Generalizing Mathematics

Yesterday I found myself, once again, trying to explain what I do all day as a graduate student in mathematics to someone who has a maximum mathematical training of pre-calculus.  I found myself describing the fact that until you are done with Calculus you are informed of “rules” of math which are you to believe without the “Why”  (ie: the proof).  When you finish Calculus you start the long and involved process of trying to figure out why calculus works (in greater and greater generality).

How do I explain how math generalizes? I mentioned the idea when we are younger we learn what a peanut butter and jelly sandwich is, then we learn to abstract this idea to understanding a PB&J is a *type* of something.  It embodies the ideas we associate with “Sandwich”.  Perhaps your working definition of a sandwich is two slices of bread with stuff in between.  Then, many years later you may learn there exists objects known as open faced sandwiches.  These only have one slice of bread and stuff.  But they still embody something about a sandwich which is undeniable.  (well you may disagree and that’s totally acceptable).  But as you get older you can categorize and understand the generalities of a sandwich/ panini/ sub/ what have you.

In math we deal with the real line until we are old enough to comprehend that there may be other ways of looking at the world without the usual basic assumptions.  (For example: You can draw a triangle with three 90 degree angles on a globe.  try it!)  Part of my work now is to comprehend these generalities- which are mostly sets of rules we require something to have.   A little like “two pieces of bread and some stuff in between”  only more math-y.  I spend my days pondering strange situations where different rules hold then I logically deduce conclusions about these worlds.  For example, in the Euclidean plane, a triangle’s angle measurements add to 180 degrees, but the example above proves that is not true for the space defined by the surface of a sphere.

Higher math is not trivial to explain to someone who doesn’t do it.  But I think it’s important for non-math folks to understand what a mathematician does all day and that we do NOT sit around finding larger and larger integers.   I would like folks to consider what I do helpful and necessary to society and not just something that’s “really hard”.  If you are a mathematician- how do you explain your job?

What are the chances…

That anyone would use probability in there daily life? I think we all use probabilities in a colloquial way. I marvel at my luck when my bill at the grocery store includes zero pennies. Or my excellent timing if I turn on the car after buying my groceries and the same song is playing on the radio that was playing when I got out of my car some unknown amount of time ago? Or I’m surprised if I roll two dice and get four 6s in a row. But am I really so lucky to have such an experience? I don’t think I would ever pause to do the calculation to determine exactly how likely these events are. …Unless of course I’m in my office casually pondering these things (But I don’t think this is a past-time embraced by most Americans). I think non-mathematicians and mathematicians alike appreciate that all things have a chance probability of happening. We all know the phrase, “anything can happen” but also we know in our hearts that some things are extremely unlikely. (Like if Angelina Jolie were to show up at my front door! The mathematician would say this has probability zero: if you will, μ(Angelina Jolie) = 0).

I think probability is quite incomprehensible. And whats more I think the more you understand about probability the less pleasant surprises you receive in life! The chance of the bus arriving within the first minute of me standing at the bus stop is “small”, so I am happy when it works out. But let’s look at some calculations: I know the bus comes every 15min, so (In a vague way) it has a 1/15 chance of arriving in the minute after I do. So the event is bound to happen every couple of weeks. That’s not so seldom! poof! my excitement diminishes. In fact I recommend that you never study probability again. you there! Put down those combination and permutation calculations. If you aren’t careful you may learn the truth about the likelyhood of things to come. You may find yourself more excited when things do work out because you know exactly how unlikely they are. And you may even avoid the excitement and fun of street-vendors probability games because you know the expected values of their profits. Stop now or your life may improve!

The big things that computers can calculate and our reliance on them.

How much computing does your computer do for you? I think we can all agree there is a big difference between a Tip calculator and getting Mathematica to do your integrals for you (Integral‘s are advanced calculus). But then again, maybe we all can’t agree.

The other week, one of my high school tutees asked me if the only calculator I had was my little scientific. Why didn’t I also have a graphing calculator if I was so advanced in math? My response: I can do everything I need with this calculator. And by the time I can’t, I need a computer anyways. This is when I realized that high school students have no idea what the upper limits of a personal desktop computer is.

I suspect most people don’t realize how much math a computer can really do for you. Computers are used by most people to write documents, balance your check book, or surf the internet. All of these things you could do in an analog manner. Pen with paper, a check log with a calculator or a library with a card catalogue. But there are some things we actually can’t do by hand. In my current research we are running 50,000 or 100,000 tests per day per computer. These tests involve 19 or 20 calculations which include ordering numbers, finding Sines and Cosines and appending numbers to list. There is no way that we could sit as humans and run all those tests by hand. It would take decades.

Computing power is undoubtedly powerful, but is that good or bad? Computers can create a 3D image that you can twist and turn in a way you could never do by hand. The downfall is research can quickly get to the point where you have so much data that we you compile it all into a picture you end up asking, “So what does that tell us? What does that mean?”

But I would like to believe that most people have a basic pocket calculator sitting in their desk drawer just waiting for the chance when the world invites it’s owner to do some calculations of their own.