Saturday, April 29, 2017

Thoughts on simplicity rules

Underlying reality itself, in SO many ways mathematics rules the world.

Theoretically as a human discipline mathematics should be the most objective, straightforward WITHOUT politics or social influence because mathematical logic does not care. But functionally have concluded that difficulty in understanding mathematical topics can bring those things BACK into the discipline.

But difficulty should only be there out of necessity, and not for social purpose.

So will admit would be so cool to flush mathematical industry of unnecessary complexity. And we should cheer simplicity and push the discipline away from worship of complexity as if it's a power to be not understood by most.

For me simplicity improves access and am sure helps reach. Am not someone who has the assumption that people will pay attention for any other reason than my math discoveries WORK.

Useless complexity though am sure can give room for charlatans to hide within mathematical field! And any such should be cleared to speed up mathematical discovery and progress. Humanity deserves that on the constant. And am sure science will need more advanced mathematical tools on the constant as well, where direction for tools with modular as a concept has unique appeal for its efficiency, simplicity and conciseness. Luckily have done lots of the groundwork over last decade plus which has a modular focus.

In number theory have been fascinated with realization that much of my own research relies on what I decided to call abstract reductionism, and how that really is about simpler approaches, including modular but not only.

For example one of my favorite results is the fastest way to count prime numbers for its size and complexity.

For those who wish to code, with positive ints or longs--where pj is the jth prime:

P(x,n) = x - 1 - sum for j=1 to n of {P(x/pj,j-1) - (j-1)}

The P(x,n) function will count primes up to and including x, if n equals the count of primes up to sqrt(x), and if as you iterate you never let n be greater than the count of primes up to and including sqrt(x), if the function receives an n greater than that value it just needs to reset it to that count.

That is the fastest algorithm for counting primes for its size.

Turns out I noticed a neat trick which lets you use old ideas with an innovation that makes for a concise algorithm, but also leads to a partial differential equation and an explanation that eluded mathematicians for over a hundred years about the prime distribution. Simplicity ruled.

And abstract reductionism is even more powerful with what I think may be the first true modular algebra, which I introduced with a tautological statement, also known as an identity:

x+y+vz = 0(mod x+y+vz)

I call that a tautological space and with that I found a way to generally reduce binary quadratic Diophantine equations, where suggest you use web search to find if don't know where is on this blog already.

Finding by web search versus me just giving a link may be more impressive AND informative as can then see competing websites. And my way improves upon techniques put forward by Gauss himself! And as he is a HUGE hero for me that's a massive big deal. Simplicity rules again.

These simplifications though to me help make number theory more fun--and accessible.

Consider the irony, if some long, difficult mathematical argument which only a few people in the world understood--is simply wrong. And I dramatically demonstrated how it could even look correct under established rules of rigor.

Have concerns about to what extent hero worship took over the math field. People chase the appearance of correctness, daring someone to catch them if they're wrong, but oh, how do you know? In a complex argument which few understand how dare you challenge?

Shredding the incentives for empty celebrity I think helps mathematicians care for truth.

There may be simple ways to remove such.

Simplicity should rule.

That shatters political maneuverings, dismisses hero worship as basis for belief, and ends worship of position in mathematical fields, as who cares who figures out some math thing? I know the math does not.

People can, and it can lead them astray.

We should care in SOME way though I think. But what way is appropriate? Main thing is: irrelevant to mathematical truth who digs up that truth for the world.

Simplicity rules.

And simplicity has other benefits too, as being able to readily play with actual numbers is so great. Like my methods lead to a thing I wonder about usefulness. Consider:

42 + 6+ 10+ 14862 = 882

Which I copied from  a post where I demonstrate a general way to sum whatever number of squares you wish to get a square.

What use is it? Um, for me is just a is fun. I guess. I've debated if should make more examples with it, but will maybe not so much fun for me. Maybe someone, somewhere out there at some time in human history will find it useful.

Main thing in my opinion is that you can just get to math. And will admit do cherish that I escaped the label of mathematician. The math does not care. Discovery should matter more than a label.

But I think it DOES matter when people can easily check a person. Overly complex and abstruse mathematical discourse in my opinion can often be about making that difficult, to force relying on people who know what they're doing--even when it may not actually be mathematically correct.

It can seem expedient to make a career over advancing your species.

I have no career as a mathematician to advance as am NOT a mathematician. And for me simplicity helps me do what I like to do.

And people should cherish valid knowledge. When you start looking at the person who discovered some mathematical truth, as if that decides if something is true or not--then you have lost mathematics.

Mathematics does not care.

Ultimately simplicity is a basic principle for obvious reasons.

James Harris