Axioms n beyond

OK so axioms and logic do seem to have some value in this. The consciousness conundrum can be formulated very simply ... this is informal I'll spend some time to make it precise later...

If we postulate that there are things, and then use those things to explain how we came to postulate that there are things then we have a theory in terms of the thing we were trying to explain. It would be impossible to have a theory not in terms of things so we can't replace that original postulate.

More generally a theory must be in terms of what it is not, otherwise we are taking it as an starting assumption... an axiom. We can't prove an axiom we can only show that it is consistent with other axioms.

The question then of how we arrive at axioms is beyond axioms then. It must be accepted by reason that we can generate propositions as a fact. That cannot be the work of the brain!

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Set theory again. On a table let us take some red beads and put them in a box and some yellow beads and put them in a box we have 2 boxes which might be described as the red box and the yellow box. The red box is not actually red, and the yellow box is not actually yellow. Our table universe (U1) has only 2 entities, namely the two boxes.

Now if we were to think about the box of boxes; in the analogy, we would create a big box for the two boxes and then we would have a new universe (U2) with 1 box (namely the box of boxes of universe U1). What we obviously can't do in the physical world is to put the big box in itself.

Now in set theory it is assumed that all subsets of a set already exist, so set theory says that in universe U1 the set of boxes {red box, yellow box} exists automatically which is the same as U1. However it is not an actual box.

When we actually build that box we get a new universe with 1 member. Only once that set is constructed does is actually exist, but to make it a member of its own construction is a problem.

If it didn't exist then it would bring its own constrcution down. If it does exist then it makes itself exist. The existence of the set depends upon itself. e.g. if T = {T} then T needs to exist to create T (so it doesn't really exist). If T does not exist then T = {} which exists.

Now we can create sets simply by chosing our ontology, and we can change subject to make sets disappear. Now with T = {T} what ontology do we have? or what universe is that a member of? To create T we must already have T.

The set of all Universes which have "blue" as a member. Which universe is T in?

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Having shown to myself today (almost like Descartes) that there is a "just is" feature of experience that at the root our experiences just are, I'm now ledt thinking how far does "just is" go. Obviously the arrangement of things in my room can be changed, it is not "just is". So what rule divides that which "just is" and that which can be investigated, understood and changed?

A full outline of the suggestion. Minds, Things and Universes

The argument is a form of John Searles famous Chinese Room argument. Here he argues that humans following rule books blindly (like machines do) do not understand the rules so how can computers be said to?

The argument proposed here is: the output of human minds, (i.e. in the form of symbols (like this email), or in the form of computing machines, or even paintings or music) cannot be understood in isolation from the mind.

Consider: if something could be understood "as independent from the mind", then you would not need a mind to understand it. Conversely if we need a mind to understand things, then things cannot be considered independent from the mind.

This has wide implications as has already been discovered and I expand below.

In advance: by "mind" I am not referring to a "thing", it is simply undefined so far. The form of the argument is natural language. The issue is one of processes.

What I was trying to show previously was the processes of the mathematician processing symbols. The symbols cannot process themselves so I was hoping to find evidence of the "mathematician" in the symbols. In particular the evidence that systems are incomplete and that this implies work outside the system. That has been done by Godel.

I'll reconstruct a similar argument.

If we define a set A = {x : x <= 3, x in N} = {1,2,3}

Now it is true that A is a subset of itself. But that doesn't mean that the definition of A can "write" itself. Clearly the mathematician has done work to understand the set constructor notation and ensure the equality. For example I've had to learn how to do this.

So let us define B = {x: x is in set constructor notation} this is like a set of all sets that can be constructed.

Unlike the precise x<=3 above, there is no formal set constructor notation that I know to define set constructor notation, I'm relying upon the reader to understand that informal set definition.

What Godel would do here in essence is to give each set-constructor a number so that he could refer to it within itself. That way you can construct the set constructor {x : x is this constructor, x is not in constructor notation}.

That is a member of the set which is a member by virtue of saying it is not a member. Thus the member can be seen to be a member (because of its form), but you try and prove within the system you get a contradiction. This shows for my purposes (and has been identified elsewhere) that "truth" cannot always be proven within the system and relies on something else.

This is the liar paradox which is expressed well like this:

The Liar Paradox. "Truth" for English sentences is not definable in English.
Proof. Suppose it is. Then so is its complement "False". Let s be the sentence "This sentence is false" . Since the phrase "This sentence" refers to s, we have s iff "This sentence is false" iff "s is false" iff not s. A contradiction.

There are many oddities all very similar: consider Richard's Paradox which I quite like.

It all points to systems not being self-sufficient and I am taking that to point to the fact that if they were then why are minds needed to work on them?

Turing machines again find limitations in the halting problem (same link as Liar Paradox is good). I am trying to write a Turing machine on a Turing machine, its proving hard - any one know if it is possible? Obviously Turing machines carry the name Turing because of the work "He" did not the machines themselves.

Another paradox comes about like this:

Take a simple set system, we must define 2 things.
The Universal set (U), e.g. the colours = {red, white, blue, green}
A subset of U obeying some rule, A = {x: x is in the Union Jack, x in U}
So A = {red, white, blue}

The Universal is a special type of set which allows the complement: A' = U - A = {green}

Now the Universal set is "the universe of discourse", or "all the things we are talking about", or the "context". A new Universal set arises whenever we change subject, clearly that is the choice and work of the operator (mind).

Let me change subject to U1 = {x: x is four letter words} and define the set B = {x: x words beginning with "b", x in U1}

So we can see that "blue" is now a member of B. What has become of "green" in this universe? it doesn't exist. This is well documented in some philosophy esp. Buddhism that the mind creates existence, or that things have no existence outside the "field of discource".

Now if we can refer to the elements under discourse U, can we also refer to the "elements not-under discourse" for example {green} in this example. Well if we did then it would be under discourse and therefore part of U which contradicts U's definition, and if we don't then what is the meaning of "changing subject". Clearly the person using the set theory must be free from the field of discourse in order to define the U from the many elements which are not in U.

That is the more general paradox that we know that there are things outside our field of "view" or discussion, but if we try to refer to them they come under our field of view or discussion. So it is impossible to refer to things outside the field of view, or discussion, even while knowing they are there. How is that possible? i.e. there are things in the room next door that I can't see, but I know they are there... so what do they look like? we sort of imagine ghostly things that are both what they would look like if they were here, but which aren't here. Well I suspect in general that is called God to some, and I call it mind. Its the unbounded limits beyond the discourse which can't be in the discourse, but we know are there anyway.

The implications:
Materialist theories posit existence as a fundamental starting point. As seen above, existence is a product of mind or at least the field of discourse. Thus any theory trying to explain mind in terms of existences is flawed. I just had a "discussion" with a American neuro-scientist on the web and the unquestionable faith he put in existence was extraordinary. On one hand he says that the brain creates our world-view, on the other hand he says that world creates our brain. Well I pointed out if the world exists already what is the brain doing? He says it is creating a "representation" of the world. I said then what you call "the world" is only a "representation" since you are your brain, a representation of what? He didn't understand. Daniel Dennett is in the same camp, as is Dawkins and their hard materialist ilk.

AI etc runs into the same problems. Clearly we could make a human, women make humans every day. Its that this would not give us insight into mind. Mind is not brain because brain is only a field of discourse of mind.

Physics. There are a lot of quantum theories to explain mind and world. Again minds must be used ultimately to operate these theories and so these theories only posit a field of discourse. If machines could be made to operate these theories (like Deep Thought of Hitch Hikers fame) they would fall foul of the same problems surely?

Self-organising systems

Self-organising systems are dynamic systems which show stable emergent properties. i questioned already whether the emergent properties are really part of the under-lying dynamics - that is a question from Buddha!

In humans there are self-organising systems (basically the series that form societies structures, and also a large part of the processes that underlie all group formation). People may believe that they can create political parties and societies but unless the dynamics are already there it won't happen. Plus it is maybe rather egotistical that one person believes that by their own power they can influence so many people!

But as seem with the Nazis such self organising groups can put the individuals who are part of the self organisation in bizarre circumstances and make them behave in ways which alone they might disagree with.

It is precisely because social self-organisation can be so dangerous that it is something we should shy from, and remain personally conscious and responsible for everything we do. That is a repeat of the arguments against society and for Anarchy before.

Some problems from Physics

I don't have a formal understanding of this yet but as much as I understand: firstly Godel's incompleteness theorum states that in systems that correspond with maths it can be shown that some theorums cannot be proven from axioms. But we may be able to show they are true by computer process. Turing non-computability theorum indicates that some problems cannot be computed.

Disregarding those problems, the issue of where did all the data (information) in the universe come from? Now we might be able to determine equalities and equations, and we might be able to write comptable processes but upon what data can we work?

The possibility from the Quine investigation is that we could work on those theories themselves. If a theory could be created which produced which working on itself created itself this is a good step forward.

Constructing a Quine

I'm a bit annoyed with myself that I gave up too quick and had a glimpse of a quine which gave me the way forward. But I can construct the argument, and the sticking point, since I only gleened a small part of the problem.

A quine is a program which outputs (prints) itself. I'll use QBasic which is free and simple. It uses " for quotations and can't print " direct u need to use CHR$(34). I'm going to use ' for simplicity, so to use the program this will need be corrected.

We know then that it must include an output function that is 'PRINT' which sends whatever follows to the output screen. So our starting program is:

PRINT

it does nothing. So we know we need to print itself so we get

PRINT 'PRINT'

which produces an output of: PRINT. But putting new PRINTs in the code while it does produce a new PRINT in the output, also puts a new PRINT in the code so we never solve the problem.

What we need to do is add a PRINT to the output without typing it directly in the program. The easiest way to do this is to put the PRINT in memory (like a second screen) and then copy that twice to the output. That way you have 2 prints in the program (like the example above) and two in the output. Thus the problem has been solved and the backbone of the quine is done.

convert PRINT 'PRINT' to

a$ = 'PRINT': PRINT a$ a$

This was the sticking point for me last night, I was stuck on the approach of thinking that the a$ must include the program so I was starting like this:

a$ = 'a$ = ': PRINT a$ a$ -- This creates another problem of the type above. You need to mix data/code between the two parts of the equation ... it will be explained.

So given the basic structure above all we need to do is fill in the syntactic details. Firstly let us get the PRINT statement right so that it outputs as close as we possible to the above. In other words lets take the program elements of the first part and feed them into the data element of the PRINT statement in th second part.

: PRINT 'a$ = '' + a$ + '':' + a$

This outputs: a$ = 'PRINT': PRINT

So now take the program code of the second part (after the :) and feed that into the a$ data of the first part.

a$ = "PRINT 'a$ = '' + a$ + '':' + a$' :

Stick those together and you got a quine

a$ = "PRINT 'a$ = '' + a$ + '':' + a$' : PRINT 'a$ = '' + a$ + '':' + a$

However this does not work because on each line the data is enclosed by " marks, while the representation of " as data is '. If they were the same the program does not know whether to treat them as string delimiters or just parts of a string. If you then start filling out CHR$(34) then the code just keeps expanding endlessly.

This method does not work.

Trying to code delimiters and then enclode them within other delimiters is bound to fail because you will always have that problem we started with of a highest level delimiter which cannot be encoded.

A brute force method to solve this is to not use delimiters at all. The CHR$(34) method suggested above is the only way in QBasic to PRINT a " symbol since the " symbol is recognised by the language so we PRINT CHR$(34). We will have to use this method anyway somewhere.

So convert the whole string into ASC codes. There is a standard method in the language using READ to assign a variable with data from a DATA list.

Simply write the READ, DATA block with a line to print the CHR$ of each number, and another line to collect the list of data. Then once the data has all been read it prints the collection of data.

So the quine program reads its own data. Turns this into character output and then prints the data that it constructed itself from.

What the program could do is find its own place in memory and read itself into another memory location. Not that amazing really.

In the Quine though I've done that work already creating the data list. Using a recursive algorithm to write the program block, then encode it, then see how big it is and adjust the program and the code parameters until they match.

To note then: obviously the program gets written first. Without the engine nothing happens. This is a general solution to the problem of outputting data. Then just get it to print the data again to produce the data block at the end of the program. Then put in the data specific to itself and voila.

Its not as neat or intriguing as the first approach and I write endlessly here trying to find a way into the problem of "encoding itself". This second solution simply avoids the problem completely ... but how?

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blurb from the first draft of this entry...

Now I'm interested in this process analogously because what I am having difficulty with, and the basis of my proof if such exists, is the implication of dualisms. For example where someone presents a limit to knowledge then this limit must be different from the knowledge itself. This is analogous to the data / program distinction. Other examples are subject / object. However if arguments of the Quine type can be constructed we have a way of expressing the "back breaking" job that dualisms create and find a way of overcoming the division.

Maybe then this investigation does not lead to a disproof of limits, but rather an personal understanding of how to construct limits within themselves, so that we can have something which full represents itself.

It is interesting that to do this we need to construct 2 copies of the partial structure, and that partial structure "represents" that dual creation. This if we were a "quine" only processing data we would know that our theorum on which we worked had two copies.

Q: how could a quine ever know that it was a quine? The process of comparing program and output is done by the user?

Q2: Next task a binary quine.

Q3: Is there a self-organising automaton version of the quine using recursive processes like above? Thesis/Anti-thesis -> Synthesis.

random jot about consciousness

remember that a flat worm trained to do a maze and fed to another flat worm somehow passes on the knowledge. Examine that because information must be encodeable in flatworm matter!!

Don't forget music and consciousness. Frequences seem important, n music has been taken as a higher state of reality than text by quite a few!

Notes as I read...

In other words true autopoesis is impossible.

Ok an autopoetic system supports itself, like butter is needed to make more butter. But you get chicken and egg because the first system must have been formed by other means. So it can't really "create" itself, only "support" itself

Alternatively its simply emergent property (almost epi-phenomalism), whereby an underlying dynamic system produces a stable property. But such a property is not involved in the system! Its a higher order.

An interesting possibility then is a system which creates an emergent property which is involved in the autopoesis. i suspect category mistake, but very interesting if this exists. i think immediately of group fitness versus individual fitness in behavioural ecology... however while these 2 levels can be studied i don't see yet how they interact.. if at all. Interesting way forward...

V. INTERSETING actually because this would test absolutely whether structure or "scale" was a product of perception or a reality. The world can be described in the level of atoms. It can also be described on the level of humans. But can it be described on both levels at the same time! This collection of atoms "died" thus leading to rapid dissipation. Does that make sense?

Just to state at this time, I've failed to see any "substance" in ideas of emergent properties. It seems to me to be simply a feature of "classification". Shall determine that belief for sure in this on going...

Just adding... regarding the old question about how remarkable it is that the world makes sense to humans... i wonder what the logical implications of it not making sense would be? What does this sentence really mean? Are we imagining a world that does not make sense? How about a world that makes more sense and in what sense. Obviously predicting the future is the test, and science does well. So in a world where we could not predict the future, or a world where the future was always known what are the logical implications? Well if the future was always known then it wouldn't be th future, and there would be no time (i suspect) and in a world where we simply could not predict the future and every moment was a complete suprise ther would be no memory or past (cos everything would be new). So time itself entails being slap bang in between the two!!! So why maths? That's the other big question... no thoughts yet...

self-organising -> basically decreasing entropy at one level, increasing at another.


Quines. Linked to compression. but first...

Compression a simple proof. If every binary sequence encodes a number, then there are as many binary sequences as N (natural numbers).

Now in compression we are saying that a number can be mapped to a smaller number and back again. In other words it is a 1 to 1 mapping within N. (btw The set of such mappings is the same size as N.) So given a number and a compression algorithm we need a bit of information to know whether the number is meant to be decompressed or not. ok that is not the proof just following a tangent, the proof was that each number by itself might or might-not be useful so compression is not an elimination of useless numbers, but rather a mapping and so it requires an algorithm to compress/decompress and that algorithm effectively generalises the information lost by compression. To develop an algorithm we extract the general information from the code, so that the code is smaller and the algorithm bigger. (We don't usually include the algorithm in the size of the result, but we should to be honest ... something need to just test with zip).

anyway the point and the relationship with Quines is that the code while producing itself is a cheat, because the binary exection code is not reproduced. A machine language Quine would be cute!! i.e. one that truely reproduced itself. And that is in Turing territory cos I need to write a Turning machine which first of all runs a set of Turing tables, and then writes itself. That would be exceptionally cute!!!

Same logic as the Turing machine on a Turing machine, consider a self interpreter... i.e. a compiler compiled by itself. Chicken and egg.

probably useless but tuppers self ref formula : http://en.wikipedia.org/wiki/Tupper%27s_self-referential_formula

note: an argument I used against Dawkins Memes thus: has a weakness. He says that ideas evolve like genes (I had the idea of viruses). The problem tho is that Memes is an idea so if follows that it has evolved like a gene. Now the fitness of the Meme meme is not what Dawkins is meaning by its truth. I can't see the idea catching on, while it might be true. So we return to truth as a measure of fitness rather than population. Its not a strong argument against it, but it shows up the fact it can't be taken to totality, cos if the Meme meme was floated and then went extinct it loses any notability. This argument needs to be strengthened cos I suspect its a form of a general argument that could be applied to all formualtions including itself!

End. brain ache! off to eat.