Immanuel Kant vs. AJ Ayer:
A Comparison
By Darren Coughtrey (B.Math)
Carleton University
Before the advent of Non-Euclidean Geometry in the late nineteenth century,
mainstream philosophy was dominated by the many ideas set forth in Immanuel Kant’s*
Critique of Pure Reason. It was thought that Kant’s quest to define truth had come closer
than any other philosopher up until that time, and set philosophy on the path to becoming
an exact science. Kant’s philosophy relied heavily on the distinction between synthetic
and analytic propositions, and it was upon the foundation of his views about the necessity
and certainty of the axioms of Euclidean Geometry that Kant built much of his doctrine.
Since the discovery of consistent Non-Euclidean Geometries by the likes of Hilbert,
Poincaré, and Lambert in the late eighteen hundreds, Kant has been, perhaps unfairly,
widely attacked and his view of truth, especially with respect to mathematics, has been all
but discredited. This paper will give a brief introduction to Kant’s perspective, but then
focus on one such criticism of his philosophy, namely that which Sir A.J Ayer set forth in
the mid 1900’s. Additional examples will be taken from the writings of famous
philosophers such as Bertrand Russell and Ludwig Wittgenstein, in order to show how
Ayer made the progression from the philosophy of Kant to what he propounds in his
seminal work Language, Truth and Logic.
In order to succeed in molding philosophy into an exact science Kant needed a
way to quantify human reasoning. What he came up with was to use the natural partition
of all statements into two categories: analytic and synthetic. He defines an analytic
*
As this paper is intended for students of mathematics, who may not have much background in philosophy,
more biographical information may be of use. For a concise account of Kant’s life and works consult the
following web page: http://en.wikipedia.org/wiki/Kant
statement as one in which the predicate of the statement belongs to the subject, and a
synthetic statement is one in which the predicate is external to the subject.
He says:
“Thus it is evident: 1. that through analytic judgment our knowledge
is in no way extended, and that the concept which I already have is
merely set forth and made intelligible to me; 2. that in synthetic
judgments I must have besides the concept of the subjects something
else (X), upon which the understanding may rely, if it is to know that
a predicate, not contained in this concept, nevertheless belongs to it”
(Critique of Pure Reason, Introduction)
The first interesting fact about these definitions is that the predicate in an analytic
judgment lies within the subject of the statement; therefore, the truth of an analytic
statement is given linguistically and not through experience. That is to say, analytic
statements are a priori, rather than empirical. For instance, when considering the
statement “All men at the Canadian Undergraduate Math Conference are men” we need
not go any further than the statement itself to observe its validity, making it an analytic a
priori statement (see Figure 1). We can also see from this is that there are no empirical
analytic statements, since the truth of an analytic statement is given without appeal to the
senses (i.e. empirical facts).
Now, it is easy to see that there are synthetic statements whose truth is found
empirically, such as “All participants of the Canadian Undergraduate Math Conference
are male”. There is nothing intrinsic to the subject “participants of the Canadian
Undergraduate Math Conference” that implies that it would be a male. Thus, we need
some experience outside of the subject to be able to validate, or refute, this statement. In
this case it may a survey of all participants, a thorough check of all of their birth
certificates, or, if we wish to go to extremes, a medical examination of everyone at the
conference. The point is that some empirical fact is needed to make sense of this
statement.
(Figure 1)
What is not clear is whether a synthetic statement can be given a priori. In fact,
before Kant, it was thought that analytic statements were the only a priori truths, and
synthetic statements were all empirical facts. However, Kant argued that there must exist
synthetic a priori statements, and that they must spring from pure understanding and
reason. What is important to the purpose of this paper is Kant’s belief that the existence
of synthetic a priori statements is drawn from his view of the nature of mathematical
reasoning. He argues that, because empirical statements are validated through experience,
they are inherently fallible, and accordingly cannot be necessary truths or absolute
certainties. Mathematical judgments must be a priori as they carry with them necessity
and certainty, which cannot be obtained by experience. Furthermore, he states that
mathematical propositions must be synthetic; his point is easily shown by way of an
example, such as the statement of Euclidean Geometry that says “The shortest distance
between two points is a straight line”*. Kant’s belief is that nowhere in the defined terms
“point” and “straight line” is there any consideration of distance, so this statement relies
on concept outside of the statement itself. Hence such statements must be synthetic, and
coupled with their presumed absolute certainty and necessity, we must conclude that they
are synthetic a priori. Kant proceeds with such arguments to build his theory of space
(using the “certainty” of the Euclidean Axioms), and time (using similar notions for the
laws of arithmetic), which set the foundation for his schema of “Metaphysics as Science”.
Kant’s Critique was a breakthrough in philosophical thought, but because it was
based on the apparent absolute certainty of mathematics, and Euclidean Geometry in
particular, it was left vulnerable to criticism when Non-Euclidean Geometries were
discovered towards the end of the 19th century. It was especially damning to Kant and his
followers when, in 1899, David Hilbert proved that Non-Euclidean Geometries were as
consistent as the geometry of Euclid. Perhaps Kant’s philosophy may have been salvaged
if it were found that either geometry was true if the other were false. However to have
them being, in some sense, equally as true could be seen as an outright deathblow to the
Kantian Doctrine.
Now, with one of philosophy’s most prized theories left in limbo, the philosophy
of Kant naturally drew the attention of some the world’s brightest minds - all of whom
wanted to know what could be salvaged, or what needed to be changed in order to rectify
these discrepancies. Moreover, since the holes left in the theory set forth in The Critique
Strictly speaking, this is not a statement from Euclid’s Elements, but rest assured it is a statement of
Euclidean Geometry. See the following for a detailed proof:
http://mathworld.wolfram.com/Point-PointDistance2-Dimensional.html
*
were of a mathematical nature, many of these scholars were some of the most famous
mathematicians of their time.
For instance it was around this time when Bertrand Russell published his Essay
on the Foundations of Geometry, in which he conjectures several interesting deviations to
Kant’s theory. He says, with regard to the synthetic-analytic distinction, that no statement
can be taken as purely analytic or purely synthetic. He argues that in making such a
distinction for a given statement we are taking it out of its original context, and thus
robbing it of any validity. For example, a given theorem of mathematics is a deduction
within a larger system, and thus analytic, but it also forms new relationships within the
system, and in this way it is synthetic.
Ludwig Wittgenstein was also that was drawn to the study of truth through his
work in formal logic, much like Russell. Although his work, at first sight, does not seem
to pertain to this study, with the knowledge we have gain thus far, it is not hard to tell that
Wittgenstein must have given some thought to the system of truth proposed by Kant. In
his famous work Tractatus Logico-Philosophicus he states:
“If a thought were true a priori, it would be a thought whose possibility
ensured its truth. A priori knowledge that a thought were true would be
possible only if its truth were recognizable from the thought itself”
(Tractatus Logico-Philosophicus, 3.04&3.05)
So without, naming Kant directly, Wittgenstein clearly states that he does not adhere to
Kant’s belief in synthetic a priori truths.
Clearly Wittgenstein and Russell have published differing views on the
distinctions made by Kant*. However, what they do have in common is that they were
two of the forefathers of modern logic, and with this they have both garnered renown and
wide spread influence. One place that their influence is directly observed in modern
philosophy is within the work of a group of scholars who came to be known as the
Vienna Circle. In the 1930’s and 40’s this group of men produced a large amount of
original philosophy by mixing their backgrounds in classical philosophy with the logical
philosophies of Russell and Wittgenstein. A young Englishman, Sir A.J Ayer, became the
Circle’s flag bearer when, in 1936, he published his inspired work called Language,
Truth, and Logic.
The philosophy of Ayer and the Vienna Circle, which came to be known as
Logical Positivism, can be seen as a sort of empiricism in the vain of David Hume that
also draws upon the logical philosophies of Russell and Wittgenstein. It was Hume’s
contention that the human mind gets confused when dealing with statements that venture
beyond the limits of experience, and thus that metaphysics is a useless endeavor. Ayer’s
main goal, however, was to show that not only was metaphysics useless, but also that
metaphysical utterances carried no meaning whatsoever, thus showing that metaphysics
as a whole was not possible. In doing so Ayer went against the doctrine of Kant in several
ways. We shall focus on their differing ideas with respect to mathematics via the
synthetic-analytic distinction we discussed earlier.
*
From the passages above it may seem that Russell and Wittgenstein endorse opposing philosophies, but it
must be noted that Essay on the Foundations of Geometry is an early work of Russell’s, and that Russell
and Wittgenstein were colleagues and collaborators. Russell’s later views were more in synch with those of
Wittgenstein, but, as we shall see, Russell’s earlier views of the synthetic-analytic distinction did not go
unheralded.
Ayer agrees with Russell, in that they both feel that Kant erred in his definition of
the synthetic-analytic distinction. Both men claim that Kant’s definitions are not mutually
exclusive. However Ayer agrees with Wittgenstein in their belief that there does indeed
exist an analytic-synthetic partition. Ayer reworks Kant’s definitions by saying that “a
statement is analytic when its validity depends solely on the definitions of the symbols it
contains, and synthetic when its validity is determined by facts of experience” (Ayer
1936, pg 73). What one should notice about Ayer’s definitions is that he has
automatically ruled out Kant’s concept of synthetic a priori truths, since Ayer believes
any synthetic is validated through experience.
Ayer also contradicts Kant by saying that analytic statements do, in a way, give us
new knowledge. He claims that an analytic statement, although showing us what was
already deducible, does bring forward new linguistic usages for the subjects of such
statements. This is an idea very closely linked to that of Russell quoted previously. For
example, what Ayer is saying is that if we are given an analytic statement like “All dogs
are animals”, then we’ve added to our knowledge because we can now use the term
“dog” in certain sentences pertaining to animals and vice versa. It is in this way, he
claims, that analytic statements “can reveal unexpected implications in our assertions and
beliefs” (Ayer 1936, pg 75).
In accordance with this result, Ayer states further that the statements of
mathematics are all analytic, and not synthetic a priori as Kant argued. Geometry, he
claims, is the one area of mathematics that is likely to be perceived as being synthetic, but
through Ayer’s eyes it is not so. He argues that Kant’s views were based on the fact that
geometry was the study of physical space, and hence the propositions of geometry have
factual content (i.e. synthetic) and are not merely tautological (i.e. analytic). However,
with the emergence of Non-Euclidean Geometry we can see, as Poincaré put it, that the
axioms of any geometry are mere definitions and the theorems are simply the logical
consequences of these definitions. Thus, a given geometry cannot not be said to be about
physical space, and indeed, as a formal axiom system, a geometry is not about anything
in particular at all. Ayer does admit, and rightly so, that in this respect Kant was a victim
of his time, as he had no knowledge of consistent Non-Euclidean Geometries. So, once
Ayer has stripped geometry of the factual basis that Kant depended on, he is free to state
that Geometry is analytic, and with this he claims to topple Kant’s theory of space.
Kant’s theory of time is similarly based on his assumption that statements of
arithmetic were all synthetic a priori; Ayer once again refutes this claim. In his famous
example Kant holds that the statement “7+5=12” is synthetic because the term “7+5”
subjectively contain the notion of the number 12, and as before he claims that such a
statement must be a priori because it carries an absolute certainty that can never be
obtained empirically. Ayer contends that logically the term “7+5” is synonymous with
the term “12” (i.e. we can always replace “12” with “7+5” and still maintain the validity
of any statement containing the term “12”), and thus the sentence “7+5=12” is merely a
tautology, which means it is still a priori, but must be analytic. The fact that the
statements of arithmetic are tautologies implies that they contain no factual content as
Kant supposed for the basis of his system of time.
Given this brief introduction to two opposing philosophical theories, it should be
obvious that mathematics, and especially Geometry, holds an essential place in the study
of philosophy. By contrasting the ideas of Kant with those of Ayer we have seen some
striking differences when it comes to the very nature of mathematical thought. By now it
should be clear that, within the realm of philosophical thought, mathematics is not as cut
and dry as many people studying elementary mathematics might believe. Both Ayer and
Kant gave very original views of mathematics in order to support their widely differing
theories of metaphysics. However, this paper has only touched upon the underlying
mathematical concepts of both theories; the interested reader is strongly encouraged to
read further about the purely philosophical aspects of these theories, as the aspects
discussed in this paper give rise to some very interesting ideas about the nature of human
thought.
Bibliography
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