### How to avoid ‘computational irreducibility’: Wolfram

##### July 2, 2012

“It seems like Nature has some secret that lets it make complicated stuff in an effortless way,” Stephen Wolfram recently told an audience at Oxford University’s Mathematical Institute.

In his talk, that you can now watch online, Wolfram, the scientist behind Mathematica and Wolfram Alpha, explored how advances in computation could benefit mathematics.

One of the key ideas he put forward was “computational irreducibility“ — the idea that some computations cannot be sped up by any shortcut, the only way to figure out what is going to happen is to simulate each step.

“People sometimes say that the reason the mathematics that we have is the way it is, is because that’s what we need to describe the natural world, I think that’s just not true,” he commented. He suggested that much of the reason mathematics covers the areas it does is historical, building on work begun by the first mathematicians in ancient Babylon.

Computational irreducibility, he said, is a “junior version of ‘undecidability’” — the idea that when you ask the question of what will ultimately happen the answer is something that is undecidable. While there are more than three million theorems in mathematics, these are all things that turned out to be decidable/provable.

There isn’t much undecidability in mathematics because maths is set up to examine those things its methods can make progress on: “Mathematics has navigated through these kind of narrow paths in which you don’t run into rampant undecidability all over the place.”

Ask mathematical questions at random, he suggested, and you would soon run into undecidability. But perhaps through exploring the space of all possible theorems, using tools such as Wolfram Alpha, you might find new paths.

He described the point of Wolfram Alpha as “to collect as much knowledge as possible and make it computable,” and that this approach could be applied to find out which theorems about a particular structure or system were interesting or powerful.

A pilot study focusing on one particular area of maths, continued fractions, is already showing that the process of organizing theorems in a way that’s systematically computable is leading to new advances, he said.

In a contrast to the days when mathematicians did all of their calculations by hand, the future of mathematical process could be that, by entering some details of a system, within seconds they would automatically see a range of theorems about it.

This would give a window on what he called a “vast ocean of unexplored generalization of mathematics that exists in this computational universe of possible systems.”

## Comments (5)

July 6, 2012by Jessee McBroom

This is an interesting approacn to generating theory fot further mathmatical selection as to viability of theory based upon mathmatecal knowns held in a data bank. The work would be in building a data bank of known and scientificaly proven data IE a compilation of all proven data relative to the physical sciences. This appears a short path; but computationl extensive approach to AI.

July 2, 2012by Gorden Russell

Yes, DeBee Corley. The nanocells will put things together according to the formuli that make fractals.

July 2, 2012by Gorden Russell

“It seems like Nature has some secret that lets it make complicated stuff in an effortless way.”

That’s why I keep repeating myself about self-assembling photo-voltaic carbon nanocells.

I was just looking at a maple tree seedling one day, just wanting it to grow faster, so that my neighbor’s pit bulls wouldn’t get their leashes tangled up in the little tree and break it off (that’s happened more than once before).

So while I was considering that, when sunlight goes into the leaf and works with chloropyll to make cellulose, I remembered that it’s been discovered that some little quantum thing is going on in the chloroplasts of the leaf. Photons of sunlight are being turned into electrons, and then the electrons flow and split water into hydrogen and oxygen, and also split carbon dioxide into carbon and oxygen. Some of this oxygen joins with carbon and hydrogen to make carbohydrates, and the rest of the oxygen is released. It’s a good thing that there is that excess of oxygen, or the world would still be inhabited by nothing but anaerobic bacteria.

Considering all that, it just hit me that some busy researcher somewhere will one day find a way to harness this happy quantum happening and use it to take carbon dioxide out of the air and reassemble those free atoms into carbon nanotubes, graphene, fullerene, fiberdiamond, and even many more carbon compounds, even the natural fibers that come from plants. If a plant can assemble carbon into flax, cotton, jute and sisal, why can’t little nanocells do the same?

July 2, 2012by star0

Sounds like a step in the right direction.

July 2, 2012by DeBee Corley

Fractals anyone?