Dualism & Consciousness: Exploring a New Perspective

[Canute]

Dick

It occurs to me that what I wrote earlier about your work may have sounded rather dismissive of it. This is not at all how I meant it to sound. I'm genuinely interested and want to understand it properly. As I say, I'd like to start a thread to discuss it in detail but don't want to do that until I know you want to discuss it. I hope what I said earlier won't put you off.

Canute

 

[moving finger]

The fact is (and I can show it explicitly) that most all of the so called "laws of physics" are no more than solutions to that equation.

Are you saying here that our so-called “laws of physics” are the ONLY solutions to that equation?
In other words, can we, with no additional assumptions, derive our so-called “laws of physics” from that equation?
If yes, can you give examples of physical laws which can be so derived?

Have you come across Gurdjieff on free-will? His view was that if we are aware of who we are then we have free-will. If we are not then we are slaves to deterministic (psychophysical) forces.

My interpretation would be : If we “own” our beliefs, if we accept “ownership” of the causal sources of our choices and actions (no matter whether those causal sources are themselves caused) then in effect we act with free will. (This is not the same kind of free will that libertarians claim to believe in – but then the libertarian kind of free will is incoherent).

Languages are part of that structure and any information encoded in languages is necessarily wrong or incomplete, so I would agree that what we call knowledge is really nothing but confusion at some level.

The same applies to the language of mathematics (either incomplete or inconsistent)

It's worse than that, I think. If the mystics are right then the truth about consciousness is actually paradoxical in ordinary language.

What paradox?

Does freewill exist? From the above you see that there are two points of view, the conventional and the ultimate. As a result the question does not have a yes or no answer. In the literature it is never given one.

There is plenty of literature supporting both yes and no. The real problem is that first one must define precisely what one means by “free will”.

Natural language cannot cope with things like PC or wave-particles.

The problem with “wave-particles” is that most of us want to call a quantum object either a particle (it has definite position) or a wave (it has definite momentum) – the fact is that it is neither particle nor wave. We need to start thinking “outside the box” of waves and particles.

The view of omniscience as knowing all that can be known is, IMHO, nonsense for the same reasons I tried to argue with MF that any consideration of "all" of any set of possibilities is nonsense.

It is only nonsense if one rejects the idea of an infinite or unbounded set. Is the “set of all integers” nonsense?

The fact is that any “unbounded set of all things” (following Godel) is either an inconsistent set or is an incomplete set – why would this make it “nonsense”? All we need to remember is that whenever we talk of an “unbounded set of all things” (such as the knowledge of an omniscient being) we are talking of an incomplete set – a set which, no matter how big it is, can always be added to (like the set of integers).

In my view, there is undoubtedly information stored in brain structures, in DNA structures, in libraries, and on hard drives. Should this be considered "known information"? Well, yes, it makes a certain amount of sense to do so. But it complicates the question of what we mean by 'omniscience'.

Information does not entail knowledge (at least not by the conventional definition of knowledge – perhaps you have a different definition?)

Sorry if I put anyone off by going into it.

Paul – sorry, but your ideas sound more and more like religion or mysticism than philosophy……

These are the same limitations that prevent you and me, or MF and me, or Dr. Dick and me, from eliminating the "almost" that persistently modifies our "agreement". I think we might be as close to agreement as we can get without "going mathematical".

Unfortunately, it seems to me that you, Canute & Dr Dick are closer to agreeing with each other than I am to agreeing with any of you. My personal philosophy is directed at explaining the world with the minimum of assumptions, and to me the assumption of the PC (or any other fundamental conscious entity) doesn’t seem to explain anything useful, but instead wraps up within itself so much that is not explained.

This may be the ideal starting point from which to disentangle our differences. Russell questioned whether human beings could know anything at all. It is impossible to demonstrate that they can. So how do we know anything? And how do we know we know we know it? Cetainly not by deriving theorems from uncertain axioms. But how can we know our axioms are true? According to Russell we can't. But Russell was a devout non-mystic, and thus could have no theory of knowledge. He assumed that knowledge was the same thing as explanation or proof by demonstration, and this assumption seems to underly some of what you are suggesting, and maybe also Dick. Can you give your views on the relationship between knowledge, proof and explanation? This might clear up some possible misunderstandings. (By 'explanation' I would mean also a description or a theory).

I think you need to be clear about your definition of “knowledge” – can you clarify what you mean when you talk of “knowledge”? (I suspect when you talk of knowledge you are thinking of “certain knowledge” – which imho would be a mistake).

As an experiment try picking one piece of knowledge that you know with absolute certainty and then figure out how you know it. It will not be anything to do with explanations, theories, formal proofs, libraries, hard drives and so on.

Do we know anything with absolute certainty?

But that conundrum appears, as I have said many times, in each and every and all attempts, by philosophers, scientists, theologians, mystics, quacks, and anyone else, to describe the ultimate beginning of reality.

Which is why the notion of “no beginning” is appealing.
If there is a beginning to time, then we need an explanation for that beginning.
If there is no beginning to time, then no explanation for any beginning is needed.

From 'Laws of Form' -G. S. Brown
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.

Could you give an example of an imaginary statement which is neither true, false, nor meaningless?

Proof - For me there are four kinds of proof. Inference by induction or deduction, and abduction (in C. S. Peirce's and Sherlock Holmes's sense - as infererence to the best explanation, ideally the only one not falsified). The fourth would be proof by direct experience, which might be called verification or ostensive proof. This latter is Aristotle's knowledge by identity.

Proof by deduction tells us nothing new about the world (ie nothing that is not already logically contained within the assumptions). All proofs by deduction are either analytic truths (truths by definition) or are tautologies.
Your “proof by abduction” is essentially a form of proof either by induction or deduction (unless you can come up with an example of proof by “abduction” which is neither induction nor deduction).
Proof by direct experience is effectively a form of proof by induction (with the single possible exception that experience tells us, by definition, that there is “something rather than nothing” – this may be the single “proof by experience” which is deductive rather than inductive).

Thus the only useful method of proof we have boils down to proof by induction – and all inductive truth is probabilistic and not certain.

Knowledge - Two kinds, relative and absolute. Relative knowledge (provisional, contingent) would be that 2 + 2 =4, or that the Earth orbits the sun. Absolute knowledge would be knowledge by identity, e.g. the unfalsifiability of solipsism, the 'I am' of the Sufis, the void spoken of by Brown).

The only “absolute” knowledge (imho) is “something exists”. What this something is, we do not necessarily know. All we know for certain is that something (as opposed to nothing) exists.

Explanation (theory, description etc) - a formal system of terms and symbols in which relative truths and falsities are demonstrated to be derivable from axioms. The axioms may be postulates or they may be known facts. ('Known facts' would have to be absolute knowledge). Explanations would normally be subject to the limits of the incompletenss theorem. No explanation could communicate certain knowledge, although they may point towards it, or explain where it can be found.

I prefer Dr Dick’s interpretation of “explanation” – an explanation is simply a way of describing one set of information in terms of another set of information (or : an explanation is a mapping, or series of mappings, between different sets of information).

Best Regards

 

[Canute]

Moving Finger

What paradox?

Metaphysical paradoxes. I would include the 'hard' problem of consciousness as one of these.

There is plenty of literature supporting both yes and no. The real problem is that first one must define precisely what one means by “free will”.

Yes, I agree. My point was just that the arguments supporting yes or no both fail. (Which is why the debate is ongoing). The suggestion is that truth about it is subtle, in such a way that the yes and no arguments are both flawed. This would explain why the debate is ongoing.

The problem with “wave-particles” is that most of us want to call a quantum object either a particle (it has definite position) or a wave (it has definite momentum) – the fact is that it is neither particle nor wave. We need to start thinking “outside the box” of waves and particles.

I couldn't agree more. Now try applying the same principle to metaphysical questions, such as the whether we have freewill, whether the universe begins with something or nothing etc. This is what Brown means by using imaginary values in our reasoning.

Paul – sorry, but your ideas sound more and more like religion or mysticism than philosophy……

I feel it would be best to call it an esoteric philosophy. But I don't know if Paul would agree.

Unfortunately, it seems to me that you, Canute & Dr Dick are closer to agreeing with each other than I am to agreeing with any of you. My personal philosophy is directed at explaining the world with the minimum of assumptions, and to me the assumption of the PC (or any other fundamental conscious entity) doesn’t seem to explain anything useful, but instead wraps up within itself so much that is not explained.

But if a single assumption explains everything else, and does not produce a reductio argument against the assumption, then would this not be a reasonable strategy to make the assumption? Still, I half agree with you, since the result can only be a theory.

I think you need to be clear about your definition of “knowledge” – can you clarify what you mean when you talk of “knowledge”? (I suspect when you talk of knowledge you are thinking of “certain knowledge” – which imho would be a mistake).

Yes, I distinguish between certain knowledge and provisional or relative knowledge. Why would it be a mistake to talk about certain knowledge?

Do we know anything with absolute certainty?

Only you can answer that. I'm certain of one or two things. The unfalsifiabilty if solipsism is the nearest I can get to knowledge that is certain and communicable. The knowledge that I'm hungry is certain, but I can't demonstrate it.

Which is why the notion of “no beginning” is appealing.
If there is a beginning to time, then we need an explanation for that beginning. If there is no beginning to time, then no explanation for any beginning is needed.

This seems true to me also. However, the idea of an eternal substance is paradoxical, so this view still leaves a question begging.

Could you give an example of an imaginary statement which is neither true, false, nor meaningless?

Well, not an imaginary statement, but I know what you mean. The statement 'the universe arises from nothing' is true or false in metaphysics. The statement 'the universe arises from something' is true or false likewise. However, in the esoteric or 'advaita' view both these statements are neither quite true nor false. The truth would be that the universe arises in a sense and in a sense does not, and that it arises from nothing in a sense but in a sense arises from something. Hence this comment by Robin Robertson about Brown's calculus.

"Anyone who thinks deeply about anything eventually comes to wonder about nothingness, and how something (literally some-thing) ever emerges from nothing (no-thing). A mathematician, G. Spencer-Brown (the G is for George) made a remarkable attempt to deal with this question with the publication of Laws of Form in 1969. He showed how the mere act of making a distinction creates space, then developed two "laws" that emerge ineluctably from the creation of space. Further, by following the implications of his system to their logical conclusion Spencer-Brown demonstrated how not only space, but time also emerges out of the undifferentiated world that preceeds distinctions. I propose that Spencer-Brown’s distinctions create the most elementary forms from which anything arises out of the void, most specifically how consciousness emerges."

Proof by deduction tells us nothing new about the world (ie nothing that is not already logically contained within the assumptions). All proofs by deduction are either analytic truths (truths by definition) or are tautologies.

Agreed.

Your “proof by abduction” is essentially a form of proof either by induction or deduction (unless you can come up with an example of proof by “abduction” which is neither induction nor deduction).

Abduction is generally considered to be a third form of inference. C. S. Peirce takes this view, as do most dictionaries. I agree that it is a form of deduction, but still feel the distinction is useful. A deductive proof would normally be more certain than a abductive one. But at the limit, say in the case where all explantions of a phenomenon except one had been eliminated, they would be effectively the same thing.

Proof by direct experience is effectively a form of proof by induction (with the single possible exception that experience tells us, by definition, that there is “something rather than nothing” – this may be the single “proof by experience” which is deductive rather than inductive).

I don't agree here. If I am experiencing pain then this is direct knowledge, not an inference. It is not possible to be mistaken, even if the pain is in a phantom limb it is being experienced.

Thus the only useful method of proof we have boils down to proof by induction – and all inductive truth is probabilistic and not certain.

This is not the case. You can see this if you consider how it is possible to prove (to yourself) that you are conscious. Still, I wouldn't object to calling this verification instead of proof.

The only “absolute” knowledge (imho) is “something exists”. What this something is, we do not necessarily know. All we know for certain is that something (as opposed to nothing) exists.

Hmm. Believe it or not I don't agree. The stament 'something exists' is not true according to many people. Mind you, the statement 'nothing exists' would also be not true. Here we meet another of Brown's complex values. There are a few proofs around that nothing really exists, although not everyone agrees they are successful.

I prefer Dr Dick’s interpretation of “explanation” – an explanation is simply a way of describing one set of information in terms of another set of information (or : an explanation is a mapping, or series of mappings, between different sets of information).

I'm ok with that, depending on how 'information' is defined. But I think this falls within my definition. The question arises of whether everything that can be known and understood can be explained.

Cheers
Canute

 

[Paul Martin]

Paul – sorry, but your ideas sound more and more like religion or mysticism than philosophy……

The way you speak of PC sometimes suggests Creationism to me, that PC creates universes by intention and lays down the laws by which they will evolve. But if spacetime universes follow directly from what is, then no intention or will can be involved.

Yes, I'm afraid my ideas do sound "religious". And I am sorry about that too. My ideas sound religious because they include intention and consciousness as operative functions in the development of the universe prior to the appearance of brains or even biological organisms. I am sorry that these ideas have the taint of "religion" because of the awful baggage accumulated by religious precepts and notions as they have developed over the millennia.

We need to start thinking “outside the box” of waves and particles.

Agreed. And with respect, I suggest we need to start thinking "outside the box" of science and religion. In my view, religion has done virtually nothing with whatever "truths" it has apprehended to help solve human problems if you net out the harm from the good. Science, on the other hand has produced astounding and wonderful solutions to most human problems even after netting out the harm from the good.

But as I see it, there are still some "gaps" in the explanations of science which naturally and traditionally fall into the purview of religion. In my humble opinion, science may be able to close some of these gaps by extending their boundaries, methods, and assumptions. I think they need to think "outside the science box". In the case they are successful, it will only extend science and diminish religion; it will not necessarily make science "religious".

What confuses me is that you agree with me far more often than I agree with you.

Unfortunately, it seems to me that you, Canute & Dr Dick are closer to agreeing with each other than I am to agreeing with any of you.

This reminds me of the duelist who was concerned because his opponent was a better shot than he was. He offered the following solution in order to make the duel fair: He said, "How about if I stand twice as far from you as you stand from me?" Maybe dualists can do the same?

I think that the reason I seem to agree with each of you more than you think you agree with me is that I can interpret most of what each of you says to make sense in my "PC scheme". To the extent that each of you accepts my PC notion, you might agree with me, otherwise my ideas probably seem like nonsense to you. As I have said before, if you can show me where my ideas are nonsensical, I will gladly abandon them. Merely labeling them as "religious", or "nonsense", however, doesn't convince me -- not that any of you do that.

This shows simply that we cannot answer all possible questions about the naïve idea of the set of all sets (ie some questions are unanswerable). Naïve set theory was superceded by various axiomatic set theories (of which Zermelo-Fraenkel (ZF) set theory is the most well known) which avoid Russell’s paradox. Godel later showed that no system of set theory can be shown to be both consistent and complete, but even this does not entail paradox. It entails only that the set of all logical possibilities cannot be shown to be complete if we also wish it to be consistent. What does this mean? It means simply that the set of all logical possibilities cannot be a complete set.

You seem to know more mathematics than you let on. I'm glad of that because maybe you can help me -- again. I have a problem with the foundations of mathematics which I tried to spell out in my thread at https://www.physicsforums.com/showthread.php?t=49732 . I was not satisfied with the response I got there so I decided that I need to go back to school at some time and study the foundations to see if I can't resolve my problem. Maybe you can help me before I do that. If you wouldn't mind, take a look at that thread and see if you can help me out.

It is my understanding that the Axiom of Choice is one axiomatic way of getting an infinite number of integers defined with only a finite number of axioms. By adding the Axiom of Choice (C) to ZF set theory, producing ZFC set theory, the infinite set of integers can be defined. Does ZF contain the infinite set of integers? If so, how are they defined? If not, then is there a largest integer in ZF?

One cannot legislate against the question that Russell asked. One cannot simply say “we will prohibit consideration of infinite sets” and then blindly hope the paradox goes away. The problem highlighted by Russell remains – there are some questions which cannot be answered. Even if legislation is introduced to exclude infinity from mathematics, one can still ask the question “is the class of all classes that are not members of themselves a member of itself?”, and the question is still unanswerable.

We can legislate however we like; we must, however, live with the consequences. If we prohibit consideration of infinite sets, by providing no way to define them in our axioms, then IMHO the paradox does not appear in the first place. Russell's approach with his Theory of Types, on the other hand, prohibits consideration of certain sets in some propositions, i.e. different rules for "classes" than for "sets", and IMHO represents the "blind hope that the paradox goes away". I would really appreciate your shedding whatever light you can on this.

I have my own thoughts about the paradoxes introduced by the concept of infinity in conventional number theory – it has to do not with the concept of infinity itself, but with the rather strange notion that mathematicians seem to have of an integer. For some strange reason, in conventional number theory we choose to define an integer as an arbitrarily large number, with every integer representable by a finite string of digits.

I'm not aware of that definition. In what axiomatic system are integers defined that way?

But it is impossible to uniquely identify every member of an infinite set with finite strings, which implies that an infinite set of integers must contain members of infinite length, which in turn contradicts the definition of an integer.

I agree with this (intuitive) conclusion. I think similar arguments can be brought to bear against Cantor's definitions and I think they also apply to Goedel's proof.

I think Feynman was trying to express the same thought when he told his nephew (I think it was) that, "There are more numbers than numbers."

No because now you must define "a mapping", various sets of information and, explain these things. They cannot be more basic than the concept of an explanation.

And your definition of explanation is “a method of obtaining expectations from given known information” – can you define “method”, “expectations”, “given”, “known” and “information” without either tautology or infinite regress?

With respect, I agree with MF here, Dick. I think you are mixing up two different methods of investigation. You want to stick to the logico-mathematical methods, but you are forced to get into an English vernacular dialog here for those of us who can't follow your math completely. And, you will have to admit, even in your paper, you include many non-mathematical English sentences which have been a source of aggravation and disagreement for many of your readers.

I think the limitations we need to keep in mind are these:

In formal mathematical development, we must start with undefined primitives, unprovable axioms, mysterious and non-specific rules of logic, a portion of some natural language in which propositions can be stated, an assumption that someone else might read the language expression of the development (This one is not absolutely necessary unless the development is to be useful at all), and an assumption that you, the developer, has enough continuity and coherence of thought to produce a sensible development. (This last assumption is, in your case Dick, your familiar assumption of the two types of mentality available to you: formal logic and squirrel logic.) That's a lot of assumptions and each one suggests some reason to question the veracity of any conclusions drawn.

In vernacular English conversation, such as we are doing in this forum and which is the primary method of philosophy, the ambiguities are not collected together in the primitives and axioms of formal systems, but instead are rife throughout the lexicon and even the grammar. With the severe limitations of natural language it is a wonder to me that we ever come to agreements on anything more significant than questions like, "Do you want fries with that?"

I am pleased and amazed that we come as close to agreement as we do here. As I have said before, I think that nearly all of our disagreements are semantic. I think we just need to be careful to realize that in our discussions here, we are involved in a vernacular conversation, not in the development of a formal system. I think we get into trouble when we talk about the formal systems of Russell, Cantor, Zermelo, Spencer Brown, Dr. Dick, etc.

A case in point is Dick's insistence that the concept of 'explanation' is fundamental to his argument. It probably is in his formal development, but it certainly isn't in our vernacular conversation here.

I think there are two approaches we could take. We could try to rigorously establish what we think are the primitive, or fundamental terms we want to use, and rigorously define the rest in terms of these, and then proceed in our discussion. But then we would be doing mathematics, and we might make better progress by studying Spencer Brown, Goedel, Schroedinger, or someone else who has already taken this approach and gained some ground.

The second approach is to continue to probe the differences in our respective understandings or connotations of the words and phrases we use, in an attempt to discover whether the differences are only semantic or whether they constitute some significant conceptual disagreement. I think that we are making some substantial progress with this approach, and I am delighted.

For example,

...experience tells us, by definition, that there is “something rather than nothing”

I consider this to be equivalent to the proposition that "thought happens". Now, to figure out whether or not we disagree on this, let me ask you, MF, do you think that the "something" that exists could be thought? Could you accept a definition of 'thought' that makes it something? Or would you prefer to consider thought as nothing?

In my humble opinion, I think that there are no iron-clad answers to those questions. I think it is simply a matter of opinion about how to express concepts in language in a way that is first satisfactory to us and secondarily has a chance of conveying that concept to another. It is "simply" that, but that is far from simple. I think we're doing the best we can.

(more to follow)

Paul

 

[Paul Martin]

Yes. I am convinced that Dr. Dick has proved this to be the case.

I haven’t seen it myself.

You can see it at http://home.jam.rr.com/dicksfiles/reality/Contents.htm .

OK, so the PC makes a choice, and thereafter the PC is constrained by the laws of mathematics and physics, yes?

Yes.

Apart from “making the choice to be consistent”, the PC does not actually create these laws, the laws follow on as a necessary consequence of the PC’s consistency decision?

No. I think that at some point PC actually "does mathematics" by choosing primitives, axioms, and definitions, which then imply, or "create" the laws. Yes, this imbues PC with a lot of anthropomorphism, but the capability to do math, IMHO, developed after a long stretch of time prior to the Big Bang. PC evolved and advanced to a huge degree beyond its extremely rudimentary, simple, fundamental primordial condition. I think this is the point you miss when trying to understand my ideas. I think there was probably a huge amount of trial and error before the precise conditions for an interesting universe like ours were stumbled upon.

This indeed seems to confirm that the PC does not actually create the laws, the laws follow on as a necessary consequence of the PC’s consistency decision. Correct?

Almost. The laws follow on as necessary consequences of the PC's consistency decision and the particular choices of primitives, axioms, definitions, and boundary conditions.

OK, so the first thing the PC did was in fact NOT that it chose to be consistent. It first had to experiment with many different possibilities, until it gained enough information about the world to then make a rational decision to be consistent?

YES! I'm sorry I didn't make that more clear earlier. That is what I was trying to say when I said that PC was extremely rudimentary and limited at the outset and that PC underwent an extremely long evolution of acquiring the capabilities and knowledge necessary to pull off a Big Bang.

It seems that your PC is becoming less and less primordial as we go along.

It seems that way to you because PC is becoming less so as we go along in this conversation and you are beginning to see what I have been trying to say all along. And, in my view, PC did indeed become less and less primordial as reality developed from its initial primordial and extremely simple and limited condition.

We are already now speculating about some kind of background environment in which the PC learns (about logic, consistency etc), and we also seem to think that the laws of logics, mathematics and physics are constrained to be necessary by virtue of consistency, quite independent of the experimentation by the PC.

Yes, exactly! I am absolutely delighted that you used the pronoun 'we' here. I have been doing exactly the speculation you described for quite some time. I am happy to learn (or at least hope) that you are beginning to entertain the same speculations. The next step is to ask you whether these speculations make any sense to you, or are they nonsense? I am sincerely eager to hear your opinions.

If the PC supervenes on, and does not create, the laws of logic and of mathematics, then it follows that these laws EITHER exist prior to the PC coming along, OR that they spontaneously come into existence at the moment of creation of the PC. Which?...
But you have just said that the PC supervenes on the laws of logic and mathematics. Now it seems you are saying the reverse.

I'm sorry that I don't understand the word 'supervene' well enough to use it in a sentence, and as a result I may be misunderstanding your questions. But here's how I see it. When PC was truly "primordial" it had no capability of contemplating laws, logic, mathematics, or anything else. These capabilities developed over a long period of time as we have discussed above. It would be a fair question to ask in what exact sequence did the various capabilities appear, such as the recognition of the notion of consistency, the rules of inference for logical deduction, the idea of logical consequences, particular choices of primitives and axioms, the ability to grind out the results of algorithms operating on specific sets of numbers, etc.

The question is fair, but I think not trivial. In my opinion, this is exactly the question that Spencer Brown and Chris Langan have attempted to answer. I think it is essentially the same question that Whitehead and Russell attempted to answer with their Principia Mathematica. I'm not sure at this point if anyone has yet come to a conclusive answer, but I do suspect it is within our reach, if not our grasp yet. I think it is possible to demonstrate logically how all of reality could have developed from nothing but the ability to know of some distinction.

But to give you my guess at the answer to your question, I'd say, Neither. The laws of logic and math did not exist prior to the existence of PC. The laws did not spontaneously come into existence "at the moment of the creation of the PC", which I take to mean the initial or primordial appearance of PC, however that came to be. The laws spontaneously came into existence the moment that rules of logical inference were adopted by PC as a deliberate choice. This is just like the fact that a bishop cannot occupy a square of a different color spontaneously comes into existence the moment the chess board and the rules of chess are defined. The fact did not pre-exist the definition of the game in any sense, and the fact is a logical consequence of the rules for the initial placement of the bishop and the rules for its legal moves.

What comes first – the constraint imposed by the rule, or the making up of the rule?

The making up of the rules of logical inference comes first. The constraint then applies to any further rules that are chosen and which conform to the rules of logical inference.

But this implies that the rules (of logic and mathematics) supervene on the PC, not the other way about (which is the reverse of what you agreed above).

I don't understand the word 'supervene', so I can't comment any further.

OK, I shall try to avoid reference to time by using supervenience instead. Do the rules supervene on the PC, or does the PC supervene on the rules? (you seem to have claimed both so far).

Well, here we have a problem we can work on. You seem to need the word 'supervene' in order to make your point, and I don't understand the word. I need to use the notion of time in order to explain exactly how I see the pre-BB evolution of reality and you want to avoid that discussion. I hope you are willing to work on one, or preferably both, of these problems.

IMHO PC "creates" by thinking that it creates, as you seem to imply.

That is interesting. Perhaps the PC only thinks that it is inventing as well?

I'm not sure how to interpret your question, but I'm glad you find my statement interesting. You taught me to ask whether PC, being the ability to know, really knows or only thinks it knows. I think that is a profound question and I have thought about it a lot. It is reminiscent of the recent question someone here asked of whether free will requires awareness. In my PC scheme, the questions of this type, in the context of humans are answered as follows: Humans only think we are conscious, know things, and have free will. We are usually aware (really think we are aware) of our thoughts only in the context of our human existence. But in cases usually labeled as "altered states", we can sometimes become aware of a higher level of existence, from which perspective, "we" know that the human "mind" is an illusion caused by PC vicariously using or driving the brain and thus limiting its attention and awareness to the physical human environment and situation. So the human mind thinks it knows things, but it really doesn't. PC (and add considerable amount of complexity here with the multiple levels of reality with their respective PC-driven vehicles, or Natural Individuals) ultimately is the only knower who really knows. But, if you go up the entire hierarchy asking exactly who (which natural individual) knows exactly what, the answer is that all the lower levels only think they know, and it is only the top level PC who actually knows anything.

But here is where the notion of time comes into play. If time is only a parameter marking PC's progress in attending to the world line of a Natural Individual, then there is no time at all when we consider what PC knows at the very highest level. So it is hard to conceive how PC can or does know anything at all at that level. Maybe PC only thinks it knows anything and there is really nothing known at all in reality. Who knows?

That's probably more than enough for now.

Very sorry to hear that – my condolences and best wishes

Thank you.

Warm regards,

Paul

 

[PIT2]

And, in my view, PC did indeed become less and less primordial as reality developed from its initial primordial and extremely simple and limited condition.

Do u also think there was a beginning to the PC? And did time exist during PC's pre-bigbang evolution?

Obvious questions, but i couldn't resist asking them.
(also forgive me if uve already mentioned it, i didnt read all the giant posts here)

If time is only a parameter marking PC's progress in attending to the world line of a Natural Individual, then there is no time at all when we consider what PC knows at the very highest level. So it is hard to conceive how PC can or does know anything at all at that level.

Im not sure what u mean with that PC doesn't know time at the highest level. Is this something like: 'when one is omniscient then nothing is relative, so time disappears'?

 

[Paul Martin]

Hi PIT2,

Sorry for being so lengthy in my posts. Thanks for reading what you do.

Do u also think there was a beginning to the PC?

I don't really have an opinion on that question. Maybe PC (the bare ability to know) sprang at once out of nothing. Maybe there always was a PC and all at once it noticed some distinction for the first time. Maybe something else accounts for its appearance. But I'd say that the starting point for reality involved a rudimentary PC regardless of how it got there.

And did time exist during PC's pre-bigbang evolution?

I'd say, Yes. I think time existed as soon as PC began pondering any sequence of anything. (Of course the only candidates for "anything" are concepts.) A dimension of time arises simply as any mark or measurement of the progress of PC's attention as it ponders elements of a sequence of concepts. The really interesting sequences are the events in the vicinity of the world line of a natural individual, but I see no reason (except for a way for PC to get the information) why PC couldn't ponder the sequence of positions of some inanimate object like a photon or a galaxy.

But...and maybe this is the question you really meant to ask, time did not exist prior to PC noticing that first distinction, because "prior" to that, there was no sequence for PC to ponder, and indeed, there was not even any pondering.

Im not sure what u mean with that PC doesn't know time at the highest level. Is this something like: 'when one is omniscient then nothing is relative, so time disappears'?

I wouldn't put it that way. First of all, I don't think there is any such thing as omniscience; in particular I don't think PC is omniscient. It's more like if PC is pondering the highest level, then there are no constructs of concepts to ponder, so there is no pondering, and thus time does not appear. It's not like it was there and then disappeared, it just isn't there at all. All concepts, and thus all time, appears only in the lower levels of the hierarchy.

I hope that helps.

Paul

 

[Canute]

Paul

Great post. By the way, when I said your ideas sounded suspiciously like creationism I didn't mean to criticize them for sounding religious. I couldn't care less whether they are religious, scientific or philsophical or none of these. My point was rather that I have a problem with the idea that the universe was created by intention.

This extract from MF interests me, and seems crucial.

Godel later showed that no system of set theory can be shown to be both consistent and complete, but even this does not entail paradox. It entails only that the set of all logical possibilities cannot be shown to be complete if we also wish it to be consistent. What does this mean? It means simply that the set of all logical possibilities cannot be a complete set.

I agree with all of this except for the final sentence. I may be misunderstanding it but it seems to assume that ordinary logic (ordinary equation theory, Boolean algebra) is the only sort there is. But we do not use this logic in quantum theory, we make use of complex values.

This may seem a small point but I feel it's central to the discussion. Suppose that PC (not my choice of term but no matter) is a contradiction in ordinary logic? Suppose it is something that is logically equivalent to a wave-particle? In this case the true 'explanation of everything' would be inconsistent and complete. It is this get out clause that prevents Godel's proof from being a disproof of the cosmology of Taoism, Buddhism etc. The structure of these cosmologies is isomorphic with quantum theory, two contradictory but self-consistent explanations of the same self-contradictory explanandum. It is this curious property of the Tao and Brown's void that I feel is missing from Paul's PC theory.

This relates to MF's point about Godel and Russell directly, since it is precisely by adopting the logical scheme of quantum theory that Brown overcomes the need for Russell's 'Theory of Types' (as Russell acknowledged) and avoids logical paradoxes in his calculus (which is a model of his cosmology). This is undoubtedy the hardest thing to understand about Brown's view (and the Buddha's, Lao Tsu's etc).

On the issue of language I agree with most of what Paul wrote. But this passage is worth some thought. It's taken from Werner Heisenberg's Physics and Philosophy(1959).

"The vagueness of this language in use among the physicists has therefore led to attempts to define a different precise language which follows definite logical patterns in complete conformity with the mathematical scheme of quantum theory. The result of these attempts by Birkhoff and Neumann and more recently by Weizsäcker can be stated by saying that the mathematical scheme of quantum theory can be interpreted as an extension or modification of classical logic. It is especially one fundamental principle of classical logic which seems to require a modification. In classical logic it is assumed that, if a statement has any meaning at all, either the statement or the negation of the statement must be correct. Of ‘here is a table’ or ‘here is not a table’, either the first or second statement must be correct. ‘Tertium non datur,’ a third possibility does not exist. It may be that we do not know whether the statement or its negation is correct; but ‘in reality’ one of the two is correct.

In quantum theory this law ‘tertium non datur’ is to be modified. Against any modification of this fundamental principle one can of course at once argue that the principle is assumed in common language and that we have to speak at least about our eventual modification of logic in the natural language. Therefore, it would be a self-contradiction to describe in natural language a logical scheme that does not apply to natural language."

If this is understood then the self-contradictory language of mysticism can be understood. This connects directly to the notion of 'nonduality' or 'advaita' that lies at the heart of the mystical doctrine. In logical terms it is the suspension of the tertium non datur rule.

Armed with this notion the problems relating to creationism, intention and so on that are IMHO problematic in Paul's view can be resolved.

Hope some of that made sense.

regards
Canute

PS. Paul - I was waiting to hear from Dick before starting a thread on your essays, but I'll go ahead anyway, just as soon as I can disentangle myself from all the different arguments I've stupidly got myself into and freed up some time to do so.

 

[selfAdjoint]

This extract from MF interests me, and seems crucial.

Godel later showed that no system of set theory can be shown to be both consistent and complete, but even this does not entail paradox. It entails only that the set of all logical possibilities cannot be shown to be complete if we also wish it to be consistent. What does this mean?

Only problem is that Goedel did not say this! He said that any logical system capable of developing arithmetic from axioms had this property. That's a big restriction because there are a lot of systems, like segments of real analysis, that don't fall under it and cannot be called undecideable.

 

[Canute]

Yeah, you're right to pick up on some sloppiness. Still, I think the general point holds once a couple of provisos are added.

 

[moving finger]

Only problem is that Goedel did not say this! He said that any logical system capable of developing arithmetic from axioms had this property. That's a big restriction because there are a lot of systems, like segments of real analysis, that don't fall under it and cannot be called undecideable.

I'm happy to stand corrected - because if anything this reinforces my point that the notion of a set of all logical possibilities does not entail contradiction.

Best Regards

 

[Canute]

What exactly do you mean by 'a set of all logical possibilities'? Do you mean the set of everything that is the case? Or do you mean that if two facts are possibly true they cannot contradict each other? Or something else?

 

[moving finger]

What exactly do you mean by 'a set of all logical possibilities'? Do you mean the set of everything that is the case? Or do you mean that if two facts are possibly true they cannot contradict each other? Or something else?

In modal logic, a proposition may be :

True in all possible worlds - ie logically necessarily true
True in some, but not all, possible worlds - ie logically contingent
False in all possible worlds - ie logically necessarily false (ie impossible).

Note that "possible worlds" refers to all logically possible worlds (ie we are not considering ourselves constrained here by physical possibility).

The set of all logical possibilities would be the set of all propositions which are either logically contingent or logically necessarily true.

Best Regards

 

[Canute]

Does this mean that the set of all logical possibilities is the set of all propositions that are tautologically true (e.g. all bachelors are single) or that may be either true or false?

 

[Canute]

Paul

I've started a thread in Metaphysics titled 'Foundations of Reality' and kicked off with a couple of questions. See you there I hope.

 

[moving finger]

Does this mean that the set of all logical possibilities is the set of all propositions that are tautologically true (e.g. all bachelors are single) or that may be either true or false?

The set of all logical possibilities is the set of all propositions, each one of which is true in at least one logically possible world.

 

[Canute]

Ok. I see what you're getting at. I wonder though. Are you not asuming that all logical possibilities can be stated as true or false propositions? What about the proposition that not all logical possibilities can be stated as true or false propositions?

 

[moving finger]

Ok. I see what you're getting at. I wonder though. Are you not asuming that all logical possibilities can be stated as true or false propositions? What about the proposition that not all logical possibilities can be stated as true or false propositions?

No, I need not assume that all logical possibilities can be stated only as either true or false propositions.

Would you care to propose a different form of logic to 2-valued (true/false) logic? We could then discuss the implications for the set of all logical possibilities (the notion of such a set does not necessarily assume 2-valued logic).

As we discussed (in another thread, or this one?), one cannot arrive at any understanding or explanation unless one makes assumptions.

One may assume 2-valued logic (in which case all meaningful propositions are either true or false), or one may assume some other valued logic.

Which logic would you like to assume for the purpose of this discussion?

(Granted that meaningless propositions may not have a truth value)

Best Regards

 

[Canute]

Yes, that's what I'm getting at, that reality may not accord with the dualism inherent in ordinary logic. In my view it does not. My suggestion is to use the modification to ordinary logic that physicists make use of in quantum theory.

(I know what you think about the necessity of assumptions, but don't forget that I haven't agreed with you yet).

 

[moving finger]

What paradox?

Metaphysical paradoxes. I would include the 'hard' problem of consciousness as one of these.

I’ve responded to the so-called “hard problem” in another thread (which, even if it were a problem, is hardly a paradox – it would be just something in need of explanation).

Are there any other paradoxes you had in mind?

The suggestion is that truth about it is subtle, in such a way that the yes and no arguments are both flawed. This would explain why the debate is ongoing.

I agree that part of the problem is defining exactly what we mean by free will. But even if we do agree what we mean by free will, the reason why the debate is ongoing (imho) is because one side insists on holding incoherent beliefs which at the same time they deny as being incoherent.

I couldn't agree more. Now try applying the same principle to metaphysical questions, such as the whether we have freewill, whether the universe begins with something or nothing etc. This is what Brown means by using imaginary values in our reasoning.

Nope. It doesn’t work (or perhaps I should say – imho one can delude oneself into thinking that one has found the answer to everything by answering nothing).

But if a single assumption explains everything else, and does not produce a reductio argument against the assumption, then would this not be a reasonable strategy to make the assumption? Still, I half agree with you, since the result can only be a theory.

Are you looking for just a single assumption? That’s called God isn’t it? Doesn’t that explain everything we need to know?

Yes, I distinguish between certain knowledge and provisional or relative knowledge.

Then how do you define “knowledge”?

I'm certain of one or two things. The unfalsifiabilty if solipsism is the nearest I can get to knowledge that is certain and communicable.

That’s an example of an analytic truth or a tautology (a truth by definition - like the certainty that “all bachelors are unmarried”) – yes one could claim it is certain knowledge, but it doesn’t tell us anything useful about the world.

The knowledge that I'm hungry is certain, but I can't demonstrate it.

Are you hungry? Or do you just think you are hungry (ie you just think that you have hunger pains, when in fact you don’t really have them)? Perhaps you are hallucinating, perhaps you have been drugged or hypnotized to think you are hungry, perhaps you are a brain in a vat which an evil scientist is experimenting with, sending signals to your brain.

This seems true to me also. However, the idea of an eternal substance is paradoxical, so this view still leaves a question begging.

Why paradoxical? Because you assume there must be a beginning?

Well, not an imaginary statement, but I know what you mean. The statement 'the universe arises from nothing' is true or false in metaphysics. The statement 'the universe arises from something' is true or false likewise. However, in the esoteric or 'advaita' view both these statements are neither quite true nor false. The truth would be that the universe arises in a sense and in a sense does not, and that it arises from nothing in a sense but in a sense arises from something. Hence this comment by Robin Robertson about Brown's calculus.

Then the question is ambiguous. It is all too easy to avoid giving “yes” and “no” answers (as many eastern philosophies do) by claiming “in one sense it is yes, in another sense it is no” – all this means is that one’s understanding of the question is ambiguous. Imho to claim that one possesses “knowledge” or “understanding” by simply uttering meaningless statements like “the universe arises in a sense and in a sense does not” (without further qualification) is to delude oneself.

In what “sense” does it arise from something?
And in what “sense” does it arise from nothing?

Anyone who thinks deeply about anything eventually comes to wonder about nothingness, and how something (literally some-thing) ever emerges from nothing (no-thing).

This explanation assumes that the “something” did emerge from “nothing” – perhaps the “something” has existed for all time (in which case there is no “emerging”)

I propose that Spencer-Brown’s distinctions create the most elementary forms from which anything arises out of the void, most specifically how consciousness emerges.

Nice “proposition” – but hardly a detailed explanation.

Abduction is generally considered to be a third form of inference. C. S. Peirce takes this view, as do most dictionaries. I agree that it is a form of deduction, but still feel the distinction is useful. A deductive proof would normally be more certain than a abductive one.

Proof by deduction is 100% certain (assuming the premises are true). Note that I am saying proof by abduction is either a proof by induction or by deduction (I am not saying that all proofs by abduction are proofs by deduction).

But at the limit, say in the case where all explantions of a phenomenon except one had been eliminated, they would be effectively the same thing.

Agreed the distinction may be useful – just as it is useful to distinguish between men and women – but they are all humans. The important point however is that (I believe) any proof by abduction is also either a proof by induction or a proof by deduction. The point I am trying to make is that fundamentally all proofs are either by induction or deduction – and only the latter provides certainty.

I don't agree here. If I am experiencing pain then this is direct knowledge, not an inference. It is not possible to be mistaken, even if the pain is in a phantom limb it is being experienced.

Pain is not knowledge – pain is a phenomenal interpretation that your conscious mind places on certain states that seem to occur within your brain. What “knowledge” do you think your phenomenal experience of pain is giving you (apart from the tautological knowledge that “experience of pain means I am experiencing pain”)?

Whatever “knowledge” you think you have from the experience of pain (apart from tautological knowledge), how do you know it is not possible for you to be mistaken? You may be a brain in a vat, with an evil scientist stimulating various parts of your brain to make you think that you are experiencing things when in fact you are not experiencing those things at all. When you feel a “pain in your foot” you don’t believe that you feel that pain in your brain – you believe that you feel the pain in your foot. But if you are a brain in a vat, you have no feet, and the stimuli that the evil scientist is providing to your brain are (perhaps) simply low-voltage electrical signals which your brain interprets as “that is a pain in my foot”.

This is not the case. You can see this if you consider how it is possible to prove (to yourself) that you are conscious. Still, I wouldn't object to calling this verification instead of proof.

How would you go about proving to yourself that you are conscious? (suggestion : first you need to define consciousness)

Hmm. Believe it or not I don't agree. The stament 'something exists' is not true according to many people

Would you care to defend this position?

Mind you, the statement 'nothing exists' would also be not true. Here we meet another of Brown's complex values. There are a few proofs around that nothing really exists, although not everyone agrees they are successful.

Would you care to support your claims with some examples?

I'm ok with that, depending on how 'information' is defined. But I think this falls within my definition. The question arises of whether everything that can be known and understood can be explained.

Absolutely everything may be explained – the real question is whether the proffered explanation is true or not (the easiest way to explain everything, which requires very little effort or thought, is the way that theists do it).

(continued in next post)

 

[moving finger]

I think that the reason I seem to agree with each of you more than you think you agree with me is that I can interpret most of what each of you says to make sense in my "PC scheme". To the extent that each of you accepts my PC notion, you might agree with me, otherwise my ideas probably seem like nonsense to you. As I have said before, if you can show me where my ideas are nonsensical, I will gladly abandon them. Merely labeling them as "religious", or "nonsense", however, doesn't convince me -- not that any of you do that.

Paul - I don’t claim that your ideas are nonsensical – they constitute (imho) just unnecessarily complex assumptions. Your ideas seem to wrap up a lot of complexity within their assumptions; complexity which I believe is emergent rather than primordial.

You seem to know more mathematics than you let on.

Naaah, I’m just an experienced bullsh**ter, who knows a lot less than he thinks.

I'm glad of that because maybe you can help me -- again. I have a problem with the foundations of mathematics which I tried to spell out in my thread at https://www.physicsforums.com/showthread.php?t=49732 . I was not satisfied with the response I got there so I decided that I need to go back to school at some time and study the foundations to see if I can't resolve my problem. Maybe you can help me before I do that. If you wouldn't mind, take a look at that thread and see if you can help me out.

It is my understanding that the Axiom of Choice is one axiomatic way of getting an infinite number of integers defined with only a finite number of axioms. By adding the Axiom of Choice (C) to ZF set theory, producing ZFC set theory, the infinite set of integers can be defined. Does ZF contain the infinite set of integers? If so, how are they defined? If not, then is there a largest integer in ZF?

Axiom of Choice, or Axiom of Infinity?

See http://en.wikipedia.org/wiki/Axiom_of_infinity

A (positive) integer is (allegedly) a number which can be obtained from adding 1 to itself a *finite* number of times.

The problem is, I have no idea how one can generate an infinite set of integers (ie a set with infinite cardinality) using this procedure. Do you? (see below)

We can legislate however we like; we must, however, live with the consequences. If we prohibit consideration of infinite sets, by providing no way to define them in our axioms, then IMHO the paradox does not appear in the first place.

But it does. Russell’s paradox has nothing to do with infinity – it has to do with unrestrained self-referentiality. Even in a finite number system, one can still ask “is the class of all classes that are not members of themselves a member of itself?”

Russell's approach with his Theory of Types, on the other hand, prohibits consideration of certain sets in some propositions, i.e. different rules for "classes" than for "sets", and IMHO represents the "blind hope that the paradox goes away". I would really appreciate your shedding whatever light you can on this.

Remember I’m a bullsh**ter. Russell’s Theory of Types did not exclude infinity (as you seem to wish to do), it excluded (as you point out) the conflation of “classes” and “sets” – this was an attempt to draw a distinguishing line between naïve sets on the one hand, and the consideration of “self-referential sets of sets” on the other hand (which latter ultimately leads to his paradox). Thus, Russell’s paradox is not a consequence of infinity, it is a consequence of unrestrained self-referentiality. THIS is why I said that legislating against infinity does not make the problem go away.

I'm not aware of that definition. In what axiomatic system are integers defined that way?

I had a long battle with the Maths geniuses on this forum a couple of years ago, in which I was basically told that I was an ignoramus for suggesting such a thing as an infinite integer –

They aren't integers. Go learn some maths

A (positive) integer is a number which can obtained from adding 1 to itself a *finite* number of times.
The strings you wrote out are not elements of the integers, nor R, with any reasonable interpretation of them.
They are elements of a p-adic system, though.

Integers in base 10 with the usual rules of presentation have only a finite number of non-zero digits. You should possibly hold back from telling some people who all have degrees or higher in mathematics or related areas things like that.

There is no such thing as an infinite integer.

There exist infinitely many integers, each of which is a finite number.

The set of integers has infinite cardinality, but each individual integer has finite magnitude.

My question (still unanswered) : If each and every integer is constructed by “adding 1 to itself a finite number of times” (this is the argument that leads to the conclusion that every integer is finite), then how is it possible to produce a set of integers with infinite cardinality?

But it is impossible to uniquely identify every member of an infinite set with finite strings, which implies that an infinite set of integers must contain members of infinite length, which in turn contradicts the definition of an integer.

I agree with this (intuitive) conclusion.

Amazing – we agree! Unfortunately, most conventional mathematicians think this is nonsense.

In formal mathematical development, we must start with undefined primitives, unprovable axioms, mysterious and non-specific rules of logic, a portion of some natural language in which propositions can be stated, an assumption that someone else might read the language expression of the development (This one is not absolutely necessary unless the development is to be useful at all), and an assumption that you, the developer, has enough continuity and coherence of thought to produce a sensible development. (This last assumption is, in your case Dick, your familiar assumption of the two types of mentality available to you: formal logic and squirrel logic.) That's a lot of assumptions and each one suggests some reason to question the veracity of any conclusions drawn.

Hey, at least someone agrees with my belief that we must make assumptions if we are to arrive at any explanation or understanding!

In vernacular English conversation, such as we are doing in this forum and which is the primary method of philosophy, the ambiguities are not collected together in the primitives and axioms of formal systems, but instead are rife throughout the lexicon and even the grammar. With the severe limitations of natural language it is a wonder to me that we ever come to agreements on anything more significant than questions like, "Do you want fries with that?"

I agree wholeheartedly with this. I believe one of the reasons that we usually believe we agree with each other is because our normal language is based on such ambiguity and uncertainty in meaning that there is plenty of room for “overlap” in both intended and non-intended meaning that we just “happen” to be able to communicate ideas with each other (sometimes more by luck than by judgment).

I am pleased and amazed that we come as close to agreement as we do here. As I have said before, I think that nearly all of our disagreements are semantic. I think we just need to be careful to realize that in our discussions here, we are involved in a vernacular conversation, not in the development of a formal system. I think we get into trouble when we talk about the formal systems of Russell, Cantor, Zermelo, Spencer Brown, Dr. Dick, etc.

OK.

A case in point is Dick's insistence that the concept of 'explanation' is fundamental to his argument. It probably is in his formal development, but it certainly isn't in our vernacular conversation here.

Understood. But even in his formal development, it seems to me that an explanation is a mapping (a series of vectors if you like) which provides a translation from one set of points in his 3D space, to another set of points in the same space. Whether the points are more fundamental than the vectors which map between them, or vice versa, is arguable. Imho the reason why I think Dick wants to believe his “explanation” is more fundamental is because he can identify a mathematical and quantum mechanical analogy in the wave equation (whereas there is no quantum mechanical analogy for the sets of points).

I consider this to be equivalent to the proposition that "thought happens". Now, to figure out whether or not we disagree on this, let me ask you, MF, do you think that the "something" that exists could be thought? Could you accept a definition of 'thought' that makes it something? Or would you prefer to consider thought as nothing?

It “could be” thought – but first (bearing in mind your very insightful words about semantic disagreement) I think we need to agree on a definition of “thought”. What do you mean by “thought”?

No. I think that at some point PC actually "does mathematics" by choosing primitives, axioms, and definitions, which then imply, or "create" the laws. Yes, this imbues PC with a lot of anthropomorphism, but the capability to do math, IMHO, developed after a long stretch of time prior to the Big Bang. PC evolved and advanced to a huge degree beyond its extremely rudimentary, simple, fundamental primordial condition. I think this is the point you miss when trying to understand my ideas. I think there was probably a huge amount of trial and error before the precise conditions for an interesting universe like ours were stumbled upon.

Sorry, Paul, but this doesn’t seem to answer the question. You seem to be saying that the PC creates the laws of mathematics, as in “the laws of mathematics do not follow on as a necessary consequence of the PCs consistency decision”.

You say that the PC “does mathematics”, but then so do most humans. But humans do not create the laws of mathematics by “doing mathematics”.

Allow me to re-phrase the question. Given the choice by the PC to be consistent, did the laws of mathematics then follow as a necessary consequence of this (independently of the PCs wishes)? Or are the laws of mathematics contingent (the PC created the laws, and could have created different laws of mathematics if it had so wished)?

The laws follow on as necessary consequences of the PC's consistency decision and the particular choices of primitives, axioms, definitions, and boundary conditions.

OK. This is true of all mathematical laws. Thus (to take an example) given a right-angled triangle in a 2-dimensional plane conforming to Euclid’s 5 postulates of geometry, the law that the square of the hypotenuse is equal to the sum of the squares of the other two sides is a necessary mathematical law. There is no way that the PC could have “created” a universe in which this law (given the postulates and definitions) would have been false. Thus in a very real sense, this law (given the postulates and definitions) “exists” independently of the PC.

Yes, exactly! I am absolutely delighted that you used the pronoun 'we' here. I have been doing exactly the speculation you described for quite some time. I am happy to learn (or at least hope) that you are beginning to entertain the same speculations. The next step is to ask you whether these speculations make any sense to you, or are they nonsense? I am sincerely eager to hear your opinions.

I have never said your ideas are nonsense (at least I don’t think I have). I think I can understand your ideas – but I’m afraid that your ideas do not appeal to me as being “reasonable”, for the reasons already explained. My philosophy is based on making the simplest and smallest number of assumptions possible, and deriving complexity as emergent phenomena from these simple assumptions. One such emergent phenomenon (imho) is consciousness. Consciousness is an exceedingly complex phenomenon, and knowledge is predicated on consciousness – your theory posits that this complexity is somehow “built-in” to the boundary conditions of our universe; my theory posits that the boundary conditions are exceedingly simple, and that both consciousness and knowledge emerge as natural but complex phenomena when the circumstances are right.

But to give you my guess at the answer to your question, I'd say, Neither. The laws of logic and math did not exist prior to the existence of PC. The laws did not spontaneously come into existence "at the moment of the creation of the PC", which I take to mean the initial or primordial appearance of PC, however that came to be. The laws spontaneously came into existence the moment that rules of logical inference were adopted by PC as a deliberate choice. This is just like the fact that a bishop cannot occupy a square of a different color spontaneously comes into existence the moment the chess board and the rules of chess are defined. The fact did not pre-exist the definition of the game in any sense, and the fact is a logical consequence of the rules for the initial placement of the bishop and the rules for its legal moves.

The analogy fails because the rules of chess are contingent, not necessary – they could have been different. But the laws of mathematics are not contingent, they are necessary. No matter what the PC does or does not do, given consistency and given a right-angled triangle in a 2-dimensional plane conforming to Euclid’s 5 postulates of geometry, it follows necessarily that the square of the hypotenuse is equal to the sum of the squares of the other two sides. This law is “true” before the PC discovers it to be true (this law always was true, right from the beginning of time), whereas the law that a bishop may only occupy certain coloured squares on a chessboard is neither true nor false until someone determines what the (contingent) rules of chess are to be.

Maybe PC only thinks it knows anything and there is really nothing known at all in reality. Who knows?

It’s important to clearly define what we mean by knowledge. When you say that the PC knows, how do you define knowledge?
(one possibility : knowledge = justified true belief, but maybe you have a different definition)

This may seem a small point but I feel it's central to the discussion. Suppose that PC (not my choice of term but no matter) is a contradiction in ordinary logic? Suppose it is something that is logically equivalent to a wave-particle?

Are you suggesting that a “wave-particle” is a contradiction in ordinary logic? Could you explain?

Armed with this notion the problems relating to creationism, intention and so on that are IMHO problematic in Paul's view can be resolved.

Can they? Could you explain the resolution?

Yes, that's what I'm getting at, that reality may not accord with the dualism inherent in ordinary logic. In my view it does not. My suggestion is to use the modification to ordinary logic that physicists make use of in quantum theory.

OK – over to you. If you don’t want to assume that any given meaningful proposition is either “true” or “false”, then what exactly do you want to assume?

(I know what you think about the necessity of assumptions, but don't forget that I haven't agreed with you yet).

I don’t know whether you agree or disagree, it seems hard to pin you down.

Best Regards

 

[loseyourname]

Canute, what is the proposition with an indeterminate truth-value that you are identifying from the theories of quantum physics?

 

[Canute]

I'm not suggesting that propositions can have indeterminate truth-values. Rather, I'm suggesting that we need to modify the tertium non datur rule in metaphysics just as we do in physics (e.g. for wave-particles and for the background-dependence problem).

 

[loseyourname]

I'm not suggesting that propositions can have indeterminate truth-values. Rather, I'm suggesting that we need to modify the tertium non datur rule in metaphysics just as we do in physics (e.g. for wave-particles and for the background-dependence problem).

I don't think that's the rule you're proposing we get rid of. Excluded middle simply asserts that
(p[itex]\vee \neg[/itex]p) is true, meaning either p is true, [itex]\neg[/itex]p is true, or both are true. It is the law of noncontradiction, [itex]\neg[/itex](p[itex]\wedge \neg[/itex]p), that states the third option cannot be true.

Both laws can be satisfied in any number of different logics, but remember, neither is actually an a priori law of logic, they are simply tautologies that arise as a consequence of the rules of logic being applied, which are simply the formal syntax and semantics, basically just the definitions of how logical connectives operate and what truth-values are available for use. In bivalent logics only the truth-values 'true' and 'false' are available for use. It seems you want to use another logic that is not bivalent, one that is used by quantum physicists because of a paradox brought about by wave-particle duality. So my question is what proposition regarding the wave-particle duality is it that you believe bivalent logic cannot properly deal with and what kind of logic is it that you believe quantum physicists use to deal with this proposition?

 

[moving finger]

It seems you want to use another logic that is not bivalent, one that is used by quantum physicists because of a paradox brought about by wave-particle duality. So my question is what proposition regarding the wave-particle duality is it that you believe bivalent logic cannot properly deal with and what kind of logic is it that you believe quantum physicists use to deal with this proposition?

I understand what I think you mean, but I think the expression "paradox brought about by wave-particle duality" is unfortunate and misleading. There is no paradox, as long as we remember that a quantum state is neither a wave nor a particle - it is something for which we have no classical analogy. A quantum state contains complementary position and momentum information "wrapped up together" as it were, so that it is false to think of it as having both a definite position and definite momentum at the same time. In trying to label it as either a particle or a wave, we are ignoring one or other of it's properties.

I don't see that any of this requires rejecting the law of the excluded middle.

Best Regards

 

[loseyourname]

Well, that's exactly the point I'm trying to make. I don't personally see any reason that bivalent logic cannot be used to make true statements about quantum physics, so I'm wondering why Canute feels this way. Quantum entities certainly behave in a strange way, counterintuitive to say the least, but they don't do anything I can think of that results in a contradiction when we try to talk about it. Certainly all of the math involved is still derivable from ZFC set theory, which relies on bivalent logic.

The only thing I can think of is, as you point out, we can only speak of position and momentum probabilistically, but the statements of probability are still either true or false.

 

[moving finger]

Well, that's exactly the point I'm trying to make. I don't personally see any reason that bivalent logic cannot be used to make true statements about quantum physics, so I'm wondering why Canute feels this way. Quantum entities certainly behave in a strange way, counterintuitive to say the least, but they don't do anything I can think of that results in a contradiction when we try to talk about it. Certainly all of the math involved is still derivable from ZFC set theory, which relies on bivalent logic.

The only thing I can think of is, as you point out, we can only speak of position and momentum probabilistically, but the statements of probability are still either true or false.

OK, agreed. (it's just that I could just envisage Canute latching onto the notion of "paradox in QM" and using this as a lever to argue for 3-valued logic).

Best Regards

 

[Canute]

I don't think that's the rule you're proposing we get rid of. Excluded middle simply asserts that
(p[itex]\vee \neg[/itex]p) is true, meaning either p is true, [itex]\neg[/itex]p is true, or both are true. It is the law of noncontradiction, [itex]\neg[/itex](p[itex]\wedge \neg[/itex]p), that states the third option cannot be true.

You almost certainly know more about formal logic than I do. However, I am always careful not to say anything I haven't heard someone who is an expert saying. Here is Heisenberg on the topic, from his Physics and Philosophy.

"The vagueness of this language in use among the physicists has therefore led to attempts to define a different precise language which follows definite logical patterns in complete conformity with the mathematical scheme of quantum theory. The result of these attempts by Birkhoff and Neumann and more recently by Weizsäcker can be stated by saying that the mathematical scheme of quantum theory can be interpreted as an extension or modification of classical logic. It is especially one fundamental principle of classical logic which seems to require a modification. In classical logic it is assumed that, if a statement has any meaning at all, either the statement or the negation of the statement must be correct. Of ‘here is a table’ or ‘here is not a table’, either the first or second statement must be correct. ‘Tertium non datur,’ a third possibility does not exist. It may be that we do not know whether the statement or its negation is correct; but ‘in reality’ one of the two is correct.

In quantum theory this law ‘tertium non datur’ is to be modified. Against any modification of this fundamental principle one can of course at once argue that the principle is assumed in common language and that we have to speak at least about our eventual modification of logic in the natural language. Therefore, it would be a self-contradiction to describe in natural language a logical scheme that does not apply to natural language."

This seems clear and straightforward to me, but is there an objection I'm unaware of?

Both laws can be satisfied in any number of different logics, but remember, neither is actually an a priori law of logic, they are simply tautologies that arise as a consequence of the rules of logic being applied, which are simply the formal syntax and semantics, basically just the definitions of how logical connectives operate and what truth-values are available for use.

Yes, I agree. This was my point. The universe need not be constrained by these man-made rules.

In bivalent logics only the truth-values 'true' and 'false' are available for use. It seems you want to use another logic that is not bivalent, one that is used by quantum physicists because of a paradox brought about by wave-particle duality. So my question is what proposition regarding the wave-particle duality is it that you believe bivalent logic cannot properly deal with and what kind of logic is it that you believe quantum physicists use to deal with this proposition?

They use a modification to the tertium non datur rule as I understand it. Here is Spencer Brown from 'Laws of Form' describing the logical scheme I'm proposing we should consider.

"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."

[However, says Brown, we need not be so constrained].

"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."

To see what he means here consider any metaphysical question. Take the something/nothing question of cosmogony for example. It contradicts reason that the universe arises from something or nothing. In other words, this question is undecidable in ordinary logic. The cause of the problem, according to Brown (and me) is that the universe did not arise from something or nothing. Rather, this distinction is ultimately innapropriate when considering such ontological questions.

Would this not be a rather neat explanation of why metaphysical questions are undecidable?

 

[loseyourname]

The application to metaphysical cosmology would be interesting indeed, though I do have to say there is a certain aesthetic dis-ease I feel at the thought of the truth values for the answers to the great questions simply being "imaginary." Even if that allowed us to use the statements computationally, it just doesn't seem very 'satisfying,' so to speak.

Heisenberg hasn't really answered my question, though, which is why a polyvalued logic would need to be used in quantum physics. I understand the problem he points out of assigning definite positions to entities. The statement 'X is in position Y' has no truth value in the quantum world. My objection is still that the statement 'X has p probability of being in position Y' does have a definite truth value. So it seems they could either choose to invoke some notion of fuzzy sets and make computations using statements of the first kind, or use ordinary bivalent logic and make statements of the second kind.

What they actually do, I have no clue, but I imagine we have quantum physicists somewhere around here that would know. That's the great advantage of being on a physics board, though I often hesitate to ask questions like this lest I get laughed at.

As far as the application of noncontradiction to reality, it agree that it does not place any absolute constraint. It seems to apply to some statements and not to others. 'X is in the bedroom,' for instance, it does not apply to. If X is standing in the doorway straddling the boundary, the statement is both true and false. They say in basic logic texts that such a statement is not truth-functional, but they can be when we use non-bivalent logics.

 

[selfAdjoint]

Heisenberg hasn't really answered my question, though, which is why a polyvalued logic would need to be used in quantum physics. I understand the problem he points out of assigning definite positions to entities. The statement 'X is in position Y' has no truth value in the quantum world. My objection is still that the statement 'X has p probability of being in position Y' does have a definite truth value. So it seems they could either choose to invoke some notion of fuzzy sets and make computations using statements of the first kind, or use ordinary bivalent logic and make statements of the second kind.

The statement 'X has p probability of being in position Y' does not always have a definite truth value in the quantum world. This is precisely where quantum physics bids adieu to classical probability just as it does to classical physics.

The properties of position and momentum are complementary. This means they cannot be measured at the same time. And in fact, the more accurately you measure one (the smaller the variance in your probability distribution for it) the less you can know about the other (broad variance; the product of the variances is a constant). If you pin down the position of something accurately, the momentum becomes completely undefined It doesn't have a probability distribution; it just doesn't exist as a measurable property. And the same thing happens if you accurately pin down the momentum - no position information AT ALL!.

For example the momentum of a photon is proportional to the frequency of the light it carries; if you find out the frequency exactly, the position becomes an undefined property.

Stick with this fact (it is as well established as any fact in physics) and you won''t go wrong about quantum physics.

 

[Paul Martin]

Paul - I don’t claim that your ideas are nonsensical – they constitute (imho) just unnecessarily complex assumptions. Your ideas seem to wrap up a lot of complexity within their assumptions; complexity which I believe is emergent rather than primordial.

Well, if you don't think my ideas are nonsensical, then I don't think you will change my mind. As for complexity, I don't think my ideas are any more complex than yours. In fact, I don't think our views of cosmogony and cosmology are very far apart after all.

My philosophy is based on making the simplest and smallest number of assumptions possible, and deriving complexity as emergent phenomena from these simple assumptions. One such emergent phenomenon (imho) is consciousness. Consciousness is an exceedingly complex phenomenon, and knowledge is predicated on consciousness – your theory posits that this complexity is somehow “built-in” to the boundary conditions of our universe; my theory posits that the boundary conditions are exceedingly simple, and that both consciousness and knowledge emerge as natural but complex phenomena when the circumstances are right.

Our assumptions seem to be nearly the same: We both assume that the ultimate origin of reality was extremely simple. We both assume that all of reality came to be what it is by a process of relatively gradual evolutionary change. We both assume that consciousness is an emergent phenomenon resulting from this evolution. We both assume that the physical universe is evolving according to some precise laws of physics, of which we have discovered some very close approximations so far.

We both believe that the complexity is emergent and not primordial.

The only difference in our views seems to be that I think the emergence of consciousness occurred prior to the Big Bang, and you think it happened (at least) sometime during the biological evolution on earth. In the big picture, I don't think that is much of a difference.

In thinking about the relative advantages and disadvantages of each of our views, it seems to me that my view has only one disadvantage: It is dangerously close to positing a "God" (shudder). But, as I have pointed out many times, this incomplete, imperfect, evolving, learning, limited, finite, error-prone, albeit powerful PC is not recognizable as anyone's description of God since Homer. I think it shouldn't be tagged with this label nor be burdened by "God's" baggage.

As for advantages, it easily explains the Hard Problem (which I know you deny) and it easily explains the otherwise highly improbable initial conditions for the Big Bang. It also, IMHO, explains many mysteries associated with humans.

I admit that my scheme is more complex to the extent that it is more comprehensive and extensive (it extends well prior to the BB). But this same complaint (if you could call it that) would also apply to Newton's extension and refinements to Kepler, and Einstein's extensions and refinements to Newton. It seems the more we know, the more complex things get. We just need to get used to complexity and accept it.

The problem is, I have no idea how one can generate an infinite set of integers (ie a set with infinite cardinality) using this procedure. Do you? (see below)

I think we see exactly eye-to-eye on this problem.

I had a long battle with the Maths geniuses on this forum a couple of years ago, in which I was basically told that I was an ignoramus for suggesting such a thing as an infinite integer –

I don't know whether you looked at my thread from almost two years ago at https://www.physicsforums.com/showthread.php?t=49732 but I was essentially arguing the same thing with the same people and was also told (gently) that I was an ignoramus and needed to go back to school. That was when I decided to study Foundations, and I still plan to do so some day.

Axiom of Choice, or Axiom of Infinity?

When I finish taking that course, I will be able to answer you better. But, for now, I'll just give you my impressions which may be wrong.

In my view, you have to account for the existence of an infinite number of integers if you are going to depend on the infinite set in any of your arguments. The Axiom of Infinity (which according to Wikipedia seems to be included in ZF theory) allows for the definition and thus existence of the infinite set. But I have the same problem that you seem to have in that the mere acceptance of the axiom doesn't explain "how one can generate an infinite set of integers (ie a set with infinite cardinality) using this procedure."

In my studies, it was explained that the Axiom of Choice allowed some mechanism for producing the infinite set of integers, and the ZFC axiomatic system is formed by appending the Axiom of Choice (the "C") to the ZF theory. (I don't know whether the Axiom of Infinity is still included in ZFC theory or not. But it doesn't matter at my level of knowledge anyway. I am just giving impressions here.)

My position, and the one I will try to defend when I take a Foundations course, is that we should develop an axiomatic system in which each and every primitive, axiom, and definition is explicitly expressed, either by the mathematician, or by a machine. I would disallow mathematical induction, because the process can't be carried out indefinitely by any known processor, human or machine, and unless it can, no infinite set can be defined.

My position is similar to Leopold Kronecker's in his opposition to Georg Cantor in that we would both deny the definition of sets of infinite cardinality. But I differ with him in one important respect. Kronecker held that the integers come from God and all the rest is the work of humans. I deny that there is any God who gave us an infinite set of integers. (PC giving us a huge but finite set, though, is a possiblity that I consider.)

Russell’s paradox is not a consequence of infinity, it is a consequence of unrestrained self-referentiality. THIS is why I said that legislating against infinity does not make the problem go away.

We may disagree here. I think the consequences of accepting infinity are fatal. I'm not sure about self-referentiality (Long ago when I read GEB I thought so, but now I'm not so sure. I need to take that course in Foundations.)

You say that the PC “does mathematics”, but then so do most humans. But humans do not create the laws of mathematics by “doing mathematics”.

But they do. See below.

Allow me to re-phrase the question. Given the choice by the PC to be consistent, did the laws of mathematics then follow as a necessary consequence of this (independently of the PCs wishes)? Or are the laws of mathematics contingent (the PC created the laws, and could have created different laws of mathematics if it had so wished)?

First of all, let me make sure you understand that when we talk about PC "doing mathematics" here, we don't mean PC in its primordial state. Instead it goes on much later after considerable mental capabilities have evolved but still before the Big Bang.

Next, there are many complex parts to "doing mathematics": There is the deductive process of proving propositions to be consistent within some system. There are some choices to be made in terms of which propositions are pursued, but for any particular proposition, its truth or falsity is a necessary consequence of the laws of that system.

But prior to that, there is the establishment of the "theory" or system itself, which consists of the primitives, the axioms, and some definitions. These are arbitrarily chosen, and different choices yield different theories or systems. These are not necessary consequences of anything and can be freely chosen.

But prior to that, there is the choice of rules of logic to be used in the manipulation of propositions and even expressing propositions. As Loseyourname has just taught some of us who weren't sure, there are several, or many, choices for the rules of logic (two-valued, many valued, etc). These too, seem not to be necessary consequences of anything and thus can be freely chosen.

But prior to that, there must be some equivalent of a natural language in which to express the choices made in the establishment of a mathematical system. People do mathematics in many different natural languages, so these seem to be arbitrary. Although it seems that the choice of natural language shouldn't affect the outcome of the mathematical system, who knows what kind of mathematics an extraterrestrial would really develop?)

And prior to that, there must be some minimal mental ability in order to even make sense of the above. After all parrots can become fairly proficient in language, but I doubt that they can develop axiomatic systems.

So, now, to your specific questions.

Given the choice by the PC to be consistent, did the laws of mathematics then follow as a necessary consequence of this (independently of the PCs wishes)?

No. The PC could still express whims and wishes in the choice of logic to use and then in the choice of primitives and axioms. (PC might choose ZF or maybe ZFC or some other.) The laws of mathematics follow from these arbitrary choices. I think the analogy of chess applies exactly here.

The analogy fails because the rules of chess are contingent, not necessary – they could have been different. But the laws of mathematics are not contingent, they are necessary.

No. The laws of mathematics are contingent on the logic system chosen and on the primitives and axioms chosen.

Or are the laws of mathematics contingent (the PC created the laws, and could have created different laws of mathematics if it had so wished)?

Yes. PC could have chosen a different logic system, and within that system, PC could have chosen from among many different sets of primitives and axioms. Many (but of course not infinitely many) different mathematical systems are possible.

We are quite fond of our mathematical system of analysis which contains the infinite set of real and imaginary numbers. It turns out that all (as far as I know) of our laws of physics fall within this system. (Loseyourname may be correct that QM does not need anything outside this system.) Dr. Dick, IMHO, has confirmed that our physical universe is built upon this familiar mathematical system since he deduces his result from its axioms and his result embodies the laws of physics.

In my view, however, which seems to resonate with some of what you wrote, the notion of infinite sets leads to contradictions and the axioms should be revised to disallow them. Whether this means dropping the Axiom of Choice, or the Axiom of Infinity, or some other I don't know. But I have sketched out a proposal for what I call Practical Numbers which are all finite. This is exactly the same set of numbers which each and every person or machine has ever used, or ever will use, to do any calculation whatsoever. Even the people who have computed the first trillion decimal digits of Pi have only produced a finite rational number and they only used finite rational numbers in all their calculations. Integers are naturally limited by virtue of the capability of the machine being used, or by the time, determination, will, and supply of paper and ink of a human calculator. The math I propose would be grainy, but then again, our universe seems to be grainy. But I digress.

Understood. But even in his [Dick's] formal development, it seems to me that an explanation is a mapping (a series of vectors if you like) which provides a translation from one set of points in his 3D space, to another set of points in the same space. Whether the points are more fundamental than the vectors which map between them, or vice versa, is arguable.

In Dick's formal development, 'explanation' is a definition he makes within the system of mathematical analysis. He starts with the assumption of that mathematical system which includes all the real and imaginary numbers as well as the notion of mapping, not to mention all the theorems that have been derived over the past several hundred years. IMHO he has proved a new theorem in that system and he has chosen his definitions so that they end up being ismomrphic to familiar entities.

I think we need to agree on a definition of “thought”. What do you mean by “thought”?

Mental activity of which the thinker can claim to be consciously aware.

Here I must be careful to avoid a mistake you taught me about. When we claim that "something exists", we might have in mind something primordial which accounts for everything else, or we might have in mind something that accounts for our present sense of the world. In our respective views of the evolution of reality, I think we agree that in the primordial state, any notion of 'thought' is far to complex to have existed. My attempts at reducing thought to its fundamental, and even primordial, essence have led me to use the notion of "the ability to know", or "the ability to realize", or the "receptive principle" as described by Gregg Rosenberg. So when I claim that "thought happens", I am referring to the present complex state of reality. When I claim that "there is something and not nothing", I am similarly referring to the present complex state of reality. In that context, I propose that the two claims are the same.

Have to stop. Warm regards,

Paul

 

[Canute]

I see the QM/logic question much more simply, in terms of the wave-particle duality.

The proposition: 'A quantum entity is a wave' is neither true nor false. Likewise, 'A quantum entity is a particle' is neither true nor false. 'A quantum entity is neither a wave nor a particle' is neither true nor false and so on. I mentioned the background-dependence problem because physicists seem to be reaching the same sort of conclusion about the fundamentality of spacetime. Some have proposed the hypothesis of duality as a solution, by which spacetime is fundamental or not depending on how we look at it. So the proposition 'spacetime is fundamental' would be neither true nor false.

The link with Brown (whose mathematics incorporates the idea of 'nonduality') is that spacetime is neither fundamental nor not-fundamental in the nondual view. If physicists were to rename their hypothesis the 'hypothesis of nonduality' the equivalence of this idea to the nonduality spoken of by the mystics would be more obvious. In this view spacetime has always been said to be fundamental or not depending on how we look at it. If this is a coincidence it would seem a very unlikely one.

Seeing as how this started out as a discussion of dualism it is worth noting that the statements 'spacetime is fundamental' and 'spacetime is not fundamental' would both be examples of dualism in the nondual view.

 

[Paul Martin]

Hi MF,

This comment had me lying awake early this morning.

Even in a finite number system, one can still ask “is the class of all classes that are not members of themselves a member of itself?”

Yes, one can ask the question, but I think the question is nonsense. I think the phrase "the class of all classes" cannot be consistently defined, even in a finite case.

Suppose the universe consisted of exactly 2 classes, say A and B. Then how could one define 'the class of all classes'? Since we are using the term 'class' in the phrase, to be consistent, we must mean the same thing by the term as we mean in the premise. In other words, the class of all classes must be a class. And since the universe consists of exactly two classes, A and B, the class of all classes must be either A or B. There are no other candidates. If the class of all classes is A, then it does not include B which is inconsistent with any reasonable meaning of 'all'. Similarly if it is B.

If we define 'the class of all classes' to be the class {A,B} then we are inconsistent with the premise because we now have a third class which is neither A nor B.

I think the problem is in the notion of "all". I think a consistent notion of "all" of anything must be time dependent. We must qualify 'all' by specifying "all at a point in time".

The problem is that the body of mathematics grows. What may be a proposition or a conjecture at one time becomes a theorem later as a proof is discovered. Similarly, if we consider a finite case, such as the existence of only A and B as above, we may introduce new concepts at a point in time which extend the "universe". So, if we have only A and B, and we form a new set with A and B as its members, we have extended the universe and from that point of time onward, 'all' has a new and different meaning.

Let's look at it in a different way. What does it mean for a set to be a member of itself? Well, let's look at an example.

Let A = {A,B}. The definition of A is recursive, but is it consistent? And does it make sense? Logically, it seems to me that there is no reason we can't define 'A' this way.

Now let's say the universe consists of exactly, and nothing but, the sets A and B and the set, C, which is defined as the set of all sets that are not members of themselves. At the time we define 'C', the only set which is not a member of itself is B. So at the time we define 'C' we have C = {B}. Then, and only after we have defined 'C', can we ask the question, "Is C a member of itself?" The answer is clearly "no" in this example.

I don't think this presents a contradiction or a difficulty. You might say that since C is a set and since C does not contain itself as a member, it must belong to the set of all sets that do not contain themselves as members. But I would say that you don't get a chance to refine, or redefine, your definitions as new concepts are added to your mathematical system. You have already defined 'C' once. That definition is consistent and you are not at liberty to change it later.

As I see it, the only problem presented is the interpretation of the word 'all' without considering the time-effectiveness of the concept. If I agreed to give you all the money in my checking account, I would reserve the right to specify exactly when the balance was going to be calculated.

Extending this notion of time dependency to the subject of this thread, I would say that time is a parameter which marks the progress of the mathematician's attention as he/she/it develops the theory by choosing primitives, choosing axioms, stating them in logical language, defining terms, and proving propositions to be consistent. The body of mathematics thus developed is time-dependent in that its state is not static but changes as concepts are added.

In other words, IMHO, you can't -- and don't -- have mathematics without a mathematician. Put another way, you can't have concepts in the total absence of a conscious mind. (The exception I have noted earlier would be if you define 'concepts' to include the symbolic representation of the mental concept as recorded on some physical medium to also be a concept, then that recorded concept may exist after the conscious originator has died and completely disappeared. In any case, a conscious mind was necessary for the concept to exist in the first place.)

I think this is the fundamental disagreement between you and me, MF. It is my conviction that "mind is necessary for concept" which compels me to accept the notion of PC existing and conceiving concepts prior to the existence of such events as the Big Bang, which very much seem to be dependent on a great many sophisticated concepts.

Warm regards,

Paul

 

[selfAdjoint]

Suppose the universe consisted of exactly 2 classes, say A and B. Then how could one define 'the class of all classes'? Since we are using the term 'class' in the phrase, to be consistent, we must mean the same thing by the term as we mean in the premise. In other words, the class of all classes must be a class. And since the universe consists of exactly two classes, A and B, the class of all classes must be either A or B. There are no other candidates. If the class of all classes is A, then it does not include B which is inconsistent with any reasonable meaning of 'all'. Similarly if it is B.

If we define 'the class of all classes' to be the class {A,B} then we are inconsistent

No your premise contradicts the definition of class. You can't restrict it that way; if you have two objects (including classes) you are free to form the class of their pair.

 

[Paul Martin]

No your premise contradicts the definition of class. You can't restrict it that way; if you have two objects (including classes) you are free to form the class of their pair.

What is the precise definition of 'class' that guarantees such freedom?

If you have the freedom to form new classes in such a way, then there can be no such thing as a finite system, as one can always extend it by forming new combinations of elements to form new sets or classes.

Paul