Philosophy Now Forum

Try chewing on this. Newton showed us that because of gravity the motion of every single physical entity in the universe is causally determined by the motion of every other. We have since figured out that the speed at which information about such relativistic motions is transmitted between such entities is the speed of light, which I have unambiguously defined as the processing speed of the cosmic computer. Therefore it is impossible, even in principle, to specify both the location and the momentum of any cosmological body without also specifying both the location and momentum of which other cosmological body this information relates to. This can then hold true ONLY for these two bodies and will not hold true for any other because Newton's simplistic notions of an absolute space and time were falsified by Einstein.

Why then is the Heisenberg Uncertainty Principle not a statement about gravity? Relativity precludes us from specifying both the location and the momentum of a galaxy at the same time because these are affected by the relativistic motions of every other galaxy in the universe. Since we do not assert that galaxies move about the cosmos at random why are we prepared to make such a claim about subatomic particles? Why wouldn't exactly the same principle apply? The problem of quantum gravity is not a problem of physics at all. it is simply a flawed understanding of the nature of determinism.

Obvious Leo wrote:Have you read David Deutsch's work on this subject?

A little Googling reveals that Deutch's idea is a quantum UTM. This idea is also (as far as I understand) the same as a nondeterministic TM. It is presently unknown, repeat it is presently unknown, whether NTMs are capable of any computations beyond those of regular TMs, although QTMs can provide a tremendous speedup. As a striking example, there is a quantum algorithm to factor integers in polynomial time. This is truly a shocking and wonderful result. However, all we're getting is a speedup. Factoring can already be done by a conventional algorithm.

If you're not familiar with Scott Aaronson's blog, you should definitely put him on your reading list. Start here. Everything I know about computer science I learned from reading Aaronson.

May I suggest that you should say that you mean QTMs, so as to avoid immediate derision from readers who know what a UTM is.

https://en.wikipedia.org/wiki/Quantum_Turing_machine

https://en.wikipedia.org/wiki/David_Deutsch

https://en.wikipedia.org/wiki/Church%E2 ... _principle

From the third link:

The principle was originally stated by Deutsch with respect to finitary machines and processes. He immediately observed that classical physics, which makes use of the concept of real numbers, cannot be simulated by a Turing machine, which can only represent computable reals.

I made the identical point earlier, pointing out that TMs can only deal with countable sets. The Wiki article continues:

Deutsch proposed that quantum computers may actually obey the CTD principle, assuming that the laws of quantum physics can completely describe every physical process.

This is of course a speculation, unsupported as yet (though a subject of intense research) by computer science.

It's now clear to me what you are talking about. It's a perfectly sensible speculation. Please for your own sake use the terminology QTM or NTM to avoid confusing people who know what a TM is.

By the way, I have my own mythopoetic and speculative idea about all this. Computations (as they are presently understood) can only express computable reals. However, almost all mathematical reals are noncomputable. These can be thought of as random bitstrings; in other words, infinite bitstrings that can not be compressed to any finite-length description or algorithm.

As is well known, there are no representations of random bitstrings in the universe, since a random bitstring requires an infinite amount of information to specify it, and contemporary physics only allows for a finite information capacity of the universe.

It's my belief that the next revolution in physics will be the incorporation of true randomness, uncountable sets, and noncomputable real numbers into physics. I argue by analogy with the discovery of non-Euclidean geometry, which was first thought to be logically impossible; then was discovered to be mathematically consistent; and then turned out to be a useful description of the actual world. I suspect that a century or two from now, noncomputability will follow a similar pattern. A mathematical truth that is physically impossible today; and the new physics of tomorrow.

Does this accord with your thinking in some way?

wtf wrote: As is well known, there are no representations of random bitstrings in the universe, since a random bitstring requires an infinite amount of information to specify it, and contemporary physics only allows for a finite information capacity of the universe.

My blood runs cold when I see the word "random" find its way into a scientific conversation. As far as I'm concerned random means uncaused and an uncaused event is a metaphysical absurdity. I agree with contemporary physics that the information capacity of the universe is finite but I also accept the first law of thermodynamics and conclude that this finite information content must exist eternally. At the Planck scale it can be encoded in a practically (although not literally) infinite number of different ways but at the fundamental level the information itself can never be destroyed. Naturally this does not apply at the emergent level since all emergent entities are beholden to the second law of thermodynamics.

'All things come from one another and vanish into one another according to necessity and in conformity with the order of time".....Anaximander.."On Nature".

wtf wrote:If you're not familiar with Scott Aaronson's blog,

I'm not familiar with his blog but I'm quite familiar with his work.

wtf wrote: It's my belief that the next revolution in physics will be the incorporation of true randomness, uncountable sets, and noncomputable real numbers into physics.

I think you don't understand the nature of chaotic determinism and are conflating randomness with unpredictability. Non-linear dynamic systems are noncomputable and yet completely deterministic, which essentially makes randomness an observer effect.

wtf wrote:I suspect that a century or two from now, noncomputability will follow a similar pattern. A mathematical truth that is physically impossible today; and the new physics of tomorrow.

Does this accord with your thinking in some way?

This bit does but your timeline is overly pessimistic. Spacetime physics is on the ropes, just waiting for somebody to mercifully deliver the coup de grace. Read my synopsis and you'll see how it can be empirically falsified.

Obvious Leo wrote: My blood runs cold when I see the word "random" find its way into a scientific conversation. As far as I'm concerned random means uncaused and an uncaused event is a metaphysical absurdity.

I only mean incompressible to a finite-length description or algorithm. Nothing more. Does that help you to understand my meaning?

Consider an infinite length bitstring. If we put a decimal point in front of it and interpret it as the binary representation of a real number between 0 and 1, then bitstrings are the same as real numbers. (The whole-number part doesn't matter for any of our considerations).

Some bitstrings have finite-length descriptions. For example 101010101010... is completely characterized by "Start with 1 then alternate 0's and 1's." The set of bitstrings (or real numbers) that can be so characterized are the computable reals.

Mathematically, if you conceptually threw a dart at the real line -- in other words picked a real randomly -- the probability that you picked a computable real is zero. The probability is 1 that you picked an uncomputable real. All this can be formalized.

So computable reals are by far an extremely exceptional event; and the normal state of affairs is noncomputability.

Randomness is (as used by me) simply a synonym for noncomputability in this sense.

I believe this would ultimately be the same as predictability, since if there is an algorithm to generate a bitstring then I can always predict the next bit; and if there isn't an algorithm, then the next bit is unpredictable.

So in the end we have the same definition; but it seems to carry more metaphysical baggage for you than it does for me. I happen to believe in ALL the bitstrings. Why shouldn't they exist along with their (extremely rare) computable brethren and sistren?

If you imagine the real number line as a model of the ancient idea of the continuum, then it's a mathematical fact that almost all the points on the line represent noncomputable numbers. Randomness is the possibility of continuity. We're back to Zeno and atomism again. My thesis is that this dilemma will ultimately be decided in favor of continuity. Regarding the timeline, Newton to Einstein was 220 years or so. That's where I got my number from.

wtf wrote: Randomness is (as used by me) simply a synonym for noncomputability in this sense.

I was hoping that's what you meant because the distinction between randomness and noncomputability is not a trivial one. In Newtonian physics noncomputability implies the metaphysical absurdity of the uncaused event and the reason for this is that Newtonian determinism and pre-determinism are regarded as synonymous terms.Unfortunately this is a logical fallacy which has led physics into a conceptual cul-de-sac from which there seems to be no escape. There can be no doubt that at the sub-atomic scale Newtonian determinism simply doesn't work but the opposite of Newtonian determinism is NOT randomness. The opposite of Newtonian determinism is non-Newtonian determinism, i.e. self -determinism, chaotic determinism, non-linear determinism, whatever we choose to call it. Our current models of physics are simply nor designed to accommodate such a notion, which is why they make no fucking sense.

wtf wrote:We're back to Zeno and atomism again.

I very much doubt that the philosophy of the quantum is metaphysically shakeable. Information cannot be infinitely divisible and still encode for a coherent physical world. In any event Max Planck proved the case beyond a shadow of doubt with his work on black body radiation. Whatever we ultimately choose to call our fundamental units of reality they must inevitably be quanta of energy of some sort. Once we chop reality it up until it can be chopped up no finer then energy is all we have. At least the physicists seem to have this part right with the big bang cosmology but the pre-determinism of the Newtonian paradigm prevents them from seeing that the big bang cannot have been the beginning of the universe. Heraclitus would soon put them straight.

wtf wrote: So in the end we have the same definition; but it seems to carry more metaphysical baggage for you than it does for me.

Applied metaphysics is my core business and I certainly don't see it as baggage. I regard the conceptual cul-de-sac that physics has driven itself into as an intellectual tragedy of Homeric proportions because it should never have happened. Consider these two quotes from the bloke who started all the trouble. They stopped listening to him almost immediately afterwards.

"Spacetime should NEVER be regarded as physically real".....Albert Einstein

"Space and time are modes in which we think, NOT conditions in which we exist.".....Albert Einstein.

It wasn't your fault, Albert. You knew that you were only offering a mathematical representation of a physical universe, not a physical one.

@Obvious Leo, I just had a glance at the article on your website. I admit that I found it hard going but also that I did not read word for word, so perhaps I'll have another go.

I do happen to be a bit of a Newton fan so I wanted to mention that Newton most definitely DID understand the difference between the map and the territory. That's what the famous quote "I frame no hypotheses" is about. Newton explicitly rejected occult explanations from science. He said that science is about modeling what we can observe.

Now it's true that in his non-scientific writings he said that God was the ultimate cause and all that, but this I attribute only to the times, just as Descartes said I think therefore I am and also since God is perfect he wouldn't deceive me and so forth. The God stuff we can safely ignore from both Descartes and Newton.

In Newton's scientific work he did not make explanations about the territory. He only showed the map; and it's clear from his writing that he understood the difference.

I am unable to understand the depth of your dislike of the man. He developed calculus and used it to build a system of the world that got us to the moon. What should he have done instead?

True he was a misanthrope, but he had a very dysfunctional childhood and his personality is understandable in that light. And he was a great scientist.

Can you say what you think he should have done instead? Does not Newtonian mechanics work brilliantly in its domain of applicability? If we have better measuring instruments now and a refinement to his ideas, that is not a mark against him. What is the source of your strong animosity toward the man? He's long dead and did his best, as we all do. And his best was very very good. His insight that stuff falls down on earth for the exact same reason that the planets are in motion about the sun was revolutionary. The same laws rule the heavens and the earth. You give him no credit for this?

I've never met a Newton hater before. Myself I'm a Newton lover. For his obsession, his solitude, and for his achievements. I don't see how anyone could literally hate the guy. He had a sharp tongue. If he were alive today he'd be right at home on the Internet. He could flame with the best of them.

Anyway, do you have perhaps a more concise version of your ideas in general? Or an angle or something I can hold on to while I'm reading your essay? I always find it helpful to have something to hang onto.

(ps) I'm making another run at your essay. You said at one point that a science must be both descriptive and explanatory. This is patently false. Science by definition is about what is measurable, and the mathematical models we can invent that explain (or match, a better word in this context) the experiments. All the better if the model has predictive power.

But all of this modeling and experimentation takes place within the physical limitations of our apparatus. All measurement is approximate. Science can never be about ultimate truth (if such a thing even exists). It can only be about measurement up to the practical limitations of technology.

It may the the case that some scientists don't realize this, and think they are talking about ultimate reality. Their opinions don't change this point. You say Newton didn't know the difference, I say he did. But that's of no consequence either way.

If we define science as the modeling of that which is measurable, then clearly science is always an approximation and can never be about ultimate explanation.

And if we define science as being about "ultimate" explanations, then it's indistinguishable from theology.

I believe this is your core error. Science is only about models. Science is not about metaphysics. You are in fact projecting your own error onto Newton. YOU are the one who thinks that science is explanatory, and you claim that to be Newton's error. But Newton did not make that error -- at least in his scientific work. In his theological thinking he was a product of his time, and in those days you had to give props to God or the Church would be pissed. I believe Newton was involved in some of the theological disputes of his day and he would have been immersed in God talk.

But in his scientific work Newton did not mistake description for explanation. YOU are the one making that error, or trying to glean explanation from description, when in fact that can not be done. All measurement is approximate. Surely you know this.

Well I'm going to read more. But you have science all wrong. And if you claim science can explain, how do you separate science from theology?

(pps) To accuse Newton of believing in intelligent design is a gross slander. You are simply misreading his work. Look at his scientific work. There are no occult causes therein. None.

(ppps) "It seemed to me that Alan Turing had offered the perfect model for a cyclical universe which programmes its own input and which could thus be defined as a reality maker."

A blatant misreading of Turing, as I've already indicated in my discussion of UTM. You know, you have got a lot of stuff simply wrong. Not just opinions, but basic facts.

wtf wrote: I believe this is your core error. Science is only about models. Science is not about metaphysics.

I think you'll find that this is the very point I'm making. Science without metaphysics can make no truth statements about reality and yet physics, in particular, makes sweeping metaphysical statements without allowing them to be subjected to metaphysical scrutiny.

I'm in the process of a complete revision of this synopsis and I'm more forgiving of Newton in the rewrite. What I didn't include in the version I refer you to is the harm that Newtonian reductionist thought has wrought in the modern world. I don't actually hold him personally responsible for this but regarding the universe as a created entity is an affront to humanity which has taken a severe toll. I accept your point about the "I frame no hypothesis" angle but countless hypotheses have been framed in his name. Einstein suffered the same fate because he didn't for one moment intend that his continuum of space and time should be awarded any ontological status.

wtf wrote: You know, you have got a lot of stuff simply wrong. Not just opinions, but basic facts.

This is a very strange statement to a philosopher of science because I don't acknowledge the existence of facts. Evidence is just raw data and the way such data is interpreted depends on the a priori narrative of whoever does the interpreting. What you probably mean is that I'm interpreting data in such a way that it contradicts current theory.

Obvious Leo wrote:I accept your point about the "I frame no hypothesis" angle but countless hypotheses have been framed in his name.

We're in agreement on this point. The fact that SOME scientists think they're possessed of ultimate truth does not indict science.

You have a lot of similar strawman arguments in your essay. For example, "The notion of life emerging from non-life is a deeply mysterious one to many folk steeped in the traditions of Cartesian dualism ..."

Well yeah. But Cartesian dualism hasn't been in favor for quite some time. [I know there are modern dualists but I don't know much about them]. So again, this is a strawman argument. Saying, "Well Descartes was a dummy because he ascribed it all to a benevolent God," is no argument at all. The guy lived 300 years ago.

I would love to see you pull out the strawmen and all the attacks on people whose ideas are obviously misguided, and also the personal stuff ("I was interested in philosophy as a tadpole," etc.) and find a focussed, concise statement of your thesis and a relatively tight argument to support it. Maybe a quarter of the length of the current piece. There's an argument in there, I'd be interested in reading it.

wtf wrote:...

Mathematically, if you conceptually threw a dart at the real line -- in other words picked a real randomly -- the probability that you picked a computable real is zero. The probability is 1 that you picked an uncomputable real. All this can be formalized. ...

Is this true? As whilst vanishingly unlikely I can't quite bring myself to believe it would be a logical impossibility as according to you it's a fact that there are computable reals in that line so there must be some possibility of hitting one? Or is it that mathematical probability is not the same as logical possibility?

Arising_uk wrote:

wtf wrote:...

Mathematically, if you conceptually threw a dart at the real line -- in other words picked a real randomly -- the probability that you picked a computable real is zero. The probability is 1 that you picked an uncomputable real. All this can be formalized. ...

Is this true? As whilst vanishingly unlikely I can't quite bring myself to believe it would be a logical impossibility as according to you it's a fact that there are computable reals in that line so there must be some possibility of hitting one? Or is it that mathematical probability is not the same as logical possibility?

Good question! The answer is that when infinite sets are concerned, probability 0 events can still happen, and probability 1 events are not certain. As an example, if you randomly pick a real (again, by throwing a dart or perhaps by flipping infinitely many fair coins) the probability is 0 that you'll choose a rational number. But clearly rational numbers exist.

Probability is based on the mathematical theory of measure. In measure theory, the measure of a countable set is zero. The rationals are countable and so are the computable numbers. Yet there are rationals. It's something that you get used to. Probability zero events can happen. Your dart might hit a rational. The probability is zero but it could happen.

(Additional technical stuff)

The Wiki paged I linked is a little too technical. Here's a brief outline of why the probability of a countable set has to be zero.

First, it's clear that if we're going to assign probabilities to subsets of the unit interval, the probability of hitting a single point has to be zero. There are infinitely many of them and clearly if we assign a single point anything other than zero, the probability of hitting any point in the interval would have to be infinite. And with probabilities we like the total of all the probabilities to be 1. Just like heads = 1/2 and tails = 1/2 and those add up to 1.

Ok. Now another thing we'd like is to say that if we have an interval of length 1/2, and another interval of length 1/4, and another of length 1/8, etc., then the probability of hitting something in ANY of the intervals must be 1. In other words we want to be able to add up countably many probabilities. So if the probability of a single point is zero, the probability of hitting any countable set has to be zero for the same reason. Otherwise we can't get an infinite version of probability theory off the ground. This is called countable additivity.

So as counterintuitive as it seems, the probability of randomly choosing a real number and picking a rational has to be zero. It turns out that this is the only logical way to make things work out.

wtf wrote:Well yeah. But Cartesian dualism hasn't been in favor for quite some time. [I know there are modern dualists but I don't know much about them]. So again, this is a strawman argument. Saying, "Well Descartes was a dummy because he ascribed it all to a benevolent God," is no argument at all. The guy lived 300 years ago.

You've completely missed the point of my argument. I'm not talking about what Descartes said, or indeed what Newton said. I'm talking about the influence these guys had on European thought and since Europe was the birthplace of modern science this influence is not irrelevant. In fact it's pernicious and anti-intellectual because the 3 dimensional Cartesian space is still being modelled as a physical object instead of a mathematical one. Even Descartes never made this claim and he invented it. Incidentally there are still plenty of people around who regard the emergence of life from non-life as a deeply mysterious phenomenon, whereas it is in fact the mandated outcome of an evolutionary process if evolution is regarded as an information theory. So too is the evolution of mind from life. Under the right external conditions this is a certainty although its precise nature is noncomputable.

wtf wrote: (pps) To accuse Newton of believing in intelligent design is a gross slander. You are simply misreading his work.

Surely you jest. His scientific work was a dilettante's sideline. His major life's work was a forensic dissection of the bible to calculate the precise date of god's creation, which he duly did whilst contradicting Ussher into the bargain.

You're obviously reading my essay and picking out the bits you disagree with as you go along. This is both discourteous and unlikely to help you grasp the central thrust of my argument because you're taking stuff out of its context. However I'd be interested to hear your thoughts on my interpretation of the so-called "constant" speed of light. The entire house of cards that is spacetime physics is founded on this fallacious assumption and it can boiled down to a simple dichotomy. Is the speed of light a constant or is the speed of light proportional to clockspeed? They can't possibly both be right but bear in mind that the speed of light was measured to be a constant before General Relativity was published and this assumption was never reconsidered in the light of the further information which GR revealed.

Obvious Leo wrote: You've completely missed the point of my argument.

I'm certain that's the case. My comments are objections, yes; but more than that, they're pitons. Pitons are the spikes mountain climbers pound into the rock face to get a hand or foot hold. I am not so much picking away at tiny little points. I'm grasping at the only things I understand. Your exposition is flowery, rambling, and erudite as well. I can't grasp the outline or structure at all, but I can tell there's something in there. So I read, and I see something I can relate to. Newton, I know a little about him. Or Turing machines, I know a little about them. Typically you say something I disagree with. Perhaps that's only a coincidence, and if I understand what you were were saying better, I'd agree with some of it. Or perhaps the more I understand the more I disagree with.

I can only say that what little bits and pieces I understand at all, I tend to disagree with. But I'm not using those bits and pieces to attack you unfairly. Rather I'm just looking for any points of contact at all between what I understand and what you're writing.

You have wide knowledge and many ideas, but I'm finding the exposition extremely difficult. But please don't take my criticisms as pickiness for the sake of pickiness. I'm just taking points of reference wherever I can. It may well be that the fault is all on my side; but I it very difficult to discern the logical structure of your ideas.

Obvious Leo wrote: Incidentally there are still plenty of people around who regard the emergence of life from non-life as a deeply mysterious phenomenon ...

I'm one of those people. I get that you think you've explained to your own satisfaction that the evolution of life is a trivial matter. But I can not discern the logical structure of any argument to support that thesis in your essay. Like I say maybe the fault's all mine. But just between you and me ... in a few short, declarative sentences ... what is your argument that the evolution of life is so simple it's barely worthy of mention?

Obvious Leo wrote:

wtf wrote: (pps) To accuse Newton of believing in intelligent design is a gross slander. You are simply misreading his work.

Surely you jest. His scientific work was a dilettante's sideline. His major life's work was a forensic dissection of the bible to calculate the precise date of god's creation, which he duly did whilst contradicting Ussher into the bargain.

Ok, a small point of history that we interpret differently. As I say, I distinguish between Newton's scientific work and everything else. He scrupulously kept occult explanations out of his scientific work. To me that's sufficient to prove that deep down he understood that science was science: observation and models. The rest of the time, who cares what he personally believed?[/] He never once put the slightest drop of mysticism into his scientific works. That proves he understood this point. Judge the man's scientific work by what he put on the page when he was doing science.

Obvious Leo wrote:
You're obviously reading my essay and picking out the bits you disagree with as you go along. This is both discourteous

No discourtesy intended. But a courtesy to your reader would be to pull out the autobiographical material, the emotionalism, the attacks on people you don't like, and the rambling disorganization; and to give me a structure I can make sense of.

Absent that, I'm doing whatever I can to get hold of this thing. And grabbing on to a phrase here and a sentence there is really all I can do at the moment. Perhaps it's my own lack of reading comprehension. Perhaps you could write up a "for dummies" version. Or more accurately, a logical version. Pull out all the poetry and just hit the bullet points in some logical order.

Obvious Leo wrote:
and unlikely to help you grasp the central thrust of my argument because you're taking stuff out of its context.

As of yet I can't discern the context or the thrust. It doesn't appear to me to be intended to be understood. You prefer to be flowery rather than clear. That's a stylistic choice on your end. I'm paying you a great deal of respect by staying with it as long as I have. I far prefer clear writers. For what it's worth I prefer the analytic philosophers to the continentals. Do you think that's the gap between your prose and my comprehension perhaps?

Obvious Leo wrote:
However I'd be interested to hear your thoughts on my interpretation of the so-called "constant" speed of light. The entire house of cards that is spacetime physics is founded on this fallacious assumption and it can boiled down to a simple dichotomy. Is the speed of light a constant or is the speed of light proportional to clockspeed? They can't possibly both be right but bear in mind that the speed of light was measured to be a constant before General Relativity was published and this assumption was never reconsidered in the light of the further information which GR revealed.

I don't know enough physics to comment at all. What do you mean by clockspeed? I know what clockspeed means in a computer. Where's the universe's clock chip? Is that part of your theory? Or Deutch's theory? Or someone's theory? From what little I do know about physics, I doubt there's a universal clock, rendering your question meaningless. In a computer system, there's a clock chip and all the other circuits are synced to the clock chip. That sounds suspiciously like a preferred frame of reference to me, but like I say my understanding of physics is weak.

Beyond that I'm ignorant of how light speed works. It's certainly strange that there should be a maximum speed limit in the universe. Where'd that come from? I regard it as a mystery.

I don't know if you regard the style of your essay as an important part of the content, or if you would consider writing a more prosaic outline of the argument so that people can follow along. Or maybe it's clear as day and I'm the one who can't understand. It's possible. I mean no disrespect. I find your style impenetrable but you certainly have something interesting in there, which is what keeps me pecking away in my picky little way.

wtf wrote: (Additional technical stuff)

Thanks for that as due to my execrable knowledge of mathematics floundering was not the word!

First, it's clear that if we're going to assign probabilities to subsets of the unit interval, the probability of hitting a single point has to be zero. There are infinitely many of them and clearly if we assign a single point anything other than zero, the probability of hitting any point in the interval would have to be infinite. And with probabilities we like the total of all the probabilities to be 1. Just like heads = 1/2 and tails = 1/2 and those add up to 1.

Unit interval was my first flounder but I think you mean the possible 'numbers' between 0 and 1 yes? Are these these 'Reals' I followed links to? If so I understand that you are saying that since there are an infinity of them the chance of picking any one of them as a rational is declared zero, or is it any real, as otherwise it would have to be an infinitely small chance and we can't work with such a thing? What I can't quite grasp is that if you add all the zeros you get 1?

Ok. Now another thing we'd like is to say that if we have an interval of length 1/2, and another interval of length 1/4, and another of length 1/8, etc., then the probability of hitting something in ANY of the intervals must be 1. In other words we want to be able to add up countably many probabilities. So if the probability of a single point is zero, the probability of hitting any countable set has to be zero for the same reason. Otherwise we can't get an infinite version of probability theory off the ground. This is called countable additivity.

Ok, now I'm in trouble. Best I get is that because it is the interval between 0 and 1 then we know there must be a halfway cut, 0.5, so we now have two halves and we can keep doing this to infinity but the problem is that within each interval there is again an infinity of points?

I'll leave it here as I've reached my limit of comprehension and maybe your reply will help clarify my thoughts.

Ta! For the time.

Arising_uk wrote: I'll leave it here as I've reached my limit of comprehension and maybe your reply will help clarify my thoughts.

I have some things to do this morning but I will post a more clear explanation later today.