Is Consciousness Truly Outside the Physical Realm?

[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\vee \negp) is true, meaning either p is true, \negp is true, or both are true. It is the law of noncontradiction, \neg(p\wedge \negp), 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\vee \negp) is true, meaning either p is true, \negp is true, or both are true. It is the law of noncontradiction, \neg(p\wedge \negp), 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

 

[moving finger]

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?

It may be clear and straightforward, but is it necessary? I could claim that we need a 7-valued logic (7 is my lucky number), but simply claiming it does not make it necessary (in the sense of being a necessary assumption to enable our understanding of the world). This seems to me like an expression of Heisenberg’s personal beliefs, but this doesn’t make such a belief necessary. Heisenberg (it seems) simply could not accept the notion of the excluded middle, therefore he chose to reject this premise. That’s an assumption of his. But does this make the assumption necessary? How could we tell?

Could you give an example where this “Heisenberg-modified tertium non datur” can be applied?

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.

Why do you say it contradicts reason? Please show that this follows. You are assuming first of all that the universe has “arisen” - presumably in time – which then assumes that time exists outside of the universe. Why can it not be the case that the universe has existed for all time (be careful how you define time)?

In other words, this question is undecidable in ordinary logic.

Not until you answer the question above. A meaningless question does not have a yes/no answer, but this does not mean we have rejected the law of the excluded middle.

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.

Saying that “the universe arose” already presumes some backdrop of time against which it arose – what if time is an intrinsic part of the “creation” of the universe? If there is no background “time” before the “arising” then there can be background “time” against which the “arising” occurs (this should appeal to a mystical mind). One can expect yes/no answers only if one poses meaningful questions.

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

It rather seems like a neat way of avoiding the question!

The whole reason why the possibility of a “third way” arises in quantum mechanics is precisely the same as the reason it arises in conventional logic – some questions are meaningless. To ask “is the King of France bald?” is a meaningless question, it has no true or false answer, because the object it refers to (the King of France) does not exist. In the same way to ask “what is the position and momentum of this quantum object” is also a meaningless question, it has no unique answer because the position and momentum of a quantum object cannot be simultaneously precisely specified (the concept of simultaneous position and momentum does not exist in QM).

Thus in both QM and logic there do indeed exist “true”, “false” and “meaningless” propositions, but there is no need to invoke a fourth class of “imaginary” propositions. This latter is imho simply mystical nonsense.

If you disagree, perhaps you could provide an example of a proposition which you believe is neither true, false nor meaningless, but is instead “imaginary”?

The properties of position and momentum are complementary. This means they cannot be measured at the same time.

Agreed – but this does not prevent us from measuring one property at a time (or assigning a probability to one property at a time).

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.

If you pin down the position of something accurately, then the momentum becomes completely uncertain, but this is not the same as saying “it doesn’t exist as a measurable property”. I can choose to measure whatever I wish, but quantum mechanically I cannot measure both position and momentum simultaneously.

Well, if you don't think my ideas are nonsensical, then I don't think you will change my mind.

Paul – I know that I cannot change your mind, and I am not here for that reason. I am here simply to explore the limits of my own understanding of the world, to find out through test and experiment and discussion whether my beliefs are rational, coherent and sound, and to learn for myself whether I need to change any of those beliefs.

All I can do is to show you the water – it is your decision as to whether you drink from it or not. Changing your mind is under your control, not mine. (It’s like the old joke about “how many psychiatrists does it take to change a lightbulb?” – answer : “None – the lightbulb must want to change by itself”)

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.

I’m a great believer in the saying “one can lead a horse to water, but one cannot make him drink”. Your PC is inherently complex – it thinks, it knows, it understands, it perceives, it makes decisions, it is conscious, it has intentions and desires, and it seems to me that you feel you need to assume such properties as “a priori properties of the world” because you do not believe these properties could arise solely from complexity if they were not already somehow “built-in” to the boundary conditions. If you genuinely believe that the assumption of this primordial PC is a “simple assumption” then you and I are talking a fundamentally different language. (It’s similar to the theist idea that the assumption of God is a simple assumption).

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."

My “solution” to the problem is that I reject the conventional mathematician’s dogma that an integer is a number generated by adding 1 to itself a “finite” number of times. Why does the conventional mathematician insist upon “finite”? To me, an integer is a number which is generated by adding 1 to itself an arbitrary (unlimited) number of times, and in the limit “unlimited” tends to infinity. Thus we arrive at the concept of an infinite integer, and thus no problem generating an infinite set of integers. I don’t see how the mathematics of integers can be claimed to be consistent otherwise. And incidentally the whole of Cantor’s ideas about different levels of infinity thereby goes out the window – the cardinality of the integers is just the same as the cardinality of the reals as soon as we acknowledge the existence of infinite integers (the only reason Cantor was able to show allegedly different levels of infinity was because of the incoherent notion of an infinite cardinality of finite integers).

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.)

See below. Your evidence for the alleged contradictions inherent in the concept of an infinite set (the set of all logical possibilities for example) was Russell’s paradox. I have shown (and continue to explain below) that Russell’s paradox has nothing to do with infinity, it has to do with self-referentiality, and the paradox occurs even in finite sets.

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.

Pardon me, but it seems to me that you are avoiding the question. Yes of course one (whether “one” is the PC or a human agent) must choose axioms (how many times in how many threads have I said that we must make assumptions before we can arrive at any explanation or understanding of the world?) – and one is of course free to choose those axioms. But given the choice of axioms, the laws of mathematics then necessarily follow. Take again my example of Pythagoras’ theorem – given Euclid’s 5 postulates (axioms) then the theorem necessarily follows – neither the PC nor any human (given the assumptions) has any freedom at all to change this.

Humans are in exactly the same position. We can choose axioms, and the laws of mathematics follow from these axioms. There is nothing “special” about the PC in this respect – the PC is thus as powerful and as powerless as any human agent.

The laws of mathematics are contingent on the logic system chosen and on the primitives and axioms chosen.

Of course. But the PC has no more “power” to choose the laws of mathematics than do humans. We humans choose the axioms, then we deduce the laws which follow. How is the PC any different to this?

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.

Can you give an example of a choice that the PC could have made which a human is not capable of making?

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 issue here is that the question depends on the context. To ask “is the quantum entity a particle?” is a meaningless question outside of the context of a measurement. All propositions can be reduced to either true, false or meaningless.

This comment had me lying awake early this morning.

I’m not sure if I’m sorry or happy

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.

Artificially restricting the universe to 2 and only 2 classes is OK, and in this case it eliminates the concept of the additional third class of all classes that are not members of themselves, but this is hardly a realistic scenario, and it does not follow from this that the general solution in a universe with a finite number of classes avoids Russell’s paradox. If I have N classes (where N is a finite integer), I can always construct another class (numbered N+1) which is the class of all classes (selected from the original N classes plus the new N+1 class) which are not members of themselves. I still have a finite (N+1) number of classes, but I now also have Russell’s paradox – and no sign of infinity.

Think of it in terms of the Barber’s paradox – this is another variant. In Seville there is a barber who shaves everyone who does not shave himself. Now ask – does the barber shave himself? Clearly at anyone time there are a finite number of individuals in Seville – thus the paradox has nothing to do with either infinity or time.

The barber paradox is EXACTLY of the same form as Russell’s paradox of the class of all classes that are not members of themselves – and the paradox arises because of self-referentiality.

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".

Nope – the problem is in self-referentiality. The “class of all classes that are not members of themselves” is class N+1, but it refers not only to all of the original classes N but also to ITSELF (class N+1). This has nothing to do with infinity, or with time. It has only to do with self-referentiality. This is precisely why Russell tried to get rid of the problem by eliminating self-referentiality through partitioning sets and classes – so that a class (ie the set N+1) could refer to a set (ie any of the sets N), but was not allowed to refer to itself (ie the set N+1). Infinity has absolutely nothing to do with it. Neither does time.

Paradox is inherent in self-referential systems – I think there is a warning there for our understanding of consciousness (which is also ultimately a self-referential system).

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.

It need not be connected with time (don’t forget that I don’t believe “the body of mathematics grows” in an ontic sense – all that grows is human knowledge of mathematics – the epistemic sense). You may see the paradox as a temporal progression, whereas I see it as a logical progression. Standing “outside of time” one can see the same atemporal paradox. The barber who shaves all those who do not shave themselves obviously has to shave one person at a time, but the paradox is not a time-dependent paradox – the barber may choose to decide whether he shaves himself or not before he decides whether to shave anyone else.

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.

What you have done is exactly what Russell did in his theory of types – he said that set C is in a different class to sets A and B, therefore set C cannot (= is not allowed to) refer to itself – thus eliminating the paradox.

Imagine that C is the barber, and A and B are two residents of Seville. A shaves himself, but B does not. Now we say that the barber (C) shaves all those that do not shave themselves – thus C shaves B. By your “rule” we are not allowed to ask if C shaves himself (thus avoiding the paradox) – and this is just what Russell tried to legislate against. But it has nothing to do with either time or infinity – it has to do with self-referentiality. Basically what you are saying is “we are not allowed to ask whether C shaves C – that is not a legitimate question”.

Indeed, I can reduce the set to 1. Imagine all the residents of Seville have left town, except for the barber. Once again, the barber shaves all those who do not shave themselves. But now there is only one person left in town to be shaved – the barber himself. We have a unique set, a single set. And we still have the paradox. Conclusion : Russell’s paradox has nothing whatsoever to do with infinity. It has to do with self-referentiality.

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.

Time is a red-herring (as time usually is – it is indeed strange how time figures in many false human intuitions). The issue is purely one of logical self-referentiality. In practice the barber could shave himself before he shaves anyone else – but you would still claim that the question “does the barber shave himself?” (ie “does C shave C?”) is a question which is not allowed.

Thus having rejected the idea that "Russell’s paradox is a contradiction entailed only by notions of infinite sets", we get back again to the question of why you think the notion of an infinite set involves contradiction. Over to you.

Best Regards

 

[Canute]

It may be clear and straightforward, but is it necessary? I could claim that we need a 7-valued logic (7 is my lucky number), but simply claiming it does not make it necessary (in the sense of being a necessary assumption to enable our understanding of the world). This seems to me like an expression of Heisenberg’s personal beliefs, but this doesn’t make such a belief necessary. Heisenberg (it seems) simply could not accept the notion of the excluded middle, therefore he chose to reject this premise. That’s an assumption of his. But does this make the assumption necessary? How could we tell?

I think you'd need to ask this question of a physicist. Heisenberg was quite a bright chap and a good physicist, and not in the habit of voicing his personal beliefs as if they were facts.

Could you give an example where this “Heisenberg-modified tertium non datur” can be applied?

A wave-particle seems a good example.

Why do you say it contradicts reason? Please show that this follows. You are assuming first of all that the universe has “arisen” - presumably in time – which then assumes that time exists outside of the universe. Why can it not be the case that the universe has existed for all time (be careful how you define time)?

Metaphysical questions are undecidable because both of their 'reasonable' answers give rise to contradictions. By 'reasonable' here I mean reasonable according to rules of ordinary logic. For example, the proposition that the universe arises ex nihilo is logically incoherent, the proposition that it arises from something pre-existent is logically incoherent, and therefore the question of how it arises is a metaphysical (undecidable) question. However, if we modify the tertium non datur rule this opens up a third possibility.

A meaningless question does not have a yes/no answer, but this does not mean we have rejected the law of the excluded middle.

Do you think Heisenberg didn't know this?

Saying that “the universe arose” already presumes some backdrop of time against which it arose –

Very true. Let us say then the the universe arises only in a particular sense.

If there is no background “time” before the “arising” then there can be background “time” against which the “arising” occurs (this should appeal to a mystical mind). One can expect yes/no answers only if one poses meaningful questions.

Exactly. To a mystic asking whether the universe arises from something or nothing is like me asking you whether you've stopped beating your wife. In Zen the repsonse may be 'Mu', a word meaning, roughly, there is nothing to say because the question embodies a false assumption and any answer would therefore be misleading.

The whole reason why the possibility of a “third way” arises in quantum mechanics is precisely the same as the reason it arises in conventional logic – some questions are meaningless. To ask “is the King of France bald?” is a meaningless question, it has no true or false answer, because the object it refers to (the King of France) does not exist. In the same way to ask “what is the position and momentum of this quantum object” is also a meaningless question, it has no unique answer because the position and momentum of a quantum object cannot be simultaneously precisely specified (the concept of simultaneous position and momentum does not exist in QM).

I gather that you're a logical positivist. I don't find that doctrine convincing, and it is now virtually dead.

Thus in both QM and logic there do indeed exist “true”, “false” and “meaningless” propositions, but there is no need to invoke a fourth class of “imaginary” propositions. This latter is imho simply mystical nonsense.

I know this is your opinion. However, I don't agree.

If you disagree, perhaps you could provide an example of a proposition which you believe is neither true, false nor meaningless, but is instead “imaginary”?

Light is a wave. Human being have freewill. Light is a particle. Human being do not have freewill. The universe arises ex nihilo. The universe arises from something. God exists. God does not exist. Bear in mind that the term 'imaginary' is used in its mathematical sense.

 

[Paul Martin]

I’m a great believer in the saying “one can lead a horse to water, but one cannot make him drink”.

I believe the same. I am eager to drink but sometimes I find it hard to swallow some of what is offered. I think the reciprocal is also true: it seems that you have a hard time swallowing some of what I offer to you. I think our objective should not so much be in the drinking, but in identifying those obstacles to swallowing.

Your PC is inherently complex – it thinks, it knows, it understands, it perceives, it makes decisions, it is conscious, it has intentions and desires, and it seems to me that you feel you need to assume such properties as “a priori properties of the world” because you do not believe these properties could arise solely from complexity if they were not already somehow “built-in” to the boundary conditions.

What you say here is true about me and my ideas as long as it is interpreted correctly. You seem to gloss over two important points of interpretation which makes me think that you still misunderstand me. I have elaborated on these two points several times in our conversations, but I don't recall you ever acknowledging them, much less confirming or denying either of them.

Point number one is that I regrettably use the term 'PC' to refer to two separate and distinct entities: first is the truly primordial entity, about which we can say very little if anything, and which existed only as prior to and until the first change of anything occurred. From that point on, what was initially this truly primordial PC evolved into the second entity, which I also refer to as PC.

This evolving PC has undergone enormous change since primordial times, and I (again regrettably) use the term sometimes to refer to the conscious entity that is driving organisms on Earth now in the 21st century, sometimes to refer to the conscious entity which I think might have consciously interfered with the otherwise unitary evolution of our physical universe, sometimes to refer to the conscious entity which I think might have been involved in the establishment or choice of the boundary conditions for our BB, and at other times to refer to the evolving conscious capabilities which I think developed over an extended period of time "prior" to the BB itself.

So, when you claim that my notion of PC is "inherently complex", I agree only as long as you mean PC in the second sense above. It was not complex at the outset.

The second point of interpretation, which I think you gloss over, is in your use of the term 'world' as in " it seems to me that you feel you need to assume such properties as “a priori properties of the world” because you do not believe these properties could arise solely from complexity if they were not already somehow “built-in” to the boundary conditions."

You would be correct in suspecting that I feel that need as long as you interpret the term 'world' to mean our familiar 4D physical space-time continuum which began its existence with the BB. In this case, I do feel the need to assume such properties as being extant sometime prior to the BB in order to adequately explain the high improbability of the BB's initial boundary conditions and to adequately explain other phenomena such as the origin of life and the origin of consciousness in living organisms.

On the other hand, if you interpret 'world' to mean all of reality, then I think you and I are pretty much in agreement. We both think things (the world of reality) started out extremely simple, and that all complexity emerged as a result of recombinant changes over long periods of time. We disagree on some details, like the lengths of time involved (my estimate being vastly longer than yours), or the sequence of appearance of various specific phenomena (e.g. consciousness vs. organisms, concepts vs. mind).

Probably the most contentious of these details is whether there is some fundamental constituent of consciousness (whatever that is) which was primordial. I say "yes" and I think you say "no". But in my view, whatever that fundamental constituent is (was), it is not at all the complex entity or idea we refer to as consciousness (whatever it is). I have speculated that the fundamental constituent is something like a rudimentary ability to know, or ability to realize, or as Lars suggested, an ability to experience, or as Gregg Rosenberg has suggested, a receptive principle.

Now to figure out whether such a primordial ability or principle can give rise to consciousness as we know and experience it, it seems to me, would take the same sort of investigation Goethe and Beethoven went through trying to identify the same thing for music (as the two of us imagined it).

But to draw a comparison, you can have vibrations without a mind, but you can't have music without both vibrations and a mind. I'd say that a similar thing is true for concepts: you can have entities or things without a mind, but you can't have concepts without a mind. And, further, I'd say that you can't have consciousness without a mind. These, of course are only my opinions, and even though I have poured them into your water glass, I don't expect that you will easily swallow them. That's fine. I just wanted to offer them to you and get your response.

you do not believe these properties could arise solely from complexity if they were not already somehow “built-in” to the boundary conditions.

Yes, you have that right except that in this case, I claim that the boundary conditions we are talking about are the truly primordial conditions, which greatly precedes the BB. And, the "somehow 'built-in'" is the "fundamental constituent of consciousness" I talked about above. I believe there was such a constituent and you evidently don't.

I should point out that I do believe in emergent phenomena. Moire patterns, and the Mandelbrot Set are examples of emergent phenomena. So is the wetness of water. But it is my opinion that conscious experience is different in kind from all emergent phenomena and that it requires something more to get it to happen or exist. I think that "something more" is ontologically fundamental. And, it seems that you can't swallow that.

If you genuinely believe that the assumption of this primordial PC is a “simple assumption” then you and I are talking a fundamentally different language.

I don't think it is that drastic. I think we are talking the same language, but that we have been using different connotations of the words 'PC' and 'world'. Hopefully, we can get over that misunderstanding. As I said before, I think our respective views of what actually happened at the ultimate "beginning" are not all that different.

(It’s similar to the theist idea that the assumption of God is a simple assumption).

I was going to categorically deny this by saying that the theists have never defined their notion of 'God'. But I realized that I have never really defined my notion of 'PC' in its primordial state. I guess you are ahead of both the theists and me by not even identifying what you think might have been the primordial constituent of reality. So without identifying it, you don't have to define it.

But, as I have also pointed out, the theists posit huge (even infinite) complexity to their God in the beginning, whereas I posit only simplicity as you do.

Russell's Paradox
Thank you for your excellent discussion of Russell's Paradox and the notions of self-reference, infinity, and time.

I think our disagreement here won't be resolved by either of us dishing up more servings of drinking water. I think the stumbling block is our different views on whether concepts can (or do) exist without first having been conceived in a mind. You say "yes" and I say "no'. This is equivalent to whether mathematics is discovered or invented respectively (and respectfully as well).

I say that no mathematical concept, thus no mathematics, exists unless and until some conscious mind conceives of the concept and integrates it into a developing body of mathematical concepts comprising a particular mathematical system or theory according to some arbitrarily but deliberately chosen rules.

I think you would say that concepts, such as the implications of logical rules, exist independently, and more importantly, prior to the existence, of any mind.

For example, we said,

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.

It is this "very real sense" of existence which I can't swallow.

But, if I did swallow it, I would agree that a static Platonic world exists populated by all concepts that might ever be discovered by mathematicians, and that that Platonic world existed at least at the time of the BB, if not before. Certainly long before the first animal appeared on earth.

But I don't swallow it. I would agree that a Platonic world does exist, at least if you claim that it is populated by any concept currently in someone's mind, or even in someone's memory, or even in symbolic form in a book. In this way, you could have a Platonic world completely resident in the physical world (assuming people's minds and memories are all in the brain, which I doubt, if not deny). But even with this type of Platonic world, there remains the question of how one "does mathematics". I think this is at the root of our problem.

In school, we are taught to develop proofs a step at a time and write them down in sequence. The correct answer is a static set of propositions listed in order, such that if a sceptic were to attend to the propositions in sequence, each should progressively dispel any doubt as it is pondered and understood, until the last one which should prove the theorem. So I ask (in the spirit of Beethoven and Goethe) what exactly is the mathematics here? Is it that static list of propositions? Or is it the time-dependent development of concepts and understanding in the conscious mind of the person either reading the list or originating it in the first place? Of course I claim it is the latter, and I suspect you would say it is the former.

If you are right, I'm wondering where that list was at or around the time of the BB, and wherever it was, how did it get to be there? Talk about complexity.

If I am right, then there is definitely a time-dependency involved in "doing mathematics".

In the context of the Barber of Seville, what you call a problem with self-referentiality, I say is due to the time-dependence. For example, in the definition, Let A = {A,B}, you would say it is self-referential because 'A' appears in the definition of 'A'. True enough. I agree.

But instead of simply legislating against self-reference because we have noticed it leads to problems, I say the problem arises because we have violated a rule I think we should abide by. That rule is that in the sequential development of the concepts of a system, any definition can be made using only terms that have been previously defined along with the undefined primitives. The "previously" gives the time-dependency away. This rule prevents recursive definitions, because at the "time" we set about to define 'A', in this example, we do not have a defined 'A' to use in the definition.

I have to eat some words here. I previously said about this example, "Logically, it seems to me that there is no reason we can't define 'A' this way." It now seems to me that there is good reason to disallow this definition. It is only when you view the list of propositions and definitions as a static entity that this definition seems to make logical sense. So, it seems to me that in your view you would say that the definition is logical, but it should be legislated against because it is self-referential. In my view, I say the definition is not legal because "at the time" the definition is being made, the term 'A' has not "yet" been defined. (The time-dependencies are in scare quotes.)

Now with regard to infinity, it isn't that I think the notion of infinity is the problem or leads to problems. What I think is that, first of all, I don't think you can, or at least nobody has yet, define the notion of infinity in a consistent way. The ancients defined it, if at all, in nonsensical ways. Cantor, with the first rigorous definition immediately noticed consequent paradoxes.

What I object to is the assumption that concepts can come into existence without first being conceived by a mind. So I object to all genetic methods of "producing" sets of infinite cardinality, such as the infinite set of integers. You seem to have a similar objection to some of the methods mathematicians have used to define the integers. As I said before, I don't agree with Leopold Kronecker who claimed that the infinite set of integers were given to us by God. The only integers we have available to us to use are those conceived by us directly, or those explicitly generated by finite machines conceived and built by us. In all such cases, the integers are finite, and there have only ever been a finite number of them defined and used. Moreover, I predict that there never will be a time when an infinite set of integers will be available for our use.

Thus having rejected the idea that "Russell’s paradox is a contradiction entailed only by notions of infinite sets", we get back again to the question of why you think the notion of an infinite set involves contradiction.

My position is not that infinity leads to problems. Instead my position is that we can't reasonably define anything infinite, and when we try to pull tricks to define infinity, those tricks lead to problems. The tricks, like Hilbert's genetic definitions, are what I think should be disallowed, just like the trick of recursive definition.


Warm regards,

Paul

 

[moving finger]

I think you'd need to ask this question of a physicist. Heisenberg was quite a bright chap and a good physicist, and not in the habit of voicing his personal beliefs as if they were facts.

None of that makes him infallible.

A wave-particle seems a good example.

In what sense? Can you express this in a form of a proposition which has neither a “yes”, “no” nor “meaningless” answer?

Metaphysical questions are undecidable because both of their 'reasonable' answers give rise to contradictions. By 'reasonable' here I mean reasonable according to rules of ordinary logic. For example, the proposition that the universe arises ex nihilo is logically incoherent, the proposition that it arises from something pre-existent is logically incoherent, and therefore the question of how it arises is a metaphysical (undecidable) question.

Why is the notion that the universe has existed for all time incoherent?

A meaningless question does not have a yes/no answer, but this does not mean we have rejected the law of the excluded middle.

Do you think Heisenberg didn't know this?

Canute, with all due respect it gets a little tiring when you simply keep referring to other people when backed into a corner, instead of addressing the issue at hand. Other people, even brilliant scientists, are not infallible, and they don’t have the secret to the universe. Again with all due respect I think you need to start forming your own opinions, supported by your own arguments. To continually refer to “other experts” without further argument is self-defeating and does not constitute a rational or logical argument in support of your position.

Very true. Let us say then the the universe arises only in a particular sense.

In what sense? What exactly do you mean by “the universe arises”?

To a mystic asking whether the universe arises from something or nothing is like me asking you whether you've stopped beating your wife. In Zen the repsonse may be 'Mu', a word meaning, roughly, there is nothing to say because the question embodies a false assumption and any answer would therefore be misleading.

What false assumption does it embody, in your opinion?

I gather that you're a logical positivist.

How on Earth do you arrive at this conclusion?

If you disagree, perhaps you could provide an example of a proposition which you believe is neither true, false nor meaningless, but is instead “imaginary”?

Light is a wave.

Whether light is measured to be a wave or a particle depends on how you measure it. Specify how it will be measured and this determines whether you will measure it to be a wave or a particle.

Human being have freewill.

Define freewill, then we may be able to debate this further.

Light is a particle.

Whether light is measured to be a wave or a particle depends on how you measure it. Specify how it will be measured and this determines whether you will measure it to be a wave or a particle.

Human being do not have freewill

Define freewill, then we may be able to debate this further.

The universe arises ex nihilo. The universe arises from something.

Define “arise from” then we may be able to debate this further. Do you assume a backdrop of time against which the universe “arises”?

God exists. God does not exist.

Either of these may be true – the fact that we do not know which is true does not make the proposition “imaginary” – it simply points to limits in our knowledge.

Bear in mind that the term 'imaginary' is used in its mathematical sense.

Bear in mind that the term “imaginary” in mathematics means something quite different to “imaginary” in normal language. A proposition based on mathematical imaginary terms is still either true or false (or meaningless).

You see – no imaginary propositions at all (excluding the mathematical kind of course – if you wish to claim that all imaginary propositions are of the mathematical kind then I do not disagree)

Point number one is that I regrettably use the term 'PC' to refer to two separate and distinct entities: first is the truly primordial entity, about which we can say very little if anything, and which existed only as prior to and until the first change of anything occurred.

Can you elaborate on the properties/abilities of this entity? What was it capable of?

The important issue (it seems to me) is to establish which properties/abilities are truly primordial (ie a priori, or in the boundary conditions), and which properties/abilities are emergent. If you are now claiming that all the PC’s properties/abilities are emergent and not primordial then that is worth clarifying.

So, when you claim that my notion of PC is "inherently complex", I agree only as long as you mean PC in the second sense above. It was not complex at the outset.

You will need to explain then just what properties/abilities you believe your “PC at the outset” possesses, before we can move on.

You would be correct in suspecting that I feel that need as long as you interpret the term 'world' to mean our familiar 4D physical space-time continuum which began its existence with the BB. In this case, I do feel the need to assume such properties as being extant sometime prior to the BB in order to adequately explain the high improbability of the BB's initial boundary conditions and to adequately explain other phenomena such as the origin of life and the origin of consciousness in living organisms.

Ahhh, now this is interesting. Which particular BB boundary conditions are you referring to? Are you suggesting that the PC is responsible for deliberately manipulating the BB conditions such that our universe would be conducive to the emergence of biological lifeforms, in some kind of PC-managed “experiment”?

On the other hand, if you interpret 'world' to mean all of reality, then I think you and I are pretty much in agreement. We both think things (the world of reality) started out extremely simple, and that all complexity emerged as a result of recombinant changes over long periods of time. We disagree on some details, like the lengths of time involved (my estimate being vastly longer than yours), or the sequence of appearance of various specific phenomena (e.g. consciousness vs. organisms, concepts vs. mind).

I wonder why you think it requires more than 13 billion years for such things to emerge?

Probably the most contentious of these details is whether there is some fundamental constituent of consciousness (whatever that is) which was primordial. I say "yes" and I think you say "no".

That is correct. I believe consciousness is emergent, not primordial. Even if yoru PC has no other properties, this is one thing that makes your premise inherently more complex (in the sense of packing more into the assumptions).

But in my view, whatever that fundamental constituent is (was), it is not at all the complex entity or idea we refer to as consciousness (whatever it is). I have speculated that the fundamental constituent is something like a rudimentary ability to know, or ability to realize, or as Lars suggested, an ability to experience, or as Gregg Rosenberg has suggested, a receptive principle.

It seems to be a matter of opinion as to whether that is a complex assumption or not. It doesn’t seem any less complex (to me) than “In the beginning was the Word, and the Word was God…….”

Now to figure out whether such a primordial ability or principle can give rise to consciousness as we know and experience it, it seems to me, would take the same sort of investigation Goethe and Beethoven went through trying to identify the same thing for music (as the two of us imagined it).

It seems that once one posits a primordial consciousness, then surely any type of emergent consciousness is possible – the sky is the limit. Nothing is really left in the explanation, since all the important properties are packed into the assumptions.

But to draw a comparison, you can have vibrations without a mind, but you can't have music without both vibrations and a mind.

That depends on one’s definition of music. If one defines music in the same sense as you seem to define concept, then it is an inherently mental thing (in which case it entails a mind, by definition). But I do not see that it is necessary to define music in this sense.

I'd say that a similar thing is true for concepts: you can have entities or things without a mind, but you can't have concepts without a mind. And, further, I'd say that you can't have consciousness without a mind. These, of course are only my opinions, and even though I have poured them into your water glass, I don't expect that you will easily swallow them. That's fine. I just wanted to offer them to you and get your response.

It seems to me that it all comes down to definitions. You seem to define music as “something experienced by a mind”, and the same with concept, and the same with consciousness. If you define these concepts in such a way then of course it follows that they require a mind. But I do not see that this is the only way such concepts can be defined.

Perhaps we should start with consciousness. How would you define consciousness? What are the necessary and sufficient conditions for consciousness?

Yes, you have that right except that in this case, I claim that the boundary conditions we are talking about are the truly primordial conditions, which greatly precedes the BB. And, the "somehow 'built-in'" is the "fundamental constituent of consciousness" I talked about above. I believe there was such a constituent and you evidently don't.

Correct.

I should point out that I do believe in emergent phenomena. Moire patterns, and the Mandelbrot Set are examples of emergent phenomena. So is the wetness of water. But it is my opinion that conscious experience is different in kind from all emergent phenomena and that it requires something more to get it to happen or exist. I think that "something more" is ontologically fundamental. And, it seems that you can't swallow that.

I see no need to “swallow it”, because I believe I can see how consciousness emerges from non-consciousness. This being the case, why posit consciousness as being primordial?

I don't think it is that drastic. I think we are talking the same language, but that we have been using different connotations of the words 'PC' and 'world'. Hopefully, we can get over that misunderstanding. As I said before, I think our respective views of what actually happened at the ultimate "beginning" are not all that different.

Of course you would like to believe this, because the one of the arguments I am using against your PC is that it is inherently complex. At the end of the day, the fundamental difference between is that you believe consciousness is an a priori assumption (built-in to the boundary conditions) whereas I do not.

I was going to categorically deny this by saying that the theists have never defined their notion of 'God'.

It’s clearly defined in the bible. In the beginning was the Word, and the Word was God. That’s a very simple assumption, surely?

But I realized that I have never really defined my notion of 'PC' in its primordial state. I guess you are ahead of both the theists and me by not even identifying what you think might have been the primordial constituent of reality. So without identifying it, you don't have to define it.

I am not so arrogant or misguided to claim that we can possibly know what existed (if anything) prior to the Big Bang, but I do claim that the Big Bang itself was purely a physical process, and all complexity in our world, including consciousness, arose from purely emergent physical processes, and none of it is primordial. I believe we can explain emergent consciousness from first principles “within” the world defined by the Big Bang – whereas you seem to think that consciousness cannot be explained as an emergent process and therefore needs to be built-in to the premises.

But, as I have also pointed out, the theists posit huge (even infinite) complexity to their God in the beginning, whereas I posit only simplicity as you do.

Infinite complexity? What’s in a Word?

I know you would like us all to believe that your premise is simplistic. But as I have said, in the end your premise packs in consciousness as an a priori assumption – that makes it inherently more complex than any premise which posits complexity as an emergent property.

I think you would say that concepts, such as the implications of logical rules, exist independently, and more importantly, prior to the existence, of any mind.

Yes, I would.

In school, we are taught to develop proofs a step at a time and write them down in sequence. The correct answer is a static set of propositions listed in order, such that if a sceptic were to attend to the propositions in sequence, each should progressively dispel any doubt as it is pondered and understood, until the last one which should prove the theorem. So I ask (in the spirit of Beethoven and Goethe) what exactly is the mathematics here? Is it that static list of propositions? Or is it the time-dependent development of concepts and understanding in the conscious mind of the person either reading the list or originating it in the first place? Of course I claim it is the latter, and I suspect you would say it is the former.

I suggest you are confusing ontic with epistemic reality. The time-dependent development of concepts reflects our knowledge of mathematics (epistemic reality). The theorems that we prove true always have been true (ontic reality), even before we discovered those theorems. We do not suddenly “make something true” by proving it true – all we are doing is discovering a truth that was always there, waiting to be uncovered.

If you are right, I'm wondering where that list was at or around the time of the BB, and wherever it was, how did it get to be there? Talk about complexity.

That complexity was always there – even in your notion of the world. Take my example again of 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. Do you believe there is any way that the PC could have “created” a universe in which this law (given the postulates and definitions) would have been false?

Now with regard to infinity, it isn't that I think the notion of infinity is the problem or leads to problems. What I think is that, first of all, I don't think you can, or at least nobody has yet, define the notion of infinity in a consistent way. The ancients defined it, if at all, in nonsensical ways. Cantor, with the first rigorous definition immediately noticed consequent paradoxes.

OK, we are making progress. The diversion on Russell came about because you claimed that the notion of a “set of all logical possibilities” leads to contradiction, and I asked you to give an example. We have eliminated the Russell example that you gave. Do you have any other example?

(I have already mentioned my belief is that Cantor’s entire edifice is built on the incoherent notion that one can have an infinite set of finite integers – which is imho nonsensical but vehemently defended by the mathematical establishment)

What I object to is the assumption that concepts can come into existence without first being conceived by a mind. So I object to all genetic methods of "producing" sets of infinite cardinality, such as the infinite set of integers. You seem to have a similar objection to some of the methods mathematicians have used to define the integers. As I said before, I don't agree with Leopold Kronecker who claimed that the infinite set of integers were given to us by God. The only integers we have available to us to use are those conceived by us directly, or those explicitly generated by finite machines conceived and built by us. In all such cases, the integers are finite, and there have only ever been a finite number of them defined and used. Moreover, I predict that there never will be a time when an infinite set of integers will be available for our use.

It depends what you mean by “available for our use”. We “make use” of real numbers like e and Pi and 1/3 and root 2, all of which are irrational hence “infinite” in length when expressed as numbers.

My position is not that infinity leads to problems.

That’s progress. Earlier you were claiming that infinity leads to contradictions (which implies problems).

Instead my position is that we can't reasonably define anything infinite, and when we try to pull tricks to define infinity, those tricks lead to problems. The tricks, like Hilbert's genetic definitions, are what I think should be disallowed, just like the trick of recursive definition.

We cannot reasonably define anything infinite?
How many digits are there in the full decimal expansion of Pi, or of root 2, or of e?
Perhaps you cannot conceive of the full decimal expansion, therefore you conclude that it does not exist. This gets back to our fundamental difference in belief in the Platonic world. I believe that the full decimal expansion of Pi exists in the Platonic world, but I guess you do not.

Best Regards

 

[Canute]

None of that makes him infallible.

Of course not. But he's more infallible on QM than you or me.

In what sense? Can you express this in a form of a proposition which has neither a “yes”, “no” nor “meaningless” answer?

I gave you half a dozen when you asked the first time.

Why is the notion that the universe has existed for all time incoherent?

If by 'universe' you mean this one then this would be obvious. If you mean 'all that there is' then it would mean that time existed 'before the BB'.

In what sense? What exactly do you mean by “the universe arises”?

Well, clearly this universe appears to exist, and clearly once it didn't.

What false assumption does it embody, in your opinion?

The assumption that the universe arises from something or nothing.

How on Earth do you arrive at this conclusion?

Because you argue that metaphysical questions are analytically or self-referentially undecidable and not statements about the world.

Whether light is measured to be a wave or a particle depends on how you measure it. Specify how it will be measured and this determines whether you will measure it to be a wave or a particle.

Exactly.

Define freewill, then we may be able to debate this further.

Maybe we've got enough disagreements on the go already.

Define “arise from” then we may be able to debate this further. Do you assume a backdrop of time against which the universe “arises”?

According to physicists spacetime came into existence with the universe.

Either of these may be true – the fact that we do not know which is true does not make the proposition “imaginary” – it simply points to limits in our knowledge.

I'm suggesting that neither of them are true.

Bear in mind that the term “imaginary” in mathematics means something quite different to “imaginary” in normal language. A proposition based on mathematical imaginary terms is still either true or false (or meaningless).

You miss the point. Brown suggests that we use complex values in our thinking. He is not suggesting that some propositions are imaginary, he's suggesting that some questions have complex answers.

You see – no imaginary propositions at all (excluding the mathematical kind of course – if you wish to claim that all imaginary propositions are of the mathematical kind then I do not disagree)

What do you mean by an 'imaginary proposition'?

regards
Canute

 

[moving finger]

None of that makes him infallible.

Of course not. But he's more infallible on QM than you or me.

When he joins the forum I’ll bear that in mind.

In what sense? Can you express this in a form of a proposition which has neither a “yes”, “no” nor “meaningless” answer?

I gave you half a dozen when you asked the first time.

And none of the ones you gave entail “imaginary” status. They can each be treated as either true, false or meaningless.

Why is the notion that the universe has existed for all time incoherent?

If by 'universe' you mean this one then this would be obvious. If you mean 'all that there is' then it would mean that time existed 'before the BB'.

It’s not at all obvious. If time was created along with the BB then our universe (ie the one resulting from the BB) has existed for all time, by definition.

In what sense? What exactly do you mean by “the universe arises”?

Well, clearly this universe appears to exist, and clearly once it didn't.

“once” means “on one occasion only, at one time, at a particular time” - the statement “once it didn’t” therefore implicitly assumes a time when the universe did not exist. But if time was created along with the BB then there was no time when the universe did not exist (these kinds of concepts should appeal to a mystic)

What false assumption does it embody, in your opinion?

The assumption that the universe arises from something or nothing.

Which is exactly what you have done yourself – in saying (as you did earlier) that “the universe arises” you have already assumed some backdrop of time against which it arose. That may be a false assumption.

How on Earth do you arrive at this conclusion?

Because you argue that metaphysical questions are analytically or self-referentially undecidable and not statements about the world.

Could you give an example where I have done this?

Whether light is measured to be a wave or a particle depends on how you measure it. Specify how it will be measured and this determines whether you will measure it to be a wave or a particle.

Exactly.

Which goes to show that the question is meaningless unless it is formulated correctly – ie to include the way it will be measured. Nothing to do with imaginary.

Define freewill, then we may be able to debate this further.

Maybe we've got enough disagreements on the go already.

Then I cannot agree that this is an example of an imaginary proposition.

Define “arise from” then we may be able to debate this further. Do you assume a backdrop of time against which the universe “arises”?

According to physicists spacetime came into existence with the universe.

That’s interesting, don’t you think? Then how can it make sense to ask what was (temporally) before the universe? Again a meaningless question.

Either of these may be true – the fact that we do not know which is true does not make the proposition “imaginary” – it simply points to limits in our knowledge.

I'm suggesting that neither of them are true.

And I’m suggesting that, if the question has any meaning, one of them is true and one of them is false. We just don’t know which.

Bear in mind that the term “imaginary” in mathematics means something quite different to “imaginary” in normal language. A proposition based on mathematical imaginary terms is still either true or false (or meaningless).

You miss the point. Brown suggests that we use complex values in our thinking. He is not suggesting that some propositions are imaginary, he's suggesting that some questions have complex answers.

I thought we were talking about whether a proposition could be evaluated in terms of “true” “false” “meaningless” or “imaginary”? Can you explain how we now get onto the issue of complex answers?

You see – no imaginary propositions at all (excluding the mathematical kind of course – if you wish to claim that all imaginary propositions are of the mathematical kind then I do not disagree)

What do you mean by an 'imaginary proposition'?

I thought we were talking about whether a proposition could be evaluated in terms of “true” “false” “meaningless” or “imaginary”. If a proposition is evaluated “true” then it would be a true proposition. Similarly, if a proposition were evaluated “imaginary” (if such a thing has any meaning) then it seems to me that it would be reasonable to call it an imaginary proposition. Would you prefer a different expression?

Best Regards

 

[Tournesol]

By 'reasonable' here I mean reasonable according to rules of ordinary logic. For example, the proposition that the universe arises ex nihilo is logically incoherent,

Not really. "Everything must have a cause" is itself metaphsyics,
not logic.

However, if we modify the tertium non datur rule this opens up a third possibility.

In Zen the repsonse may be 'Mu', a word meaning, roughly, there is nothing to say because the question embodies a false assumption and any answer would therefore be misleading.

A false assumption like "everything must have a cause" ?
But that is again a modification to
metaphysics, not logic.

 

[Eric England]

Not really. "Everything must have a cause" is itself metaphsyics, not logic.

I would think it depends on the definition of logic. I think Homer and Bart said it best, when Bart said "duh"and Homer said "doh"!

 

[selfAdjoint]

Kant remarked that to say the universe came into existence at some moment, and to say that there is an infinite regress of prior causes, are both incoherent. You pays your money and you takes your choice.

 

[Tournesol]

The definition of logical truth is "that which cannot be denied withut
contradiction". "Every event has a cause" can be denied without
contradiction.

 

[moving finger]

In Zen the repsonse may be 'Mu', a word meaning, roughly, there is nothing to say because the question embodies a false assumption and any answer would therefore be misleading.

In science, Mu would be translated as “meaningless”.

Example Question : In the absence of any kind of measurement, is the quantum object a wave?

Answer : The question is meaningless, because it embodies a false assumption (that a quantum object has either a wave or a particle property in absence of measurement), and any answer would therefore be misleading. Presumably a Zen Buddhist would in such a case answer Mu.

Similarly, to ask "Is the King of France bald?" is meaningless (Mu), because it makes a false assumption that there is an existing entity which is "the King of France".

As I’ve pointed out earlier, “True”, “False” and “Meaningless’ are all we need.

Best Regards

 

[moving finger]

Kant remarked that to say the universe came into existence at some moment, and to say that there is an infinite regress of prior causes, are both incoherent. You pays your money and you takes your choice.

If time is created along with the universe, then neither of Kant's alternatives are applicable. In such a case, the universe would not come into existence at any "moment", because the very idea of associating a moment with the creation assumes a backdrop of time against which the universe is created. Neither would there need to be an infinite regress of prior causes (since this also assumes a time prior to the creation).

Did Kant address this possibility? That it perhaps makes no sense to talk of time existing prior to the universe "coming into existence", because time is created along with the universe?

Best Regards

 

[Tournesol]

No, several of K's anomalies are undermined
by modern develoments.

 

[Canute]

As Self-Adjoint mentions, Kant was clear about the incoherence of both these ways of conceiving of the origin of the universe. But all philosophers reach the same view as far as I know. This is why the questions of its origin is an undecidable metaphysical question. As S-E says, we pays our money and we take our choice. Unless, that is, we modify the tertium non datur rule.

When he joins the forum I’ll bear that in mind.

Ok. I prefer to assume he knew more than you and I do about physics and formal logic even though he isn't a member.

And none of the ones you gave entail “imaginary” status. They can each be treated as either true, false or meaningless.

Or they can be treated as having complex values for answers, as I am suggesting.

It’s not at all obvious. If time was created along with the BB then our universe (ie the one resulting from the BB) has existed for all time, by definition.

Yes. Here we meet a language problem. My view is that time is not fundamental, that it has no inherent existence. However, as Huxley points out in his book on mysticism, the words we use are designed for use in time and this causes all sorts of problems. You have to allow a bit of slack on this topic, otherwise it becomes impossible to discuss it. The universe exists in some sense so it arises in some sense. But by saying this I do not mean to imply that time is fundamental, or that the universe exists inherently.

Which is exactly what you have done yourself – in saying (as you did earlier) that “the universe arises” you have already assumed some backdrop of time against which it arose. That may be a false assumption.

I completely agree. In the Buddhist view the phenomena of this world are 'evanescent thing-events' rather as they are in physics and exist, inasmuch as they do, now and only now. There's probably a better term than 'arise' I could have used, but I suspect all the alternatives would also imply time.

Could you give an example where I have done this?

No need. I must have misjudged you position from what you've been saying about solipsism. I though you must apply the same logic to all metaphysical questions. Obviously not.

Which goes to show that the question is meaningless unless it is formulated correctly – ie to include the way it will be measured. Nothing to do with imaginary.

I think you'll see this is not the case if you rephrase the question as 'What is it that, when we observe it, appears to be either a particle or a wave?

This is not a meaningless question unless we assume that particles and waves are two things rather than two aspects. If they are two things then yes, the question would be meaningless. But if they are the dual-aspects of something ontologically prior then the question is meaningful and has a complex value, in Brown's sense of the term, for an answer. The answer, whatever it is, requires that as well as particles and waves there is something that is not a particle or a wave. Thus, as well as these two logical possibilities there is a paradoxical third option. We have no term for this third thing so just call it a 'wave-particle' or 'wavicle'. It's the best we can do in natural language. The concept of a 'wavicle' is pardoxical precisely because it contravenes the 'no third option' rule.

Then I cannot agree that this is an example of an imaginary proposition.

I'm not suggesting there's any such thing as an 'imaginary proposition'. But see below.

That’s interesting, don’t you think? Then how can it make sense to ask what was (temporally) before the universe? Again a meaningless question.

It is not meaningless to ask what was prior to the Big Bang as long as we are careful not to make assumptions about what we mean by 'prior'. If we mean previous in time then I'd say that it's not a meaningless question but an irrational one. But it is possible to ask what is prior to spacetime without descending into meaninglessness as long as we accept that to do so we have little choice but to use rather inapropriate terms.

And I’m suggesting that, if the question has any meaning, one of them is true and one of them is false. We just don’t know which.

Yes, I know this is what you're suggesting. What I'm suggesting is that we should introduce complex values into our reasoning in order to overcome the paradoxes the result from your assumption, which is the assumption that gives rise to all the undecidable questions that plague metaphysics.

I thought we were talking about whether a proposition could be evaluated in terms of “true” “false” “meaningless” or “imaginary”? Can you explain how we now get onto the issue of complex answers?

Sure. Brown gives examples of contradictions that arise is ordinary equation theory that we currently solve by the use of complex values (such as sqrt-1). He goes on to suggest, consistent with the idea that mathematics can model reality, that we adopt the same approach to solve the contradictions that arise in metaphysics. He claims that the need to take this approach follows from the nature of reality. (In other words, this is not just a question of adopting a convenient methodology).

For example. In Brown's view solipsism is neither true nor false. We've had enough battles over this one so pick any metaphysical of your choice. Take the question - Is the universe caused? Metaphysicians are stumped completely by this one. Neither answer makes sense. Brown's suggestion is that this is because we refuse to suspend the tertium non datur rule in metaphysics, even though we modify it all the time in quantum mechanics and mathematics. The 'complex value' here, which would be the answer to this 'first cause' question, is 'yes and no'. Here we find ourselves back with the 'causeless cause' of Taoist cosmology and of the perennial philosophy in general.

You may not agree with Brown but it is worth noting that some understanding of his ideas will make the mystical cosmology much more comprehensible. In this view metaphysical questions are undecidable because neither of their answers is correct.

I thought we were talking about whether a proposition could be evaluated in terms of “true” “false” “meaningless” or “imaginary”. If a proposition is evaluated “true” then it would be a true proposition. Similarly, if a proposition were evaluated “imaginary” (if such a thing has any meaning) then it seems to me that it would be reasonable to call it an imaginary proposition. Would you prefer a different expression?

I see what you mean but yes, personally I find this a slightly confusing expression. (Even though your's may be a mathematically correct expression). I prefer to think of this in terms of questions and answers. This amounts to the same thing but avoids the phrase 'imaginary proposition' which sounds a bit peculiar in everyday language.

The question 'Does God exist?' would be meaningful - the general view these days is that the logical positivists failed in the end to show that such questions are not meaningful - but would not have a yes or no answer. This is the view of the mystics and of many modern theologians. It is not that the answer cannot be known, so it is said. The question does not have a yes or no answer precisely because its answer can be known and it is not yes or no. Those who claim to know the answer say that the truth is too subtle to express in a on/off, yes/no, wave/particle kind of way.

In science, Mu would be translated as “meaningless”.

Yeah, that's close enough unless you want to be picky.

Example Question : In the absence of any kind of measurement, is the quantum object a wave?

Answer : The question is meaningless, because it embodies a false assumption (that a quantum object has either a wave or a particle property in absence of measurement), and any answer would therefore be misleading. Presumably a Zen Buddhist would in such a case answer Mu...

Similarly, to ask "Is the King of France bald?" is meaningless (Mu), because it makes a false assumption that there is an existing entity which is "the King of France".

This isn't a correct understanding of the use of the word, but it's something like this.

As I’ve pointed out earlier, “True”, “False” and “Meaningless’ are all we need.

Yes. However, what you've said does not determine what is the case.

Cheers
Canute

 

[Eric England]

How about "mass, space, and time can neither be created nor destroyed – but are infinitely existent within something greater in principle"?

(maybe the "big bang" is just a metaphor for "geeks wanting a date"?)