on Nov 10th, 2007The Problem of Math and the Law of Identity
This post will question whether Math is a properly basic belief. I have what I hold to be a reasonable argument so please hold with me dismissing the topic.
In epistemology there are a number of strong theories of knowledge. These range from saying that you can not know anything to saying you know something as long as you can answer the question of why to the people around you (provided they even ask). Most theories allow though for knowledge of math.
xkcd.com had a fabulous web-comic about this very topic.
Most everyone, even many philosophers would not hesitate to say that they know that 2 + 2 = 4. Another popular axiom is that of the Law of Identity, A = A. Both of these statements appear at face value to be reasonable premises. However I think it is possible to show that with these two apparently reasonable premises we are able to apply some apparently reasonable reasoning and arrive at an unreasonable conclusion, basically a paradox. Unless you subscribe to a Dialetheistic theory of knowledge paradoxes are unacceptable and if a belief generates a paradoxical outcome then it must be rejected. All of this sounds really nice, but is meaningless unless I can show the paradox.
Provided Math is True.
Provided A = A.
Then A – 1 ≠ A
Let A = ∞
∞ – 1 = ∞
Therefor A – 1 does and does not equal A!
This is a counter example to the theory of identity when taken in context of a true belief in math. There may not be a good counter example to the law of identity outside the presence of math, this may mean that the problem is with the math and not the identity. Let us look a little closer at the problem of math.
Provided that math is not a properly basic Axiom, I think it is very reasonable to say that math is an incredibly self consistent theory. However, there are points where math breaks down and math rules become ad hoc. Such a basic math rule might be that any number divided by itself equals one. There is an ad hoc rule added here though and that is “except when the number is zero.” We can show at least two places where the basic rules of math break down (math operators at infinity and division by zero). There also happens to be the existence of what is known as imaginary numbers but that is outside the scope of my thoughts here.
So where does that leave us? Can math be considered a properly basic truth, and axiom? How about the law of identity, does the same apply to it as math? If someone asked you why you believed in math could you answer them? Can you Know math? I would argue that the answer to these questions is No. You can not know Math. However it is worth noting that even within a skeptical view of Math, math is very reliable. That leaves me topic I might try and address tomorrow: Skepticism versus Reliablism.
You might read your great great great grandfather’s statements on theory in Chronic Disease. I think you would find it interesting.