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I read a post by JDJ yesterday about a distinction between facts and mathematics ^ that I found quite thought-provoking. I'm not well-versed in philosophy, much less in formal philosophical discussions about the concept of truth or the mechanics of logic--my background is in pure (as opposed to applied) mathematics. As such, my exposure to the concept of mathematical truth is almost entirely from the point of view of research within mathematics. Regardless, I've thought about the topic a lot, and it's something I think is worth talking about.
A fact is any statement that is true. What does that mean? People much smarter than me have devoted extensive treatises to the subject, but in common parlance, a defining property of truth is that it is consistent, regardless of human experience. An example of a fact in JDJ's post is that 1+1=2. This statement is true whether or not you know about numbers or how to add.
This isn't a straightforward as it seems, though. A common issue in these kinds of discussions is that of naming or defining things. For example, if I define the following words thus:
"One (1)": This many things: | "Three (3)": This many things: ||
suddenly I can claim that 1+1=3. As I've defined the words, that's correct. In everyday life, we usually ignore problems like this, because we assume the same words will carry the same meanings even in different contexts. But the underlying issue is still there: facts are dependent on definition.
In pure mathematics, the issue of definition is an important topic in almost every field. There are always questions about what should be considered an "axiom" (a statement taken to be true without proof) and what can be derived as a "theorem" (what can be shown to be true using axioms and their properties). Even the term "mathematics" itself struggles to yield a strict meaning; Wikipedia points out in its opening paragraph about mathematics that "[t]here is no general consensus about its exact scope or epistemological status." ^^
If I were to give a one-sentence definition of math, I would say:
Mathematics is applied logic.
I've noticed a common thread running through all mathematical pursuits, from the most elementary schoolroom arithmetic to computational complexity to imaginary numbers to vectors to category theory. That common thread is that certain fundamentals are simply taken to be true (the aforementioned "axioms"), and based on those assumptions, other truths are derived using only rules of logic. Mathematics strives to use logic to build a completely-rigorous internal consistency: given the same set of initial truths, any other truths that CAN be derived from those base truths WILL always be derived from those truths.
Whether mathematics can be considered "fact" is a tricky question, because it depends very strongly on how one views the concept of truth as it relates to our real, physical world. If I hold up two pencils in one hand, and I hold up two more pencils in my other hand, I will be holding four pencils in my hands. That, we could say, is a fact. But the mathematical statement "2+2=4" is an abstract statement that does not exist in a tangible way. Is that a difference between these two statements? Does the abstract statement have some kind of philosophical "truth" to it that corresponds with, but is not the same as, the "truth" of a fact manifest in the real world?
At the heart of this discussion seems to be the question of whether logic can be considered a fact. Logic tries to provide a mechanism by which true statements can consistently yield other true statements, but this property of logic doesn't automatically mean that logic itself has the same kind of truth that physical objects do.
This may seem like a superfluous or even meaningless point, but consider an example. A central concept in analysis and many other areas of mathematics is the imaginary number i: the square root of -1. Imaginary numbers are used in everything from the study of waves to electrical engineering, but the actual concept of an imaginary number itself has no physical manifestation or significance--its utility only comes from its mathematical properties in relation to other numbers. Can we say that imaginary numbers are not a fact, or have a kind of truth different from a fact, because they cannot exist tangibly?
In my (perhaps philosophically naive) view, logic can itself be considered a fact, just as much as tangible facts in our world. Its ability to produce true statements based on true statements is, if you will, a method of "truth preservation", and that to me seems to be a hallmark of something that is itself true. On that basis, I consider mathematics--a set of logical statements based on fundamentals taken to be true--as true. And if it's true, by our original definition, it must be a fact. In short, I think math is the same thing as facts.
I will concede that I think a fact manifest in the real world and a fact expressed in pure mathematical terms are not the same fact. These facts may correspond to each other in a way meaningful to us, but that does not make the facts themselves the same. However, just because two facts are not the same facts does not mean they are, in a categorical sense, the same type of thing: facts.
As I said, I have read essentially no philosophical material on this matter. I might be entirely wrong, or there might be other nuances to the discussion I'm not aware of. These are just my feelings based on my direct study of mathematics.
^ Math doesn't mean the same thing as facts
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