Dear david f,

Thanks for the reaction. I agree that “instantaneous transmission” does not make much sense. Note that I referered to entanglement as being either instantaneous (no transmission) or associated with speed. Probably I should have referred to the standard “non-local" (connection) instead of the clumsy “instantaneous”.

I don’t see the relevance of experiments with unconscious processes, except that the weirdness of the Copenhagen interpretation of QM hinges indeed on its seeming involvement of consciousness, i.e. conscious observer/measurer.

Perhaps the following is a good non-technical explanation of why quantum entanglement is not a transmission (http://www.razorrobotics.com/knowledge/?title=Quantum_entanglement):

Quantum entanglement, also called the quantum non-local connection, is a property of certain states of a quantum system containing two or more distinct objects, in which the information describing the objects is inextricably linked such that performing a measurement on one immediately alters properties of the other, even when separated at arbitrary distances. Specifically, such a system is said to be in an entangled state if it cannot be written as the tensor product of its constituent subsystems. This discovery posed a serious conceptual challenge to physicists of the day, because faster-than-light influences were assumed to be prohibited by special relativity. In that framework, it was thought superluminal effects would lead to causal contradictions because a change of reference frame can reverse the order of the events. It is now understood that while these nonlocal correlations do occur, they cannot be used to transmit information and thus do not violate causality.

I really think that we cannot solve here the questions raised by the two papers about the “speed” of entanglement.

Dear Banjo,

I can understand what “very small” means, I have only problems with “infinitely small”.

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Dear George,

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"I can understand what “very small” means, I have only problems with “infinitely small”.
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I sympathise with you.

May I suggest that for there to be quantity of something, that something must exist. If something does not exist it cannot have quantity.

For something which exists, I call "infinitely small" the minimum limit of its existence.

In other words, “infinitely small” means the smallest quantity of something which exists.

The difficulty lies in identifying, apprehending and measuring such a quantity, but that is probably true whatever the quantity.

Can we measure anything exactly?

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Dear Banjo,

Sorry, I still do not understand how to interpret what you wrote within accepted physical theories. Existence of entities at the very basic level of physical reality depends on the theory you use to represent reality with, and cannot be handled by using “common sense” approach to these abstract concepts, as I tried to argue - referring to Hawking and Mlodinow - in http://www.onlineopinion.com.au/view.asp?article=14464.

>>“infinitely small” means the smallest quantity of something which exists <<
Space (and time, and other physical entities) are usually represented as continuum, hence “smallest quantity” does not make sense (like there is no smallest positive number). Unless you hint at the unusual (and still speculative) Lee Smolins’ Loop Quantum Gravity theory (see the article Atoms of Space and Time by Lee Smolin in Scientific American, January 2004, pp. 56-65):

“Loop quantum gravity predicts that space comes in discrete lumps, the smallest of which is about a cubic Planck length, or 10^(-99) cubic centimeter. Time proceeds in discrete ticks of about a Planck time, or 10^(-43) second.” (Here ^ stands for “to the power of”).

Thus if Smolin is right, it is more useful to represent spacetime as a discreet set of events, not a continuum.

Mathematically speaking infinity is a limit. Consider the infinite series 1, ½, 1/3, 1/4, 1/5, …… Each term of the series gets smaller and smaller, and terms get very small. Let us consider the nth term of the series. Its value is 1/n. No matter how big n is, its inverse is very small but greater than 0. The value of the number is infinitesimal but not infinitely small mathematically speaking.

However, Banjo was not mathematically speaking. In natural language an infinitely small number can be one that is so small that it is 0 for all practical purposes.

Banjo and George are both right. They were merely using different languages. Banjo was using natural language, and George was using mathematical language.

Dear david f,
Thanks again, for providing an opportunity for further attempts to explain my position.

“Mathematically speaking” neither “infinitely small” nor “infinitesimal” makes any sense, although the latter is used in popular language as “a value approaching zero”. As I used to tell my students, a value cannot “approach” (or “go to” anything), because it has no legs, meaning that these verbs depend on the concept of time, that belongs to physical, not mathematical, reality. You might remember the epsilon-delta definition of continuity to avoid the intuitive, but imprecise “f(x) goes to f(a) as x goes to a”. As we used to say, Newton would not pass our first year (pure) maths exams, exactly because of these matters.

Of course, Newton’s approach to calculus was based on intuition, the same as Leibniz’s who allegedly was the first to use the term infinitesimal. Also, a physicist’s evaluation of observations/experiments must start from intuition, after which he must incorporate them in some existing theory, or suggest a new one, which at the end cannot avoid the language of CONTEMPORARY mathematics.

I can understand what “a number can be one that is so small that it is 0 for all practical purposes” means, provided the context - all practical purposes e.g. when speaking of elementary particles - is known.

I am afraid the times have gone - latest with Einstein - when one could build physical theories (about the structure of matter or physical reality) using only non-mathematical, imprecise “natural language”. Of course, one has to use natural language when explaining the matter to non-specialists. That is why I did not argue against Banjo’s suggestions using the term “infinitely small” (or infinitesimal) quantities; I just did not understand what in my limited knowledge of physics he was referring to.

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Dear George,

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[>>“infinitely small” means the smallest quantity of something which exists <<

"Space (and time, and other physical entities) are usually represented as continuum, hence “smallest quantity” does not make sense (like there is no smallest positive number)".]

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Apparently, the precision "other" indicates that you consider space and time also to be physical entities. Rightly or wrongly, I do not. I consider that neither "exists", the latter being purely a human concept.

I was not referring to either space nor time but to something which "exists" and the smallest quantity of that something.

Even if the "something" could be described as a "continuum" I do not see why it should not be quantifiable and divisible. Perhaps this differs from the mathematical definition of "continuum" which you may have in mind.

I see, for example, a one meter wooden ruler as a divisible continuum. Perhaps not so in mathematical language.

Also, I note that you indicate that space and time and "other" physical entities "are usually represented as continuum" - not "continua". Do you mean that each is a separate "continuum" or that they, considered as a whole, form a single "continuum"?

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"Existence of entities at the very basic level of physical reality depends on the theory you use to represent reality with ..."

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It is my view that "reality" pre-exited mankind and will continue to exist post-mankind. I consider that it is independent of what mankind thinks of it and not in the least influenced by his opinion.

Btw, I do not see "infinitively small" as “a value approaching zero” but the final value before zero.

Thank you, George, for the link to your very informative article on "The nature of reality". I think you often underestimate your ability to render complex mathematical and scientific concepts comprehensible to uninitiated persons such as me.

I truly appreciated your copy and hope there will be more.

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