[NKS for mathematics] - A New Kind of Science: The NKS Forum

A New Kind of Science: The NKS Forum

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NKS for mathematics

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Posted by: Todd Rowland

Last January, I wrote the attached essay on NKS and theoretical mathematics. Hopefully, the readers on the forum will find it interesting.

I am interested in hearing what other people think about the implications of NKS for mathematics.

Todd Rowland


Posted by: Tony Smith

In your essay you say
The intuition that proof-based methods can solve any problem is still widespread.
The way mathematics is presented to non mathematicians and prospective mathematicians encourages belief in its potency, in its ability to prove things, to separate right answers from wrong answers. Mathematics is thus set up as an ideal to which all would be sciences should aspire, as an idealised source of truth and understanding.

The discussion in other threads in this forum of the distance between mathematics and economics is informative here. Mathematics and economics, and a few other, but my no means all other disciplines, are on a par as far as the totalising claims of their adherents. They each aspire to be the grand narrative in an era when optimistic pluralists like me would have consigned all grand narratives to the dust bin of history.

The NKS book, in so much as it represents the state of the are in complex systems research, is a real threat to totalising views. You say, with respect to mathematics
Only in a few fields has undecidability had a significant impact.
but that applies equally far beyond mathematics. The second threatening world that children learn is "Why?" In a world in which the intuition of NKS replaced the intuition of mathematics, more adults might be willing to answer honestly "I don't know."

I would like to think that the much more profound truth and understanding which NKS points the way towards will eventually put mathematics in its place as just one more useful tool kit. I fear that that intimate connection between NKS and Mathematica will instead sustain a wider perception that NKS and complex systems in general are subsidiary to mathematics.

Others have come from other directions to one very important position you ascribe to NKS and are asking the same questions:
NKS says that underlying all of space and time is a discrete network. What geometry can be derived when all one has are points and connectivity data? How can this network limit to the observed continuum?
I see what NKS says about human perception in being its most important contribution to those questions. Even when we know that is how the world must be, it can still be extremely hard to see it that way.

It is one of the paradoxes of NKS that if the principle of computational equivalence really is true, how is it that we are able to so quickly recognise higher level implications amongst the results of millions of experimental runs?

I still can't help suspecting that the limitations of our first round of experiments with recursive grammars, be they "natural" languages, mathematics and other symbolic systems, computer languages or even markup languages, is positioning us like the Ediacarian fauna sitting around waiting for the Cambrian explosion. NKS might point a way out of the recursive grammar trap.




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