Philosophy

With the help of some colleagues, I am beginning to understand why I am bothered by most discussions of the philosophy of math.  Philosophers have a stance. Examples:

• "Math objects are real but not physical."
• "Mathematics consists of statements" (deducible from axioms, for example).
• "Mathematics consists of physical activity in the brain."

And so on.  They defend their stances, and as a result of arguments occasionally refine them.  Or even change them radically.  The second part of this post talks about these three stances in a little more detail.

I have a different stance:  I want to gain a scientific understanding of the craft of doing math.

Given this stance, I don't understand how the example statements above help a scientific understanding.    Why would making a proclamation (taking a stance) whose meaning needs to be endlessly dissected help you know what math really is?

In fact if you think about (and argue with others about) any of the three, you can (and people have) come up with lots of subtle observations.  Now, some of those observations may in fact give you a starting point towards a scientific investigation, so taking stances may have some useful results.  But why not start with the specific observations?

Observe yourself and others doing math, noticing

• specific behaviors that give you forward progress,
• specific confusions that inhibit progress,
• unwritten rules (good and bad) that you follow without noticing them,
• intricate interactions beneath the surface of discourse about math,

and so on.  This may enable you to come up with scientifically testable claims about what happens when doing math.  A lot of work of this sort has already been done, and it is difficult work since much of doing math goes on in our brains and in our interactions with other mathematicians (among other things) without anyone being aware of it.   But it is well worth doing.

But you may object:  "I don't want to take your stance! I want to know what math really is."  Well, can we reliably find out anything about math in any way other than through scientific investigation?   [The preceding statement is not a stance, it is a rhetorical question.]

Analysis of three straw men

The three stances at the beginning of the post are not the only possible ones, so you may object that I have come up with some straw men that are easy to ridicule.  OK, come up with another stance and I will analyze it as well!

"I think math objects are real but not physical."  There are lots of ways of defining "real", but you have to define it in order to investigate the question scientifically.  My favorite is "they have consistent and repeated behavior" like physical objects, and this behavior causes specific modules in the brain that deal with physical objects to deal with math objects in an efficient way.  If you write two or three paragraphs about consistent and repeated behavior that make testable claims then you have a start towards scientifically understanding something about math.   But why talk about "real"?  Isn't "consistent and repeated behavior" more explicit?  (Making it more explicit it makes it easier to find fault with it and modify it or throw it out.  That's science.)

"Mathematics consists of statements".  Same kind of remark:  Define "statement".  (A recursively defined string of symbols?  An assertion with specific properties?)  Philosophers have thought about this a bunch.  So have logicians and computer scientists.  The concept of statement has really deep issues.  You can't approach the question of whether math "is" a bunch of statements until you get into those issues.  Of course, when you do you may come up with specific testable claims that are worth looking into.   But it seems to me that this sort of thinking has mostly resulted in people thinking philosophy of math is merely a matter of logic and set theory.  That point of view has been ruinous to the practice of math.

"Mathematics consists of physical patterns in the brain."   Well, physical events in the brain are certainly associated with doing math, and they are worth finding out about.  (Some progress has already been made.)  But what good is the proclamation: "Math consists of activity in the brain".   What does that mean?  Math "is" math texts and mathematical conversations as well as activity in the brain.   If you want to claim that the brain activity is somehow primary, that may be defendable, but you have to say how it is primary and what its relations are with written and oral discourse.  If you succeed in doing that, the statement "Math consists of activity in the brain" becomes superfluous.