[I am new to NKS and this forum] - A New Kind of Science: The NKS Forum

A New Kind of Science: The NKS Forum

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I am new to NKS and this forum
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Posted by: tomjones

Sorry if this a dumb question to ask but I just recently finished the NKS book and was wondering, if mathematics is universal as Wolfram says it is, and universality is the sort of upper bound of complexity (you don't have to go farther since your system is maximally complex) where NKS fits? So to clarify if universality is the end-all and traditional math makes it there then why do you need NKS? It seems to me that then all NKS is doing is looking at what math does from a new angle.

I don't know maybe I am wrong but this is what struck me reading NKS...

Thanks

Posted by: Jason Cawley

Mathematics is an ambiguous term, with wider and narrower meanings. In its widest sense, mathematics includes any abstract formalism, and includes anything computational as a proper subset. In that wider sense, NKS is certainly within mathematics, and is just emphasizing a different subfield within it as a promising area to develop further. But mathematics in narrower senses typically refers to the areas of formal thinking that have been most fully developed already, which are basically generalizations of arithmetic and geometry, or algebra and analysis. And these are much more specialized, even if theoretically one can translate into them.

One might for example consider every possible state of a digital computer as being a single large integer, specified by its memory state in binary, and then think of every possible update of a computer's state as an integer function. But one would hardly use this way of thinking about programs to write "hello world". It would be pointless. Translating problems into arithmetic is a sensible procedure only when the arithmetical functions that result are simple in some sense. An arbitrary map from a huge space of integers to itself is generally not simple. For much of traditional mathematics, some form of smoothness or continuity provides unifying simplicity, but that means one is considering a narrow slice of possible maps.

One of the points of chapter 12 of the NKS book is that part of what has made mathematics successful and relatively general in the past is the computational universality embedded within it. But it also claims that was not the focus of traditional mathematics itself. When one considers typical powerful theorems of analysis, say, they try to show how to approximate something with some sort of smoothness arbitrarily closely. But programmatic thinking is about discrete states in which there is generally no such smoothness, or "nearby" states have widely different trajectories. Analysis has noticed such cases but generally treated them as pathological counterexamples to be avoided, not as typical. There are areas of overlap or close approach to NKS issues, though, in iterated maps, recurrence relations, and the like.

As for the idea that once one has one universal system, one need not be interested in any others, it doesn't work that way. The phenomenon of universality doesn't mean that everything is a Turing machine, any more than the universality of arithmetic means everything is an addition problem, or the ability to draw a continuous curve through any set of points in a arbitrary number of ways means everything is a continuous curve. Quite the opposite - the generality of such formal theorems tells us that being able to draw such a curve in a specific case tells us nothing about that case, because we could do it in any case whatever.

Instead we need a reason like simplicity, choosing between the formal possibilities. You want the low order curve through the data, not the infinite family of arbitrarily squiggly ones. Similarly with NKS systems, you want the simplest program that plausibly matches a phenomenon seen, not a arbitrarily complicated one hacked into a single unchanging enumeration of possible inputs to one fixed machine. NKS is telling us more than that universality exists aka that there is a program somewhere (arbitrarily far out in some enumeration) that might produce X; it is telling us that we can get quite complicated behaviors with relatively simple programs, and therefore there may be quite simple explanations for a whole class of pretty complicated-looking Xs.

You can consider that a branch of "math in a wider sense" if you like, certainly.

Fine question BTW.

Posted by: tomjones

Thanks for the reply

So to use a specific example let's say I am a mathematician who is an expert in group theory (which is universal), and I am reading NKS, what is supposed to make me want to use NKS methods? Since according to the definition of universality once you reach it you can emulate any system. You mentioned simplicity but it seems to me that simplicity is sort of a vague term for example the traditional mathematician would find group theory simpler then NKS methods most likely. Isn't it the case that simplicity as you are speaking of to some degree a relative thing? From there it would seem to me there must be some-other reason like making science accessible that NKS proves useful? Or am I am completely off base here?

Thanks

Posted by: Jason Cawley

Simplicity isn't vague, and it isn't relative to the state of your expertise in a given area. That's an accident of biography. Any given real system or problem you are examining has a real way it functions, and if you explain it in terms close to the real way it functions, you will get a shorter and coherent explanation. If instead you shoehorn it into your system of epicycles, you will get a needlessly complicated monstrosity. So in any applied modeling situation, you want to have a whole battery of formal methods and scan for the one that fits the real system under examination.

Further, if we want to understand complexity theoretically, we need to know its necessary and sufficient conditions. Satisfying the axioms of a group are neither, but orthogonal (there are groups too simple to support universality, there is universality in systems that do not satisfy the axioms of a group, ergo the group axioms and universality are independent). Yes you can find complex groups; you can find heavy lumps of iron without it making iron-ness the essence of gravity, and hot lumps of iron without it making iron-ness the essence of heat. If you want to understand what causes complexity, formally speaking, you have to do sensitivity analysis on the formal conditions, and under such analysis group axioms are simply not relevant to the presence or absence of complexity.

The same is true of a whole series of formal phenomena previously studied and sometimes touted as essential to complex phenomena in practice. If you follow the argument of the NKS book, where complexity is first notice in cellular automata, you will see all the specific accidental features of CAs stripped away one by one, while a low threshold to complexity remains. That is an instance of such sensitivity analysis, and by its nature it has to be done across formal system types rather than within one of them. As long as one stays in a single formal set up, one risks mistaking accidental facts about that set up for essential causes of behaviors seen within it.

There is of course a reason for studying group theory - it is to get as much mileage as possible about of general theorems known to apply to anything that satisfies group axioms. There are just some questions you can't reach that way, like how relevant or not those axioms are to this or that formal phenomenon. You can't check sensitivity to a variable without varying it. And we know the answer to the NKS question "what is the specific cause of apparent complexity in nature?", has nothing to do with the axioms of groups. (Those actual causes just show up in groups as in other things).

Now, all that said, there is a distinction between what science needs to do to investigate its formal and empirical subjects, and what one scientist (or mathematician) needs to do to make helpful contributions to that overall program. Specialization is fine. One researcher may study groups and another CAs and another recursive functions. Not because nothing else is needed but because it is easier to learn one discipline than all of them. Asking where (in some internal resources scale) complexity onset occurs in groups would be a fine NKS question.

Posted by: Roger Kowalski

I also am new to the NKS forum.

I have a question to the people who
have read the book... After viewing
all the graphics and pictures in the
book.... Just by looking at black and
white triangles or graphic images or
whatever.... I sometimes think most
of those were a waste of time and
effort... They do prove that simple
computer programs can create very
very complex output... I would have
designed the book much differently....
You could have saved a lot of trees in
the process..

I would have created a book of about only 250 pages...
I would have created hundreds of thumbnails of the images.... I would have cut down on the text as well.... But that is me.. Some of the images if looked at for 100 years would have no meaning
to the average person.... So why
even print it.... The idea of an
image or picture is to communicate
something between viewer and
the work of art.... Complexity from
simple concepts is being shown...
But are we communicating anything ????
I say we are not.... We are showing
black dots and millions of black dots...
and hundreds of rules....
and asking the viewer to come to
a conclusion.... about what ????
I would conclude that 95% of the
people opening this book for the first
time, said to themselves...What the
heck is this ???? I'm going to buy
and carry around this heavy book,
of which I don't know what these
images represent in the first place...
Hynmmmmmm.... I know the author
has good intentions about presenting
his ideas.... But for the average
person, who needs all this stuff....
It is a complete waste....of time,
material, and money....But I am not
the author... If I have a big bank
account, I will do what I wish....
(and he did )......

Posted by: Ray Donald Pratt

The fine scale iterations in the CA are specific to each step, but then they scale up to a large scale pattern with interesting breaks, and if you have ever spent long hours watching the changing patterns in Blackjack or Keno or Poker, etc., and tried to discern the causes, you would probably see the NKS patterns as being deeply moving and interesting. They are models of similar systems with deterministic influences.

If you don't care about such systems, if you don't gamble in any sense, then I suppose that you may have no interest in such patterns, or at least that you don't want to devote the effort required for such an interest.

I'm interested enough to regularly set aside time for self-education, even though I am 50 years old and extremely far from my goal.

Very Respectfully,
Ray Donald Pratt

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