wtf

No that is false. Newtonian gravity, for example, is not computable. The jury is still out on whether quantum physics is computable. There are many easily human-conceivable problems that can never be computed by an algorithm. The Halting problem is one such, and there are many others.

Solve the Halting problem. Equivalently, generate the decimal digits of Chaitin's constant. These are tasks no computation can do, even given unlimited resources. https://en.wikipedia.org/wiki/Halting_problem https://en.wikipedia.org/wiki/Chaitin's_constant

We have more automation than at any time in human history, and also way more jobs. Of course there are always dislocations. The buggy whip makers had to "learn to code" as they say. You can train an algorithm to recognize a tumor, but a human being has to break the news to the sufferer. Of course there was a story recently that a guy was told he was going to die by a robot. https://www.bbc.com/news/world-us-canada-47510038. But in the future there will always be a need for "high touch" jobs. Remember that although the masses will be taught by machines, the elite will send their kids to schools staffed by gifted teachers. Humans aren't done yet. I hope.

Not sure what the paper is about. But any reversible process may be represented as a group; and every group may be represented as a group of permutations. That's the famous Cayley's theorem. https://en.wikipedia.org/wiki/Cayley's_theorem

Sure. https://en.wikipedia.org/wiki/Balanced_ternary

No. Consider the computation that maps every input to zero. That's not reversible and is not a permutation.

That's not the answer to the question the OP asked.

Irrational numbers can be the output of perfectly rational (in the sense of reasonable) processes. Suppose that by a rational process we mean a process that can be implemented as a computer program. There are computer programs that generate the digits of pi to arbitrary accuracy, or the square root of 2, or the base of natural logarithms e. The real numbers whose digits can be cranked out by a computer program are called the computable numbers. What's interesting is that there are lots and lots of irrational numbers that can NOT be so computed. These are the non-computable numbers. They have no names, no uniquely characterizing properties. The noncomputable numbers are indeed worthy of being called irrational in both senses. They are crazy numbers in the sense that there is no rational process that generates their digits.. They are perfectly random.

I find it unlikely that a solution to RH could be found without the use of mathematical analysis; that is, without the use of limiting arguments. RH involves quantifying over real and complex numbers, including noncomputable ones. That's my sense of the matter, not being a specialist in RH.

I've got three artificial joints in my and not a day goes by that I don't thank my wonderful orthopedic surgeon. I'm all for whatever modern medical science can do.

You made some good points, let me take another run at a reply. > Well, isn't it confusing? You say it already: most adults get confused when discussing infinity, even if it is in one of the simplest examples, mathematically, with natural numbers. This was in reference to my claim that Hilbert's hotel would confuse a five year old. I guess I don't agree that we should tell a story that confuses the issue. But I don't know anything about five year olds. So that part of the conversation I should try to avoid. > I don't agree. The rubber sheet and bowling ball are real things that can be shown, but are a bad illustration of relativistic gravity. Yay!! I'm glad you agree with me. At least about the bowling ball. How many times have we seen the picture of the bowling ball and rubber sheet. It doesn't stand up to scrutiny though. Now I do take your point that the Hilbert hotel story is at least mathematically accurate. I just object to the extraneous details of a hotel with guests in it, because I have seen many people get confused about those physical elements. > Infinity in natural numbers is mathematically real, but the Hilbert Hotel isn't. However, I still think it is a nice illustration to show that our conception of 'infinity' as 'a great number' is wrong. And there is a direct relationship with the examples of the Hilbert Hotel (the hotel is full, but you still can add (1) one guest, (2) an infinite number of guests, etc) and the mathematical operation (1 + infintiy = infinity, infinity + infinity = infinity, etc). You do not need a mathematical argument: you only need a 1 to 1 relationship between the mathematical operations and the example. (In this of course the rubber sheet analogy miserably fails: it already presupposes gravitation). Well here I really disagree with you. Let me first admit that almost everyone agrees with you and not with me. Hilbert's hotel is very popular. My problem with it (for adults) is that it brings up all kinds of irrelevant objections like, If the hotel is full how can everyone move to a new room? Do they all move at once or what? What happens to the last guy? Oh there is no last guy? Why not? And if there are infinitely many guests where do you get new guests? I've personally seen these types of questions completely derail discussions of Hilbert's hotel. I much prefer a simple mathematical example, bijecting the natural numbers to the even natural numbers. The evens are, on the one hand, a proper subset; but on the other hand, clearly in one-to-one correspondence with the whole set of naturals. From which we conclude that, unlike a finite set, an infinite set can be in bijection with one of its proper subsets, Which we then turn around and make that the very definition of an infinite set. A set is infinite if it happens that it can be placed in bijection with one of its proper subsets. I would MUCH rather have to explain and defend and explain again that paragraph, which is actual mathematics; than to have to parry questions about hotels and guests and how many maids does it take to clean all the rooms. (Only one. She cleans the first room in half a minute, the second room in 1/4 minute, and so forth. She cleans the whole hotel in exactly one minute). I personally don't like clouding up the explanation of what is an infinite set, with confusion about how such a hotel could exist and whether the guests all move and once or one after another, and where do they move to. Those are good questions, but they are completely off in a wrong direction if you're trying to explain mathematical infinity. That's my opinion on the matter. But truly, if anyone's bothered by it, please be relieved to know that everyone agrees with you and not me. The story is very popular. > In what way exactly then is the Hilbert Hotel confusing? Everything implied by "hotel." It's a building, it has rooms, there are people in it. Leads to many irrelevant issues. > That is, more confusing than the abstract concept of 'infinity'? I think simply noting that the even numbers are in in bijection with the natural numbers is a far simpler observation to make. > Then it should be easy to explain to a 5 year old... I honestly I don't believe that. But as I'm on record saying that the Hilbert hotel story shouldn't even be told to adults; I feel even more strongly that it's not suitable for children. Just show people the sequence of natural numbers and show how you can biject the evens to the whole set. THEN when they understand that, you can tell them the story about the hotel. If you do it the other way around, confusion inevitably results.

I stated the points of confusion in my earlier post. I have nothing to add. And needless to say, my feelings about the subject won't alter the popularity of Hilbert's hotel.

I can only say that I myself have a clear and strong intuition of infinity. Whether I acquired that at an early age, or only feel that way because I had it beaten into me in math class, I honestly can't say. As Wittgenstein might have said: Whereof one cannot speak without snark; one must thereof put a sock in it. I shall heed his advice. Uh ... which is it? Are there twice as many or the same as? Perhaps it's beyond YOUR comprehension. Damn, the snark leaked out anyway. Only happened because I read your post a second time and realized you yourself are fuzzy on the example. The story of Hilbert's hotel is beyond the comprehension of most adults. I don't know the five year old referenced by the OP.

Do you claim to have no intuition of the endlessness of the counting numbers? Sticking to infinity for the moment since I'm not qualified to talk about the physics of gravity. But note that even though I have little knowledge of the physics of gravity; I have a strong intuition and experience of gravity from birth and probably from before that, since a fetus probably knows up from down. Just as one has an intuition of infinity or even an experience of it from contemplating the counting numbers; even without having technical knowledge of the theory.

Useful, sure. But potentially misleading. Why do bowling balls distort a rubber sheet? It's a pretty good question actually. One not addressed by the analogy. ps -- It's CERTAINLY not gravity! After all a bowling ball does not distort a steel plate, and the gravitational attraction between the ball and the plate is STRONGER than between the ball and the sheet because the steel plate is more massive. No, the reason a bowling ball distorts a rubber sheet is because of gravity AND the molecular structure of rubber, whatever makes it stretchy. That's MUCH more complicated than gravity! You may have seen that Feynmann video on Youtube where he talks about why his chair holds him up, and in the end it really comes down to the weird forces acting inside the atom.

I hope you understood my point, which was: The bowling ball and rubber sheet story is not physics, just as Hilbert's hotel is not mathematic, and sometimes popularized metaphors can confuse as well as enlighten. Your post was interesting about gravity. I'm aware your bolded words pass for conventional wisdom among the physicists. but "A causes B and B causes A" does not strike me as much of an explanation of anything. On the contrary. Can you search your mind and your soul, and tell me whether you have an intuitive idea of the endless sequence of counting numbers 0, 1, 2, 3, 4, ...? If yes, you have comprehended infinity. It's more sensible to imagine the sequence going on forever; than to imagine it suddenly stopping at some point. Infinity is an idea that's baked in to our brain. Agreed that the formal study of mathematical or physical or philosophical infinity is complicated. But the intuition is obvious even to children. You can count forever. And even if the physical world couldn't allow it; you can still count forever in principle. And I would contend that this is obvious to virtually everyone who thinks about it, even for a moment, unless one is a committed ultrafinitist. https://en.wikipedia.org/wiki/Ultrafinitism

For a five year old? Maybe, if you want to confuse them. Most adults have trouble with that fable. How could there be such a hotel? If all the rooms are full, how could everyone move to a new room? If there are infinitely many guests, where do new guests come from? What about the propagation speed of signaling? How could you notify the people in the faraway rooms to move? Etc. etc. etc. I've seen all of these issues causing confusion in discussions of Hilbert's hotel. Hilbert, by the way, only mentioned the story once in his life, in a lecture given to the general public. He never wrote about it or mentioned it again. It was only resurrected by George Gamow in a popularized math book. And Gamow, let it be noted, was a physicist and not a mathematician. Hilbert's hotel is a fable, a story for the tourists. It's not a mathematical argument. It's like the rubber sheet and bowling ball visualization of relativistic gravity. What makes the bowling ball push down on the sheet? Meta-gravity? No, the answer is that the rubber sheet and bowling ball picture is just a story; a visualization for amateurs that is not actually a physics demonstration. It's not meant to be thought about too deeply. Likewise Hilbert's hotel, which generally causes more confusion than enlightenment. Not to mention that sophists like William Lane Craig have used Hilbert's hotel to make theological points to credulous and naive audiences. And for a five year old? What are they supposed to learn from such a distorted and confusing story? You really think they're going to understand that the story is not about hotels and guests, but is rather an illustration of the fact that an infinite set can be in bijection with a proper subset of itself? I think not. Well now you've heard my rant about Hilbert's hotel, and thanks for asking!

The octonions have a subfield isomorphic to the reals in which all the usual arithmetic properties apply. We identify the real numbers with that subfield and say that "the octonions contain the reals" or "the octonions contain a COPY of the reals," depending on how formal we want to be. It's no different than asking if the integers are rational. Strictly speaking, the integers are only isomorphic to a subring of the rationals; but the identification is always made unless we are being EXTREMELY picky. So if you deny that the octonions contain the reals, you'd also have to deny that the rationals contain the integers and that the reals contain the rationals. The identifications are standard. It's interesting that we're in the Philosophy section. Because this is indeed a point of philosophical interest. Are the rationals a subset of the reals? Or are they only isomorphic to some subset of the reals? Should we care, or should we just make the natural identification and not worry about it? We're always told that "math is based on set theory," but that is falsified by this example. As sets, the integers are not a subset of the rationals and the rationals are not a subset of the reals. But nobody ever cares to make that picky point, which shows that even though we give lip service to founding everything on set theory, we don't always mean it literally.

Good question! 4, 6, and 8 are natural numbers, we all agree. Are 4, 6, and 8 integers? Are 4, 6, and 8 rational numbers? Are they real numbers? Are they complex numbers? Are they quaternions? Then why aren't they octonions? Discuss.

taeto, Have you heard about Cantor's theory of transfinite numbers? Mathematical infinity is basic to modern math. As @studiot asked, it's not clear if you're asking about mathematical infinity, or how to explain philosophical infinity to a five year old. If the former I can be of service; if the latter, probably not.

A joke that occurred to me in the several hours I spent away from the computer this afternoon. Great minds think alike. Here's an interesting take on the question. What is the probability that a random subset of the reals contains pi? Well, for each set that contains pi, its complement doesn't. The pi-containing and the pi-not-containing sets are in one-to-one correspondence, or bijection. So the probability must be 1/2. How about that!! However ... that argument fails. Consider the question of what is the probability that a random positive integer is even. The evens and the odds are in bijection, so the probability must be 1/2. But the multiples of 3 are in bijection with the non-multiples of 3, and the multiples of 4 are in bijection with the non-multiples of 4. In fact there are very sparse sets of positive integers such as the primes that are in bijection with the non-primes. So the naive idea of 1/2 doesn't work. By the way what IS the probability that a random positive integer is even? Well, it's NOT 1/2. Why is that? Well, it turns out that there is no uniform probability distribution on the natural numbers. And why is that?? The rules of probability theory as formulated by Kolmogorov include the requirement that a probability measure must be countably additive. We want that so that we can say that if we have pairwise-disjoint intervals of length 1/2, 1/4, 1/8, etc., their union has length 1. But now what should be the probability of picking a random positive integer? If it's zero, countable additivity says that the chance of picking any integer at all is zero. That's absurd. But if the probability is positive, then the total probability of all the integers is infinite. Either way we are stuck. We are force to conclude that there is no uniform probability measure on the positive integers; and in fact there is none on any countably infinite set. That's a hard fact of life. [Uniform just means each number has the same chance of being picked]. https://en.wikipedia.org/wiki/Probability_axioms Now there IS a way out of this. We can relax the requirement of countable additivity and require mere finite additivity. Then there IS a finitely additive probability measure on SOME special subsets of the natural numbers. For example if we pick some finite number n, about half of the numbers below n are even; and as n goes to infinity, the proportion of even numbers goes to the limit 1/2. We call this the asymptotic density of the even numbers. Similar reasoning shows that the asymptotic density of the multiples of 3 is 1/3, and so forth. This seems promising! BUT! Asymptotic density only applies to very specialized sets. For example the primes get very thin as you go further out; and the asymptotic density of the primes is zero. And again in terms of probability, asymptotic density is only finitely additive and is therefore not a full-fledged probability measure. https://en.wikipedia.org/wiki/Natural_density With these intuitions in hand, let's get back to the pi-containing sets. What we need now is some kind of measure or weakly additive or vaguely useful notion of a measure that would apply to the powerset of real numbers. I don't know anything about any such measure. I'm sure mathematicians have probably studied such a notion, but I can't get any insight into the question of how many sets of reals contain pi. It seems like a question that would involve some advanced math I'm not familiar with, if a solution exists at all. My guess is that the question doesn't have an answer but I've gotten curious about this so I'll look around. ps -- I did a little Googling. Questions about measures on the powerset of the reals quickly lead to questions about measurable cardinals. Those are the next large cardinals after the inaccessible cardinals required by Solovay's model. These are all cardinals whose existence is independent of standard set theory. I found one quote in one Mathoverlow page that's understandable and on-point. Mathoverflow is a site for professional mathematicians. This is from Stefan Geschke's comment underneath Andrés E. Caicedo's green-checked answer: "... you cannot construct a sigma-additive measure on P(R) without the help of some strong additional axioms." https://mathoverflow.net/questions/103583/finite-measure-on-the-power-set What this means is that the question of what proportion of subsets of the reals contain pi is a question of higher set theory at the professional level; and not answerable within standard set theory. As far as I could follow this page. pps -- Another interesting MO thread relating to this discussion is: "Is a random subset of the real numbers non-measurable? Is the set of measurable sets measurable?" Of course by "interesting" I mean, "Interesting to see that professional mathematicians talk about this stuff, even if I can only understand a word here and there." https://mathoverflow.net/questions/102386/is-a-random-subset-of-the-real-numbers-non-measurable-is-the-set-of-measurable

I'm perfectly well-aware of Solovay's model. You are replying too fast to be giving any thought to any of this. You want to discuss inaccessible cardinals? Why are you throwing in out-of-context red herrings?

One has to be careful of Wikipedia. Solovay's model assumes an inaccessible cardinal. The problem with Wiki is that it doesn't provide context. Solovay's model is "inside baseball" in advanced set theory. It's not grist for the web forum mill. In any event, questions about the measure in the powerset of the reals of subsets with various properties are meaningless. There's no measure to work with that I know of.

No, I said the probability may not be defined at all. You are replying to me way faster than it would take for you to give thoughtful consideration to my posts. Again: * In order to have a probability space, which is required to have a total measure of 1, we need to restrict our attention to the unit interval of reals. * Therefore we have to replace pi with a proxy in the unit interval. Pi minus 3 = .141... is adequate for the task. * So now we ask: If we pick a random subset of the unit interval, what is the probability it contains pi - 3 ? * The problem is that there's no total probability distribution on the powerset of the reals. Some sets are not measurable at all. * So this problem is insufficiently specified. * Probability theory doesn't apply to the powerset of the reals. You can't ask questions about what percentage of the uncountable subsets of the reals have such or so property. I know of no such theory. What is the probability that a random set of reals is open? Closed? Compact? Connected? These questions have no answers. They can't be well-posed at all.

I wish you would carefully read what I'm writing. I'm perfect well accounting for uncountably many monkeys in each of my last several posts. Which part is unclear? You are asking if a randomly selected uncountable subset of the reals contains pi. I clearly answered that the question's not well-posed, since not all uncountable subsets of the reals have a measure assigned to them.