  # wtf

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## Everything posted by wtf

2. You are asking a legal question, not a metaphysical one. If some court rules a washing machine a person, then legally a washing machine is a person. Modern washing machines have chips and "make decisions." That's a lot different than asking if an "AI" in quotes, since there is no true AI, can be sentient or creative or can invent things. We already have programs that write other programs, for example compilers and 4GL languages. Doesn't mean anything. The program in question runs on conventional hardware and is in principle no different than any other conventional program. Machine learning and neural net approaches are a clever way to organize datamining, but all they can do is identify patterns based on what's already happened. By definition they can't create. But again, those are arguments for the metaphysical position that contemporary computers are not sentient/creative/self-aware/whatever. You are asking a legal question though, and the answer to that is, "Whatever a court decides is the law." Are you suggesting that my hammer owns the chair I built with it? Such a view is nonsensical. Does your computer own the words you typed into this forum? A program doesn't invent anything, it just flips bits according to an algorithm. An algorithm written by a human. A computer program is a tool.
3. ## Proof there are as many numbers between 0 and 1 as 1 and infinity?

I found a picture of the mapping from (0,1] to (0,1). Instead of using the inverse powers of 2, it uses the sequence 1/2, 1/3, 1/4, 1/5, ... See how this works? The 1 at the right end of (0,1] gets mapped to 1/2. Then 1/2 gets mapped to 1/3; 1/3 gets mapped to 1/4, and so forth. The mapping is reversible so this is a bijection.
4. ## Proof there are as many numbers between 0 and 1 as 1 and infinity?

Walk through it step by step. Convince yourself that the mapping is: * Injective: Different inputs go to different outputs; and * Surjective: Everything in the target set gets hit. By definition, the mapping is a bijection. Stepping through the details will make the idea clear. To see exactly how it works, I suggest using the method I showed at the end (instead of my solution). Map 0 to 1/2, map 1/2 to 1/4, 1/4 to 1/8, etc. What happens is that 0 gets absorbed and all the powers of 2 get slid down one space, mapping [0,1) to (0,1).
5. ## Proof there are as many numbers between 0 and 1 as 1 and infinity?

1/x is a bijection from (0,1) to (1, infinity). There's a bijection from [0,1) to (0,1). How do we do this? The rationals in (0,1) are countable so they can be enumerated as $r_1, r_2, r_3, \dots$. Now you define a bijection $f: [0,1) \to (0,1)$ as follows: If $x$ is irrational, then $f(x) = x$. That is, $f$ leaves all the irrationals unchanged. $f(0) = r_1$. And $f(r_n) = r_{n+1}$. In other words $f(r_1) = r_2, f(r_2) = r_3$, and so forth. Now you can see that every rational in (0,1) is in the range of $f$, and $f$ is reversible; that is, it's both injective and surjective. So it's a bijection. Graphically, we've mapped the rationals like this: $0 \to r_1 \to r_2 \to r_3 \to r_4 \to \dots$, in effect sliding 0 into the enumerated rationals in (0,1), which do NOT include 0. This trick amounts to embedding Hilbert's hotel in the unit interval. Initially the hotel is full, with guest 1 in room 1, guest 2 in room 2, etc. Now an "extra guest" named 0 shows up. We move guest 1 to room 2, guest 2 to room 3, and so forth, leaving room 1 empty. Then we move the extra guest 0 into room 1. In effect we've made room for the extra point at 0 by sliding everyone else up one room. To biject [0,1] to (0,1) we just do the same trick twice. Map 0 to $r_1$, map 1 to $r_2$, map $r_n$ to $r_{n+2}$, and leave the irrationals alone. Then to map [0,1) or [0,1] to (1, infinity), just compose the bijections. First map [0,1) or [0,1] to (0,1), then map (0,1) to (1, infinity). There are some other clever solutions in these Stackexchange threads. https://math.stackexchange.com/questions/28568/bijection-between-an-open-and-a-closed-interval https://math.stackexchange.com/questions/213391/how-to-construct-a-bijection-from-0-1-to-0-1 The checked answer by Asaf Karagila in the first link gives a completely explicit solution by sliding along 1/2, 1/4, 1/8, etc., without the need for a mysterious enumeration. That is, to map from [0,1) to (0,1) you map 0 to 1/2, 1/2 to 1/4, etc., and leave everything else alone. To map [0,1] to (0,1) you map 0 to 1/2, 1 to 1/4, 1/2 to 1/8, 1/4 to 1/16, etc. I like that one, clean and simple.
6. I've wondered the same thing, but about the scientific rather than philosophical effects. We looked at the stars and drew pictures in the sand and worked out trigonometry and early theories of astronomy. What if everything else had been the same but the earth were covered by clouds? We'd still have gravity, but we couldn't look at the heavens. How would science have progressed? Hope this isn't too much of a thread hijack but it's a question that's long been on my mind.
7. Yes exactly. I suppose you'd have to subtract the two integers and see if the result is integer-divisible by the modulus. That seems like the most sensible way. Probably not the only way.
8. Great question. Yes they are "the same but a little different." In math, mod is an equivalence relation on the integers. We say that $a \equiv b \pmod n$ for two integers $a, b$ if it happens to be the case that $n | a - b$, where the vertical bar means "divides." So for example $5 \equiv 3 \pmod 2$ and $17 \equiv 5 \mod 3$. You can verify that this is an equivalence relation: It's reflexive, symmetric, and transitive. It partitions the integers into $n$ pairwise-disjoint equivalence classes. It's a fundamental concept in number theory. In programming languages, mod is a binary operator that inputs two integers and outputs a third: 5 % 3 = 2. It inputs 5 and 3, and outputs the remainder when 5 is integer-divided by 3. The math and programming concepts are closely related. $a % n$ is the remainder when $a$ is integer-divided by $n$; that is, the result of $a % n$ is the smallest positive element of the equivalence class of $a$ under the $\pmod n$ equivalence relation. This turns out to be the same as the remainder under integer division. The tl;dr is that in math, mod is an equivalence relation that inputs a pair of numbers and a modulus and returns True or False; whereas in programming, mod is a binary operator that inputs an integer and a modulus and outputs the smallest positive member of the equivalence class of the integer mod the modulus. The difference is shown by, say, the fact that $17 \equiv 14 \pmod 3$; but $17 \ \% \ 3 = 2$ and not $14$. That is. even though mathematically $17 \equiv 14 \pmod 3$, in a programming language, $17 \ \% \ 3 == 14$ would evaluate to False. That's an example that illustrates the difference.
9. I don't think they have names, but there's a famous theorem of Fermat that says an odd prime is the sum of two squares if and only if p = 1 (mod 4). https://en.wikipedia.org/wiki/Fermat's_theorem_on_sums_of_two_squares
10. I think it's interesting that this question always gets asked in terms of the Internet, because that's the only complex computer system people have a daily experience of. But it's far from the most complex and mysterious computer system. If any computer system were to become self-aware, my bet would be the global supply chain. The system that moves raw materials from here to component factories there to integration sites somewhere else to distribution points somewhere else and ultimately puts a finished consumer good on the shelf at your local big box store, at a price point attractive to buyers yet high enough to ensue a profit for every single actor along the chain. The global supply chain is an immensely complicated system, far more complex than the Internet, whose architecture is generally well understood. It involves maintaining just-in-time inventories, tracking taxes and tariffs across international and local borders, integration of air, sea, and land transportation, predictions of consumer demand and raw material supply, and all the rest of it. It's a system that nobody sees but that affects literally every physical thing around us, from the furniture we sit on to the food in the fridge, and the fridge itself. It touches everything. You can turn off the Internet in your home, but not the global supply chain. If the thesis is that a sufficiently complex system can become self-aware, the global supply chain would be my candidate. Not the Internet, whose architecture is simple by comparison.
11. Yes, the Leibniz series is known to converge extremely slowly. Fun beginning programming project. I think he was continuing the series, not any previous conversation.
12. Aha. I did help you. I am pointing out your specific error. You are not asserting the question of whether one set is an element of another. You are interpreting the left hand side as a string that specifies a set. That's your error. You're equivocating "n in N such that n is prime" as a specification, on the one hand, and as a set, on the other. That is YOUR error, not an error in logic books. The logic books DON'T make that error. If it's a string, then you have to take its Godel number to ask if it's prime. If it's a set, you don't. But you can't have it both ways at the same time.
13. You replied before I edited that out. I actually don't think I can help. I agree with you that $\{n \in \mathbb N : n \text{ is prime}\} has a Gödel number, which in fact is not prime (since it will be the product of a bunch of prime powers). Aha I have a bit of insight. if [math]S$ is a statement in the language of set theory, let $G(S)$ be its Gödel number. Let $P$ be the set of primes. Then $G(\{n \in \mathbb N : n \text{ is prime}\}) \in P$ is a valid statement with a definite truth value, which in fact we can determine to be False. But $\{n \in \mathbb N : n \text{ is prime}\} \in P$ is NOT a valid statement, because it mixes up the metalanguage with the model. I think if you rewrite your argument, including the notation $G(S)$ whenever you intend to take some sentence's Gödel number, the error in your argument will be more clear. So I can restore my edit. I think I did help you. Let me know when you get the million. My contribution is worth at least a few bucks.
14. Strings talking about sets are not sets. Well the set of primes is a set. But the string "n in N such that n is prime" is not a set! I agree that the string has a Gödel number, and you can assert that collections of Gödel numbers are sets, but I don't think you can ask if "n in N such that n is prime" is an element of the set of primes. Well yes all the statements have Gödel numbers but I don't think it works this way. I'm out of my depth on Gödel numbering here so maybe you can explain why you think it works this way. I hope you get the million dollars. I confess that I'm annoyed at myself that I can't say exactly why your argument is wrong. If only I'd worked harder studying Nagel and Newman back in the day.
15. Looking at your paper, I see some issues. 1. What is a "set definition?" Do you mean a predicate? But predicates don't define sets, that's Russell's paradox. So what do you mean by a set definition? 2. You have the expression $X \notin S(X)$. But $X$ is a "set definition," whatever that is, and NOT a set; and $S(X)$ is a set. So your expression makes no sense, it's neither true nor false. 3. Likewise $M \in S(M)$, has the same problem. $M$ is a set definition and not a set; and $S(M)$ is a set. Perhaps you can clarify, starting with what you mean by a "set definition." Or perhaps a set definition is a specification like $\{n \in \mathbb N : n \text{ is prime} \}$. But in that case a set definition is a piece of syntax, a string in the formal language. It can't be an element of some set that the language is talking about. You're mixing syntax with semantics, formal strings with models. It's a plot by the publishers to sell more books. Or: Have you seen viXra lately?
16. This popped up on my radar this morning. "Finding primes using a parabola." Looks like it might be of interest to you. https://www.cantorsparadise.com/finding-primes-using-a-parabola-a939c6e8be53 First time I clicked on it the story appeared, then after that it said I needed a paid app. Not sure how this works but perhaps you'll be able to view it.
17. You have not demonstrated this relationship for 5, 17, and 85. The best you can do is compute the sector area for 5 and 17 and then adjust the spiral constants after the fact. You'd then have to pick different constants for every pair of numbers. And you could do exactly the same for 4, 6, 24; or 8,10,80. First, the primality of the first two numbers isn't needed; and secondly, for each pair of radial lengths you need to adjust your spiral constant after the fact. Don't you see this? What are the spiral constants for 5 and 17?
18. It doesn't have any truth OR any falsity. You haven't stated any clear proposition that would have a definite truth value. You haven't said why radial lengths of 5 and 17 are meaningful. I could do the same exercise with lengths of 4 and 6. But I don't mean to argue with you. I learned a little about the log spiral and perhaps helped you to clarify some of your thinking, or not, as the case may be.
19. That's exactly what I did earlier when I computed the sector area between radial segments of lengths 5 and 17. That's when you asked me why I didn't account for the angle. It's because the formula doesn't involve the angle, because given the radial lengths, you can compute the angle. That's this post: https://www.scienceforums.net/topic/124453-simple-yet-interesting/?do=findComment&comment=1178970 Oh I see what you mean. I interpreted $r(\varphi)^2$ as functional notation. $r(\varphi)^2$ is the square of the radius associated with the angle. I didn't take it as multiplication. It doesn't make any sense to square an angle. I see your point, the notation's a little ambiguous but I think they mean to input the angle to obtain the radius, then square that radius. I could be wrong. Either way it doesn't matter. There's nothing about primes or semiprimes about any of this. You pick two angles and get a pair of radial lengths and compute the sector area, but it doesn't mean anything with regard to primes or semiprimes.
20. The primality of 5 and 17 have nothing to do with anything, nor have you reached or stated any conclusion. You take any two angles and you will have radial segments with a sector area between them as computed by the formula on Wiki. It doesn't seem to mean anything. What happened to the radial segment of length 5? What point on the spiral corresponds to a radius of length 17? If the spiral constants are $a = 1, k = 1$, the spiral is given by $r = e^\varphi$. What value of $\varphi$ gives a radius of 17? $\varphi = \log 17$, right? That's the angle, in radians. It comes out to around 161 degrees. $\log 17 \approx 2.8]$, and 2.8 radians is around 160 degrees. That's the angle it would make with the line from the origin to (1,0) if you plug in $\varphi = 0$. How are you making your calculation?
21. It's the sector area formula from wiki. https://en.wikipedia.org/wiki/Logarithmic_spiral#Properties
22. 17 is supposed to be a radial length. What is 0 here? Can you clarify? What you wrote doesn't seem to make sense in the context of the problem. Did you mean the angle between radial lengths of 5 and 17? Well if the spiral is $r = e^\varphi$ then the angles are $\log 5$ and $\log 17$, respectively, $\log$ meaning natural log. Is that what you're doing? From the sector area formula on Wiki, the sector area between radial segments of length log 5 and log 17 is (17^2 - 5^2)/4 = 264/4 = 66. That's with spiral constants a = 1 and k = 1. You could adjust k to make the area come out to 85, but I'm not sure what point is being made. Have you looked at the Wiki page on the log spiral? Your log spiral changed into a parabola? As Galois said as he was frantically writing up his mathematical discoveries the night before he was to die in a duel, "I have no time. I have no time."
23. Ok referring to https://en.wikipedia.org/wiki/Logarithmic_spiral the log spiral is given by $r = a e^{k \varphi}$ where $a > 0$ and $k \neq 0$ are real constants. The sector area is $A = \frac{r(\varphi_2)^2 - r(\varphi_1)^2}{4 k}$ where $\varphi_1$ and $\varphi_2$ are your two angles. So: What values of $a$ and $k$ have you chosen in order to get radial distances of 5 and 17; and what are the corresponding angles; and does the sector area work out to 85? You should work this out and see what makes sense. You need to find the constants and the angles that make your picture valid. And the primality of the radial lengths doesn't appear to be significant in any way, so can you explain why you think it is? If there's anyone here who knows anything about spirals, I'm confused about something. In the picture on the Wiki page, they show the sector area as going between the origin and two points on the spiral. But clearly the radial lines pass over other parts of the spiral more near to the origin. What defines the sector, exactly? How do you know which arm of the spiral defines the sector? I don't know anything about spirals, I'm confused by this. ps -- Oh I see what's going on. The radial line segment is defined from the origin to the point $(r, \varphi)$. So for example take a log spiral with $a = 1$ and $k = 1$, so the spiral is given by $r = e^\varphi$. Then for an angle of $0$, the point on the spiral is $(1,0)$ in polar coords. If you go around the circle again with $\varphi = 2 \pi$, the corresponding point is $(e^{2 \pi}, 2 \pi) \approx (535, 0)$. In other words the angle is always taken mod 2 pi, but the radius keeps growing (or shrinking towards zero as the angle decreases to minus infinity). So if two angles are close together (within 2 pi of each other) the sector makes sense; but if not, it's unclear what the sector is. Anyway I'm beginning to get some feel for the log spiral. "Today I learned!"
24. Well if the angle is zero, the area is zero. And if the angle is 180 degrees, the area is infinite, depending on how you calculate the area, because you'd be crossing the spiral multiple times. Maybe you only count the area till the first crossing, in which case it's finite. So there's probably some angle such that the area is exactly 85, unless you only count to the first crossing, in which case there might not be. But so what? The same can be said for any two radial lengths whatsoever. I don't see why this is meaningful. Explain please what the primality of the lengths has to do with anything.
25. Ok I believe you if that's true. What's special about primes? Wouldn't it just depend on the properties of the log spiral? You could just as easily use 4, 6, and 24, if that multiplicative property is true. But even though this isn't drawn to scale, clearly the distances of 5 and 17 are wildly off. Are you sure any of this is accurate? Are those supposed to be distances from the center? Or arc length along the spiral? The formula for sector area is given on the wiki page. https://en.wikipedia.org/wiki/Logarithmic_spiral#Properties I find it somewhat unlikely that all these would be integers, given the formula for sector area. I haven't worked any of this out, I'm wondering if you have and if any of this is actually true.
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