wittgensteinian position holds. This position states that
mathematical "truths"and "falsehood" and "warrant" and "diswarrant"
are explained by our dispositional states . Namely, they are explained
in how we behave with respect to the statement in question.
Consider the statements that "there is no number x such that
160 (Given that people took sufficient care to
make sure the proofs were correct in Peano arithmetic).
Putnam argues that statements such as "peano arithmetic is
10^20 consistent" are mathematical facts and that these facts can not
be explained by our 'nature' or 'form of life'. The Wittgensteinian
believes that mathematical facts (1) do not express objective facts
and (2) their truth and necessity (or appearance thereof) arise from
and are explained by our nature. According to Putnam this is false.
These facts are obviously not a priori in the sense that no
conceivable universe would hold in which they are false. There does
though, seem to be a factual element involved. There is an objective
combinatorial fact in this consistency.
What can be considered rational also appears to be a
comprimise. There still seems to be an element of the a priori that is
important in mathematics and the rest of science. It's true nature and
limitations though are not at all clear.