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What Is Math, Really?

What is math? Numbers, shapes, logic, stuff like that, right? Sure, but what do all of these things have in common? In other words, what is math...really? Keep on reading to find out!

By
Jason Marshall, PhD
May 31, 2014
Episode #200

Illustration of mathematicsToday we're celebrating the number 200. Why? Perhaps because 200 is the amount of money you get when passing "Go" in Monopoly?

Or maybe because 200 is the status number for successful HTTP connections on the Internet?

Or mightn't it be because 200 is the smallest base 10 unprimeable number (which, I just found out, means that it can't be turned into a prime number by changing just one of its digits)?

While those are all certainly lovely reasons to celebrate the number 200, they're not the reason we're celebrating today.

Nope, we're celebrating because this is the 200th episode of the Math Dude. Which, although I don't necessarily recommend doing so, means that you could now listen to Math Dude episodes for well over 24 straight hours.

And it also means that I've gotten a bit nostalgic over the past week and have been looking at everything we've talked about over the past 200 episodes. One thing I've noticed is that we've talked about a lot! And I've also noticed that we've really never talked about the most basic—but yet surprisingly difficult to answer—question: What is math…really?

So I invite you to sit back, relax, and celebrate 200 episodes with me as we think about this question together. Oh, and before I get carried away with that, I'd like to say a big "Thanks!" to all of the math fans out there who have been supporting this podcast over its first 200 episodes. Here's to another 200!

And with that, let's get on to today's show.

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The Many Faces of Math

I recently stumbled upon a blog post on ScienceBlogs linking to what turns out to be a fascinating thread on Reddit asking: 

"Can someone explain mathematics to me?" 

The person asking the question kicked things off by writing:

"Throughout elementary and high school, I got awful marks in math. I always assumed I was just stupid in that way, which is perfectly possible. I also hated my teacher, so that didn't help.

A friend of mine got his PhD in math from Harvard before he was 25 (he is in his 40' now). I was surprised the other week when I learned he isn't particularly good at basic arithmetic either. He said that's not really what math is about.

So my question is really for math fans/pros. What is math, really? I hear people throwing around phrases like "elegant" and "artistic" regarding math. I don't understand how this can be. To me, math is add, subtract, etc. It is purely functional.

Is there something you can compare it to so that I can understand?"

I absolutely love this Redditor's questions…and I think they're questions that many, many people have—even if they haven't yet realized it. And, and is turns out, the seemingly simple question—What is math, really?—is surprisingly tough to answer. But let's give it a shot.

The first thing that comes to mind might be that math is all about numbers. But that's a very narrow (and incorrect) way to think about it. Saying that math is all about numbers is akin to saying that food is all about pizza. Yes, pizza is one (very important) component of the world of food, but it's certainly not everything. And likewise—as you probably already know to one degree or another—there's a lot more to math than numbers.

The Meaning of "Math"

Before going any further, it's interesting to take a look at where the word "math" comes from and what it actually means. The exact etymological trail varies slightly depending on the source (here's a nice summary), but common to all is the fact that the word "mathematics" comes from a Greek word—roughly translated to "mathema"—meaning something like "that which we can know."

The word "mathematics" comes from a Greek word meaning something like "that which we can know."

Obviously, this is an overly broad statement since there are many things we can know that have nothing to do with math. But this idea of math being "that which we can know" actually points us in the direction of a definition that is much more informative. And to find that definition, all you have to do is ask Google. Yep, that's right—just type "define mathematics" into Google and you'll find that math is "the abstract science of number, quantity, and space."

What Is Math, Really?

The key word in this definition—the word that in my mind ties all of math together—is "abstract." If you go back to the blog post I mentioned earlier that links to the Reddit thread that got us on this topic, you'll find what I think is a great description of math. The author, Mark Chu-Carroll, writes:

"To me, math is the study of how to create, manipulate, and understand abstract structures[....] Math can work with numbers: the various different sets of numbers are examples of one of the kinds of abstract structures that we can work with. But math is so much more than just numbers. It’s numbers, and sets, and categories, and topologies, and graphs, and much, much more.

What math does is give us a set of tools for describing virtually anything with structure to it. It does it through a process of abstraction. Abstraction is a way of taking something complicated, focusing in on one or two aspects of it, and eliminating everything else, so that we can understand what those one or two things really mean."

As I said at the outset, it's actually very hard to define exactly what math is. But I think these couple of paragraphs do a pretty darn good job. The big take away message is that at its heart math is not just about numbers, formulas, or shapes, it's about studying the underlying web of abstract ideas (which indeed can often be expressed using numbers, formulas, shapes, etc.) and figuring out their consequences.

Math isn't about the leaves of the tree. It's about the tree.

It's easy to get caught up in a problem and not realize that you're looking at one small part of a much larger picture. The different branches of math are like leaves on a tree. But it's important to realize that math isn't about the leaves of the tree. It's about the tree. Or the forest. Or the entire planet. And once you see that, you see that it is indeed quite elegant.

Wrap Up

OK, that's all the math we have time for today.

Please be sure to check out my book The Math Dude’s Quick and Dirty Guide to Algebra. And remember to become a fan of the Math Dude on Facebook where you’ll find lots of great math posted throughout the week. If you’re on Twitter, please follow me there, too.

Until next time, this is Jason Marshall with The Math Dude’s Quick and Dirty Tips to Make Math Easier. Thanks for reading, math fans!

Math image from Shutterstock.

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