Russell's Paradox

[protheory]

I still think there's something in 'Pro-theory', albeit that it doesn't quite seem to be a theory yet. However, I have to say that the way it's being described is currently incomprehensible to me. I see a buried connection between the idea of a neutral truth-value and certain meanings of the phrase 'Middle Way' in Buddhism, 'advaita' (not-two) in esoteric Hinduism, and more generally the 'nonduality' so often spoken of by the mystics. However, I'm struggling to understand protheory's idea of neutrality and maybe I'm way off track.

Canute

Hi Canute, I'm sorry I'm still not able to describe my theory as well as I'd like, I'm thinking about it every second of the day at the moment in the hope that I might soon be able to make some sense to you.

This is a rough idea of how it all started for me...

What originally led me to the idea of the three potentials was that I imagined atoms, just balls of energy in my mind, glowing and somehow creating everything.

I thought this was gospel physics for a long time and as I kept searching and pushing the boundaries of my own thoughts, I realised that if everything was made of atoms that by the same token everything must behave the same as a single atom repeating itself over and over to create the universe and my own life somehow.

I looked into atomic structure and thought that if I were to take the words away (I was still a reductionist at this point) I would be left with an example atom containing negative, positive and neutral energy somehow.

At this point I'd not really realised what I was onto with this idea, I just thought "that's interesting to me as a thinker" rather than my thoughts now which are more like "I've written a TOE and it works like this..."

So basically I imagined that somehow the bricks making the wall I was beside at that point in time were somehow composed of these three energies, I didn't know how it worked or formed, but I knew that everything was made of atoms and that atoms contained three potential energies so logically every atomic creation (literally everything, nothing etc) should follow this same pattern somehow.

I continued my mental search and realised that perhaps symmetry was the answer for the bricks, I imagined a single brick with two opposite sides and a middle, then I imagined that each corner had three points in a triangular pattern, I used the bricks example to try and apply this pattern somehow to other things.

I looked at the window in a room I was in and noticed that it was perfectly in balance, symmetrical, with two opposite sides and a middle, I continued this process for months and months while writing down all my thoughts about everything being based on atoms.

It wasn't until maybe a year later that I started to take this as literally as I do now, I mean by saying plus this plus that, and when I realised how literally this idea showed a pattern within everything I could think of I changed my previous thought 'everything is atoms' to 'everything is atoms plus the opposite plus neutral ad infinitum' which showed a definite shift in my thoughts.

Eventually I gained in intellectual confidence and started to read physics theories to see if and how I might be able to explore them through my new pattern.

My journey towards what became Pro theory was long and arduous but I think it's important to note that I only decided to call it a TOE when I realised its completeness.

I'm still not sure whether this will be adequate for you to see this idea as I see it but I hope it will go some way towards how I came to think about all problems in the way that I do

[protheory]

I agree with Buddhists, Taoists and their like about Nature, or the nature of reality.

I still think there's something in 'Pro-theory', albeit that it doesn't quite seem to be a theory yet. However, I have to say that the way it's being described is currently incomprehensible to me. I see a buried connection between the idea of a neutral truth-value and certain meanings of the phrase 'Middle Way' in Buddhism, 'advaita' (not-two) in esoteric Hinduism, and more generally the 'nonduality' so often spoken of by the mystics. However, I'm struggling to understand protheory's idea of neutrality and maybe I'm way off track.

Canute

I pretty much agree with you, the Buddhists, and the Taoists too Canute, I've been reading the Tao Te Ching and it's just like my own idea in parts, it's also a good way to see nature and the universe.

And what you say about the buried connection of the middle way, I think you're right there, knowledge is always simple and I think that it just gets recycled throughout time and human history so it's no wonder that Pro theory and Taoism have so many things in common.

If my idea truly shares things with ancient wisdom then I think it helps to add weight to the idea itself regardless of who wrote it, it also gives us a new method for interpretation as we may look to the ancients for information too and adapt this idea into a true TOE.

I stumbled upon my three potentials idea through atoms but if you can see it in a different way through ancient wisdom then go ahead, it all helps to understand everything doesn't it which is our goal.

And so I guess I am actually re-inventing this knowledge and calling it a TOE rather than a Tao.

It does seem perfectly logical to me that the idea of Pro theory has been seen in history before but in a different form (the Tao, nirvana etc) and if this is the case then maybe it will help us to interpret my own ideas in a different way.

[protheory]

I see the definition of Russell's Paradox as "is the set of all sets a member of itself?"

This kind of paradox shares its characteristics with other paradoxes too, such as "is the word word a word?" and "Is this a question?"

I also see "which came first, the chicken or the egg?" in the same way and I think that a common root runs within all paradoxes of this kind.

Millennium Problem number 4 is the P versus NP problem which is a paradox with the same similarities as Russell's Paradox and the others I mentioned above.

If there is really a common form within all paradoxes, and therefore a universally applicable method for understanding them (a TOE) then it should follow that if we can understand and explain one paradox through Pro theory, we should be able to explain them all.

4. P versus NP

Introduction

This problem is about efficiency.

Problems of class P can be solved in a finite amount of time, and problems of class NP require an infinite amount of time to solve.

Questions

It seems to be theoretically possible to create some questions that would take us millions of years to solve using our current technology.

Atom

Imagine trying to calculate the amount of atoms in the universe.

Infinite

Any answer requires an infinite amount of explanations, or at least an infinite amount of time to complete.

Better

We need to know if there is a more efficient way to solve these problems, or if we are stuck with the method of eliminating all steps individually, which could take us an infinite amount of time.

Amount

The amount of possible combinations seems to be so large as to defy any direct attempts to solve the problem using our current technology.

Proven

Nobody has yet proven whether these tasks can actually exist, or whether they are illusory.

Summary

Do questions or tasks exist that could take us an infinite amount of time to solve?

Is there a formula or shortcut that will allow us to solve them in a finite amount of time?

The Millennium Problem

The Millennium problem is to explain the nature of this kind of infinite problem and to prove whether or not they can actually exist.

For the exact problem description please refer to Claymath.org

The Answer

Conjecture

The P versus NP problem presents us with two options, either we can solve these questions in a finite amount of time or we cannot, and asks us to choose one of them.

Dealing

We are dealing with opposite and neutral potentials, regardless of the specific details of each individual question.

Remember

All questions have three simultaneous answers.

Details

All details are relative and subject to change in three ways simultaneously.

Three

We are dealing with three possibilities at all relative times therefore we have three possible types of task or question.

1. Finite.

2. Infinite.

3. Neutral.

Simultaneously.

Possibilities

There is the possibility that each type of task may either exist, not exist, or nothing.

Answers

1. Some tasks require a finite amount of time to complete.

2. Some tasks require an infinite amount of time to complete.

3. Some tasks are neutral.

Simultaneously.

The paradox and the answer within all of these similar ideas is that we need to use three answers to account for the three possibilities, at least that's what I think about it.

[Canute]

I see the definition of Russell's Paradox as "is the set of all sets a member of itself?"

I don't think this is quite right. If the set of all sets is a member of itself this is not necessarily a contradiction. Usually the starting point is the set of all sets that do not contain themselves. Call this W. Is W a member of itself? If it is, then it is not. If it is not then it is.

This kind of paradox shares its characteristics with other paradoxes too, such as "is the word word a word?" and "Is this a question?"

Again, I think this not quite right. Yes, the word 'word is a word. There is nothing paradoxical in this idea. 'Is this a question? is a bit trickier, but the answer seems to be a simple yes.

I also see "which came first, the chicken or the egg?" in the same way and I think that a common root runs within all paradoxes of this kind.

I agree. The common root, in my opinion, is epistemilogical/cosmological dualism. That is, the idea that meaningful statements must always be true or false.

If there is really a common form within all paradoxes, and therefore a universally applicable method for understanding them (a TOE) then it should follow that if we can understand and explain one paradox through Pro theory, we should be able to explain them all.

I agree again. In the same vein, Martin Heidegger argued that all metaphysical problems are the same problem and therefore have the same answer. This seems obvious to me but, needless to say, as yet not everyone agrees.

The paradox and the answer within all of these similar ideas is that we need to use three answers to account for the three possibilities, at least that's what I think about it.

I sort of agree. But your explanation of this idea suggest to me, pardon me for saying this, that you are trying to reinvent the wheel from scratch when it might be better to focus on a literature review.

You may like these extracts. They all concern what I've called 'epistemilogical dualism', and they suggest that the 'tertium non datur' rule of ordinary logic is the cause of metaphysical paradoxes. The first, for instance, suggests that the ancient One and Many paradox that baffled the early Greek philosophers (and most ever since) is a product of our assumption that in cosmology either monism or dualism must be true. Sorry about the change of font, can't put it right in quick reply mode.

"Reality is one. It must be single, because plurality, taken as real, contradicts itself. Plurality implies relations, and, through its relations, it unwillingly asserts always a superior unity. To suppose the universe plural is therefore to contradict oneself and, after all, to supose that it is one. Add one world to another, and forthwith both worlds have become relative, each the finite appearance of a higher and single Reality. And plurality as appearances (we have seen) must fall within, must belong to, must qualify the unity.

We have an idea of this unity which, to some extent, is positive. It is true that how in detail the plurality comes together we do not know. And it is true again that unity, in its more proper sense, is known only as contra-distinguished from plurality. Unity therefore, as an aspect over against and defined by another aspect, is itself but appearance. And in this sense the Real, it is clear, cannot be properly called one. It is possible, however, to use unity with a different meaning."

Francis H. Bradley
Appearance and Reality (1893)

"Where knowledge is of a dual nature (as between subject and object) then the self hears, sees, smells, tastes and feels: it knows everything.[But] where knowledge is not of a dual nature, it transcends cause, effect and action [of any kind], [it is] beyond speech, nothing can be likened to it, one cannot tell of it."

Maitri Upanishad
V, 7

"The position is simply this. In ordinary algebra, complex values are accepted as a matter of course, and the more advanced techniques would be impossible without them. In Boolean algebra (and thus, for example, in all our reasoning processes) we disallow them. Whitehead and Russell introduced a special rule, which they called the Theory of Types, expressly to do so. Mistakenly, as it now turns out. So, in this field, the more advanced techniques, although not impossible, simply don’t yet exist. At the present moment we are constrained, in our reasoning processes, to do it the way it was done in Aristotle’s day."

G. Spencer Brown
Laws of Form (1969)

"What we do … is extend the concept to Boolean algebras, which means that a valid argument may contain not just three classes of statement, but four: true, false, meaningless and imaginary. The implications of this, in the fields of logic, philosophy, mathematics, and even physics, are profound."

G. Spencer Brown
Laws of Form

"A certain caliph, wanting to test an idea on an unsophisticated person, asked his guards to range into the desert and bring him a bedouin Arab. They surrounded the first one whom they met, who happened to be a Sufi. ‘The Commander of the Faithful requires your presence,’ said the captain of the guard. ‘Who are the faithful, and how do they come to have a Commander?’ he asked. The soldiers concluded that this was indeed an unsophisticated man, and they brought him before the Caliph.

‘I have been told,’ said the ruler, ‘that bedouins are so ignorant that they do not know the simplest things.’
‘Who has told you?’
‘It was during a discussion with my intellectual advisers’.
‘If it is intellect you want, the problem is easy enough. Ask me anything.’
The Caliph ordered a dish of porridge to be brought. The Arab sniffed it and began to eat. ‘What is that?’ asked the Caliph.
‘Something that can be safely eaten,’ said the bedouin.
‘Yes, but what is its name?’
‘Adopting the methods of formal logic, applied to the knowledge available to me, I say that this is pomegranates.’
There was a laugh from the assembled scholastics who had told the Caliph that the bedouins were fools.
‘And how, pray, do you come to that conclusion?’
‘By the same methods that your scholastics use. I have heard the phrase "Dates and pomegranates" used to describe tasty foods. Now I know what dates are, as I live on them. This is not dates. Therefore it must be pomegranates.’

From ‘Esoteric Research’ (Tahqiq-I-Batini).
Reputedly written by Sir-Dan (Knower of Secrets) Daud Waraqi.

[lodestar]

Thanks for this comment Canute. I would like to ask dleviwing how he can explain the fact that time itself is timeless if it were not for the truth, the irony that is, and the just contradiction, of everything.

dleviwing

I'm no mathematician but I think Lodestar has a point about mathematics. It's hard to see much difference, for example, between the task of axiomatising the number line and axiomatising the universe.

However, as you say, "you can’t treat mathematics as a philosophy, it is simply a collection of rules that dictate the system." This is true, but imo misleading. Mtaphysics is also constructed from a set of rules that dictates the system.

You say that paradoxes only occur when "the system does not reflect the true behavior of nature." I agree, but this idea would have far reaching implications. How, for example, would we explain the innumerable paradoxes that arise in 'Western' or 'scholastic' metaphysics?

Would we say that metaphysicians (and by implication physicists) make assumptions about Nature that do not reflect Nature? This is my view, since I agree with Buddhists, Taoists and their like about Nature, or the nature of reality. Is it also your view? If paradoxes do not occur in nature then I don't see any other way of explaining the undecidability of metaphysical questions other than by saying that the assumptions that lead to them mischaracterise Nature.

Feynman famously remarked, 'the way we have to describe Nature is incomprehensible to us'. If paradoxes are all in the mind then one has to wonder whether he was not confusing an incomprehensible and paradoxical explanation with what is actually the case.

I still think there's something in 'Pro-theory', albeit that it doesn't quite seem to be a theory yet. However, I have to say that the way it's being described is currently incomprehensible to me. I see a buried connection between the idea of a neutral truth-value and certain meanings of the phrase 'Middle Way' in Buddhism, 'advaita' (not-two) in esoteric Hinduism, and more generally the 'nonduality' so often spoken of by the mystics. However, I'm struggling to understand protheory's idea of neutrality and maybe I'm way off track.

Canute

[protheory]

If the set of all sets is a member of itself this is not necessarily a contradiction. Usually the starting point is the set of all sets that do not contain themselves. Call this W. Is W a member of itself? If it is, then it is not. If it is not then it is. Yes, the word 'word is a word. There is nothing paradoxical in this idea. 'Is this a question? is a bit trickier, but the answer seems to be a simple yes.

I see what you mean actually and looking at it that way I think you're right and I agree, the answer does seem obvious at first but I see all of these questions as paradoxes requiring more than one answer.

The contradiction within all of these paradoxes comes when you try to prove why the set of all sets is a member of itself or not, as I stated in my original web page introduction Russell noticed that it was impossible to prove one answer as any more accurate than another.

I know it seems obvious at first to say that the word 'word' is a word but after we've said that what then? What if I say "well if it's a 'word' then why is it?'

We can't prove one answer as any more correct than another because providing one answer such as "yes" still leaves the two other potentials open and unanswered according to my theory.

The 'Is this a question?' one I got from a friend at Cambridge University, apparently it was a question on a philosophy paper and the accepted answer was 'Only if this is the answer.'

Even this answer given above for full marks on the exam is not totally accurate if you see what I mean as it is singular and so we need to add two other answers to view it through the TOE perspective, which is my persprective and so I say "this is a question plus the opposite plus neutral."

Perspective
If we look at it from a singular answer (non-TOE) perspective instead then obviously I agree with you, the word 'word' is of course a member of itself as is the set of all sets in turn a contradictory proof of itself.

I also agree when looking through this perspective that 'this is a question' because this is the singular (non-TOE) perspective, I didn't quite realise that we were seeing different things in quite this way before but I think that now I understand you better, I hope so anyway.

So that explains it really, you seem to be seeing one answer to these paradoxes in the logical sense, and I am seeing three potential answers because I am treating the possibilities of the words more literally.

I agree. The common root, in my opinion, is epistemilogical/cosmological dualism. That is, the idea that meaningful statements must always be true or false.

I do agree that meaningful statements must and should be true or false but I still think about three potentials at all times when I imagine the true TOE.

I do understand that perhaps you mean it's ok to say all is 123 but by its very nature you can't know absolute neutrality and that neutrality can't mean anything so to speak.

I agree again. In the same vein, Martin Heidegger argued that all metaphysical problems are the same problem and therefore have the same answer. This seems obvious to me but, needless to say, as yet not everyone agrees.

I agree here, I think that everything is everything and this is the whole reason that I'm here writing this in the first place, I wrote my idea and realised that it answered everything, all questions about all things.

I just thought when I started writing that if I can see everything in my mind's eye then everything must share a pattern, or in other words 'all metaphysical problems are the same problem and therefore have the same answer' with which I agree.

I sort of agree. But your explanation of this idea suggest to me, pardon me for saying this, that you are trying to reinvent the wheel from scratch when it might be better to focus on a literature review.

No problem, I understand what you mean about the re-invention here, I think that all knowledge is simple and everlasting so no wonder previous thinkers have seen this pattern before, I find it quite comforting in a way, perhaps I'm not quite as crazy as I thought and the only flaw with my theory is my inability to explain it clearly

I'm not so much meaning to start from scratch here, it's more the case that I came to the idea of a TOE from scratch when I originally wrote Pro theory, I started from atoms and increased the scale from there.

What I meant about other literature was that if it helps us as people to understand how and why everything forms in a pattern of threes then I think we should adapt it to the present day.

Every bit of knowledge helps doesn't it.

You may like these extracts. They all concern what I've called 'epistemilogical dualism', and they suggest that the 'tertium non datur' rule of ordinary logic is the cause of metaphysical paradoxes. The first, for instance, suggests that the ancient One and Many paradox that baffled the early Greek philosophers (and most ever since) is a product of our assumption that in cosmology either monism or dualism must be true. Sorry about the change of font, can't put it right in quick reply mode.

Thanks very much for the quotes, I've copied them to my desktop so I can re-read them later again.

I like the way they illustrate the often paradoxical nature of knowledge and reality, I particularly like the Tao Te Ching though over all philosophical texts, it's the best down to earth and non-religious text I've ever read.

Thanks for the information and if I come across any good quotes I'll be sure to let you know about them too.

I'll reply about the monoism or dualism dabate later but I feel as though things become clearer to me with each post I make.

I'll keep searching for a way to explain better.

PRO

[protheory]

The TOE versus non-TOE view

I'll try to sum up more clearly my point about the different views on Russell's Paradox and the others.


View one (non-TOE)

When for example we look at a paradox like "Is this a question?" and say "the answer is obviously yes" I see this as the non-TOE view as it is singular.

It's akin to my everyday life view of things, for example I don't meet new people and say "hello, I'm Pro, not Pro plus neutral" I just say "hello I'm Pro" because in that specific (non-TOE) context I don't need to use that level of accuracy.


View two (TOE)

The TOE view is different to the previous and obvious singular interpretation as it proposes the potential for three possibilities not just one.

When we look at the paradoxes this way we notice that if we wanted to be totally accurate beyond all possible doubt of a TOE, we must use three simultaneous answers to account for the three potential answers, we literally leave nothing unanswered.

To ignore the other two potentials and say "yes, it definitely is a question" would have been ok in the non-TOE example because we didn't need this kind of pedantic accuracy, but when we look for the TOE we need to be absolutely accurate and truthful.

To follow the example further, if I were meeting someone and I was thinking with TOE accuracy, I would say "hello, my name is Pro, not Pro plus neutral" as this is the most possibly accurate answer I could give according to the TOE.

Please forgive me for saying this but if anybody doubts the accuracy of my three answers then you can try to prove me wrong, right, or neutral.

Whatever you say I'll simultaneously oppose, agree with and neutralise your argument which further reinforces my three potentials argument plus the opposite plus neutral ad infinitum.

I'm just saying that this is ultimate literal and TOE accuracy, literally everything answered.


The Difference

The difference between the two views is only that one says one thing, and another says three things, in other words it just depends how accurate you want to be depending on what you happen to be answering.

The whole reason that I view Russell's Paradox and the others I mentioned through the TOE (three potentials) argument is because before I wrote my theory I didn't understand them at all.

I was scared of paradoxes as they made no sense, and I hadn't a hope of answering them accurately either, but when I realised my idea I started to answer them through the threes view and so I've always approached them all in three ways.

I didn't realise before that they could be seen as non-paradoxical, as in "yes, obviously the word 'word' is a word."

I only studied Russell's Paradox after the fact of Pro theory and so I've always imagined it as a paradox rather than a simple yes or no answer.

It's like how I come across as a person on this board for example, I seem to make no sense and talk rubbish but on other boards online that I visit, I'm a normal member of the community, I don't talk in threes as it's not needed when I'm chatting to someone.

The only time I use three arguments is when I'm talking about the TOE, such as when I'm on Toequest, this context forces me to use total accuracy because if I didn't use it and somebody spotted a chink in my intellectual armour, my theory of everything fails.

Pro

[Canute]

R's paradox does not have a yes or no answer. The question 'Is this word a word ?' is not paradoxical so it does have one. My current favourite book on paradoxes is 'Travels in Four Dimensions' by Robin Poidevin (OUP). I highly recommend it. It runs through all the well known paradoxes of time, space, motion, change etc. so is a useful reference.

Canute

[protheory]

R's paradox does not have a yes or no answer. The question 'Is this word a word ?' is not paradoxical so it does have one. My current favourite book on paradoxes is 'Travels in Four Dimensions' by Robin Poidevin (OUP). I highly recommend it. It runs through all the well known paradoxes of time, space, motion, change etc. so is a useful reference.

Canute

Thanks for the book information Canute, I'll look into it further on Amazon

[lodestar]

I do agree that meaningful statements must and should be true or false but I still think about three potentials at all times when I imagine the true TOE.

This right here shows what is wrong with your theory PRO. You say you think of three potentials when you think of the "true" TOE. Well what makes it the "true" TOE then if it's supposed to be about 3 potentials? If it's simultaneously false and neutral, then it's not really the "true" TOE now is it???

I think what you're doing is trying to impart meaning to your theory by calling it true even though the theory itself proves that it is false and neutral at the same time. A theory that is simultaneously true, false, and neutral is by definition undefined.

Now often times you might say stuff like "...therefore my theory is true (and false, and neutral, just to cover mysefl)" but even in that case you put the false and neutral parts inside of the parantheses, as if they are just tacked on to the larger conclusion which is that the theory is primarily true. So it is very apparent to anyone looking at your theory that you are always trying to give it a positive slant even though it should have no slant at all.

But your attempt to ascribe meaning to the theory by labeling it true even though it is simultaneously false and neutral goes to show that my theory of the optimistic principle is indeed true. The optimistic principle guides destiny and all life and it is what caused you to discover a true/neutral/false theory and label it as simply true. This proves that even in your theory you accepted the axiom of choice and chose to choose truth, instead of false, or neutral. And this is the same reason that mankind has named the positive direction as the direction in which our time flows. It is the positive direction, the direction of optimism, of valor, of justification for all. That is the optimistic principle which you try obscure within your theory, even though IT is the overriding principle, which your true nature brings out without you even realizing it. I think that's really why it ended up being called pro theory, instead of anti theory, or neutral theory, or pro/anti/neutral theory.

I agree here, I think that everything is everything and this is the whole reason that I'm here writing this in the first place

sorry if I took your quote out of context but I just had to because what you said right there solves Russell's paradox once and for all. You said specifically "everything is everything" which means that everything is part of everything which means that there is not anything which is not part of everything. Now let's translate all of what I just said into set theory terms. All sets contains all sets. That's what we mean when we say everything is everything. So all sets that contains no sets is the no-set. That is the empty set. The no set contains nothing and thus it is nothing. So the set of all sets that do not contain themselves is nothing and it is simply nothing. The set of all sets that does contain all sets, is simply everything which is also part of everything. So that is the answer to Russell's paradox. It is as complex and simple as that. 1/0=1/0 and 0=0, or rather, 0 is unequal to anything and 1/0 is equal to everything. That is the contents of the set of all sets and the set of no sets.