Dear Banjo,

Thanks.

>>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.<<

Well, maybe “entity” was not the right word to describe space and time. In philosophy (Immanuel Kant) “category” is used instead. However, if you want to take into account the progress in physics, that somehow bypassed that in philosophy, you need conceptual models or representations that cannot do without non-trivial mathematics. So e.g. time is modeled by the real line (mathematics) - unless you accept Smolin’s model.

Here “the final value before zero” does not make sense. (Whatever positive number you would suggest I can always name a smaller, but still positive, one).

Neither do I understand what are “purely human” concepts. What other concepts are there?

>> I see, for example, a one meter wooden ruler as a divisible continuum. Perhaps not so in mathematical language.<<
Mathematical language does not deal with wooden rulers, and I do not understand what “divisible continuum” - in distinction to “indivisible continuum” - means. We know from elementary physics that the smallest piece of "wooden ruler" obtained by mechanical means is a molecule. Beyond that you need chemistry and nuclear physics and to understand the latter you need the non-trivial mathematics of quantum physics.

>> Do you mean that each is a separate "continuum" <<
By continuum I simply mean spacetime modeled by quadruples of real numbers, in distinction to “discrete” (see Smolin) modeled by quadruples of integers.

>> 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 <<
What you describe here is the BELIEF in the existence of a (physical) reality independent of how we perceive it, a belief shared by practically everybody although it is not something that can be obtained as a “logical” consequence of scientific enquiries, as I indicated, following Hawkins and Mlodinow, in my article. We try to understand it by representing that reality through concepts, theories etc

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Dear George,

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[Here “the final value before zero” does not make sense. (Whatever positive number you would suggest I can always name a smaller, but still positive, one).]
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We humans have developed and continue to develop greater intelligence than all other living species.

Our capacity for abstract conceptualization is greater today than it was 5 to 7 million years ago when we broke away from our common ancestor with the chimpanzee. There is nothing to suggest that it will not be even greater in 5 to 7 million years time.

What we have difficulty conceptualizing today may not be a problem tomorrow. The fact that something "does not make sense" today does not mean that it will not make sense tomorrow.

Quite a large number of phenomena "did not make sense" to many scientists and mathematicians before new concepts provided the appropriate explanations.

You correctly quoted me as saying “the final value before zero”. For the reason you indicated, I did not hazard a guess as to what that final positive number might be. I shall leave that important detail to the next Euler, Gauss, Riemann, Euclid or whoever.

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[Neither do I understand what are “purely human” concepts. What other concepts are there?]
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I understand that all living species equipped with a brain are capable of conceptualizing (making mental representations) to varying degrees. This would include all animals except certain invertebrates such as sponges, jellyfish, adult sea squirts and starfish.
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"I do not understand what “divisible continuum” - in distinction to “indivisible continuum - means.”.
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This looks like a false problem. I thought you may have a definition of "continuum" which excluded its divisibility so that "smallest quantity" made no sense to you.

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"What you describe here is the BELIEF in the existence of a (physical) reality independent of how we perceive it ..."
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Yes. I call it "faith" - but not "blind faith". This is faith based on circumstantial evidence, with a high degree of confidence in the truth or existence of something not immediately susceptible to rigorous proof.

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Dear Banjo,

Sorry, I cannot better explain elementary properties of real numbers. There is no such thing as the smallest positive real number and nobody can or will change that, unless you redefine the meaning of the words involved. I am sure david f would confirm that.

Also, I spoke of the belief, (acceptance, assumption) of something that is the basic building block of most everybody’s world-view. Not of faith, blind or not. Circumstantial evidence, high degree of confidence etc, are things that depend on, well, circumstances, your state of mind, and other subjective factors. These would be relevant to justify religious belief (or “unbelief”) that certainly not everybody shares, but not in the case of belief in the existence of a reality outside and independent of my (and your) mind that, as I wrote, is shared by practically everybody, I suppose even by those who for philosophical reasons (to stay within what can be scientifically established) prefer the Hawkins-Mlodinow working assumption.

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Dear George,

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"There is no such thing as the smallest positive real number"

I am not sure that we are talking about quite the same problem.

You will recall that I defined the “infinitely small” as the smallest quantity of something which exists.

The problem is not whether there is such a thing as "the smallest positive real number" or not, but whether there exists such a thing as the smallest quantity of something or not.

If the answer to that question is "yes, then perhaps my definition of "infinitely small" may be considered as making sense.

Perhaps "the smallest quantity of something" depends on the nature of "the something". If the something were a gas, for example, perhaps its minimum quantity may be a molecule. If reduction (or should I say decomposition) of the molecule results in zero quantity of the gas, then a molecule of the gas is what I referred to previously as "the final value before zero" of that something (the gas).

If we wish to go a step further and measure "the smallest quantity of (that) something which exists" we would need to be able to measure the molecule of that particular gas.

The problem is not to measure the smallest of all positive real numbers - which you say does not make sense - but to measure "the smallest quantity of something which exists".

I have no idea if physicists and mathematicians are capable of identifying and measuring that quantity. If, today, they are not, I see this as a difficulty, not an impossibility.

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" Circumstantial evidence, high degree of confidence etc, are things that depend on, well, circumstances, your state of mind, and other subjective factors ... belief in the existence of a reality outside and independent of my (and your) mind that, as I wrote, is shared by practically everybody ..."
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All those "subjective factors" which, as you wrote, are "shared by practically everybody", backed up by objective circumstantial evidence, certainly provide a solid base for belief (faith) in the existence of a reality independent of mankind.

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Dear Banjo,

Sorry, but I really do not know what “"the smallest quantity of something" is. I am glad you agree that if that “something” are positive real numbers, then it does not exist. If it is a material object then "the smallest quantity" is a molecule or atom, or some elementary particle etc, depending on what you call “smallest”, but we have been through that already. Otherwise, I don’t understand, so please just have to leave it at that.

>>All those "subjective factors" which, as you wrote, are "shared by practically everybody”<<

I never said that. On the contrary I said those subjective factors were relevant only in the context of RELIGIOUS beliefs “that certainly not everybody shares”, whereas you do not have to resort to these factors in the case of the belief in a reality independent of our mind, because it “is shared by practically everybody”.

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Dear George,

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"Sorry, but I really do not know what “"the smallest quantity of something" is. ... If it is a material object then "the smallest quantity" is a molecule or atom, or some elementary particle etc, depending on what you call “smallest”".
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Yes, I was referring to "matter" (thing that has mass and occupies space), the smallest quantity of which being that which is the smallest in all its physical dimensions while remaining capable of stable independent existence without undergoing a change of nature (without modifying its physical or chemical properties).

My definition of “infinitely small” as the smallest quantity of something which exists applies to all forms of matter:

- element (a substance that is made up of a single type of atoms)

- compound (a pure substance that is made up of two or more elements chemically combined in fixed proportions)

- homogeneous mixture (matter composed of two or more components in which the compositions are variable and the components are indistinguishable from each other)

- heterogeneous mixture (matter composed of two or more components that are visibly distinguishable from each other)

I have done my best to speak your language and hope you find it comprehensible.

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