## Sunday, August 10, 2014

### Is mathematics a science?

Finding functional definitions intrigues me as I try to understand my reality, and recently I came back to ideas about science, where I posted about it on one of my other blogs. For here I'll note I've seized upon the following simple statement:

Science is the art of prediction.

That is a functional definition in my opinion in that you can use it to determine if something is a science or not. For instance, medical science allows doctors to predictably set simple fractures. Geological science allows us to predictably tell the age of rocks. Bureaucratic science allows executives to predictably manage companies of a certain size.

Well where might one predict with mathematics?

How about with the classic equation: x2 - Dy2 = 1?

Its fundamental solution, which is the smallest set of positive nonzero integer values for x and y with positive integer D greater than 1, can vary in wild ways.

The equation is commonly called Pell's Equation by mathematicians, and you can quickly find a paper considering the problem with a search on: Pell's equation size

Doing that search now I found the following paper with open access at #2:

### The size of the fundamental solutions of consecutive Pell equations

Now then that has a predictive aspect which should be readily understood, so I say that paper represents a mathematical science inquiry.

Reliably predicting the size of the fundamental solution would be a scientific triumph, and the 'art' to my statement above can be understood with this example! As how would you approach getting this answer?

That is, if you're a math person and I suggest to you the scientific investigation into the 'why' of the fundamental solution, how would you proceed?

If you're quite brave I suggest you try, as you may notice if you read that paper. I scanned through it for this post and noted something I can explain as the authors are puzzled by the very large size of the fundamental solution for D=1621 versus D = 1620.

I can explain it has to do with D being prime, and with D-1 having a lot of small primes as factors, as well as 4, and you can check: 1620 = 4*3*3*5

My full explanation which gives you the full predictive science behind the fundamental solution and connects to derivations is on this blog:

somemath.blogspot.com/2011/09/two-conics-equation-size.html

My predictive ability can be compared to the prior paper, which I think helps explain science for those who wonder, as the certainty given can be used to DO something.

So then, to the extent that mathematics gives you predictive ability, you have a science, but when it does not, you do not.

Practical usage of mathematics is often about prediction, and mathematics is often used in science for that purpose. But my example above shows how even with a "pure math" result you can have prediction as well, though it's also possible the method is practical.

What about a counter-example? Well I also have a technique for reliably reducing binary quadratic Diophantine equations! Hmmm...that sounds like it's still a science.

I also have my own way to count prime numbers. That predictably counts prime numbers.

Interesting. Kind of hard to remove prediction here, or am I missing something?

Oh yeah, I have a prime residue axiom which allow you to...why bother? It allows you to reliably predict prime gaps, like the occurrence of twin primes for example.

Well it looks like functionally I'm looking at example after example of prediction given by mathematics, and have an example of a mathematical scientific investigation where you can even challenge me! As you can go forth and try to make predictions about fundamental solutions to x2 - Dy2 = 1.

Starting this post I wasn't sure, as the post itself is, for me, a scientific inquiry, or maybe a metascience one.

My own conclusion is that from what I can tell mathematics is indeed a science, which allows one to gain certainty about numbers. And there is the 'art' in the science which is figuring out how to get to a solution!

You can make a hypothesis, test your hypothesis, and come to a conclusion.

But even better you can have mathematical proof to give absolute certainty, which then is gratifying in supporting of the statement by Gauss that mathematics is the "Queen of the Sciences".

Of course Gauss was a preeminent scientist as well as a mathematician, so no surprise there. Nice to be in agreement with him, of course.

So am I saying that mathematicians are scientists? I guess so, though I wonder how often they act like I tend to think they should! To me scientists are excited by expanding zones of prediction. I'm not sure mathematicians in general behave in that way, as the predictive aspect of mathematics is not emphasized.

So to me, scientists would be intrigued by the greater prediction inherent in my approaches, but I'm still laboring for mainstream acceptance, which indicates to me that the power of prediction is not a driving force with the mathematicians of today.

My conclusion then is: as currently trained mathematicians de-emphasize the importance of prediction in mathematics and so do not behave like scientists not so trained.

But I need to be more careful here, and remind myself that the need is to explain why mathematicians have not behaved as I originally expected.

In my opinion mathematicians don't seem to think like scientists though mathematics itself clearly is a science. That is I'm guessing about training, which emphasizes building on the known, while science emphasizes expanding zones of prediction.

Building on the known can expand zones of prediction, but is not the primary goal, and without the power of prediction to tell you when you err, there isn't an important failsafe.

Prediction has power even in "pure math" areas, as it allows you to say how numbers should behave, and see what happens.

Instead mathematicians built on what came before trusting that no mistakes were in it, which I've found left room for error.

My guess is that mathematics will evolve as a discipline, and that future mathematicians will be scientists like all the others, probably discounting the value of established positions like physicists who tend to try and attack them.

My ability to conclusively demonstrate with classic number theory that prior work was insufficient to explain, may be considered a watershed moment in mathematics.

Intriguing.

The current refusal to properly acknowledge a problem is not surprising to me though, as without a true scientific perspective mathematicians today have reverted to typical human behavior valuing belief over scientific reality.

They would prefer to be confidently wrong, rather than predictably right, showing that a scientific perspective is necessary here.

James Harris