A.M.D.G.
Kevin Kolker
TAMU: MATH 482H
05-08-13
Mathematics and Philosophy
Both mathematics and philosophy have long histories throughout human existence; indeed,
from the earliest human records, there have been gestures at the infinite, questions of how things work
or why things are a certain way, and efforts to solve simple problems through the help of mathematics.
Looking at these two intellectual subjects now, one may at first see no similarity whatsoever since
philosophy deals with questions about why and how things are, while math helps to solve many
problems in the physical sciences. However, despite their major differences, I believe these two
subjects bear much in common.
These subjects’ greatest similarity is that both mathematics and philosophy claim to be efforts
to seek truth. First, philosophers from all ages have always practiced philosophy in an effort to answer
questions about the way things are. Whether trying to explain our interactions with the world, deciding
whether humans have free will, pondering the morality of eating meat, trying to show that there is or is
not a divine being, or asking why anything exists at all, philosophers have sought answers to questions
that help us to better understand our existence. While mathematics cannot typically answer questions
of this type, it does deeply enrich our understanding of the world around us; indeed, by providing the
framework for physics, chemistry, and even biology, among many more nuanced sciences, mathematics
has allowed us to understand the way the world and everything in it works. This makes it very clear that
both mathematics and philosophy seek what is true and provide answers to many difficult questions.
Furthermore, both subjects use many identical methods of proof, including arguments by
contradiction, by cases, by contrapositive, and the use of definitions. First, the general idea behind an
argument by contradiction is to begin with a premise that may or may not be true and then use the rules
of logic and other known facts through a series of steps to lead to an obviously false conclusion. Since
the conclusion is false, we know either that one of the steps in the argument is incorrect or that the
original premise is incorrect; yet, since we used logic and known facts to ensure that each step of the
A.M.D.G.
Kevin Kolker
TAMU: MATH 482H
05-08-13
argument was correct, it must be the case that the premise is false. This gives the desired conclusion
that the negation of the premise is true. Indeed, math uses this type of argument quite often, such as in
the proof that the square root of two is irrational. That proof goes as follows:
Suppose √2 is a rational number. Then we can write √2 = a/b where a and b are whole
numbers, b not zero.
We additionally assume that a/b is simplified to the lowest terms, since every rational
number can be written as a fraction in lowest terms. Notice that in order for a/b to be in
its simplest terms, a and b cannot both be even (or else they would share a common
term of 2). Thus, one or both must be odd.
From the equality √2 = a/b it follows that 2 = a2/b2, or a2 = 2 * b2. So the square of a is
an even number since it is two times something. From this we can know that a itself is
also an even number. Why? Because if a were odd, then a * a would also be odd since an
odd number times an odd number always yields an odd number.
Now since a itself is an even number, then a is 2 times some other whole number, or a =
2k where k is this other number matter. Also, since a is even, b must be odd (as
suggested above).
If we substitute a = 2k into the original equation 2 = a2/b2, this is what we get:
2 = (2k)2/b2
2 = 4k2/b2
2b2 = 4k2
b2 = 2k2
This means b2 is even, which, as shown above, means b is also even. Now we have that b
is odd and that b is even, which is a contradiction. Since we followed all the rules of
math and logic in our proof, the only thing that can be incorrect is the original premise
that √2 is a rational number.
Therefore, √2 is irrational.
Also, philosophers similarly use the method of contradiction; Peter van Inwagen’s discussion of the issue
of human free will is an example of this type of argument in philosophy, but we will wait to examine that
A.M.D.G.
Kevin Kolker
TAMU: MATH 482H
05-08-13
argument until we look at other types of proof since his discussion models multiple different proof
methods.
Next, consider the method of proof by cases, which uses exhaustive possibilities to reach the
desired conclusion. The general idea is this: when trying to prove a hypothesis about something in
general (such as integers or functions), one breaks the general category down into subsets that exhaust
the entire group (such as even and odd integers or functions), and then proves the hypothesis for all
these subsets. This leads to the conclusion that the hypothesis is true for the entire original group. As
an example of this in math, consider the hypothesis that if n is an integer, then 3n2+n+14 is even. The
proof is as follows:
First, since n is an integer, n is either even or odd. These two possibilities exhaust the
entire set of integers.
Case 1: Suppose n is even. Then n = 2k for some integer k. Substituting this form into
the given expression we get
3(2k)2+(2k)+14 = 3(4)(k2)+(2k)+14 = 12k2+2k+14
Factoring, we get 2(6k2+k+7), so the result is even (since it is written in the form 2m for
some integer m).
Case 2: Suppose n is odd. Then n = 2k+1 for some integer k. Substituting this form into
the given expression we get
3(2k+1)2+(2k+1)+14 = 3(4k2+4k+1)+2k+15 = 12k2+12k+2k+3+15 = 12k2+14k+18
Factoring, we get 2(6k2+7k+9), so the result is even (since it is written in the from 2m for
some integer m).
Thus, since in both the even and odd case, the resulting integer is even, we can
conclude that if n is any integer, then 3n2+n+14 is even.
Again, this same type of proof is used by philosophers as well, and since van Inwagen’s free will
discussion also utilizes a proof by cases, we will now examine that argument.
A.M.D.G.
Kevin Kolker
TAMU: MATH 482H
05-08-13
First, let us clarify the important terms used throughout this discussion. First,
determinism is the belief that based on the laws of nature and the way things are at a
particular time, there is only one way forward. Under this view, all actions are
determined by circumstances prior to each action. Secondly, a person has free will if at
least sometimes in one’s life, one can actually choose between different possible
courses of action.
Now, since everything obeys the law of the excluded middle, determinism is either true
or false.
Case 1 (Compatibilism): Assume determinism is true and humans have free will.
Obviously, this immediately seems to contradict itself because determinism says there is
only one way forward, while free will requires at least two. Therefore, the compatibilist
must accept the mystery that although there is only one future that is physically
connectable to the present, there can still be open futures that are not physically
connectable to the present.
Case 2 (Libertarianism): Assume determinism is false and humans have free will. Here
we have that there are multiple authentically open futures since determinism is false.
However, without determinism, what causes one to choose option A over option B?
Since nothing in the way things are at the moment before a decision determines the
outcome (since determinism is false), it appears that nothing causes a particular choice,
so then the agent doesn’t cause that choice, meaning that it is not up to the agent.
Thus, the agent is not free. Here we again see our premises leading to the contradiction
of one of the premises, namely, free will. In order to resolve this problem, the
libertarian must appeal to agent causation, which is the idea that although nothing in
the way things are at the moment before a decision causes the outcome, the agent
himself causes the action. This notion of agent causation is a great mystery, however,
since it claims that the agent himself and nothing else causes an action.
Case 3 (Disquieting Possibility): Finally, since (short of accepting a great mystery) both
compatibilism and libertarianism contradict free will, and these exhaust the realm of
possibility (since determinism is either true or false), the expected conclusion is that
humans have no free will. However, van Inwagen believes this also leads to a great
mystery, which is that if we have no free will, then we have no understanding of the
world whatsoever, since we have always acted as though we were making legitimate
choices.
According to Peter van Inwagen, one must accept one of these mysteries, and since he
believes the disquieting possibility to be the greatest of them, he rejects the idea that
humans lack free will.
A.M.D.G.
Kevin Kolker
TAMU: MATH 482H
05-08-13
Allow me to briefly comment on the above “proof.” First, it is clearly an argument by cases in that cases
1 and 2 form an exhaustive list that both conflict with the idea of free will. Thus, the conclusion from
these two cases should be that free will does not exist. However, van Inwagen spins this argument in an
unexpected direction by saying that since free will is basically a “given” in human experience, there must
be something wrong or missing in the argument. In this way, he treats the conclusion of the first two
cases as a contradiction, showing how this argument can be considered a proof by contradiction.
Similarly, both math and philosophy make use of proofs by contrapositive, which prove the truth
(or falsity) of the contrapositive of a conditional in order to conclude the truth (or falsity) of the original
statement (since a conditional and its contrapositive are logically equivalent). The proof of the
statement “if n2 is odd, then n is odd” would be an example of this type of argument in mathematics and
goes as follows:
The contrapositive of the statement “if n2 is odd, then n is odd” is “if n is even, then n2 is
even.” By showing this latter statement to be true, we will also prove the first
conditional.
Assume n is even. That is, n=2k for some integer k. Then
n2 = (2k)2 = 4k =2(2k)
Thus since n2=2m for some integer m, n2 is even. Therefore, we conclude that the
contrapositive is true, so the original conditional must be true.
In philosophy, one area where this type of argument is used is in attempting to illustrate what is known
as the “Ought Implies Can Principle.” This principle attempts to bound our moral obligations by using
the contrapositive of the statement “if I can’t do something, then I’m not obligated to do it” to say that
“if I ought to do something, then I actually can do that action.” Now as long as someone agrees to the
first statement, which seems intuitively obvious, then by the contrapositive, they agree that the only
actions we are obligated to do are those that we are capable of doing.
A.M.D.G.
Kevin Kolker
TAMU: MATH 482H
05-08-13
Lastly, both math and philosophy stress the importance of definitions in proofs. For instance, to
show a fact like “14 is congruent to 5 modulo 3,” one simply uses the definition of congruence mod
some integer, which is that a is congruent to b mod n if and only if n divides a-b (with a, b, and n being
integers). With this definition in mind, we can see that it is true that 3 divides 14-5=9 since 9 divided by
3 is a whole number, 3. Similarly, in philosophical thought experiments, which are often used to make a
point about the consequences of a particular view, the important thing is that nothing violates its own
definition. To clarify, in these thought experiments, it would be okay for zombies to exist since zombies
are logically possible (they do not violate anything that is true by definition), but it would not be okay for
there to be a triangle with four sides (since this violates the definition of a triangle).
Finally, despite all these similarities in styles of proof and despite the fact that both mathematics
and philosophy seek the truth, there is a salient difference between the two subjects that lies in the use
of axioms. Indeed, in math all axioms for a particular area of study are agreed upon by mathematicians
and then these axioms are used in accordance with the rules of logic to arrive at numerous conclusions.
For instance, the Peano axioms of the natural numbers are held as true by all mathematicians and these
axioms are used to prove everything we know about the natural numbers. In contrast, in any given area
of philosophical study there is no consistent set of axioms; rather, each philosopher brings his own
biases and inveterate beliefs to the table, such as Peter van Inwagen’s belief in free will, which for him
trumps the arguments against freedom. Certainly, nothing in philosophy can be taken to be universally
agreed upon by all, for one can always find philosophers who deny even what seems most obvious, such
as the fact that we have bodies or that the physical world exists at all. Furthermore, due to this lack of
consistency in philosophical axioms, there are convincing arguments for nearly every philosophical view
and deciding what to believe becomes a matter of deciding which axioms you agree with most. Of
course, speaking about “deciding what to believe” is completely foreign to the mathematician in his field
of study because the truth of mathematic axioms guarantee the derived conclusions.
A.M.D.G.
Kevin Kolker
TAMU: MATH 482H
05-08-13
References
Van Inwagen, Peter. "The Powers of Rational Beings: Freedom of the Will." Metaphysics. 3rd ed.
Boulder, CO: Westview, 2009. Print.