Old Skool SRH is back!

Rethinking the SRH can now be reformulated with a better understanding.

SRH Definition

A system cannot refer to itself without using aspects of a meta-language that cannot be exressed within the system.

This is why the SRH makes the claim that self-reference is impossible, because actually within a system it is impossible. It is only by using aspects that the observer (outside the system) can see that we can interpret anything as self-reference. Thus to self-reference we by definition are not entirely ourself - that is what the SRH saw but seeks to prove.

Godel numbering maps the formulii of Principia Mathematica to the Natural Numbers. These are coded within PM (as I understand it) as repeated applications of the successor function on the symbol 0. So the number 5 is represented by sentence "0SSSSS".

Now I failed a few posts ago to find a contradiction with the conception of such a formula within PM. But then wondered what was gained by mapping expresions in PM onto the subset of expressions {"0","0S","oSS","oSSS", and so on} also in PM. It means that there are two names for any expression: itself and "0SS..S" the Godel number of times. Is this all that was required?...

Think some more...

The problem maybe is that while we know which mapping is being used, can PM?

Quines again...

A quine is defined as a fixed point of a particular execution environment.

Let E(x) be an execution environment - say C language. It uses the french system of « » for quotes. We give it a program:

E(«printf("%s","hello");»)

The function produces an output of:

«hello»

1.0 E(«printf("%s","hello");») -> «hello»

if Q is the string of a quine then:

2.0 E(«Q») -> «Q»

Now suppose we had a 2 variable function in C code:

3.0 E(«

float multiply(float a, float b) {
return a * b;
}

»)

Our execution environment has code but no data. We can add data by using the function say with values 2.1 and 2

4.0 E(«

float multiply(float a, float b) {
return a * b;
}

printf("%f.0",multiply(2.1,2));

»)

Our execution environment now outputs 4.2. We could use this function to create the 17 times table from 1 to 20:

5.0 E(«

float multiply(float a, float b) {
return a * b;
}

for(float i=0;i++<20;)
printf("%f.0\n",multiply(i+1,17));

»)

The point is that the Execution environment takes only 1 string of data input (the program) and all the data required is hard coded into that string. But the data that E takes in each of the examples above is hardcoded into the next higher level of operation within the computer. So that once loaded the computer takes no input. E5 for example is the compiled/interpreted program which takes no input and outputs the 17 times table from 1 to 20.

Any function taking data is just one level of thinking and is the same as the higher level function that does the same thing (at the same time) without taking data.

So returning to E. The fixed point of E is the quine Q. E(Q) -> Q. So while we are thinking in terms of E running the program Q, there is the higher level (the computer itself) F which has the program hardcoded into its memory.

Computer F Program -> Execution environment E

Now with this thinking what happens with a program that takes data from the environment? The function F lying outside the data includes the human operator! If the human operator is just data inputting (and they do this accurately) then the function F is fairly easy to picture. But if they are typing like me here then the data comes from all over the universe! This simple blog application that is passed data from the keyboard is part of the function that is the universe! This F is huge! It is a much more sophisticated program than the ones above which are hardcoded and easily closed up in a function taking no input. The question of when those functions are executed is interesting - that is data real meta to the system!

Now this is why I think A.I. has mostly missed the point. The intelligence is mechanical yes, the ability to move around a room is a machine task yes. But the holy grail of A.I. the creating of a person doesn't need to be done! You simply need to link a system of sufficient capability into the universe and it becomes "a person". The "person" does not reside in the processing or the brain or the machine at all but in its ability to interact with the universe. This blog program is not classically intelligent, but what it is taking down (the data) I hope has some elements of intelligence to it - it is part of an intelligent system. And it is me being part of a much larger intelligent system which makes me a human. Shut me in a sensory deprivation environment my whole life and see how intelligent my brain is! I am not in the wiring or programming!

Returning to the Quines: Now a real quine is not E(Q) → Q but E(E) → E !!!

That is an execution environment like the C compiler that takes the C compiler and produces the C compiler.

I had a bit of a think about this. A through program that simply printed its input would work. Feed it itself and it would print itself. This is like y=x. This E needs to be a proper execution environment that doesn't quine every input.

Shop so you can Work so you can Shop...

http://www.moneymarketing.co.uk/regulation/bank-deputy-governor-tells-savers-to-spend/1019205.article

Charlie Bean, BoE deputy governor, has called us all to spend more. Why? Because this will create jobs. Why? So we can work more. Why? So we can spend more? Why? So the economy can grow. Why? So investors have something to invest in. Why? So they can get an income without working. Why? Because people don't like to work. So taken as a whole it is rather crazy to have a system which reduces the amount of work, by increasing the amount of work. Of course we know the reason this insanity exists its because only "a class" of people are allowed not to work, and they do this by forcing work onto another class of people. Endlessly fascinating how mad this species is.

But I suppose it is consistent. We like private property and we hate public spending and taxes. So now the Conservatives have put the money in our pockets by cutting public spending it is only natural that we have to do the spending that previously the government was doing. And what we the sheep can't see is that 60 million people buying flowers for their garden at 3.99 a flower is going to create a swathe of Britain covered in identical cheap plants - while a central park committee with a budget of just £240 million would be able to provide a very much better swathe of Britain for everyone to enjoy - larger, cheaper and better. Apply that to everything and we see how jobs are created by private ownership and fragmenting society into small inefficient family units who then have to work harder.

And no-one is going to listen to Charlie Bean because the Right wing in the West have been busy turning everyone into small time owners and capitalists. Now we have a right to invest our savings so that we can earn back the dividends that many people think righly belong to the people who did the work in the first place. The tabloids (owned by capitalists and preaching the message of capitalism) fail to note that while the tax payer takes money from us to spend on us, the dividend paymenst are taken from us to spend on investors for their own amusement. This never gets criticised for some reason - wonder why ;-)

The standard retort is that without dividends what incentive is there to invest? Well a friend recently sent me a BSc thesis to read on Capital structure and it was pointed out that creating too much debt in a company will frighten investors off so the share price will drop and this will hamper the ability of the company to borrow more. So actually the banks are the real supply of money anyway and investors hold the company to ransom threatening to sell in times of trouble. Quite what the investing public are supposed to see that a bank seeking to find investments itself can't see is a mystery to me. The whole stock market still seems extraneous and essentially the betting shop for upper class people. It doesn't work either as the crashes show. And while talking of investment banks that can lend upto 14 times more than they have (even after the new Basel III rules) don't really have any right to charge interest on that money because they don't own it. But given that such free money is so available why do companies need investors at all?

And it is even madder than this because the banking system even with all this free liquidity has actually been lent by the common man the eventual estimated sum of £80,000 for each worker in the UK. We are lending the banking system almost as much money as most of us will ever be lent. I hope people get the cheque for the 5% interest they are due! If we had all this money in the first place why do we ever need to go to a bank? Next time I want a house I'll just go to the treasury and ask for the money that they so quickly found for the banking sector...

Except of course it doesn't work like this because we are "us" and they are "them". It is the same old structure as has always been of ruling elite and the common people who support them. Why anyone ever thought things would ever change is a mystery to me. This is what humans are like: face it. It is the same now as times of war, the ruling elite get threatened and they ask us to bail them out. I've never seen why we should? They take the position as ruling elite and so having built the system up they create their own problem in keeping it propped up. It is not our problem at all. If it meant that much to them I suggest they took it all a bit more seriously when they had to the chance. Mr Bean I suggest would do better to use his authority not to prop up a system that doesn't work, but rather to create an economic system that actually did worked. But that wouldn't be playing the game and he wouldn't be in the ruling elite for long. If he ever needs a starter in this project I suggest he builds the system around the "Bernanke Helicopter". The problem is the distribution of wealth not (while we have oil and machine labour) the supply of labour. But the elite would never accept this because they wish to keep the right not to work to themselves; so be it, my view is sink with the ship if you won't jump over board.

A stat I wanted to find a while ago: In UK we use 165 Gj/person/year. Calculated from http://en.wikipedia.org/wiki/List_of_countries_by_energy_consumption_per_capita

A high sustainable output rate while cycling is about 200W (200j/sec). Using this rough value we use the equivalent energy each per year of 26 slaves cycling at a good rate non stop all year.

Also http://en.wikipedia.org/wiki/Orders_of_magnitude_(power)

Actually 200W is the useful output; 500W is the total energy consumed so better calculation is 10 slaves working 24hours. So 30 slaves working physical labour for 8 shifts/day.

The brain is estimated to use between 20-40Watt. Say 40Watt for hard thinking jobs. So an intellectual army of 400 slaves doing thinking for us!

The only problem is that slaves don't eat oil... but imagine if they did!

Oh and of course there is me contributing my labour to this slave work force!

That is how much work gets done for us by oil. That is why it is a nonsense to say that we really have to work at all. Increasingly jobs are created by the ruling elite chosing an economic model that fragments the efforts of the masses and forces them into petty private ventures. As long as we think in terms of just 1 person and what 1 person can do, we create inefficiencies that create jobs.

This is a different analysis from the paranoid people who think that the elite do this to control us. I take Thorstein Veblen's argument that what the ruling elite fear most is not revolution but the sheer productivity of the masses. Somehow this much be controlled and directed toward the rich to ensure that social hierarchy is maintained because as argued in this blog the human species needs social hierarchy (for some reason) just as rabbits, mice and primates do.

It is not an orchestrated conspiracry. There are people pulling on all sides of the rope. For now however we know that Charlie Bean has sold out to the city to support the economic status quo. Ed Miliband on the other hand has not and it is a relief to see him speak up for the other team for once - been a long time!

My mum & accounting + SRH

I'm beginning to give up on criticising Capitalism it is so ostensibly nonsense that it's not even worth it... but another silly case just popped up...

My mother has started to charge me £20 rent each time I pop home, to cover costs mostly.

Had the wicked thought just now that I do work for her in the garden. Just spent 4 hours this w/e and about 5 hours last weekend. My cousin has a gardener who charges $15/hr and I could quite reasonably charge £8 so mum owes me £70 against her bill of £40. So really I should be asking her for money if we are going to start accounting...

But it gets silly because she can charge me for my share of the washing - she washes the house clothes in one go at w/e's and I often throw some things in the load. It's just practical rather than running the washing machine 3 times for each of us in the house.

But then she can play the trump card and charge me for the 18 years she and dad brought me up. Think that runs to about £180,000. Plus she could charge me for the lost income she could have got if she had worked instead of bring me up so that is more like £400,000.

But my argument against this has always been that she owes her parents the same so we are quits. Actually this isn't true because parents spend more on their kids each generation. The very ancient ancestors of the human genetic line didn't even have parental care - things like fish and reptiles.

Anyway even if my argument fails it would mean my parents would have the right to charge me interest on that £400,000 which would ensure that I never got out of debt. Even at 5% per year that sum would be 560k by the time I was 18 which is a further 28k per year. Then I would have to have children and pass onto them not only my parents debt but my own charge as well. But when my parents die I would inherit the debt, which if they lived another 35 years after I was 18 and left home would mean I would owe myself £3million. It is a strange thing property that when we owe ourselves money actually we suddenly don't owe anything. So it would never work.

So it all falls apart and no-one ever gets paid (sound familiar, viz recent credit crunch ;-).

So accountancy can't ever totally invade human interactions, it is impossible. Some kind of formal proof available here and like all things the formal system of money and exchange only works because of the informal system of "giving" that props it up and from which it steals (using the word from the formal accounting system, since in a system of giving there is no stealing).

Accountancy was the origin of counting and maths. Only through people trying to make equal trades were the scales created and the concept of equals and congruency created and then addition and subtraction properly formulated then debt and negative numbers and then fractions, ratios and percentages and only then did its branch into maths and real numbers and complex numbers. And, now with computing we are back to counting and natural positive numbers and I reckon all the contradictions and problems that arise in computers exist equally in accountancy and trading. The "real" world, as the SRH assures, is always outside any particular paradigm we may hold and of which it mocks loudly for those who will hear!

I've a growing boredom with logical systems and sensible things. It seems all the rules we humans manufacture and fight to conform to are all easily subvertible. Even the powerful formal logical systems, designed to be water tight, turn out to easily punctured; all that power being used by Godel et al. like skilled martial arts fighters against their opponent. It is the SRH belief that this is always true. Humans, like Icarus, will always get burned when they try to approach the sun.

In the Enlightenment period science was discovering the absolute truth that God had laid down. But as we looked we found that it was not so clear after all. Revolutions shook Aristrocratic society, Heisenburg demolished the idea of every knowing the exact place and speed of a particle, Godel turned over Russel's search of formal mechanical proof. Nietzsche won over where Kant and others had sort to lay certain foundations. God indeed seems dead. Yet by irony it is his death and the loss of certainty which is the SRH. God would never allow himself to be held onto as a fixed image, as a mere creator of Order. What a trivial being that is who eschews Chaos. God is Transcendent what does he care of order or chaos? This is patent the thought of man, and worse man who is fearful of losing his possessions and himself - that is the worst kind of godless man! So the Godless, by irony, have had it proved that god is dead and are thrown into uncertainty - where actually certainty never lay cos why should God be synonymous with the certain? Was the Certain ever the God of God? No! God doesn't need to be certainty at all.

I am beginning to make sense of the Jewish conception of Yahweh. The Being beyond human understanding. The fearsome being beyond reason, rationality and certainty. I say being in the loosest sense: who ever would say that they "knew" God. And I only say fearsome because we see in God our own fears. If we are afraid of losing certainty then God will be exactly this

and when we want God to be certain we will be shown uncertainty by the Being beyond all comparison and accountancy. It is our limitations that we will always find reflected back on us until we finally have the bravery to step beyond ourselves into the meta spaces outside our own trivial world beliefs. This is the teaching of All religions that I know of: each one implores us to reach beyond ourselves in Love, Kindness, Generosity, Compassion, Bravery and Selflessness. Yet each one of us does the opposite seeking our own place in Heaven and seeking comfort by the fire of Gods generosity.

"Eat, pray, Love yourself" the new film by Julia Roberts is excellent because it shows us exactly how not to live. Not seen it, this opinion is based upon the Review Show on BBC TV. It completely negates what God teaches us. I suppose for the religious it is the closest thing to the devil's teaching yet. An interesting point was made that we all pursue "self realisation" while actually the religions teach us "self abnegation". I am certainly guilty of not seeing this contradiction. Enlightenment is often viewed as self-realisation. It is not. But at the same time it is realisation of our True Self, that is realisation of God. Yet God is always outside ourselves. The religious path is finding the way to step forever outside ourself. A difficult undertaking since we always create a new self when we grow and so a new Diagonalisation or Godel statement needs to be made. What it is not however is aiming for a self that is truer or better than all the others! True-Self is an oxymoron - the Godel statement of Godel statements only true when you are outside the system and never derivable from within the system. Of course we are already outside the system it is just that like Godel (maybe I'm, not sure) we can't see that yet.

Godel numbering is the SRH.

Godel famous statement G to recap says:

Given formula:

F: ~(∃x)(∃y)(Pxy · Qzy) with Godel number f and free variable z

If we apply the formula to itself by binding f to z then we get the famous Godel statement

G: ~(∃x)(∃y)(Pxy · Qfy) see

If we say that the godel number provides a name for the sentences so sentence X has a godel number x whose physical numerals can be used as a name like "x", So it states that there is no string "x" which proves the statement "y" when y is the godel number of this formula (that is what the Qfy is a way of determining) i.e. "y" is a shorter name of "~(∃x)(∃y)(Pxy · Qfy)". If this were ever derived within PM, the system would have found a string "x" which proved "y". This proof of a contradiction would invalidate PM and allow for any string to be proved (this being the danger of contradiction). On the other hand if it can't be proved then it is true. But this extraordinary result shows there are things which are true which lie outside the transformation, derivation or computation rules of PM. A Godel statement can always be made for any system which can provide a Q function so it's quite powerful.

I've noted before, and note here again, that Truth of a contingent truth is different from proof by contradiction. To say that "birds cannot fly" is false because I can find a flying bird. But to say that "1 + 1 = 3" is false because "3=1+1+1" so "1+1 = 1+1+1" is different. "1+1=2" is then not true because I can find an instance of it, but because to accept anything else would upset our definitions and accepted system. Counting (and therefore maths) is based upon thousands of years of trading and accountancy culture - this is what is at stake. Inconsistent truth and contingent truth are different and I think this is the bit about logic that leaves a bad taste in the mouth. We can argue that "this person must have done the murder because there is no other way" but this is different from actually seeing them, or being murdered by them. There is always that possibility that out system is at fault. A friend said something at work which proves to be one of the wisest things I ever heard "Assumption is the mother of all cock-ups." When we get a contradiction the system doesn't decide it is true or false but rather we change our assumptions. Contradiction breaks systems and is undecidable for semantic rules. False however is defined by semantic systems. Contradictions aren't false, so Tautologies therefore aren't True. That is the point. Godel's statement isn't True then! Anyway the point of this blog is elsewhere... Before that: Semantic rules are simply rules that split the sentences of a logic into two piles (True and False usually). Tarski proved that these rules can't be part of the system because it creates a contradiction. So the "use" of a system must lie outside the system. I'm saying that actually there are 4 piles. True, False, Null (for contradictions) and whatever I decided to call the opposite of Null when I discussed this before. Anyway that sort of leads into the point. IN PM we can represent any number as 0SSSSS..S where S is the successor function. I presume we can derive any number from repeat application of thus formula.

So if there was a formula $(some number) then PM could derive any formula it wanted both nonsense and Godel statement! So the proof of Godel statement in PM would simply be: "$(0SSSSS.....S)" as many times as needed to get the Godel statament. Let us call this proof of Godel's statement "$(0G*S)" for G times S or simply X for chaos as well as big x. If "$(0G*S)" is the string which proves G then we have found an x. So X proves a contradiction. But it was obvious that the function $() could prove a contradiction cos it can prove nonsense as well like the statement "y))))Qf)))))". So it must be that there must be no formula in PM that can do what $() does on pain of death!

3/10/10: I have read the Godel paper now and indeed the Godel formula is not expressible in PM. Neither would the $() function above either. Firstly this must be true as argued above because it would enable the proof of any string, and secondly because there is no way in PM to handle characters. PM formulae take numbers as values not characters like "(". Godel goes to a lot of trouble in the paper to produce the function Bew which is the function which is true if the encoded sentence it is fed has a proof. But it relies on the "user" to do the encoding. I thought modern computer systems might be more powerful than PM but of course a modern computer (like the one running this blog page) works in numbers also. What we as users take to be characters of an alphabet are really ASCII codes which are converted into arrays of pixels for the screen. At no point in a computer is there a "character". Even BASIC functions like Mid$() don't return a character they return a number that is handled to produce a character. Characters are extrinsic to logical systems, they are meta the system!! So how do characters arise at all? Do characters even exist? Godel numbering looks simple but hides something tricky...

17/5/2022 note that numbers are abstract. Computers use switches! Numbers are encoded into switches.

investigate the relationship between characters and logic systems (do we go Searle here?)

SO that proves what I suspected at the beginning ages ago that Godel numbering is a necessary meta feature of a logical system. To talk of Godel numbering is already to step necessarily outside the system, so capping it by finding a sentence in the system which Godel numbering can talk about but which the system can't is really only a specific issue of something much bigger. For me the very fact that stepping outside a system is possible proves the rest already. Godel and "us" sit outside PM because we can Godel number. Who cares whether PM know this, we are already outside by necessity. I can't believe that Godel didn't realise that he was already outside PM - at least that Dangerous Ideas program seems to suggest he spent a lot of time trying to get outside.

Boredom with logic and computing looming - SRH about the meta strike the SRH search!

Disappointed in Godel's argument for God :-( He is actually trying to argue that there exists an entity which... God created all things how can he be an entity which?

http://en.wikipedia.org/wiki/G%C3%B6del's_ontological_proof

Feeling today that there is a huge flaw in logic at its very inception - the same reason why many people can't get their heads around it (myself to some extent too)...

"This sentence can't be created by a sentient being."

Well it just has, so it can't really have any meaning. Yet by simple symbol crunching and knowledge of grammar I've just created it. Can't any system create a Godel statement simply by syntax manipulation? I seem to gone through the argument and come out the other side missing the whole point. Time to re-examine at some stage... it is interesting how climbing the mountain one often wanders off the side missing the path to the top. But I am of growing certainty and expectation of disappointment because what seemed so solid and profound is becoming just a puddle of water. In many ways this illustrates the SRH that whatever we find ourselves bounded within we very quickly find ourselves stepping over the boundary - this is why I thought I would be interested in philosophy ... but even here the boundaries come in too tight sometimes.

That said some respect for the collection of ideas put together here ... v. nice and in the spirit of the SRH so I like for that reason too :-)

http://www.authorstream.com/Presentation/FunnyGuy-38280-Limits-Computing-G-del-Turing-Chaitin-Edge-Naturalism-Things-Computers-Points-thelimitsofcomp-Education-ppt-powerpoint/

Diagonalisation & Systems & proof#1 of SRH

The general pricipal is simple:

Take any list:

1234

2345

1357

4567

and create a certain new entry to the list by making a new number composed of the first digit of item 1, the second digit of item 2, the third didgit of item 3 and so on. This gives us :

1357

Now change all the digits to anything other than what they are, say +1

2468

This must be a new number, not in the list, because it differs, by definition since we just changed them all, from item 1 in digit 1, differs from item 2 in digit 2, and so on.

Suppose we add that number to the list, how then do we get a new number?

1234

2345

1357

4567

2468

We can't use diagonalisation as above because each number only has 4 pieces of information in it to change and we need 5 - that is, there is no diagonal. I imagine this solution works +"0" to the start

01234

02345

01357

04567

02468

Then our diagonal is 02368 and changing all the digits say +4 Mod(10) gives us 46702 which is not in the list.

We could use the other diagonal of course. In fact there are n! (n factorial) diagonals to chose from. In the case of 4 that is 24 diagonals.

And taking any one of these numbers there are b^n where b is the number of digits to chose from (i.e. the base - 1). In the decimal cases above that is 9^4 = 6561 possible numbers.

Not sure how to prove the next bit, but fairly obviously going through all the diagonals and changing to all the possible digits (with a lot of duplication) this method could find any of the missing 9996 numbers in the list of 4 (taking 0000 as a number also).

In reality numbers of countably infinite so this method can just add a 0 each time and go on forever always discovering a new number of greater length for the list.

Even at infinity the method works because 0 + infinity is a new digit (unlike infinity + 0!). This is the conundrum of the fully booked infinite hotel which can never-the-less always take a new guest! Not however by handing out the key to the infinite + 1 room, but rather by getting everyone to move to the next door room (quite within the rules of infinity) and thus freeing room #1 for the new guest.

So anything that can be listed automatically has exception by definition of Cantor's extraordinarily simple and powerful argument!

Hofstadter in GEB makes the splendid point that Godel, Tarski and the host of other proofs are versions of this argument, proceeding by making a list and then finding the new term that is not in the list. For example in Godel once the list of all provable statements is made, proof is already doomed ;-)

So turning to my own obsession the +1 hypothesis. This was the real start of the SRH quest, and it revolves around the sense that a "system" must be incomplete, in that an observer can "know" more about the system than it can itself. The outside view always being larger and more substantial than the inside view. What Hoftsadter calls the Meta level. Tho i'd call it exo versus endo levels. The problem is how to characterise something so general as a system.

Any system will have a "state". That is to say a particular collection of data that determines the stage of processing. A static system like this sentence has just a simple value. But a looping program in a computer will step from state to state as it works through the calculation. If it were possible to group all the states of a system together in a long string (like Godel's proofs in Principia Mathematica) then we could represent any finite system at all as a countable number. Turing's never terminating Turing machines (NTTMs) just become real numbers, and the halting problem isn't a problem anymore.

The damage then is already done because we can always find exceptions to this list of all systems, altho oddly the exceptions to the NTTMs are very huge of size Aleph 2! Quite what non terminating systems which have infinite solution spaces are like I have no idea i.e. computers that work in continuous data ... except don't we know one... The brain? We assume it works by discrete (finite) neuron triggering, but we also know that neurons work by summing action potentials and this data is not discrete but continuous in nature. Maybe a brain can't be simulated digitally and only in analogue... does analogue computing exist? check it up. What is analogue logic? Analogic ;-) ... apparently the astrolabe counts as an analog computer taking us back to Herpatia and Alexandria. How odd if there were hidden depths to this ancient computing after all?

Anyway I have digressed and this doesn't shed any light on the +1 hypoithesis... try again another day... off to fix the patio... tho 1st update on thoughts on train last night (from written notes)...

The Komogorov complexity of a system would be the shortest algorithm that simulates the system.

Now an algorithm takes some data and some combination rules and recursively applies them until the answer is achieved... how does a system know it has got the answer? Obviously the test to end the alogithm is independent of the "nature" of the answer (that incidentially is a very good example of SRH!) cos we can't know it's the answer until we have the answer!

So in Principia Mathematica, for example, all the provable statements have k-complexity set by the relevant axioms and the shortest list of derivation rules.

But what then of the Godel statement for a particular set up of PM? It can't be derived from the axioms by the derivation rules. So there is no algorithm to compute it (within PM) so it has an undecidable k-complexity.

Doesn't the Godel statement prove that self-reference is actually impossible?

Just thinking... if PM has the ability to Godel encode and also to reverse Godel encode then can't it just reverse Godel encode the Godel number of the Godel statement? Which proves what I suspected in my naivete that actually PM can't do Godel encoding itself and Godel encoding is a meta event... in other words the so called "self"-reference relies upon a perspective that is already meta the system! By suprise doesn't this prove the original form of the SRH that true self-reference is impossible because if it were possible then any system could decode the godel number of the Godel statement and thus create a contradiction.

===

If we look at a derivation of a sentence in logic as analogous to the "explanation" of that sentence (i.e. taking what we know and showing how something we don't know can be derived from it) - which suggests that learning a new thing must come before it is explained!! (as I have learned from trying to tutor kids - they need to discover it first and then once they know it, it can be explained - you can't teach new things by explanation). So Godel stataments can't ever be explained but once found always become the basis of new explanations. They also mean that a system cannot be defined in totality.

Now above I used the idea of the string of discrete states to define a system.

Would the PM system be the string of all possible combinations of derivation rules on axioms? But the PM system doesn't then have the Godel statement as a state because that statement is not derivable!!! So do we then define the system as the string of all possible wff (well formed formulas) defined by the grammatical system. But then we know this can't be defined recursively (using PM).

So a problem defining the complexity of a system because it has two forms!

Now Turing Machines versus Logical systems - what is the difference.

It seems to me that there is a Turing machine to produce every number. This means that there is a Turing machine to compute anything (finite). The only thing they can't computer are the infinite sequences (mentioned above) which are the Turing Halting problems. But why not include infinite sequences (defined recursively) as above?

Anyway in a Universal Turing machine which can compute any number the analogous Godel statements must be machines that output infinite numbers.

Why can't PM produce any number however? And if it can therefore the Godel number of every statement? So actually PMs limitation is the same as a Turing machine... I don't understand :-(

Need think some more... why didn't I do a degree in IT or maths then I'd have the background for all this then! And if anyone who knows ever reads this (which is 1 reason I put it on the web wishing for some serendipity) give us a gentle nudge in the right direction save me the searching in the dark ;-)

Diagonalisation & +1 Hypothesis

Note...
The proof that the power series of S is always greater than S means that the set of all sets is a contradiction. This is analogous to the SRH that there is no absolute theory

Diagonalisation

Turning to Hofstadter he gives an excellent account of diagonalisation in GEB and offers his proof using diagonalisation that some functions taking 1 input cannot be computed.

If a table is made of all functions against their output for all the natural numbers (computers can only work in natural numbers - the decimals they process are really just natural numbers and an index) then the output of a function not in the list can be created very easily. Take the output of program 1 for input 1 and make the new non-computable function output some other value (say +1). Then for program 2 take its output for an of 2 and make the new function output something else (again say +1). Do this for all functions in the list. The new function is then not in the list because it differs from all the functions in the list at, at least, the value we have defined it to be different. This is diagonalisation and means that there is always at least 1 function that is not in the list.

So the +1 hypothesis was that any system would need to be larger than itself to model itself and the notion of itself hence +1. The basic idea that the meta level must be larger than the level, if only because it needs to accomodate (encode) the new idea of the self. If we make the level self-referential is this true... need think...

Is it not true that we can construct a Turing machine not in the list of all Turing machines using the Diagonalisation argument? Take the output of each Turing machine for each input, and +1 to the output ... what if it doesn't output ok the halting problem... enter Turing himself... what if we had a special code to say it didn't output (if we had an infinite amount of time) so it did effectively output? say {0,0} if it didn't output and {1,output} if it did. Creating a pair is like creating a meta-level interesting...

So a three level system would be {1,2,3} where 2 could refer to system 3 and 1 refer to system 2... whatever

still to read... got lunch
http://plus.maths.org/issue5/turing/

Diagonalisation & Paradox

http://www.uea.ac.uk/~j108/diagonalisation.htm
Interesting Russels argument

There is a greatest of all infinite numbers, which is the number of all things altogether, of every sort and kind. It is obvious that there cannot be a greater number than this, because if everything has been taken, there is nothing left to add. Cantor has a proof that there is no greatest number… But in this point, the master has been guilty of a very subtle fallacy, which I hope to explain in some future work.
Russell, ‘Mathematics and Metaphysicians’, 1901

Within a system of thought we always feel like this: surely this is everything... but that is because the system limits our view... it is the escaping the system (of Natural numbers here) the aufhebung in German which Russel over looks the possibility of.

SRH - next stage

Just exploring the meta-meta level which would open up the possibility of producing isomorphisms on a level removed...and so on ad infinitum proving there was no outer limit

Actually is that is... if there is always a meta level by the diagonal lemma and these can be shown to compund indefinitely then by a wierd sort of induction there is no ultimate-system.

The other thing to develop is the difference between formal and computable systems and human thought. Following Turing's insight, human thought must be finite because we use discrete things and ideas - an infinite system can't carry information... that said an ordered infinite system can like the Natural Numbers can can't it? Dennett also has the idea that what we call human thought is really just a "computer" being run like software within the brain hardware. Hofstadter also makes this point that human thought is not done on the brain hardware - we don't think in neurons and synapses. Rather this level of organisation has emergent properties that have emergent properties, and the levels are not stacked on one another but intersect with strange loops. So what we (and Turing when he designed the computer, which was based upon how a human would compute in their head) what we are conscious of is actually caught up in the levels of human brain activity.

An insight as a child was that we do not experience objective reality. What science says is the truth and what we see must be different - reference the blog entries on vision and brain activity. It is quite obvious that for us to see anything light must "enter" our eye. The object doesn't enter the eye, only the light. We see light then and not objects. This is why cinema and TV works. Objects are thus made up by the brain from the light that enters. Yet when I look around this room in which I sit everything looks normal and solid. The problem is then to ask well what is a "real" object like, that is the ones the light is bouncing off and going into our eyes, the ones that are being recreated in my brain? We can never know! You can't get outside your brain. This kind of blew my mind as a kid. This is the noumenal level as opposed to the phenomenal level.

Some ways of dealing with that are:

• The Cartesian approach: the existence of perceptions is proof that a brain exists, so the brain's existence is a different level from the perceptions of that brain (SRH) and is logically prior to those perceptions: it makes them possible. Thus our existence is not the existence of a perception of ourselves but the fact that a perceieved world exists at all. Had that realisation washing my hands before a big staff meeting in the Food Giant in Wathamstow in 1995 of all wierd places.
• The Hegelian/Buddhist approach: the idea of a real world is just that, an idea. The true reality we infer objective things to have is just an idea. Phenomenon are all that exist.
• Holistic approach: Our perception of things is not like an arrow hitting a target, rather it is the whole process taken as one. The arrow must be mounted in the bow, fired, make the journey and hit the target for it to hit the target. Any break in the process is a failure. To see something is not the light getting to the eye or the electrical stimulation hitting a part of the brain. So to see something requires the whole process and it is not just brain activity that is important but everything. I always thought as a child that we place too much importance on "brains" they are material like anything else so why focus only on them when we talk about perception?
• A recent innovation is this view. The noumenon is rather a "potential". We only know what the noumenon is when it has become a phenomenon, or when the mind has categorised it, but we can categorise it many ways so the noumenon is not only One way. Rather it is a "potential" to be many things. Systems of phenomena are not random however and have logic like the idea of "harmonic structuralism" (see blog).

So if it can be shown that human thought is finite and computable then it is fairly easy to show that human thought has no limits.

This is all hugely self-referential now because human thought is just a thought itself and if we can show that human thought has no limits then the question of what is reality (which is also a thought) has no limits either. This then would be the SRH with icing on top and an eternal candle.

It is kind of obvious, but there are those of the belief that we are progressing toward a certain truth, which may be true but the point of the SRH is that we can never arrive because the diagonal lemma (I believe this is the active ingredient) means we will always break out of any theory that tries to go for totality.

NOTE: correct the blog on Tarski - he does use self-reference. It is by self-reference that all the problems and inability to become complete and total arise. The Aboslute will always escape computing for this reason, so there will always be God outside the theory - hence the rather grand name for the SRH the God Principle. Put another way God cannot be brought into the theory by definition and he can't be pinned down by definition - chasing God is like a dog chasing its tail and all those "in the know" will see people who try to theorise on God involved in such a pointless activity. Most up for the scaffold in my opinion are the growing number of people who assume God to have been refuted, who don't realise that in the same breath they are saying that Man (whatever that is) or Reality (whatever that is) are sufficient concepts to explain the universe. It is their self imposed narrow mindedness that I find self-defeating. Why if you wish to know about things and have opinions on things would you take a dogmatic position that actually limits your outlook? I can only imagine it is the poor image if "religion" that has emerged where people focus on the failing on churches but I note always while ignoring the immeasurable good they have done. It is interesting that war is often cited in connection with religions - yet a better case for war is put down in this blog that really it is just people vying for power and too often religious divisions are exploited for this purpose, rather than the religions actually being the cause.

Islam for example is being used at the moment to incite division between powerful Arab leaders and Western leaders who (I agree) interfer too much in Arab affairs. But this has gone on for centuries, Britain did win some important wars in Arabia a century ago which is the deal when you fight and the Arabs are rich because of the oil sold to the West so it is all very incestuous. If America and Britain became Islamic I bet the Arab leaders would just find another way to fight for power in the Middle East. Bin Laden is a complex character he is on the surface fighting for power in Saudi-Arabia but I wonder to what extent he is so caught up in U.S. foreign policy that he is actually working for the Americans now. If congress wish to cut funding to the terror war the military just need a need produce a video from him. Entirely speculative but this is what i would do. I imagine that Mohammed understood the importance of battle to protect his nation, but I doubt very much if he wished to spread his wisdom at the end of a sword - hardly wisdom if you have no choice but to accept it! The other great error in the minds of people who wish to use violence on behalf of God (or believe that people do) is the simple question: if God wants something why do I need to do it for Him? He made the universe and Man I'm sure he would remove the infidels off the face of the Universe by Himself if he wanted. When people for world gains like power and land (on Earth or Heaven) it is the act of a Man - only when they act for sublime purposes is it inspired by God. For the meaning of sublime see the SRH.

Tarski and Arbitrariness

The theorem is obvious because if the notion of truth was definable with in a system, then the system would then able to define the truth of its own formula for truth... and if it could do that then it could just as easily have said that its own truth was false.

The issue is one of Arbitrariness.

Arbitrariness is removed when a system does not determine itself, but when a system can determine itself then it can determine itself either way and so the definition becomes arbitrary.

So the difference between either-way and this way is to do with the difference between self and other.

I my life and philosophy the great problem has been to escape the argument that everything is arbitrary and it doesn't matter which way things go.

It seems it may be to do with whether we take ourselves as originary or other people and the outside world.

This is the main teaching of religions that we should put other people first. This teaching removes arbitrariness. Failure to do so creates arbitrariness which when it takes hold means things (for us) can go any way and is called madness.

Seems all things (for me at least) are connected in these very simple ideas I'm trying to explore in the SRH thread.

Levels of Self-Reference

As I understand at the moment:

To reference one needs a thing and its name. It is this difference between a thing and its name which enables reference. When these are clearly separate as in "The dog ran after the stick" there is no confusion.

When names name themselves however things get tricky and paradoxes arise.

A thing followed by its quote is a quine. "A thing followed by its quote is a quine."

This loose (since it is not completely grammatical) but easy to see example is both doing the referring to "a thing" and is also a quine itself. It is what I'll call an indefinite self reference since it refers to the group of which this sentence belongs.

Of more interest are definite self-references that actually refer by name to the thing doing the referring.

"I am going out."

and

"Alva is going out", said Alva

would be me definitely referring to myself. This is also direct self-reference since the name is given explicitly: a pronoun "I" or actually my name. "This" also acts as direct self-reference, as in:

This sentence has five words.

In a definite self-reference such as these the object referred to does not belong to the group referred to but is actually the object referred to.

Another type of self-reference is indirect self-reference. It is still definite self-reference because it makes reference to the particular entity that is itself, but indirectly.

"preceded by its quote makes the sentence." preceded by its quote makes the sentence.

Which sentence we are not told but we know that the the sentence being referred to is made from the phrase being quoted before itself... which is what happens in the sentence so we know that it is referring directly although indirectly to itself. This is a direct version of the opening quine. This is the process of Quining, or making a Quine which is the means of indirect self-reference.

GOOD ARTICLE ON REFERENCE

http://en.wikipedia.org/wiki/Indirect_self-reference

Following links Strict Evaluation is what I was complaining about in self-defining entities (which passed into the discussion on impredicativity http://en.wikipedia.org/wiki/Impredicativity where something can never be evaluated e.g. T:={T}. Yet it seems that set-theory still allows this definition to my knowledge.)

Quotes

The usual way to create a name from a piece of text is to put it in quotes.

The dog ran after the stick. We now know that a dog once ran after a stick. It is proposing (though we don't know if actually happened) that a particular dog did this thing, and the object being referred to is that dog. If I wish to refer to that the words in that sentence then I would write "The dog" which no longer means the furry thing with a wagging tail but rather the 6 letters that make up the name used to refer to the furry thing with a wagging tail.

Hierarchy

The world is hierarchical which is to say that any object is composed from other objects and at the same time belongs to higher groups. Objects may be members of many groups at the same time so it is not a clear ordering.

If we refer to a table as "My table" and say that I have moved the table to my new home, it can be assumed that all the components of the table also went. It might be however that a screw from the table was lost during the move. In reality selfhood is a very inexact science.

If I say that I am a member of the rugby team and the rugby team just played in Coventry then

one can automatically assume that I just went to Coventry. So I am indefinitely referring to myself when I refer to the rugby team. I might then say that I scored a try. Now I am definitely referring to myself and we are sure that I went to Coventry. But then I might say that I had a bike accident after the game and lost my leg. This is shown in the venn-diagram to the left.

We are accustomed to thinking of the leg as part of me and myself as part of the rugby team. But a little contemplation shows this up and leaves us with a complex hierarchy of overlapping things. The leg I lost was useful and a part of the rugby team just as much as it was to my job or whatever else I do. Likewise I might lose a lot more in the accident, but as each thing is cut off in which part do I remain - in what remains of my body or in the pile of amputated tissues? Some would say the brain, but the start cutting that into parts! Self does not reside at any particular level. Yet mysteriously in the world of mass interconnecting things there are those that are somehow more relevant and those that are not, like the difference between Alva's leg and yours. This is another issue, and I don't understand it yet.

Hierarchy shows itself in self-reference because when we refer to a self which aspect of the hierarchy are we referring to? "I am right" could mean that sentence, but this is pointless. More usually it means the discussion in which it is uttered is right. But a discussion has many parts and often some are more right and some are more wrong.

The issue of translation.

The true statement "This sentence has thirty one letters." seems quite straight forward but in french "Cette phrase a trente et une lettres." is false. The equivalent true sentence in French is "Cette phrase a vingt a neuf lettres." which has 29 letters. This self-reference is making a statement about the form of a sentence but the two sentences are understood in different semantic systems so their truth changes: the first commonly makes sense in English the others in French. Choosing the semantic system can't be done within the syntax (basically the marks on the page) [or can it?]. So translations traditionally compare the semantics between languages and create the syntax in the new language to simulate the semantics in the old language. This doesn't necessarily preserve truth however as this example shows and if preserving truth is more important then the semantics may need to change also!

Meta

The difference between name and thing (which interestingly the knowing deeply of being the goal of Buddhist vipassana meditation) is what ensures the meta level. However through self-references this distinction can become very blurred which is how paradoxes arise. This sentence is false. The previous sentence is the famous paradox being both the object of discussion and also being a meta statement saying about the object of discussion that it is false. People complain about this but fact remains it is a sentence within the semantic system of English that we understand and get stuck in infinite evaluative loops with.

In Passing a note to say that link between thing and name is subtle as any answer needs to talk about the link itself and so depends upon link itself. A constructivist approach is clearly out of the question and we can only hope for a phenomenological/hermaneutic approach of accepting the hermaneutic circle of prior referencing at the outset of the discussion.

Levels of Reference.

A system can reference when it has names.

It can self-reference when it has a name for the system or anything in the hierarchy above the entity doing the referencing.

References can be definite where they pick out an exact entity e.g. the object itself or indefinite where they pick out an entity to which the object belongs; or direct where they make the reference explicitly or indirect where it is implied.

Things within the entity doing the referencing are referred to relative to the above hierarchy. For example the nineteenth word in this paragraph is actually the word "nineteenth" but the referenced entity was given relative to the paragraph not the statement making the reference.

Self is completely hierarchical and I need to think about this more and the idea that inside all the circles is a homunculus giving them a centre and a "relevance". Shades of what Herpatia and the ancient world were thinking about in Agora... and while cosmologically we've moved on, in terms of our deep world view still no progress!

SRH - final expression & proof of God

The Self Reference Hypothesis is not that self-reference is impossible, quite the contrary. It is precisely because self-reference is possible that things can contradict themselves, and it is precisely because self-referential things must contradict themselves that the self is never enough.

The search for a problem with self-reference itself was misguided. Things cannot know themselves ironically because they can refer to themselves. It is the making statements about themselves which creates the contradictions that make things incomplete (in the normal sense of the word). And it is this incompleteness and which demands a larger system (a meta system) must exist.

The necessity of a recursive larger self brought about through self-reference is the proof of God that the SRH was seeking. God is not a thing Himself, but simply the awareness that there are more things in Heaven and Earth than are dreamt of in our philosophy (Hamlet to his friend Horatio in the eponymous play). If we look carefully at the teachings of religion this is a much more accurate description of God than is often offered by believers and non-believers alike (to believe and to not-believe both entail a belief about God).

Tarski's Undefinability Theorem is the formal expression of this idea. The recent turn about in SRH thinking was not that formal systems prove the principle but rather that the more general principle is expressed where ever there is self-reference, which includes formal systems.

Suppose, in some logical system, that there was a formula with one free (input) variable that was only ever true when that variable was bound to a true formula and only ever false when bound to a false statement, that is the formula True(x) is true when x is true and False when x is false. This is written:

True(x) ⇔ x

But then we can use the True formula to create a statement by negating this:

¬True(x) ⇔ x

This states that if x is True then ¬True(x) is true in other words True(x) is false, which means that the True formula doesn't work, and can't work.

This argument works for any attempt to create a formula to identify a predicate in a system.

AND INTERESTINGLY IT uses the weakest form of self-reference with the system only referring to statements in general... if doesn't need to deal with statements that refer to themselves!!! That changes a lot... stuff to rethink... SRH not over yet....

RECONSIDER...

Two aspects of systems cause of problem for self-reference. One is the existence of statements of the form where a self-reference is negated such as "It is not the case that I am truthful". By negating such statements a self-contradiction is created - essentially modelling the self and then saying not-me within the model which is isomorphic with something outside the model itself. This I wish to explore as a conclusion to the SRH myself. The other is the Halting problem. Systems which depend upon their own output form infinite loops and so never return a value. They cannot thus determine whether they are themselves an infinite loop, which implies "infinite loop" must be a term from a meta-level. So self-reference in statements is analogous to recursion, and contradictions analogous to infinite loops both which imply a meta-level.

Full Employment Theorem

Can't believe I have never visited here before...

http://en.wikipedia.org/wiki/Full_employment_theorem

This is the SRH. Interesting that it turns out to suggest that there is an endless amount of work to do which contradicts another thread in this blog. Accepting an infinite scope for human activity then endless economic expansion is possible but progress could just as easily be a change in socioeconomic rules. The analysis that progress is relative is still valid and that any place in the line of progress will always be ahead of one place and behind another.

This argument is based from the outside in: that no compiler is optimal because there are tasks it can't optimise (like the Halting problem) which can only be see by escaping the system stuck in the halting loop, and Godel's Theorem which creates a statement that can't be discovered by the rules of proof working from any axioms, but stepping outside the system can be seen to be true and valid.

Maybe the SRH needs to work from the outside in, rather than looking for an immanent finger pointing out, rather a finger outside pointing in.

Bike stolen

Blimey my bike just got stolen from outside the library ... kind of predictable as I was chancing it without a lock ... but just in case it was time to believe that people aren't that bad really here's proof otherwise. Still I wonder whether to report this (which is enormous hassle) or just give it to whoever took it - they obviously want it and it will make their life a bit easier... and the person who buys it off them so it's all good really... except it causes a tiny bit of deflation in bike prices and devalues the labour of people in the industry a bit.

Was thinking anyway how is it that the demand for things still so massively outstrips the supply even with oil, machines and relatively cheap labour (in third world) making things already so easy to come by? It is as though we haven't progessed at all. That bike is worth only 2 weeks labour at the minimum wage - and you only need buy one bike in a lifetime so it is very easy to come by. But this is one of the points in this blog - we haven't progessed relatively at all.

There are two driving forces to poverty. One is population: we will have more sex and children to always fill any supply or resources until poverty is once again reached. The pressure that person felt, and the pressure I feel having the bike stolen is partly the pressure of sex driving up population and driving down the availability of things.

The other force is greed. Greedy people will always feel poor regardless how much they have. This bike is a good test for me. It will be exciting to go back to life before the bike to re-explore a poorer world and become less greedy and so happier with it than I was with the bike!

But this does open up the other side of this which I think I had resolved before but evidently forgotten. The person who stole that bike is unfortunately going to feel rich for a while now he has got it, but sadly he will only stoke his greed so that he will feel even poorer in the future. It is such a waste that it is me who will feel richer having coped with my poverty while he will feel poorer for this transfer of wealth. This is why I was supposed to be locking my bike more securely to make sure that I didn't give weak people the opportunity to make themselves poor. But really this whole state of affairs is irksome that I must take responsibility for other peoples weaknesses... certainly one thing I have lost is patience, my attitude very much is let them learn the hard way.

===

I was in a quandry about whether to report the bike. On one hand my purpose in not locking it was it would involve a cycle home and back again to get the lock... stitch in time etc... so there was more effort generated by leaving it unlocked not just for me but for the police if I reported it.

But at the same time this bike was a present and I should treat it with respect. So I decided to report it.

I have to say that the police are beyond my best expectations - I said I wasn't claiming insurance and he said don't worry it shouldn't have been taken in the first place... so the police are actually pushing for a moral society not just a practical, legal, institutional society. He certainly made me feel that theft under any circumstances should not be tolerated ... unlike my mother for example who would blame me for leaving it unlocked. That said I have certainly made a lot of work for everyone.

My next concern is the person who took it. He will be poor, maybe a drug addict and somewhat on the outside of society. My first thought is that he needs the bike more than me. But actually may be if the police are really seeking a better society he is better in their hands than on his own. Plus reducing theft raises the value of people work in the legitimate bike industry and most important of all increases the trust between people which is what this is all really about.

Sadly between now and people trusting each other is a very great deal of work. I wonder if there is ever a way to stop people becoming greedy and feeling poor and losing the respect of other people. The police today, once again, have made me think maybe there is.

But once again there is the war in Afghanistan and the one before in Iraq and this weighs very heavy on my view of politics. The police may be working for the Good but they have only a very limited influence - they can't decide how they are funded or how schools are run or how communities are supported or how people are treated by the economy and investors - it is these which will influence society most.

Beware Pharma!

It is odd that the industry which is supposed to be the one we can turn to for help is actually one of the biggest wolves in sheep's clothing.

2008 I visited the dentist and after a brief check up was told that I grind my teeth in my sleep. First time I was ever told this in 30 years of visiting a dentist. The solution offered was a specially moulded orthodontic shield at some considerable cost. I said for that money I'd work on my subconscious and stop grinding!

Returning a week later for x-ray results I over heard the patient before me being told exactly the same thing.

Listening that summer to Russel Brand's radio show he was reading out a letter to the papers by some ex-girlfriend in which she said he even wore a gum shield to bed... orthodontic shield Brand insisted and explained it was to protect his teeth from grinding...

Ah ha I thought even the famous get scammed.

Obviously a company that sells Orthodontic shields had sent an army of representatives out to dentists across the South East offering dentists a slice of the commission for every one that was sold.

Unfortunately in a Capitalist country it is profit and sales that matter not service to customers. I notice (possibly unconnected) that I never had this problem with my previous NHS dentist in 20 years.

I am writing this now because a friend has just been told at the end of the 2 years treatment required for his daughter's photo-epilepsy that in fact she never had the condition at all. In this case the drugs did help with her condition, but it is odd that she was never accurately diagnosed before treatment started and why should the specialist only inform them after the end of treatment? Was he getting a percentage from this drug company?

I have friends who work in the pharmaceutical industry and basically this is exactly how it works... and worse. One pharam giant a few years ago stopped research into oral vaccines despite being ready for clinical trials so they could divert funds to the far more lucrative new viagra market. Oral vaccines would all buy eradicate the diseases that plague the third world - but there is little money here so little attention from the industry. What is worse is that they protected all their research so that no-one could carry on from where they had left off.

This is all basic economics but it shows how all the supposed benefits we are endlessly told come from capitalism are actually over shadowed by endless costs. The WHO has done huge amounts of work to improve the health of the world's people, why does this body not wield the money of the pharma giants and fund such research? It is because it is not in the interest of investors, even while it would be in the interests of billions of non-investors.

One doesn't need to look far to see the problems that we have to accept (and agree with?) if we support the dominant way of life.

Health may start to become a feature of this blog. A friend is exploring with some suprising positive results homeopathy. I also need to digest at some stage Hans Georg Gadamer's "Enigma of Health" that I read a few years ago.

SRH notes

Fixed point theorem and proof:

http://www.columbia.edu/~hg17/naming-diag.pdf

Let T be the theory and, for each wff φ, let ¬φ¬ (that left right corners symbol) be the term that serves as its name. (Clarify that notation!).

===
http://thalesianfools.blogspot.com/2007/04/gdels-diagonalization-function.html

In a previous post, I introduced Gödel's famous incompleteness theorem: it is impossible to capture all arithmetical truths in a single, consistent formal system.

That is SRH.

===

http://www.math.hawaii.edu/~dale/godel/godel.html

Shutter Island - Violence

Exellent demonstration in thsi film of why non-violence is non-negotiable.

The film is split between a narrative (1) where De Caprio is an FBI investigator seeking to uncover the truth behind a mental asylum which ultimately leads to the suggestion that he has been part of brain experiments and the narrative (2) where he is a patient at that asylum undergoing a radical form of therapy where he is allowed to play out his delusions.

Both narratives (I hope) work until beyond the end of the film where he is seen being taking to the light house to have brain surgery either to pacify his mental illness or to continue the research. I hope the film is this complex though it does look like Scorcese wants the film to conclude with him being delusional.

The point is that he commits a lot of violence during the film. The film could have continued along the lines of narrative 1 and it would have been an exciting FBI investigation. His punching of people all part of the films we are used to seeing. But if we take it from the standpoint of narrative 2 then he looks like a crazed killer living out paranoid delusions and we absolutely don't accept exactly the same violence.

What struck me very hard is that almost all US films could be seen as the actions of paranoid delusional people with the camera just following their delusion - we are used to this type of thinking where we are oppressed, in danger and need to protect ourselves.

Yet from the outside the US individual looks like a paranoid and dabgerous lunatic (as the US looks to the rest of the world in Iraq and Afghanistan).

It showed me that we can never use violence, or even harsh language, because we can always be seen as delusional from the other persons perspective. Only peaceful solutions work whether we are delusional or not. Which lens support to the idea that De Caprio is indeed delusional in the film.

Private versus Public

I get the sneaking suspicion that the debate is all but over these days in the West... but unless I missed something it never really started. Had a short discussion at the weekend with a friend who is a supporter if the private paradigm and decided that really we don't know either way yet.

I was just about to do a pros and cons summary of the two arguments but its not even that clear cut.

The private system is designed to raise money from the general public. The incentive is a return on that investment. As I understand it this raises money only once when shares are sold - unless there is a further share issue. Yet the dividends are paid indefinitely. I don't see reasoning behind this.

This makes especially no sense when the private banking system is designed to be able to lend out virtually unlimited cash (new Basel III rules yesterday are reinforcing limits on this at last!). Why can't companies get money the old way from banks, which are lending private money anyway?

Competition is often cited as the magic regulator of the private free market. Yet competition is universal and it exists for state contracts equally - look at defence contracts. It is true that the state is more corrupt and more likely to offer contracts based on nepotism and favour rather than "value". But again competition is universal and competition within the state - especially in a multiparty system - is just as fierce as within the "free" market (this leads I suspect to the paranoid behaviour which creates dictators). But within a democratic type system the state is answerable to the public so there is extreme competition here. I don't see how the private sector has ever had a monopoly on competition.

Efficiency. The public sector is regularly criticised for its inefficiency. I suspect it is a matter of how things are measured. Has anyone ever added up the cost in time and resources of transporting people by public transport and compared it to private transport? Does 30million people owning their own cars and driving themselves each day really compare with a few thousand trains and buses being driven by a single driver each. It must also be pointed out that a private/free-market model can't even apply by definition to national industries (despite efforts). In public health the idea that experts, machinery and information needs to be duplicated in each private health company doesn't seem as reasonable as the system in the UK where expertise, equipment, knowledge and patient records are shared across the country. That is efficiency. It is the simple (and somehow forgotten) economic principle of economy of scale.

Certain things escape the envious eyes of the private sector; in particular the government and the military. I mention regularly in this blog the smile the thought of the private paradigm trying to accomodate these brings to my face. If these are deemed successful then why can't everything else work by the same principle? The builders of tanks probably don't get many contracts outside the government so why haven't they failed for the all the reasons cited by the private sector?

The other obvious inefficiency of the private sector is the supply of money. Those with money to invest also gain the right to returns on that investment. Thus a percentage of the economy's money flow feeds into the hands of the investors. Yet we already know that they have money to invest so they are unlikely to spend this money: it is invested further. Capitalism creates a growing snowball of investment money. The economy needs to grow in order for the investors to find somewhere to invest that money, yet they have starved the economy of the sales it needed to grow. Governments put in place a taxing system that recycles money from the rich but it hasn't proved to be enough and the rich grow richer and the economy grows weaker.

Now the other great crisis is that the economy is built upon a paradigm of employment, which is ultimately how the investors make their money (in Classical thinking anyway). Yet the salaries to support labour are more expensive than machines. It is natural for economies then to replace workers with machines thus increasing profits and returns to investors. Yet with workers being made reundant the money flow once again flows into the hands of investors rather than spenders and the market shrinks actually reducing profits on the global scale. Again growth is needed to provide jobs for workers so they can be paid, so the market can be kept liquid.

Provided there is infinite growth then the private system ought to be viable. Yet in reality there are very obvious limiting factors to growth: space, energy and other raw materials. Plus there is the natural environment when it is perceived as something more than just raw materials for exploitation (which was my own entry point as a child into this argument).

So if growth must give, then investment must give and investors must have their excess money recycled through the system more efficiently. The best way that I know to do this is to cycle money through a central investor who manages investment on a national scale. This way Bernanke's helcopter runs as a daily service avoiding problems rather than just fixing them in emergencies. The welfare system for example becomes economic centre stage and a necessity rather than an ill understood backwater covering up for economic failure. The walfare system also solves the issue of machine efficiency since it redistributes the added wealth of the system to the market through people, rather than into investment.

Another thing I read all the time is that state spending has been unsustainable. The fact that the private sector just spent too much somehow escapes notice. But that is the only similarity. Public spending has created 70% of the jobs in the UK in last 10 years. Private spending seems to have only created a series of speculative price bubbles - like house prices - whilst creating only 30% of the jobs. In the current climate people think that tax is theft and buying the trapping of the private lifestyle is freedom. But for me, at least, leave the buying to the professionals - I'd rather have the transport industry decide how to manage the states' transport so that I can turn up and use it rather than have to make the decision on car, insurance, maintenance and driving all myself. And the centralised efficiency (lower cost) would free more people from having to be employed at the same time, providing another wealth: more free time. So I speculate (and it is something to investigate) that for every % our taxes increase there is a greater reduction in the amount we spend and have to earn. It seems odd to realise that things like "choice" and "freedom" are actually the noose that is starving us of exactly these things... is there a reason for that ironic feedback? One needs only be reminded that with the exception of the industrial revolution in the West (which are the only figures normally published to prove the contrary) we are working the longest hours mankind (indeed any primate) has ever worked now (9 hour/day average figure includes "work" such as transport and personal grooming).

One thing I am not sure about is how the banking system ought to work. I have argued that really lending at interest should be outlawed, but if people wish to raise capital I suppose there should be a way to do it. The obvious way is to have the government as the lender. They have the countries taxes at their disposal and as we have seen they are indeed still the lender of last resort when capitalism fails (as it seems to recurrently do). So why do we have a private banking system at all? Why in a democratic country do our elected leaders have to ask permission from unelected banking magnates to borrow money? Actually how on earth did a handful of unelected people gain the right to write almost infinite cheques out of thin air? Indeed why in a democratic country do unelected people control government at all? This as a side issue is one major problem with capitalism by itself (altho many say it is a problem of that species called corporate capitalism only). It seems that putting lending under the control of the electorate (since it is them who ultimately underwrite lending) is essential. But there are problems with handing such enormous power to leaders as there is a tendency for them to buy favour with the people by creating unservicable debts... but actually this doesn't work cos we vote them out as we just did with Gordon Brown... something we can't do with the banking sector... so the argument for private banking seems far from on any solid ground.

All that said and I hope not militantly for either side, it stuns me (to the point of inaction) why the economic policies seem to always head in the opposite direction... even the mind set of the non-capitalist class seems to accept their fate and points in the opposite direction!

I can only understand that it is psychology that drives this. I believe that people don't like working. If they did they why do people struggle to become rich so they can pay other people to do things for them? Given this simple insight the psychology is simple. Investors are seeking a free ride by getting an income without doing any work - essentially they are buying people in a company and taking a percentage of their productivity. The workers on the other hand who work only because they have to, and deep down resent that fact, vent their anger on anyone who gets a free ride. It is down to packaging that investors aren't seen as free-loaders while people on the dole are. It might also be down to the old class system where an upper class person, while resented, it understood to deserve an easy life unlike a lower class person isn't. Since the investors and owners of companies (especially the media) get to control the image they have and also the policies of the governments it is no suprise that the system is bent in favour of them - evn while in the long run this attitude will be the ruin of everyone because it simply doesn't work. Greed doesn't pay - if it hasn't been said before I say it here.

The answer
In answer to this I have my book Anura still waiting inception. Within myself I know the consciencious attitude which takes plasure in helping other people and trying to make things better. I believe we all have this and at some stage we decide that while we don't like work, the reward of making the world a better place is a far deeper pleasure. This is a spiritual transformation. Once that transformation is made we are on the road to the greatest riches - but they are not material and can't be bought and sold. Inner peace, tranquility and happiness is something we can't buy or sell - it is ironically a private quest but it is gained through entirely public endeavours. We can and mostly do work for the welfare of our fellow people; money is actually the least of our interests required only to service the lifestyle that enables us to live well with our fellows.

The great fear of the above system that has been instilled in us throughout the bourgeoise era is that without strong social control we would return to a "state of nature" where our lives would be "nasy, brutish and short" (Hobbes 1651). Running from this image brought to the popular imagination in Lord of the Flies (William Golding 1954) we accept the rigorous establishment without question. In particular the structure of capitalist employment, salary, shopping, private ownership. But these are on a path to failure if the above analysis is correct, and an alternative is required. Apocalyptic images courtesy of the state of nature and also the Aryan religions - who believe in a final great apocalypse of ragnarok - fill our minds when we think of anything other than this. Lots of films like Madmax portray this common belief. Yet a simple argument somewhat undermines this thought. If humans really are based on evil and are unable to do good then how did this system we have now evolve? There are forces at work that bring humans together outside of establishment and social structures (this is SRH btw).

Certainly there are aberrations and illnesses of our species. People become violent (Desmond Morris has an interesting take on this with his "Human Zoo" concept linking the types of abnormal behaviour seen in zoos to those in human society). Societies become violent also as we have been in Iraq and Afghanistan. But these are the exceptions rather than the rule and the vast majority of human interactions are friendly and supportive (viz the Christmas football game between the trenchs of WW1). If there is research to be done it is not on how humans become supportive and good but simply to find out how conflicts and destructive behaviours arise.

Now the problem of how society would organise itself outside of "labour" and "shopping" becomes more addressable. We know already that humans can have enormous fun outside of the labour system because the aristocrats of the previous economic system showed us. All this "work" done by them needs to be dug up again because I believe the wrong side won the revolution and the working class should have become aristocrats (in stages as machines took their jobs) rather than the aristocrats become workers. Big mistake in Marxism there, and a spilling over of the negative sentiment mentioned above that ultimately poisoned the workers and society.

Education, politics, art, sport are just a few of the endeavours that humans have done and lauded in the past. Aristotle even argues that free-time is essential for a democratic political system because how can we make intelligent choices without considerable time in education and discussion of the choices available. There is perhaps a critical mass where once free-time has increased to a certain level it is self supporting because we have more time to make sensible choices about our future and create even more free time. Certainly historically the increases in technological efficiency brought about by free-time have generated whole classes (priests, rulers) based upon freedom from labour.

I don't accept there is an Orwellian conspiracy to do the opposite it is just an unfortunate outcome of greed that the investment classes seek more investment without actually thinking out what they are investing in and how it can possibly exist, and then a simple self-justifying attitude portayed in the public consciousness.

The selling of the state services by Thatcher in the UK seems non-sensical to me, especially the low price that they were sold at. It is pointed out by Walayat that this was most probably to get everyone excited about capitalism because of the huge gains made by the first investors. Of course no-one notices that those gains were made at our own expense (losses that continue to be felt today as the price of services continues it inexorable rise). Another case of the money of the masses falling into the pockets of the few - like a mass lottery we all play along because we might be the next lucky ones. It is remarkable how short sited we are and how the state can use that narrow-mindedness against us. We will steal a pound from the state because everyone only loses a penny.

So there is a very clear path open to us it is just in the opposite direction to where we are going. But my purpose here wasn't to change anything - if we want a ragnarok then so be it - it was just to reopen the debate that no-one seems to be having anymore.

Self functions

A Quine is a function that takes no input and produces itself in some pre-established execution environment (a program which takes input of another program and executes it producing its output).

This struck me as not the best example of "self" because the program does not actually identify itself - it is up to the observer to realise that the output is the same as the program and so identify that the program is a quine. A program which just counted would eventually output itself. What is more interesting is a program which knows that the output is itself. In a quine the programmer has found the program and confirmed that it output itself. Interesting programming task, but not really an example of "self" (whatever that is).

Logically then a better quine takes input, and outputs a unique value only when the input is the same as itself, thus establishing itself as a Quine. Let these be called Self functions. Like Quines they are entirely created by the programmer but what the programmer must create is one level removed as the program does nothing but confirm itself in the input (something the quine programmer previously had to do).

However how does it know what itself is? The only way to do this would be to read its own source code... but it will still need a pointer to that memory. The problem to contemplate here is just what "self" means and whether it can be established immanently (that is the main point actually).

Anyway... to classify Self functions...

Say a species outputs zero unless their input is themselves where upon they output a function. This way a Self function can do anything (not requiring parameters) as long as the resulting function is executed.

Let S be the set of all Self functions. Let s1 be the first member of this set.

s1(some number != #s1) = 0
s1(#s1) = #some function

where the number of the function s1 is #s1 (actually the numbering can be designed to be the same as program) so that can be rewritten as:

s1(some number != s1) = 0
s1(s1) = some function

Now imagine a super self function that takes two inputs, S(a,b). The first is the number of a self function, and the second is the input for that function. S is essentially an interpreter. The above can be rewritten:

S(s1,some number != s1) = 0
S(s1,s1) = some function

So S(a,a) establishes whether the function "a" is a self function (does it output 0 or not) and outputs the function that "a" would if it was.

Is this of any thought experiment use... i will find out...

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Just for fun was thinking about another type of self function. The 'nelf' is the a less strong version of the self function that simply produces a negative result on its input:

n1(n1) = 0

Now imagine an alternative to the S function (say the F function) that takes only 1 input (the function) and determines (somehow) whether the input function is a nelf i.e. does the input function output 0 when fed itself. It outputs some function if the function is a nelf and 0 otherwise:

F(a nelf) = some function
F(not a nelf) = 0

Now what happens to

F(F)

if F is a nelf then it outputs some function which makes it a self function. If F is not a nelf then it outputs zero which makes it a nelf function. Yet F was defined as the function that could determine whether a function was a nelf. Such a function is therefore impossible.

This conclusion below only works if the nelf is a strong version of the self function - but since the F function outputs 0 whenever the input function is not a nelf it doesn';t qualify as a strong nelf (which only outputs 0 when fed itself). I keep the conslusion cos its interesting...
Since a super nelf function (F) is just the negation of a super self function then a super self function is impossible also. There is no function that can determine whether its input function recognises itself.

So we are limited to functions that can recognise themselves, but not determine whether what they recognise itself will recognise itself!!!

Now we in the meta-world can clearly see that if a function recognises itself, then what it recognises is a function that recognises itself... but within the system a function can't determine this.

This vindicates the insight that AIME97 was never going to become conscious by some magical feedback loop feeding on itself. Dennett and Hofstadter it seems (if I understand them correctly) do think that consciousness is magically produced in this hall of mirrors.

Looking at the F algorithm it can be seen it is an probably an infinite regress (unless there was some direct way to determine the nature of the code other than running it... on itself ad infinitum) so it would never halt.

does an inf. regress always imply a contradiction then?

===

Thinking then back to the discussion on T={T} (ages ago!!). Here there was an infinite regress.

What about T = {T'}, T'=U-T, T = {U-T}, T = {U-{U-{...{U-T}...}}}

T = {U-{U-T}}

Now this suggests to me that U is a power set on some prior elements e.g.

U0 = {1,2} and T = {1} then T' = U0 - T = {1,2} - {1} = {2}

Now using the definition above T = {U1-{U0-T}} = {U1- {2} }

So if U1 = P(U0) + U0= { {1},{2},{1,2} } + {1,2} = { 1,2,{1},{2},{1,2} }

Then T' = { U1 - {2} }= { {1},{1,2},1,2 }

And again,

T = { U2 - T' } which suggests that U2 is a powerset of U1 + previous universals

U2 = { {1},{1,2},{1,{}},{1,{1}},{1,{2}},{1,{1,2}},{2},...,{{1}}...,{{2}}...,{{1,2}...,{{}}...}

which has 2^6 terms which already includes the previous universes except U0 so 2^6 + 2 in total = 66.

So U3 has cardinality 2^66+2 = 7.37869763 × 10^19
and U4 2^7.37869763 × 10^19 + 2

And this is just getting an idea of how big the cardinality starting from a universe of 2. Imagine starting from a countably infinite starting set.

So the definition T = {T'} only exists in a U populated by an infinite power set... which is a very large universe indeed!!! Find out how big???

The U in which a T = {T'} exists must have cardinality 2^(...(2^(2^inf + inf)+inf) + inf) !!!

If the first universe is countably infinite (B0 - i.e. beth zero) then each recursion shifts the size of the universe up an beth number so the size of U is B(B0), otherwise if the first universe is countable then U has cardinality B0. Either way it is massive.

Does this suggest that T={T} only works in such a vast universe also?

And what does this say about these bizarre sets?

to be continued...
===

Can Godel be seen as an infinite regress? ~(x)(y)(Pxy · Qfy). No he has found/created a statement that says that there is no other statement which acts as a proof of this statement (all referenced indirectly with statement numbers). If there was then there would be a proof of something which said it couldn't be proven which would prove a contradiction (inconsistent they call it), from which everything could be proven so it must be accepted that the statement can't be proven... which is what the statement says so it is infact true that it can't be proven. Truth arising outside proof and the logical system of proof clawing at the edges of a system that extends beyond it - incomplete they call it. But infinite regress this isn't.

Why do I always end up with infinite regress which is not logically dull and conclusive :-(

Anyway here we have a function trying to decide what it is. Clearly if it doesn't know what it is, then it can't decide what to do and so can't decide what it is! The SRH at play :-) got the SRH a bit more under the microscope.

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Kolmogorov Complexity

Got me thinking about complexity. If there is such a thing as KC then it would provide an absolute measure of functions. An absolute but imperfect ordering since two functions could have the same complexity.

Also a function KC() which calculated the KM complexity of a function would be able to calculate its own complexity. Now what do we know about the KM complexity of the function to recursively calculate KM complexity?

===

Any function has a "self" that the observer can see (namely its form and number). Yet normally we consider self to do with the system in some way incorporating its own form or number so that it doesn't need an observer. The SRH I believe is trying to argue that the latter is impossible but apart from lose observations myself I can't see it is a logical problem... so far.

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A Principa-mathematica version of the self argument
FINISH ITS WRONG
Presumably there are formulas in the PM with 1 free variable that are True iff the variable is bound to their godel number. These are self function.

And there are also formulas that are False iff the variable is bound to their godel number. These are nelf functions.

Now suppose there is a super nelf formula (SNF) with one free variable that is True iff it is bound to the godel number of a nelf formula.

What happens when it is bound to its own godel number?

If it is a nelf formula it will be True which makes it a Self formula. If it is not a nelf formula then it will be False.

Either way it contradicts itself.

Ergo: There is no SNF. And since this is just a negation of s super self formula there is no SSF either...

===

http://en.wikipedia.org/wiki/Full_employment_theorem

Been looking for this for a while. No algorithm can solve all problems, or explain all datas. My argument is that it would need to explain itself... but it seems there are better arguments... explore down this avenue instead of SRH maybe. 2Do