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Mathematical Knowledge: Where Truth and Proof sit in the Analytic Tradition

Over the course of the semester we have been introduced to a variety of
philosophers who produced a great portion of their work in the analytic
tradition of philosophy. We focused on specific works from Bertrand Russell,
A.J. Ayer, Ludwig Wittgenstein, and Wilfrid Sellars. The project that I am
embarking on can be extended to the authors we read in this course but didn't
focus on; indeed, there are some interesting things to point out about their
positions on knowledge and analytic truth, which we don't fully see in these
four philosophers I will be focusing on. For some, such as Mill or Frege, the
examples are obvious if anyone has studied the responses to their work. For
others, such as Quine or Carnap, it perhaps isn't so obvious; indeed, my
examples against them are far less 'set in stone' and far more exploratory. But
for the sake of brevity and clarity, I will focus on 'the big four'. 

The primary books by these authors that we've looked at are 'The Philosophy of
Logical Atomism' by Bertrand Russell, 'Language, Truth, and Logic' by A.J.
Ayer, 'On Certainty' by Ludwig Wittgenstein, and 'Empiricism and Philosophy of
Mind' by Wilfrid Sellars. While these authors seem almost overwhelmingly
interested in issues relating to empirical truth, they have comments on what it
means for something to be analytically true. This is true especially when they
begin making comments about such things as mathematical truths, and these
comments offer an entirely different and perhaps frequently overlooked avenue by
which one can question, probe, and prod the espoused viewpoints. In this paper,
I will assume at least a moderate level of familiarity with the works of each
author. Indeed, for someone like Russell who made a name for himself in part for
his contributions to philosophy of mathematics, it would be almost redundant to
delve into his work in a rigorous way. Instead, I will attempt to offer examples
to the philosophies argued for by the authors from the field of mathematics, to
clarify what might be called gaps in their work. 

If anyone has ever studied the philosophy of mathematics, Bertrand Russell is
perhaps the first name they would come across. Russell's project was in some
ways an extension of Frege's.  His major motivation was to resolve his now
famous Russell's Paradox that arose from Frege's work, and so the Russellian
position can be properly characterized as logicism: an attempt to reduce
arithmetic truths to logical truths. Russell's development of class theory is
perhaps one of his greatest achievements in the field. However, it can also
point out to us what might be called major issues with his other work. In
grounding mathematical truths in logical truths, Russell was attempting to
demonstrate that mathematical truths were analytic ones. That is to say, they
were true just in virtue of their logical form. In developing type theory,
Russell found that the paradoxes he pointed out to Frege about impredicative
definitions were also proving difficult for his own work. As a result, Russell
introduced a principle of reducibility, which allowed him to produce a class
which was predicative for any class. This principle is almost entirely necessary
for producing anything beyond basic, natural number arithmetic. By this token,
one must ask how it is that the principle is justified. Indeed, given the
development of type theory, it seems ad hoc, instituted only to solve problems
revealed after the fact. Indeed, it isn't knowable a priori, and it cannot be
grounded in logic - this much Russell admits of the principle. However, a
problem exists beyond this flaw. If one considers numbers to be certain classes,
even predicative ones, then the levels of classes become almost cumbersome to
navigate - one can easily get lost in the hierarchy. For the Russellian
development, negative integers are, in one sense, simply relations on their
positive counterparts. Herein lies the issue: any number beyond natural numbers
in the usual construction is simply a relation, and not in itself a class of
classes (as the naturals are). Indeed, '+2' is wholly distinct from '2'; yet
this seems wildly difficult to accept, on at least a common-sense understanding
of numbers. The difficulties presented by type theory and class hierarchies
makes mathematics unwieldy and complicated, and the principles we have to invoke
to properly do so without contradiction seem to make mathematics, in part, not
wholly analytic, or at least the ways in which we talk about it isn't. In
itself, the logicist project is a worthwhile consideration. But the problems
presented to us by type theory make doing mathematics on any level from
arithmetic to real analysis almost entirely unbearable. Indeed, it is unclear
exactly how Russell might aim to reduce something beyond real analysis to logic
and speak of it in wholly logical terms.

Jumping ship from Russell's logicism we arrive at what might be considered an
extension of the view: logical positivism. At the very least, the logical
positivists attempted to ground analytic truth in facts about meaning, rather
than a wholly logical position like Russell. For Ayer, an analytic truth would
be one in which its truths depend entirely on the meanings of the symbols it
uses, and for Ayer this is all that mathematical truths amount to. This position
seems rather enticing. Indeed, mathematical truths exist outside of our
experiences of fact, and are not verified by some sort of observation of the
world. Instead, mathematical truths exist wholly separately from proper
experiences, and are true in virtue of their meanings alone. This move
eliminates many objections to Russell's position on the matter. No longer do we
have types and classes and relations to worry about. Instead, we need only focus
on the meanings of our words, and the symbols we employ. As such, knowledge of
mathematical propositions is simply knowledge of the correct use of mathematical
language. But this identification of knowledge of mathematical propositions with
knowledge of its use seems quite problematic. If this is the case, it seems
entirely unlikely that there would be many unsolved problems in mathematics at
all. There are a few cases for why these problems might be unsolved. For
instance, it might be that they aren't mathematical propositions. Alternatively,
it could be the case that we don't know what we mean when we use these
propositions; we don't have a proper understanding of their meaning. However,
both seem wholly unlikely. The Riemann Hypothesis [1] or the Golbach Conjecture
[2] seem very much like mathematical propositions: they are written in the
language of mathematics, they have mathematical consequences, and they wholly
arise out of mathematical facts. It also seems unlikely that mathematicians
don't know what it would mean for something like the Riemann Hypothesis to be
true. Indeed, we know that if it is the case that it's true, then the Continuum
Hypothesis [3] is true. As it turns out, the positivist tradition put forward by
Ayer does not provide us a method by which we can make determinations of the
truth of mathematical propositions. There is no algorithm by which we can check
their validity. The fact that this is the case for such simple statements about
natural numbers and primes in Golbach's conjecture is a proof of fact of this.
Indeed, Fermat's Last Theorem [4] was not solved until only recently, yet anyone
who reads the statement of the theorem knows exactly what it means - it is a
statement simply about addition, integers, and natural numbers; some of the
simplest things in mathematics.  However, the mathematics used to prove its
truth extend far beyond what any nonmathematician (and even many mathematicians
themselves) would ever know. From this, it seems difficult for the positivist to
make claims about the analyticity of mathematics in virtue of meaning alone. 

Wittgenstein might be the first author we read this semester who both
simultaneously fails to provide a particular system which accommodates
mathematical truths and formally accepts mathematical truths. The trouble with
Wittgenstein is that he is not to be regarded as a mathematician, classically
understood. Wittgenstein's philosophy of 'meaning as use' and things being
descriptive of a language game as part of logic entail that classical
mathematics might not be the only system of mathematical formalism. Indeed,
Wittgenstein himself was influenced from lectures he attended by L.E.J. Brouwer,
a famous mathematician and philosopher. Brouwer led the development of
intuitionist mathematics, which sought to do mathematics in an intuitionistic
logic, diverging starkly from the usual method of classical logic which we see
employed by those earlier authors like Russell, Frege, or Ayer. For the
intuitionist, a proposition A is true just in case there is a proof of A; the
negation of A is true just in case there is a proof of not-A. In this way, the
intuitionist does away with the law of excluded middle being a tautology,
something that most modern-day mathematicians would not accept. However, it
would be quite incorrect to say that Wittgenstein himself was an intuitionist.
From what I have read, it isn't entirely clear what sort of logic Wittgenstein
would have preferred. That being said, there are certainly interesting points
offered by mathematics that allow us to perhaps better understand what
Wittgenstein's theory of language games would produce for us. Take for instance,
the relation 'is less than' [5].  One of the most fundamental things one learns
about numbers is that 1 < 2, 2 < 3, etc. The idea that there is an ordering -
indeed, a total-ordering - on numbers is oftentimes taken for granted. We can
extend this notion from real numbers to things like the integers. After
introducing zero, we can say things like -2 < 0 and 0 < 27, therefore -2 < 27.
We can further generalize the notion of <, by saying that if a < c, a + b < c +
b, given the proper framework surrounding addition. What's curious to note is
that, for almost any 'normal' language user - someone who would use mathematical
facts in a layman sort of way - these points are almost trivial. But take for
instance, the complex numbers. Those numbers which have the form a + bi, where a
and b are real numbers and i is the square root of negative one. Suppose you
were to ask someone whether 1 + 2i < 3 + 4i was the case; what would they say?
Perhaps they would be inclined to point out that, because 3 and 4 are larger
than 1 and 2, it seems clear that this statement is true. On the face of it,
it's unclear how they might be wrong. But consider the following: is i less than
or greater than zero? This is an almost elementary exercise for those who pursue
a degree in mathematics - given either assumption, there will always be a
contradiction [6].  As it turns out, this is a demonstration that the complex
numbers are not an ordered field like the natural numbers, the integers, or the
real numbers. There are other 'trivial' facts that likewise fail for larger
number systems. Division in the complex field works differently than it does for
real numbers. One of the more humorous things to note is that 'ordinary' numbers
cannot describe rotations; indeed, appeals to complex numbers must be made to
rotate things in two-dimensional space [7].  The question we are forced to ask
ourselves given these revelations is, who was right? Given a language game, it
seems that mathematicians know something different about the ordering relation
than the layman does and therefore is perhaps using it differently, and is thus
speaking a different language; they could be said to be using the relation
differently.  Yet we continue understanding ourselves and each other with no
real issues arising from it, as if we weren't saying anything different at all
[8].  The point at the very least is an interesting one, and while it isn't
necessarily a strike against the 'meaning as use' idea itself, what we once
considered to be immutable and a priori truths certainly shed a new light on the
sorts of problems one might face in using a Wittgensteinian framework. 

From this avenue, we arrive at the Sellarsian picture of knowledge. The question
of whether someone can be said to have knowledge of mathematical truths on
Sellars' view is an interesting one. Due to its close relation to Wittgenstein's
philosophy about meaning and the interrelation of concepts, Sellars seems to
avoid any sort of serious qualms one might have with the idea of rule following
giving rise to knowledge. The crux of the point here is that within the space of
mathematical truths, the question of whether it is analytic or not is
unnecessary. As these beliefs sit in a logical space, the premier question is
whether there is a logical justification for a certain belief or fact that is
known. What this means for something like the prior example of relations on
numbers is that if a person can justify their claim, they can be said to have
knowledge. As a result, both individuals - those who are aware of the relation
as it applies to real numbers and those who realize how it doesn't apply to
complex numbers - can be said to have mathematical knowledge. As a result, it
appears that the only thing required of someone to have mathematical knowledge
is that they are able to provide some sort of demonstration that it is the case;
in general, this might be called a proof. The only foreseeable problem to arise
from this is that individuals might be hard-pressed to prove basic mathematical
truths in a general sense. We might be welcome to the idea that, if someone
claims to know that 2 + 2 is in fact 4 and they can show this fact in some sort
of way, they know it is the case that two and two make four. However, the
question arises of whether we should demand such proofs for every claim about
mathematical facts if they haven't demonstrated the principle in general. It
seems that, at this point, the demand for justification becomes too extreme, as
we might be tempted to ask everyone who espouses mathematical facts, even in
passing, to become casual mathematicians.  However, it seems that we should be
more than allowed to waive such a requirement in general; our justificatory
practices need only extend so far if we share common beliefs. In this way,
Sellars has a position very similar to a sort of Wittgensteinian, language game
holism he can pull from to deny this sort of line [9].

[1] The Riemann hypothesis is a conjecture that the Riemann zeta function has
its zeros only at the negative even integers and complex numbers with real part
½. The Riemann zeta function is itself in one part a way to study the
distribution of prime numbers.

[2] The Golbach conjecture states that every even integer greater than two can
be expressed as the sum of two primes. It dates back to the 18th century and has
yet to be solved. 

[3] The continuum hypothesis states that there is no set whose cardinality is
strictly between that of the integers and the real numbers. That is to say,
there is no set bigger than the integers that is also smaller than the number of
real numbers.  The set of real numbers is itself thought to have a cardinality
equal to that of the integers as a power of two.

[4] Fermat's Last Theorem states that there are no integers a, b, c such that
an + bn = cn for integers n > 2. 

[5] From here on, this relation will be denoted by <, its opposite, >. 

[6] Let 0 < i.  Then 0 < i2. Thus, 0 < -1, a contradiction. So it must be that 
i < 0. But then i4 = 1 < 0, a contradiction. 

[7] Even more strikingly, rotations in three-dimensional space require
quaternions, we might think of them as 'four dimensional numbers'. 

[8] This all hints at a larger issue for Wittgenstein as it relates to rule
following in general, and there are some excellent papers about it. One of the
more interesting points is about the 'plus' function versus the 'quus' function,
by Kripke.

[9] A great deal of thanks for this is due to Stewart Shapiro's 'Thinking About
Mathematics'. For those interested in reading more on this issue as it relates
to realism and antirealism, Shapiro's work offers an excellent covering of these
problems historically and contemporarily.


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