[a mathematical NKS foundation] - A New Kind of Science: The NKS Forum

A New Kind of Science: The NKS Forum

Pages:1


a mathematical NKS foundation

(Click here to view the original thread with full colors/images)


Posted by: Philip Ronald Dutton

When I think about NKS and it's potential for augmentation or reformulation of the foundation of mathematics, I somewhat feel the urge to eliminate a certain phenomenon- namely, the paradox. Does thinking in terms of the mechanical nature of NKS require the end of the paradox? The paradox does not seem a natural fit within the algorithmic universe.

Yet, from what I remember, many mathematical "proofs" rely on the use of paradox.

If it was possible to arrive at a theory of everything which was built on NKS then I can not see where a paradox could find room to breathe.

(my mention here of NKS is strickly related to simple programs, algorithms, etc. ...and not necessarily related to emergent behavior or complexity, etc.)


Posted by: Jon Awbrey

Philip,

I think you are confusing paradox with contradiction.

Jon Awbrey


Posted by: Philip Ronald Dutton

Jon,

Yes there is a distinction, perhaps. The two are close cousins. I think it is safe to talk about either one within the context of my original post. Using the term, 'contradiction', I still find it to be one of those things which does not seem to be able to fit inside an algorithmic 'ether' (world, foundation, etc.).

(ASIDE:Within the many worlds of human arithmetical thought, are there any disciplines which have, from the beginning, sought to remove any possibility of contradiction or paradox? One in which such entities are 'illegal'? )

Surely, a purely algorithmic foundation of arithmetic will find it hard to allow contradiction or paradox.??? Contradiction and paradox just do not seem to be able to fit into a mechanical foundation.

Thanks for the input.

Philip


Posted by: Jon Awbrey

Philip,

Thanks for the additional information.
Perhaps what you are worried about is
more the issue of constructive versus
non-constructive proof, the contrast
between intuitionistic and classical
logic, especially with regard to the
use of the excluded middle principle
in application to infinite sets.

In that context, there are widely held to be
many close connections between constructive
mathematics and effective computation, for
example, via the calculus of "combinators"
S, K, I.

These support a symbolic calculus that is
roughly equivalent to lambda calculus, and
their functional types (in the typed version)
are linked with a logical system known as the
"positive intuitionistic propositional calculus"
via the so-called "propositions as types analogy".

There's a really enjoyable presentation
of all this in Raymond Smullyan's book,
'To Mock a Mockingbird'.

Jon Awbrey


Posted by: Philip Ronald Dutton

I was thinking (in physical analogy) of a set of gears. Let us say that they are perfectly arranged, symmetric, etc. They could work and work forever and never jam.

Then there is also another set of gears just like the first. One of the gears, however, has a badly positioned gear tooth. Let us say that the set could operate for a long time before the bad tooth caused a problem- more specifically a jam.

I think about a mathematical foundation rebuilt using NKS and I think of the first set of gears which will continue to churn out correct configurations.

Now we can see using our analogy that the system will produce only good configurations and not ever a bad one. Why not a bad one? Because the gears keep turning and working. There is never a jam because everything was arranged correctly. Or, to put it another way, there is not a gear which has bad teeth, crooked teeth, or misplaced teeth. No mechanical bugs!

If we then discover that the whole universe is operating in such a mechanistic way, we can also assume (similar to pure naturalism) that human thought is also abiding by the same mechanistic algorithms.

Now, we still have the problem that we can arrive ( in certain analysis of the mind) at points of contradiction or paradoxical conclusion. Do we stop? Do we complain that "whoa! this is an unprocessable conclusion!" ? Do we reformulate our analysis?

Remember that (sticking to pure naturalism) the point of contradition or paradox had come about by the "good set" of "gears"! The next immediate gearset rotation (one will come because there are NO jams) is a new, good configuration and therefore, the contradiction or paradox was indeed processable. Or, in other words, the gears continue and do not JAM because this really was not a bad configuration!

I am tempted to say that the recognition of the existence of this paradox or point of contradiction was simply a meta-mathematical statement by the one doing the analysis. If so then the gear set's configuration which represents the statement, "this is a paradox", is also a "good gear configuration."

But you can not actually GET AT or GET TO this thing which we desire to call a paradox or a contradiction.


Posted by: Jon Awbrey

Philip,

All this talk of gears reminds me of Babbage.

I still think that it is crucial to reasoning
to sort out paradox, which is by nature not a
well-defined term and covers a motley crew of
diverse dilemmas that appear for the moment to
baffle our power of reason, from contradiction,
which is a very well-defined term and which we
can reason quite well about.

In my experience, all paradoxes arise from assuming
a false statement, the catch being that we adopt it
in such a way that we are unaware of having done so.

I have to be on the road the next few days,
but I will leave you with an illustration
of what I mean by this, posted elsewhere
in this area.

Jon Awbrey


Posted by: Philip Ronald Dutton

In my experience, all paradoxes arise from assuming
a false statement, the catch being that we adopt it
in such a way that we are unaware of having done so.

Apparently, our rudiments of thought have allowed us the ability to ACCEPT a false statement. There is nothing that prevents me from accepting a false statement. The laws and rudiments of this universe let me do it and end up with paradox. The system does not really care if I do it or if I do not. Accepting a false statement is a LEGAL action within this universe.

In other words: If I start with accepting a false assumption and derive some related statements only to end up with a paradox, then the system is satisfied because "you used a legal mechanistic procedure." Almost as if the end result does not matter as much as the fact that an algorithm was used.

Which means, possibly, that nothing (no configuration of this universe) is bad-- INCLUDING human thought. The "gears" of the universe keep working and working and working and can never get to a bad configuration (except for the big crash! whatever that is.... doubtfully caused by some human who thinks up some hyper paradox!). Surely whatever the human thinks is a legal configuration of the universe.


Posted by: Jon Awbrey

Philip,

Yes, it's even worse than that.
A critter may act just "as if" it
accepted or believed a proposition
that happens in reality to be false,
without ever having acknowledged it in
any conscious, critical, deliberate,
explicit, reflective, or wary way,
as a critter that writes might be
forced to examine it as it came
to write it down, for instance.

But this is not unique to logical statements.
The laws of creation allow whole generations
and entire species of critters to go extinct
if they have too many quasi-beliefs that go
without the critter's critique ...

... and eventual correction.

Jon Awbrey


Posted by: Philip Ronald Dutton

Once a "critter" realizes the capacity it has to indulge in quasi-beliefs, it should then realize that it can no longer employ it's previous ability to trust in the power of logic to deem something true or false. For then what is true and what is false in light of the fact that the system allowed establishment of quasi-beliefs? The system (or universe) is still in a valid configuration upon establishment of a quasi-belief.

It seems that no matter what someone proves to me to be true, I can always reassign it to be false, regardless. I can actually apply a mechanistic equality statement which, in and of itself, does not care about what I am assigning to each other.

So how does one relate quasi-belief "stuff" back to the business of "truth and falsity"?


Posted by: Philip Ronald Dutton

Please allow me to restate my thought problem, as, after all, it appears to be deeply connected to the viability of NKS foundations of _________ (fill in your blank... I will stick to 'mathematics').

Simply put, how can the business of Logic (or another system), which, apparantly, has some capacity for the business of telling truth or falsity, retain its merit when in fact it also has the capacity (via inheritance from whatever you call that "system" which rests above it on the heirarchy) to couple to Quasi belief operations?

As soon as you "make" a foundation based on simple programs or algorithms... then you loose the ability to value your truths and your falsities. Why? Because the "foundation" is higher on the hierarchy (than the logics) and, therefore, it is churning out, not only the "true(s)" and the "false(es)", but also the quasi-beliefs! We can waste away the quasi beliefs rather simply upon inspection, BUT we can not get rid of the algorithm that let us ASSUME that particular quasi-belief.

It appears to be more of a philosophical question than a mathematical question. No doubt the logics will continue to do their thing, but the algorithmic foundations that we are carving out will in turn carve away our ability to value the truths which it is actually producing (or which we think it is producing).

(will try to reexpress my thinking in clearer language later- if possible)


Posted by: Jon Awbrey

Philip,

I'm using the term "quasi-belief", literally, "as if belief", for
a type of situation where Agent_1 observes some action of Agent_2
and then describes the action in these terms: "Agent_2 is acting
just as if it believed that Statement_3 were true". For instance,
I might watch a moth flitting about a flame and say that the moth
acts precisely as if it believed that the flame were good for it.
The prefix "quasi" comes in because I have no idea what the moth
believes or even whether it's the sort of agent that can be said
to entertain a proposition at all.

If Agent_1 = Agent_2, we describe this as a question of reflection.

Some questions that we need to ask up front are these:

  1. What does it mean to believe or even to consider a proposition?

  2. What sort of system can be said to be take positions
    with regard to propositions, that is, to be in states
    of doubt or belief with regard to logical propositions?

  3. How complex does a system have to be before it has the
    capacity to reside in or pass through states of question
    about an issue, or to contemplate various proposals with
    regard to answering a question?

I think that the question of what it means for an agent to act as if
a proposition were true is one that we have to face whether or not
the agent is algorithmic and no matter whether the proposition is
a logical law or a contingent statement.

Jon Awbrey




Forum Sponsored by Wolfram Research

© 2004-2008 Wolfram Research, Inc. | Powered by vBulletin 2.3.0 © 2000-2002 Jelsoft Enterprises, Ltd. | Disclaimer
vB Easy Archive Final - Created by Xenon and modified/released by SkuZZy from the Job Openings