OK, computation

Jul 02, 2012 By Pete Wilton
Credit: Stephen Wolfram via Wikimedia by Stephen Wolfram PR/Stephen Faust

"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 , 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. Whilst there are over 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 generalisation of that exists in this computational universe of possible systems."

Explore further: Math modeling handbook now available

More information: www.stephenwolfram.com/

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User comments : 7

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Vendicar_Decarian
3 / 5 (4) Jul 02, 2012
Translation - Modeling.

A mathematical tool that Climate change denialists just can't accept.

"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." - Article
Tausch
1 / 5 (2) Jul 04, 2012
"One of the key ideas...the idea some 'computations' cannot be 'sped up' by any 'shortcut'...! - Article.

Nature has a language.
Will our 'translations', 'models', or 'maths' always fall short to the original language of Nature?

Ja. Die Hoffung auf die vorigen Antwort bleibt.
(Sonst gabe es keine menchenlichen Aufgabe mehr!)

AtlasT
1 / 5 (1) Jul 04, 2012
Before some time, the mathematicians did see the world clearly. Max Tegmark, a MIT teacher: The Mathematical Universe

but now something changed:

Alan P. Lightman, a MIT teacher: We are living in a universe uncalculable by science.

A "slight" paradigm shift, so to say...;-) But the change of formal education is the same problem, like to force the physicists into acceptance of cold fusion under the situation, when majority of them are already engaged in research of alternative methods of energy production/conversion/transport and storage. In similar way, the high school teachers will never admit, that the teaching of reality trough math equations has its apparent limits, because they would just threaten their tediously occupied social position.

For additional reading: Dehumanized: When math and science rule the school.
Tausch
1 / 5 (2) Jul 05, 2012
Well, yes.
All humans learn.
The gap between teaching and learning garantees the justification for the existance of all institutes of teaching.
Man made institues of learning are nonexistance.
No teaching or teacher can replace Nature provisions for learning.
Vendicar_Decarian
5 / 5 (1) Jul 06, 2012
Yup. There are very few problems in science that can be solved with simple equations.

"Will our 'translations', 'models', or 'maths' always fall short to the original language of Nature?" - Opie
xen_uno
not rated yet Jul 06, 2012
Atlas - "But the change of formal education is the same problem, like to force the physicists into acceptance of cold fusion under the situation"

LOL .. is that it? Lack of education or some bias? I think it was Terriva or possibly Otto that posted a list of 10 companies working on it. Haven't heard squat from any of them .. have you? IMO they are a lot like someone finding religion when on the death bed .. you know .. covering all the bases in case there is something to it.

Hot fusion is only hindered by plasma containment and stability but slow and steady progress is being made. The physics behind it are solid and proven. Cold fusion or LENR however, is hindered by lack of independent testing, vacuous physics, and con men. So I'll make you a bet ... 100,000 quatloo's says hot fusion will be viable long before cold fusion.
Job001
1 / 5 (1) Jul 06, 2012
Irreducible add to undecidable, unknowable, and ultimately paradoxical? Nature does what nature does well while our models fail to adapt?

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