Thinking too much about infinity is very bad for your health. Just say no. Or, as is popularly expressed, "Shut up and calculate !".
... one single real number can contain the answers to all (binary) questions one can formulate in any human languages. To see this it suffices to realize that there are only finitely many languages, each with finitely many symbols. Hence, one can binarise this list of symbols (as routinely done in today’s computers) and list all sequences of symbols, first the sequences containing only a single symbol, next those containing two symbols, and so on.
When they represent a question whose answer is yes, we set these bits to 01 and if the answer is no we set them to 10. This procedure is not efficient at all, but who cares : since a real number has infinitely many bits, there is no need to save space! Hence, one can really code the answers to all possibly (binary) questions in one single real number. This illustrates the absurdly unlimited amount of information that real numbers contain. Real numbers are monsters!
That's an elaborate, though interesting, way of saying you can write down an answer to any question but it might be wrong.
Furthermore, everyone knows also that each stored bit requires some space. Not much, possibly soon only a few cubic nanometre, but definitively some finite volume. Consequently, assuming that information has always to be encoded in some physical stuff, a finite volume of space can not contain more than a finite amount of information. At least, this is a very reasonable assumption.
Consider a small volume, a cubic centimetre let’s say, containing a marble ball. This small volume can contain but a finite amount of information. Hence, the centre of mass of this marble ball can’t be a real number (and even less 3 real numbers), since real numbers contain - with probability one - an infinite amount of information.
Which is a variation on asking whether reality is continuous or discrete.
But the fact that a finite volume of space can’t contain more but a finite amount of information implies that the centre of mass of any object should not be identified with mathematical real numbers. Real numbers are useful tools, but are only tools. They have no ontological reality - no physical reality.
The way I see it, mathematics is a language. Like all languages it describes physical and other (i.e. conceptual) realities, but it is not the same as those realities themselves. It's a description : sometimes startlingly accurate, but at other times perhaps flawed. Maybe it's not mathematics itself that's flawed, but our poor linguistic skills : e.g. it can take infinite or finite time to cross and event horizon depending on your coordinate system. As to whether physical reality has infinite continuity and precision or is somehow discrete, I've no idea.
The view I am suggesting is that the first bits in the expression of x are really real while the very far away bits are totally random... One may object that this view is arbitrary as there is no natural bit number where the transition from real to random bits takes places. This is correct, though not important in practice as long as this transition is far away down the bit series. In the classical case, one may imagine that the very far down the series of bits of “real” numbers in binary format are totally random
I'd say that's tremendously important ! You can't just arbitrarily decide that very long numbers are so inconvenient that they must actually have a finite but unspecified length.
The main idea here is to introduce randomness into classical physics. Some finite length of the infinitely precise real numbers describe actual physical parameters, whereas the rest (if I understand this correctly) in some way provide a random element. Like the idea that reality is a simulation, I place this firmly in the category of, "genuinely very interesting, but I hate it."
https://arxiv.org/abs/1803.06824
https://www.sciencenews.org/article/real-numbers-physics-free-will
I recently used bc to perform calculations on the difference in the volumes of spheres where one is a single Planck Length smaller in radius than the other. I wanted to know the size of a sphere that could hold the volume of the skin of a thin balloon.
ReplyDeleteOkay, I'll comment before reading this paper, as well as after.
ReplyDeleteHow I think of infinity is this; imagine everything that is known, that we know, that we can see, and that we can think of...this is all inside of a box, well... to be more precise, ...your skull. Infinity is all of that, everything in everyone else's skull, including every dinosaur, insect, spider, birds, etc. as well as every other thinker, predator, and herbivore that ever lived, and everything else that is, ...as well, ...that is both known, ...as well as the unknown. ...and then there is more! much, much, much, more. In fact, more than your finite self here on this tiny, tiny rock swooping around this tiny, tiny little star, would be able to comprehend, unless of course, you happen to be your infinite self, ...which, as far as I know, seems very, very, unlikely.
There are some interesting well documented cases though of our awareness extending beyond just one lifetime as we currently know it, but as far as I know, these few well documented cases of anyone's awareness do not extend beyond more than two lifetimes. Curious people can find such thought chains about extending ones awareness through multiple lifetimes in ancient Buddhist scripture, as well as in traditional Talmudic/Islamic/Christian scripture, as well as the Zoroastrian tablets, and in the Vedas of the Maharabta as well.
I'm going to argue, even before I read the paper referenced here, that real numbers do exist, even though they are simply a concept, and further, immaterial, as well as intangible. Numbers are extremely useful for comparing the various aspects of our known universe, and allowing us to better define our concepts and comprehension of the nature of our physical universe as we observe it, thus enhancing our chances of surviving beyond the present incarnation of this universe, in said universe.
Infinity is more than this universe though, and I'll get back to that in awhile... For now though, what we can measure, and the tools we use to measure with, one of the most important being, ...well, ...numbers.
Well numbers aren't physical and nobody has claimed that they are. It is also already well known that truncations in computer simulations destroy determinism and time reversal symmetry. (As do unmeasured/unmodeled fluctuations occurring in a real system.)
ReplyDeletePhysics depends on empirical measurements, which certainly only have a finite number of digits. It's also rather anthropocentric to talk about the particular digits of a number expanded in base 10.
I'll be somewhat surprised if this passes peer-review and I'm not sure why it's being reported on as a preprint.
Gopindra Hannigan I don't think it's a paper, it's a summary of a talk. The arXiv comment says "Presented at the David Bohm Centennial Symposium, London, Octobre 2017."
ReplyDeleteRhys Taylor I see. This seems like somewhat of a "water is wet" or "dog bites man" topic. What is novel about it?
ReplyDeleteCertainly a simulation should not and cannot be carried out to infinite precision. So, what is new here?
Gopindra Hannigan I don't think there's anything new here, though the author probably disagrees.
ReplyDeleteOn the topic of enigma and paradox or unthinkable thoughts and health: Georg Cantor suffered multiple nervous breakdowns and was eventually to die in an asylum. Kurt Gödel suffered from depression and an anorexia which eventually killed him. A question which might surface here is that of the relationship between paradox and insanity, but the causal or generative side of the equation may be difficult to determine. Does such study degenerate psychological well-being or does psychological dissonance find itself attracted towards such study ? On a sample population of two, there are no solid conclusions to be drawn but these two thinkers are certainly not common examples necessarily subject to any metric analysis of standard deviation...
ReplyDeleteRelated (and in passing): David Bohm had some fascinating ideas of extrapolated consequence in holistic/global systems analysis. I found some of the holographic organisational principles to be profound in their capture of aspects of distributed complexity and information in a broad spectrum of kinds of systems...