A definition of mathematics

In my web-tacular adventures, I stumbled across a math professor and writer, Reuben Hersh out of the University of New Mexico.  While he is, of course, a mathematics professor, he is also looks at the social impact of mathematics and the philosophy of mathematics, which is pretty excellent.

I read his short Definition of Mathematics (which can be found on his site, listed above).  It is very interesting because he looks at what mathematics is.  I found this extremely useful because, well, I never really thought about the definition of mathematics.  Math was...well, math was math.  You did stuff like functions, addition and subtraction, calculus, just to name a few.  It is actually fairly ridiculous that a math teacher doesn't really have a working definition of what mathematics actually is.



Reuben Hersh talks about mathematics as both a science and a humanity, talking about how it is about "mental objects with reproducible properties."  Mathematics can be verified by practice and agreed upon by experts, but the subject matter are ideas and concepts.

The old definition of mathematics was "the study of the properties of numbers and geometrical figures," and Mr. Hersh says that that was all fine and dandy then, but it isn't today.  He says, "mathematics includes abstract algebra, logic, and probability, none of which is part of traditional arithmetic or geometry."  So mathematics grows and changes as our understanding continues to grow and change.

He makes mention of actively discovering or creating mathematics.  I think that's terrific.  Something so old as to have been around in the time of the Ancient Greeks, in the time of Alexandria, that which wars and cities depended on, is still growing and changing today.  I've always enjoyed the word "active."  Someone is trying, they are doing.  An "action word".  I just thought I'd ramble about that for a bit...

Another interesting reference he makes is to the book Where Mathematics Come From by George Lakoff and Rafael Nunez.  To quote: "[The authors] showed that mathematical proof often can be understood as based on 'embodied metaphors.'" I was quite liberal with my exclamation marks beside that comment, mostly due to the reading we had read before in class on metaphors - how metaphors are a part of our daily language and conversations, that we naturally think in terms of metaphors.  I thought when I was reading that: "But hey, what about math?  Does the language of math have metaphors?  Do we perhaps block off an area of our student's brains (the cultural part) when we put them into a mathematics classroom and speak to them using a mathematical language and they are unable to follow because of that lack of metaphors."  Anyways, I'd like to look further into this idea of embodied metaphors; if not within this 5 week (5 week?!) span but sometime in the future.

(Just as an aside, I'm finding as I'm becoming more of an individual in the professional community, I'm finding many of these books that my professors are recommending and particularly the books that I'm finding during the course of my research for this project, fascinating.  What is this?!  I was hoping to be reading pulp after I was finished school, not learning!!  Nah, I'm kidding).

Finally, he asks what applied math is and this quote is really took good to give up:

Mathematical activity or behaviour includes: thinking, wondering, dreaming, learning about mathematics; solving math problems, at all levels, from pre-kindergarten up through postdocs and Fields Prize winners; and teaching mathematics, at all levels.


I thought it was terrific.  Just fantastic.

So, according to Mr. Hersh, mathematics is neither purely science nor purely a humanity.  It's interesting.  We here at By Hand are, however, in the business at looking at math as a language so I'm going to ask - how does this definition fit into that idea of math as a language?  It kind of indicates that I have to look into what a language is.  Boy, this inquiry is getting far more complex than I had thought!

I am in the process of reading his article in the American Mathematical Monthly on Math Lingo vs. Plain English.  It's very interesting what he has to say and I will be following up on that most likely tomorrow. Like Mr. Holmes (I'm in the process of reading The Adventures of Sherlock Holmes so I apologize if I constantly make reference to him), I'm going to take time to take out my "old and oily clay pipe" and take some time to think, re-read and think some more.

Until then.