Sunday, October 21, 2012

Does math need science? Is mathematics discovery or invention?

These are two questions I have been asked lately and I feel they are sort of related to each other, so I will attempt to address them in a single post.  I didn’t really have a well thought out response when one of these discussions was started, but I feel I have formulated a good idea of what’s actually going on.  My professor says my view of math is a lot like that of G.H. Hardy, so I apologize with anyone that heavily disagrees with this and sees things in a different light.  I will try my best to explain why I think my view on mathematics is quite accurate as it interrelates to other subjects and why it is both invention and discovery at the same time.

First off: Does math need science?

I firmly say no and most mathematicians will heartily agree with this assertion.  Does science need math?  I say yes and again many will agree.  It’s this interplay between the disciplines that generates some confusion on the nature of math.  Non-mathematicians, for the most part, seem to view math as an extremely powerful tool for explaining observed phenomena in our world.  I completely agree with this view and I honestly can’t see things being done any other way.  However, does this necessity work the other way?  Certainly, mathematics’ level of abstraction does not require our world, our universe, or anything else physical in nature to define its world.

Most of the tactics used in this discussion deals with the nature of how science proves things versus how mathematics proves things, but I will go with a different tact, because I think it is far more interesting.

In science, scientists spend time doing experiments and gathering data.  Sometimes this data will lead scientists to generalize the data with a mathematical equation.  We see this in physics all the time with things like F = ma.  But does F=ma have anything to say about any mathematical truths?  No, not really.  Please note, I am not saying that scientists can’t prove things in mathematics, this is patently untrue.  Physicists have developed wonderful things in mathematics, but they were not doing physics, they were doing math and these are different techniques.

Here’s the real crux of the problem, scientists do not get to choose their world.  Reality is what it is whether they want it to be that or not.  They do experiments to see what they can find out about the universe.  Sometimes this requires the creation of a mathematical model that helps explain a phenomenon.  This is surely invention on the part of the scientists and they are using rules of mathematics as a tool to delve deeper into this creation they have made.  However, any truths they claim to find about reality by doing math will obligate them to do an experiment to verify it.  If the experiment does not confirm the mathematics, then their mathematical model is simply not the right one.  That is all, so they move on and choose a better one.  This is how science tends to trudge along.

Second: Is mathematics invention or discovery?

It is both!

Mathematics is entirely different.  In mathematics we choose our world and we have no choice over the results.  Mathematics always begins with laying out definitions; this is the piece that is pure creation.  However, we then see what logical implications the definitions behoove us to make; this is why it feels like discovery when a new Theorem is “discovered”.

For example, if we set up an ordered algebraic field for the Real Numbers, then we further come up with definitions for surpemums and infimums… the Mean Value Theorem will result!  As long as all of the conditions exist this theorem is true and it isn’t just true it is an absolute truth.  You are guaranteed this as a result of the definitions.

Science clearly never works this way.  They always have “here is the universe, here is the earth in the universe” etc, these are the givens they must work with and now it is up to them to construct mathematics to help explain what they observe.  The other wrench in this is that there is also some percent error, so things are never absolutely true.  They may be very close, with 99.995% certainty, but there is always a degree of uncertainty.  Furthermore generally accepted ideas in science can be overturned if some other fact presents itself, or some experiment shows it to be false.  Mathematics does not need to worry about this.  Once a theorem is proven there is nothing that can ever come along to show the theorem is false.

I am not trying to downplay science.  Science is very practical and there really is no other realistic way to study the world around us.  Mathematics is a very powerful concept, but you pay a price for that power.  You are no longer bound by reality.  If mathematics were bound by reality then we would never be able to talk about things like right triangles.  There would always be some atom partially out of place throwing off that proper 90 degree requirement.  Mathematics talks about a purely mental idealization of things.  This idealization lets us talk about things that may never exist in reality and that’s all good fun, but it really is just head games.

Hopefully this has made sense and if it hasn’t, I may spend time revising it later.

Thursday, May 17, 2012

First Year of Grad School

Currently Reading: “I Want to be a Mathematician” by Paul Halmos

I wish I had more time to update blogs, but I never seem to once I start one.  This is honestly the third one I’ve attempted to start and I figured a themed blog would make me more motivated, I guess not.  On the other hand Graduate School is incredibly time consuming.  I have also found it to be incredibly disheartening.  In some respects I think the main problem is that I’ve done the undergraduate degree far too quickly so it feels like there is a lot I don’t know.  Becoming good at math is more an experience oriented thing than a “natural skill” oriented thing.  I find that it doesn’t seem to matter how good you are at the subject, the length of time you spend doing it is what’s important.  Mathematics is a hugely time consuming subject and those that put the time into it have a major payoff.

I am just worried that I have done things so fast, that I will never be able to achieve the goals I want to achieve.  I don’t see myself doing very well in a general corporate environment.  The only schedule I think I could keep up with is being something like a research mathematician at a University.  I don’t think I even care if it’s a prestigious university… just a university in general would be nice.  But that job market is so utterly competitive that finding tenure track jobs might be out of reach for someone as mediocre as me.  University’s don’t care much about someone’s ability to teach the future generation, what matters is doing original research.  I am certainly not opposed to this since I think doing research would be quite exhilarating.

I feel that I could be up to the challenge.  I just hope I don’t get crushed under the weight of the academic machine that has been put into place for years.  I don’t always have the strongest grades compared to my peers, such as my friends Brendan and Eric, who I believe have far more ability than I.  However, I hold out hope that this will not hold me back for getting into other programs. As I’m reading “I want to be a Mathematician” by Halmos, I am given some comfort that he also did not have stellar grades in mathematics and also found Analysis quite challenging.  I worry my time-line is too fast for the system, but I am reminded of the story of Leibniz who only studied math for a mere five years before turning to original work.  However, Leibniz wasn’t obligated to have a thesis advisor, take a Math GRE, and things of that nature.

Matters have become more depressing, because I have also recently lost my job at the lab I worked at.  I am going to try and look at the bright side of this.  I am going to buckle down and try to solidify the math I’ve already learned at my professor’s recommendation.  My professor, who we affectionately call Kiwi at his insistence, has pointed out that I should know things more quickly than I do, and I really believe that is true.  Hopefully I will have the diligence to amass more skill with this newfound free time.

Also I am hoping to add much more to the blog.  I have recently finished a course on Partial Differential Equations and whenever I searched for useful examples on the web, it was impossible to find anything that had a lot of detail.  Many steps were frequently skipped or not even explained.  I want to spend some time writing up my own solutions and post them on here.  I am debating if I want to teach myself LaTeX in order to publish these or if I should write them up in Mathematica and then just convert the document to LaTeX as I learn that language.  At least in the latter case the brunt of the document writing will be finished… we’ll see.  A decision for next week maybe?