The deepest and most profound insight in science is that mathematics somehow models reality. There is no known reason WHY this should be

so – it just IS. Sometimes it is said that discovering extraterrestrial life, especially of an intelligent variety, would be the greatest discovery in the history of science. This is not true. Once we figure out a mutual language of communication – almost certainly based on mathematics in general and probably prime numbers in particular – the first thing we will ask aliens is whether or not they have figured out why mathematics models reality. If they say yes, and tell us, THAT would be the greatest discovery in the history of science. More likely, that’s what THEY would ask of US – and be disappointed when we said no because we didn’t know.

Over the centuries, humans have achieved greater and greater insights into how mathematics models reality. We discover whole numbers model sheep in a field of grass, and so the human endeavor of accounting is born. We discover calculus models the motion of air flowing over the wing of a jetliner taking off from an airfield, and the human endeavor of aviation is born. We discover matrix algebra models the interaction of subatomic particles in a quantum

field, and the human endeavor of quantum physics is born. In every field throughout history, NEWLY DEVELOPED math is the key to reaching deeper and deeper levels of understanding – and mastery – about the reality which surrounds us.

There is no reason at all to believe this process of developing new types of mathematics and finding new ways it models reality has stopped, and every reason to believe it will continue. Arithmetic and counting sheep is pretty well figured out by now (we’ll leave a discussion of Godel’s theorem for another day) but there are new branches of math puzzling us today as much as the mentally taxing switch from a vague “many” to precise counting of sheep puzzled our caveman ancestors.

Like so many other current areas of human endeavor, the pace of change and progress in mathematics development is accelerating. Just as microscopes have provided biologists with a new window into a previously unknown microworld, just as telescopes have provided astronomers with a new window into previously unknown deep space, so have modern computers provided mathematicians with a powerful new window into a previously unknown world not just of numbers, but of new unsuspected relationships about math and the nature of reality. And these new relationships mathematicians are discovering between numbers and reality are just as philosophically profound as the implications of DNA complexity or the vastness of the universe.

Example: fractals. At first glance, the equations of fractals don’t seem that different from a 2+2=4 equation you would write down on a sheet of paper. But when you write down the “4” in that arithmetic problem on a sheet of paper, you’re finished. When you write out a set of simple-looking fractal equations, you’ve only just begun. To solve a fractal equation, your first answer gets fed back into the equation and you compute its answer all over again. Then you take this second answer, feed it back into the equation yet again, and compute the third answer. And the fourth. And the fifth. And the millionth. And it goes on like that forever. You can’t possibly write out all of these different answers by hand; it would take a lifetime just to get started. But by using your personal computer as a type of microscope and as a type of telescope, you can generate a lifetime-long list of fractal equation answers in a fraction of a second.

So what? Well, fractal math is so new we aren’t really sure what it all means just yet, but it is virtually impossible to deny that somehow fractal mathematics have a profound relationship with reality that we have never appreciated before. As an example, the millions of fractal numbers that are churned out of one set of fractal equations somehow mimic how the millions of plant cells in a fern leaf form such a distinctive fern-like shape. These fractal equations are like a recipe for making a fern, but you won’t find them encoded in a fern’s DNA. So…how does a fern spore DO that??? It’s like an incredible magic trick, only the magic is not fake or coincidental but REAL. It’s like this certain fern-leaf shape or structure is mathematically encoded into the very fabric of space or reality.

If it works like this for fern leaves, could it work like this for brain cell layout? Is the key to the mystery of consciousness not in DNA at all but in fractal mathematics instead?

Stay tuned and in the next Part of this article I’ll make a leap-of-faith jump from fractals to prime numbers…

Most excellent summary of why math is cool :)

Nice article. I have always been amazed at how mathematics, often extremely abstract and seemingly devoid of any ‘real-world’ application can turn out to be very appropriate as a model for some physical situation. Riemann’s work in non-Euclidean geometry jumps out at me: developed in the mid 1800s, it was a mathematically interesting, but not incredibly practical, branch of math; until Einstein was able to put it to use in General Relativity in 1915 (or was it 1917?).

And Dirac (I think), who posited the existence of the anti-electron (nee positiron) because the theory had a mathematical symmetry that suggested it should be there. He was laughed at, at first. Eventually there were experimental data whose only explanation was a positron. Anti-matter is now very well accepted (indeed, even created).

Math is amazing. As someone finishing up a BS in math and physics, I find that physics is often trivial — it is always the math that is hard. Well, the math and finding the patterns among the experimental data that might lead one to the math. Our favorite geek joke among my physics friends is that we are mathematicians with realistic boundary conditions. Indeed, the term theoretical physics really implies a branch of math: that of making mathematical models to try and model the physical world.

As others, Im interested, and astonished at the same time, to see how mathematics seems to describe, much better than words, “what is happening here”.

My feeling is that match “describe” better because it reaches deeply, but not precisely in the nature of the external world, but in the nature that is both “inside” and “outside” our minds.

I will not attempt to explain more this, as I lack time at this moment, but what Im trying to say is that we see (always) through a set of concepts instead of seeing with just our senses. Those concepts rely on our language, and deep inside our language resides the language of mathematics, maybe the first set of abstractions we developed as a race.

What we see “outside” or what we want to explain is always more in our minds than in the so called external universe, in this sense, math not only deal with that external world, but with our most intimate self.

Take a look at the Number theory and physics archive for some more relations between pure maths and physics.

In recent years, a rapidly expanding body of work has been making unexpected, seemingly unrelated connections between the mysterious distribution of prime numbers and various branches of physics. Note that in general, mathematics ‘informs’ physics, but not vice versa. That is, mathematicians have traditionally been able to provide physicists with useful insights and techniques, but this has been largely a one-way process. What we are considering here is the reverse process, where insights and techniques derived from physics are shedding new light on pure mathematical (in particular, number theoretical) concerns. The number theory and physics archive pages are an attempt to document and archive this material as comprehensively as possible.

These pages are part of the more general site “Inexplicable secrets of creation” of Matthew R. Watkins. The author provides a good summary of current research on number theory as the ultimate physical theory, including his own research: “In some previously unexplored context, the familiar ‘shape’ of the sequence of prime numbers is the result of a kind of dynamic or evolutionary process.”

Fractals are great aren’t they? The way you say they are infinite is a lot like the brain or should I say the mind – it’s all confusing, but we can remember things like a supercomputer, and I imagine our thoughts and memories mapped out like fractls.

Some people believe that fractals are proof that we were created by an intellegence and NOT simply by evolution or ‘chance’.