• [deleted]

Rob,

If you want to discuss the Lamport paper, please read it first.

You write, "Your whole argument is based on the false assumption that everything behaves like memoryless particles - when given a symmetric input, they have no way to 'break the symmetry' (except via random, equiprobable statistics). But every entity, more complex than an electron, can exploit its memory (internal, alterable states) to break the symmetry, and that cannot be modeled by random equiprobable statistics."

Sheer gobbledegook. I do not assume memoryless particles; I simply assume physical laws. Not only statistical -- continuous functions as well.

"I may have gotten all too philosophical for your tastes, but Albrecht's argument is too pseudo-scientific for mine."

Whatever you think pseduo-scientific means, it does not apply to Albrecht's framework. Better be more specific here if you want to sound convincing.

"There is no objective science in assuming that the behaviors of non-elementary particles lack any mechanism for "symmetry breaking", and can therefore be modeled via trivial, random probability theory."

Complex systems science is loaded with examples of order from randomness. Try Gell-Mann's *The Quark and the Jaguar* or Steven Strogatz's *Sync* for easily accessible reading.

"I *stated* that only the world itself, in its entirety, can determine some outcomes, and even it can only determine those outcomes, by actualizing them, in 'real-time', and no faster. If you wish to call it a computer, then it is an analog one, not a digital one, and, most importantly, it is an 'analog' of itself. It 'computes' its next state, by the simple act of entering its next state."

You don't seem to understand that if the world cannot be described by an algorithm shorter than itself (Kolmogorov-Chaitin complexity) -- i.e., is not algorithmically compressible -- then it *is* random at foundation. As I said.

Tom

Tom,

"I simply assume physical laws. Not only statistical -- continuous functions as well."

You have also assumed that the decisions based on your continuous function inputs, must also be continuous.

That is the assumption I object to. If you give me a continuous function, of finite bandwidth, I can digitize it. I can then use the entire bit-stream as a single serial number to look-up a set of instructions that dictate my subsequent behavior. My behavior is not a continuous function.

Rob McEachern

  • [deleted]

Rob,

You write, "My point is that the multiverse hypothesis does not forbid the 'laws' to be such that *any* set of initial conditions, acted upon by those laws, always produce the same result - a universe that looks like ours."

Yes it does. Because the bit is fundamental -- one binary choice from among an infinite set of initial conditions implies continuously branching probabilities. Think of the symmetry breaking at a table setting of perfectly symmetric silverware -- the first person's choice of left or right determines all the other choices.

"The rules of logic and avoidance of self-contradiction also do not forbid such laws. Hence, since the multiverse hypothesis assumes that all possible laws are instantiated, this peculiar law and its peculiar universe, must exist."

Nothing compels any universe to exist; existence is always framed by domain and and range -- this peculiar domain and range does not obviate other such. The universe is possibly self limiting, as Wheeler said, "The boundary of the boundary is zero."

"It is a universe in which it is *Necessary* that any and all initial conditions result in the universe we actually experience. You may believe that you also live simultaneously in another universe, whose existence is *contingent* upon having just the right initial conditions. But that universe is not *necessary* to the existence of the other. Nor are any of the other universes, within the multiverse."

So what? Existence of itself, any way one describes it, is not necessary. Objective existence is dependent on domain and range, and that what we are describing here when we discuss a multiverse. Hawking is reported to have said that the many-worlds hypothesis is trivially true -- I agree.

"The definitive assumption of the multiverse hypothesis, that all possible laws be instantiated, is *sufficient* to cause the hypothesis itself, to be *unnecessary*, because, hidden within it, is at least one copy of our universe, whose existence is *necessary* as a result of a very peculiar law, rather than the law's initial conditions - a law that has "chosen" to ignore the initial conditions."

As I told Peter, logical consistency alone is no test of a physical theory.

Tom

  • [deleted]

Rob,

You write, "You have also assumed that the decisions based on your continuous function inputs, must also be continuous."

No. The range of variable values is continuous. Read the Lamport paper.

"That is the assumption I object to. If you give me a continuous function, of finite bandwidth, I can digitize it. I can then use the entire bit-stream as a single serial number to look-up a set of instructions that dictate my subsequent behavior. My behavior is not a continuous function."

The signal is, though. Buridan's Principle is quite profound: "A discrete decision based upon an input having a continuous range of values cannot be made within a bounded length of time." The implications for physics go far beyond the arbiter problem in computer science.

Tom

  • [deleted]

That's not physics, it's philosophy, and bad philosophy. You should be on a philosophy site - but even there, unlike you, they will see the need for answers that you don't give, and that your view can't give. You don't understand the questions others are trying to answer, and at the same time, you think you've solved it all.

Well here's a question. Imagine you're looking through a microscope at two bacteria (or it could be two particles). They're in the same place, right under the microscope, very near each other. But one is not moving, while the other is moving to the right at 0.00005 cm/sec. The two bacteria are ageing at different rates, and in a measureable way - this has been effectively proved by experiment, because measurements have been made in equivalent situations. Do you not see the need for an explanation for this? As your fingers type on the keyboard, you're seeing them in slow motion, because they're moving in relation to you. They are ageing slightly more slowly than your face. Later on the cells will be older than those in your face. If you can't explain this, then you should face it, and admit that there are unanswered questions, instead of saying that you know it all. If you knew a little more, you'd know what you don't know.

  • [deleted]

PS. Sorry, correction - 'later on the cells will be younger', not older

  • [deleted]

PPS. I'm amazed - I just read your last post more slowly. Yout think we don't allow for the Doppler shift, and light travel times. We do.

"But, when relative distance is altering (ie there is changing relative movement), then the perceived (ie received) rate of change alters, because the delay is ever increasing (or decreasing) at a rate which depends on the rate at which the distances are altering. To the observer this gives the impression that the rate of change is slowing/speeding up, over time, but is an optical illusion, as the actual rate of change does not alter."

No, we allow for the Doppler shift, and light travel times. You can't explain time dilation that way, by deciding we're all too stupid to see that! Do the reading.

Tom,

"A discrete decision based upon an input having a continuous range of values cannot be made within a bounded length of time."

Start listening to a sound with a "continuous range of values", like a sinusoid. Then attempt to make a discrete decision - that you have tired of listening and wish to stop and reply to this comment. If what you say is true, I suppose I will never hear from you again, since you will be unable to ever make the decision.

"I do not assume memoryless particles".

You have, whether you realize it or not. An entity with sufficient memory can *always* use the contents of that memory to modify an input signal, to avoid all the problems you have discussed. If nothing else, it can digitize it, so that it is no longer continuous.

"You don't seem to understand that if the world cannot be described by an algorithm shorter than itself (Kolmogorov-Chaitin complexity) -- i.e., is not algorithmically compressible -- then it *is* random at foundation. "

I understand that quite well. However, I was not discussing how to formulate algorithmic descriptions of the world, I was discussing the "hardware" requirements for running any such algorithm; the entire universe does not seem to contain enough "hardware" to predict some events before they occur. That is the ultimate "cause" of free-will.

"logical consistency alone is no test of a physical theory"

I did not know we were discussing a testable physical theory. I was discussing the untestable, metaphysical multiverse hypothesis. What have you been discussing? If you have a testable physical theory that you wish to discuss, please identify it and the test.

Rob McEachern

  • [deleted]

Tom,

Generalities often push reasoning to the fringes of reality where the meaning of words get diffuse and ambiguous. I'll try to avoid such 'blackholes of reasoning' where no light escapes! And narrow the scope of our discussions to specifics.

Specifically to the Pythagorean Theorem, you write "the Babylonians ... applied the Pythagorean theorem in measuring land. ... there was no mathematical necessity for these applications; the phenomena were known long before being described in formal language."

Counterpoints:

1)The Pythagorean Formula at the time of the Babylonians and the Egyptians was NOT a proven mathematical theorem. Rather, this was a 'physical law' since it was discovered in the measurements of physical distances. Its validity at that time was found only in the empirical evidence. The Babylonians realized this as a 'fact of Nature' and not a 'fact of Math'.

2)No mathematical necessity is requires to apply any mathematical identity to anything. Only the will and judgment of man. The 'mathematical necessity' is in the proof of its validity.

3)The Pythagorean Theorem is not the formal symbols that describe this "phenomena". It is the mathematical reasoning and proof that makes this a 'mathematical identity' and not a 'physical law' per se.

4)In an analogous way, I have shown Planck's Law to be a 'mathematical identity' and not a 'physical law' as is currently believed by our own 'Babylonians'.

5)I am arguing all Basic Law of Physics likewise can and should be formulated and derived as 'mathematical identities'. Take this as a call and not an established fact. Were this to be done, as I believe it can, it would solve the philosophical enigma why Nature should follow our mathematical calculations and derivations. And it would also keep physics from morphing into metaphysics. As we now see happening, with such metaphysical beliefs as 'backward causation', 'multiversies', 'time travel', and 'spooky action at a distance'.

You write, "No mathematics describes what is". We agree. The problem is many physicists believe it does! Spacetime, for example, is not real yet believed to be real. On the other hand, "the moon really is there when no one is looking"!

As for "domain" and "range", are you saying "Domain" is Math while "Range" is Nature? More reasoning at the fringes of reality!

Constantinos

  • [deleted]

Paul,

Try to understand what I am saying here. I am arguing for an analogy, not an equivalence: "knowing 'what is' the Universe is no more possible than knowing another human being truly".

Constantinos

  • [deleted]

Peter,

The cat is dead. It's been in the box for at least 54 years.

CIG Theory is alive and well... well...causality is alive... and...well, just needs a little nudge..and understanding.

last line: The Spacetime Curvature becomes the Matter. Why not include that too?

The probability is real, as it is the spatial manifestation of traveling mass. The collapse is real.

Buy a round on me.

& thank you for understanding CIG Theory

Your thought on the Fourth Law? (i.e. Gravity & radiation follow the same force)

As excerpted from CIG Theory & relevant to time (applicable to this topic)

THX

doug

FOURTH LAW OF SPONTANEOUS MOTION:

ANY accumulation of matter (i.e. pencil) occupying space and without motion with respect to its directly accompanying surrounding frame of field reference, which when allowed to be free of another holding field (the hand in this case being the holding field), and which then moves (the pencil in this case) with respect to that initial surrounding frame of field reference, are ALL THEMSELVES ABIDING BY the SAME PHYSICAL FORCE (tendency). That force being and the motion of the object also being according to the following:

The Spontaneous Motion will proceed from one pressure gradient {field densities with high Time "T" value variations per unit volume}to another pressure gradient {field densities with a different Time "T" value variations per unit volume}, {and/or the same "Time" density per unit volume space but with different volumes observed}, {and/or equal volumes observed (i.e. same volume but different pressures = more time per given space!)} until such time that they ( the reactive parties) reach equilibrium with one another (in this case, the pencil reaching equilibrium with the desk and now resting on the

desk). The pressure difference being the relative field pressure between the two fields, AS PERCEIVED ONE FIELD TO ANOTHER. The field itself being the amount of matter per given volume of space (or, per MT=S, this represents the amount of Time per given volume of space). Here, the concept of pressure is being applied not only to gaseous elements but to all of matter, and in a field-like manner. This Spontaneous Motion holds true for: objects released in air (at my desk) (Gravity), objects released in space (no movement is equal to equilibrium between perceived fields), electricity through copper wire, light through fiber optics, a rubber band stretched and released, Big Bang, metal fillings allowed to be free within a magnetic field until their

movement is so stabilized, beta radiation, a burning match, a bubble of air under the water and released to allow it to "float" to the surface, the combining of elements and all of chemistry, and

ALL ELSE, WHEREIN PARTICLES/FIELDS/OBJECTS WITHOUT MOTION RELATIVE TO THEIR INITIAL SURROUNDING ENVIRONMENT THEN FIND MOTION, UNTIL SUCH RELATIVISTIC FIELDS OF THE ENVIRONMENT HAVE BEEN STABILIZED TO REACH EQUILIBRIUM (TIME!) AS

PERCIEVED ONE FIELD TO ANOTHER. This equilibrium appears consistent with entropy, however, "The Theory" views entropy as a sate whereby things are moving from a state of disorder to a state of order, or equilibrium, a very ordered state, a very very ordered state. Entropy - disorder to order.

Using I and II above, along with Sir Newton's three Laws, and substituting the Law of Spontaneous Motion for Sir Newton's Law of Gravity, all things may be better understood. The four laws of motion now include gravity, radiation, etc. etc., and can be applied to everything. Part of "FLEA" in four simple laws of motion. The concepts of gluons, electrons, protons, leptons, quarks, elements, etc. as these particles have been described as such to "explain" our world, are still simply variations on the underlying MT=S theme. In the

current Standard Model, there is no end to the number of particles, simply keep simply colliding, and for that subject, why not simply extend the standard model upward and outward to the cosmic scale, where the planets themselves can be considered particles to the solar system and solar systems particles to the galaxies, for this is what the Standard Model offers, but on the micro way down, not the macro way up. "The Theory" rather states that there are only GODROPLETS in a GODROP, following Newton's Laws 1, 2, and 3, and the above Law of Spontaneous Motion, now Law Number 4. The pressure differences explain why one particle needs an anti-particle, why a south pole needs a north pole, etc., since things are being viewed with respect one pressure gradient to another. From the above, one can see that there are no gravitons; rather only Godroplets (Things Two) trying to reach equilibrium with each other and within the Godrop (Thing One). The Standard Model is Not Wrong, but the One Particle Aspect of The Theory Consolidates!

The law of gravity & the law of charges are not arbitrarily so very identical, rather they are explaining the same thing, though on different scales. The Theory looks at "m" and "q" not as "m" and "q" but as time-field densities. And the further these fields are apart (r^2), the less direct they "see" each other.

b)

On Motion Alone

All motion is simply the rate at which equilibrium is attempting to be reached. The greater the mis-equilibrium, the greater the rate of motion (the reason for catalyst in chemistry) to have the constituent particles/fields reach equilibrium. The ball dropped on the moon (as perceived by the ball to the moon) has less of a mis-equilibrium than the same ball looking at the earth, consequently, the resultant motion is also less. The greatest

mis-equilibrium possible is that of the electromagnetic spectrum, wherein the rate at which equilibrium is met is the speed of light (c). My offer, though rather obscured, is to the effect, can someone create a mathematical model/formula which determines the rate at which motion will arise, when one knows the equilibrium difference between fields (mis-equilibrium). All motion is a direct result of equilibrium being attempted to be reached (i.e. a thrown football, gravity, catalyst reactions, and speed of light). Perhaps with respect to gravity, and synthesis with strong force, weak force, and

electromagnetism, monopoles etc. one can consider the fact that a body resting on the earth's surface, having been dropped accordingly, that that body itself does not proceed directly to the center of the earth, to the gravitational center that is, that this represents a link analogy to a repulsive force, as is viewed by a positive nucleus, with a surrounding field of negative electrons. In this case, the body so dropped would be the electrons, and

the center of earth's gravity, the very center of earth, would be the positive nucleus. Here justification through this analogy may view gravity as a repulsive force, thus further strengthening the notion that one is seeing the same thing in electric fields as we are in gravity - The Theory simply states it is the way we look at it.

c) More

Consider this input:

Find two things (particles, fields, softballs, ants, etc.) going at the same rate (speed), and even though these things may be totally different (i.e. ball dropping, car moving, hurricane movement, etc.), and we can positively state that the proportionality of these two different systems attempting to reach equilibrium, (ball to ground, car to stop, hurricane to New Orleans), as perceived one equilibrium to the dis-equilibrium (misequilibrium

& dis-equilibrium are used interchangeably in this technical paper) they are

attempting to reach equilibrium with, are the same (when they at the same speed - obviously when the car stops their proportionality of equilibrium-disequilibrium is now different & all bets are off!). This leads to:

Lipp's Law of Proportionality

The states of any two systems having the same rate of motion see equal states of equilibrium to dis-equilibrium. (Internal note: The tie in with constant motion appears elsewhere in this paper) So the above is the reason for all motion, and their rates, & explains Gravity! END -Synthesis of Gravity into the normal realm: Fourth Law of Motion

  • [deleted]

Constantinos

Now it depends whether you meant it is not possible to know another human, because as with many physical entities it is too complex to know acurately and comprehensively (which I do not think you did mean). Or whether it is not possible because the human has certain attributes which are difficult to 'pin down' (ie know), which is not correct because you are referring to features which are not physically existent (ie temperament), but these do have a physical basis (which is what I thought, and indicated, you meant).

Furthermore, a definition of Universe is required, something I alluded to in the first post. Is this an allusion to what is 'really' there (ie what existence 'really is'), or is this meant to refer to existence as in what is knowable to us? The former being unattainable, existence might be anything, we can never know. The latter being what must be presumed to be a particular form of existence, but we can never know whether it is or not so that presumption is irrelevant. As long as it is examined on a bassis which ensures its independent identity, then that is it/real/correct, for us.

In short, in physics, the subject of investigation is the physically existent state of whatever is referred to. Humans have a tendency to refer to entities which are not actually physically existent. Indeed, the whole way in which physical existence tends to be conceived is incorrect, but that is another level of detail....

Paul

  • [deleted]

Anonymousse

"That's not physics, it's philosophy, and bad philosophy"

As per previous responses to such an assertion, the question is, what specifically was, and why?

What experiment has proven that, given two entities, which are both moving, but are moving at different rates, age at different rates? And how can this be so? The duration of their physical existence is not dependent on movement.

"As your fingers type on the keyboard, you're seeing them in slow motion, because they're moving in relation to you. They are ageing slightly more slowly than your face"

This is incorrect. Movement does not determine duration of existence. Furthermore, it is not the physical entity which is 'seen' but a physically existent representation of it, commonly known as light. Indeed, in my last post, somewhat anticipating your flawed way of thinking, I explained an optical illusion associated with differential movement.

"No, we allow for the Doppler shift, and light travel times. You can't explain time dilation that way"

How? And what is 'time dilation', in proven physical terms, if is not the time differential of receipt of light. I do not want to hear about perceptions, they do not create physical existence.

Paul

  • [deleted]

Constantinos

"Your "factually correct statements" may be factually correct to you. But they are "blatant misunderstandings" of my arguments"

Ironically, this assertion has made the very point I am making. There can only be one form of existence which is 'correct', for us. We can only know what it is possible for us to know. Whether this is then expressed in the form of a mathematical identity/theory is irrelevant. The reference for validity can only be what is knowable (obviously that is limited to what has become known at that time, but that is a practical point associated with the development of knowledge, not a metaphysical one).

You might notice that others are arguing against what you are asserting, except that they are using specific example. I am pointing out your fundamental confusion over what can constitute valid/invalid, which is dependent on a particular reference, given our existence.

Paul

Tom,

Does Buridan's principle not assume an isotropic homogenous universe?

We now (mostly) seem aware it is not. It is handed and anisotropic. The CMB is both anisotropic and asymmetric, we have Chirality, There is intrinsic rotation in space, and it rotated a body one way not the other.

The ass then would surely not need to make any choice (apart from NOT to move!), as the mountain (one side or other) of hay will come to Mohammed.

I understood Joy's thesis was also founded on such a principle. When do the thoughtful police let him out again by the way?

I have to say as an observer that Rob's above case seems rather more watertight than yours. I hope it's because you're diverting some intellectual effort to keeping your promise about reverting on my proposition.

Best wishes

Peter

  • [deleted]

You're quite simply wrong. People have believed that time dilation exists for a century, having derived it from theory, and then having confirmed it by experiment.

You consult your intuition, and decide that we're all less smart than you, and decide this is wrong, without even studying it first. And yet physics is often not intuitive, we know that. You admit you haven't done the reading.

In the many experiments that confirmed time dilation (see for example Chou, C. W. et al, Optical clocks and Relativity, Science 24 Sept 2010: Vol. 329 no. 5999 pp. 1630-1633), we of course allow for other effects, though your ridiculous attempt to explain it depends on us having failed to allow for the Doppler effect. You also try to use light travel times to explain it sometimes.

If you could explain it away using intuition, we wouldn't have an unexplained effect, now would we? A century of thinking dismissed with a wave of the hand from someone who hasn't even read about it! This conversation is now closed - I'm terribly sorry, I didn't fully realise where you were at until now.

  • [deleted]

Rob,

You write, quoting Buridan's principle (Lamport): 'A discrete decision based upon an input having a continuous range of values cannot be made within a bounded length of time.' And comment:

"Start listening to a sound with a "continuous range of values", like a sinusoid. Then attempt to make a discrete decision - that you have tired of listening and wish to stop and reply to this comment. If what you say is true, I suppose I will never hear from you again, since you will be unable to ever make the decision."

As I said before, you are trying to apply anthrocentric reasoning to a general problem. Buridan's principle applies first to the arbiter problem -- how to assure allocation of computing resources to a fair decision at any particular moment. As you allow, *some* decision can always be made. What Lamport found, is that in any *bounded length of time* there is no way to prevent some 'ass' from starving, i.e., to prevent accidents: "The status of continuity in quantum physics is less clear than in classical physics. The laws of quantum mechanics (such as Schroedinger's equation) are continuous, and the Uncertainty Principle, like random noise, seems to prohibit only the deliberate starvation of the ass, not its accidental starvation."

Applied to the multiverse and assuming an unbounded length of time, Buridan's principle makes an accidental universe not only plausible but inevitable.

Quoting me: 'I do not assume memoryless particles'.

"You have, whether you realize it or not. An entity with sufficient memory can *always* use the contents of that memory to modify an input signal, to avoid all the problems you have discussed. If nothing else, it can digitize it, so that it is no longer continuous."

Right. See above. The problem is not that a decision cannot be made, it's that there is no schema to guarantee that any decision in a bounded length of time won't lead to "starvation."

Quoting me, 'You don't seem to understand that if the world cannot be described by an algorithm shorter than itself (Kolmogorov-Chaitin complexity) -- i.e., is not algorithmically compressible -- then it *is* random at foundation.'

"I understand that quite well. However, I was not discussing how to formulate algorithmic descriptions of the world, I was discussing the 'hardware' requirements for running any such algorithm; the entire universe does not seem to contain enough 'hardware' to predict some events before they occur. That is the ultimate 'cause' of free-will."

A program can run on any substrate (that's the source of Minsky's description of humans as computers made of meat). What Chaitin found, is that the choice of program (equivalent in our context to Albecht's choice of clock) affects the computation; Chaitin's number - the halting probability of a Turing machine -- outputs a different result depending on the computer language running the algorithm. There's randomness even in arithmetic. (If you are interested, my ICCS 2007 powerpoint and paper discuss this subject.)

As to free will, I don't find it to be based in the inherent randomness of the universe; rather, I find that Wheeler's hypothesis of a participatory creation consistent with both relativity and quantum theory, in the free choice of initial conditions. I am also encouraged by Joy Christian's research in continuous measurement functions and quantum correlations -- because if nature has a free choice ("God had a choice in creating the world") so do we.

Quoting me: 'logical consistency alone is no test of a physical theory'

"I did not know we were discussing a testable physical theory. I was discussing the untestable, metaphysical multiverse hypothesis. What have you been discussing? If you have a testable physical theory that you wish to discuss, please identify it and the test."

Working on that. We haven't even gotten that far in this discussion, however. One has to accept that given an unbounded length of time' Albrecht's hypothesis *is* testable. Einstein relativity describes the universe as "finite but unbounded" -- meaning bounded in time at the cosmological singularity and unbounded in space by Riemannian geometry. I maintain that the theory does not suffer any loss of meaning when describing the world as finite in space and unbounded in time. Can one disprove my conjecture? -- if not, we have the physical basis for Albrecht's clock ambiguity.

Tom

  • [deleted]

PS. The above cited experiment was on particles observed from very close by, and moving comparatively slowly, rather like my example with the bacteria. There are many other experiments that confirmed SR listed here

http://www.aei.mpg.de/~mpoessel/Physik/RT/srtest.html

In your last post you say 'how can this be so?' That's ironic - I keep telling you, we don't know, we're trying to find out.

  • [deleted]

Constantinos,

You write, " ... I am arguing all Basic Law of Physics likewise can and should be formulated and derived as 'mathematical identities'. Take this as a call and not an established fact."

I have tried to explain to you that it is not possible to make it a fact (i.e., a theorem). Here is the simplified reason: For any self-consistent mathematical formulation that describes a physical phenomenon, there are any number of equivalent formulations that do not. Conversely, for any physical phenomenon described by mathematics, there are any number of phenomena that remain unexplained by mathematical language. We only say that we have understood a phenomenon when we do formalize it in mathematical language.

" ... Spacetime, for example, is not real yet believed to be real. On the other hand, 'the moon really is there when no one is looking'!"

Spacetime is most certainly physically real in the theory of general relativity, as the eclipse experiment -- and the accurate prediction of the precession of Mercury's orbit -- demonstrate, and which are validated by physical measurement.

"As for "domain" and "range", are you saying "Domain" is Math while "Range" is Nature? More reasoning at the fringes of reality!"

It's truly astounding to me that you would make such a statement -- domain and range, along with other concepts such as limit and function, are simply fundamental tools of mathematics.

Tom

  • [deleted]

Paul,

Easy to get lost in your own thinking. Let me rephrase!

Are you married? Do you truly know your wife?

Constantinos