wtf

I agree this leaves unanswered the question of whether Euclid regards the intersection of two lines as a point. But I'm still baffled as to why you (a) think this will be helpful to pengkuan, who's confused about the nature of the real line; or (b) is relevant as a response to my questions. If you tell me you're a scholar of Euclid in the original Greek language then perhaps you can elaborate on your claim that Euclid thinks the intersection of two lines is something other than a point.

https://en.wikipedia.org/wiki/Line_(geometry) I do not have sufficient knowledge of Euclid's original work to know if this is something he states or prove. I'd be very surprised to know that Euclid thought two lines can meet in something other than a point. Have you got a reference? I genuinely don't understand why you are pursuing such a seemingly wrong idea. Maybe you know something I don't, in which case I'd be grateful for a reference.

Studiot you know very well that the intersection of two lines is a point. This is true in ancient Euclidean geometry and in modern analytic geometry. So I ask again, what are you talking about?

Studiot, two lines intersect in a point. That was true for Euclid and it's true in the modern Euclidean plane. Can you explain what you are trying to say?

Depends on what you're vaping. Nicotine is a poison regardless of how you ingest it. Perhaps there's some harm reduction from vaping rather than smoking, but vaping is too new for data to be conclusive. In ten or twenty years we'll have a better idea.

Let me outline the flaw in this viewpoint. The world's most clever neural net or learning algorithm, running on the world's most powerful supercomputer, is still nothing more than a physical instance of a Turing machine. A basic fact about TMs or any type of computer program is multiple realizability, also known as substrate independence. A program's capabilities are independent of the physical implementation. The sets or functions computable by a given TM do not depend in the least on the physical details of the execution of the instructions. A practical version of this idea is familiar to every programmer who gets stuck and "plays computer" by working through the algorithm using pencil and paper. This means that if a program is self-aware when running on fast hardware, it's already self-aware if I execute the same logic with pencil and an unbounded roll of paper. Given your assumption, that a sufficiently complicated TM can be conscious, then suppose I get a copy of the computer code, a big stack of pencils, and an unbounded roll of paper. (For example a big roll of TP, or Turing paper). Then as I execute the code by pencil and paper, there must be self-awareness created somewhere in the sytem of pencil and TP. Please, tell me where this consciousness resides and how it feels to the pencil. And how many instructions do I have to execute with my pencil in order for the algorithm to achieve self-awareness? After all, any program running on conventional computing equipment executes one instruction after another. (And a single-threaded TM can emulate multiple threads, so modern multi-core processors don't extend the limits of what is computable). Even if I grant you a supercomputer, you would not say your algorithm is self-aware after executing the first instruction, or the first 40 or 50. But at some point it executes just one additional instruction and suddenly becomes self-aware. I hope you can see the deep problems with this idea. In short, digital computing systems seem extremely unlikely to be able to implement self-awareness. Multiple realizability defeats every argument for computer sentience that depends on complexity of the algorithm or the power of the physical implementation. Any self-aware TM executing on fancy hardware is already self-aware when executed using pencil and paper. This poses a big problem for those who stay that algorithms executing on sufficiently fast hardware can become self-aware.

How do we know it's difficult? Because so many people have been unable to solve it! OP has a good question. What is it, exactly, that makes RH a difficult problem? Why have FLT and the Poincaré conjecture been solved, but not RH? That's way above my pay grade. But at heart it's a very good question IMO.

Why not take a few months and study the mathematical formalism of real analysis? The subjects you are thinking about have been studied for thousands of years. What is the nature of the mathematical continuum? And how does it relate to the ultimate nature of the real world? Standard mathematics, the modern theory of the real numbers, is humanity's best answer yet (at least to the first question, the nature of the mathematical continuum). It's probably not the final answer. If you seek to overthrow the conventional knowledge, shouldn't you take the time to learn the conventional knowledge first? Newton, whose work created the modern scientific world, got his start by fully mastering the work of the ancients.

Yes, every real number has a decimal representation consisting of infinitely many digits (to the right of the decimal point). There's one digit for each of the real numbers 1, 2, 3 ... Not necessarily. You can't list the real numbers. That's Cantor's diagonal argument. Any list of real numbers must leave some out. https://en.wikipedia.org/wiki/Cantor's_diagonal_argument A more precise statement is that any function from the natural numbers to the reals can not be surjective; that is, it can not hit every real.

I don't know much physics and I don't know what you're talking about. I am sincere in wanting to understand your point. As far as I know, no theory of physics posits an actual infinity in the real world. 10^80 atoms and all that. How can a "light path," whatever that is, have an infinite length? How does any infinite length fit into a finite universe? Perhaps you think I know more physics than I actually do. I have no idea what you're talking about but I am interested in understanding. I don't know what you mean by proper path or proper length. I don't know what you mean by lightpath. The only thing I know about infinities in physics is that they need to be renormalized so that they go away. And the bit I mentioned about Hilbert space. I know Hilbert space only as an abstract mathematical structure. I don't have any idea how quantum physics is reconciled with a finite universe. Love to find out though.

I have no idea. Perhaps you can explain your remark. I'm aware that quantum physics takes place in Hilbert space, an abstract infinite dimensional function space. I don't know anything about how philosophers of physics reconcile the apparent contradiction of using vast infinite spaces to model the real world, when there is no evidence for infinite collections. For example there are 10^80 hydrogen atoms in the known universe. If you know something about how this apparent contradiction is reconciled, I'd be happy to learn.

In binary, .0101010101... = 1/3. On the other hand some other bitstring might be irrational. This is just basic binary notation of real numbers. It's no different than decimal, in which .14159... is the decimal representation of pi - 3, which is irrational; and .3333333... is the decimal representation of 1/3, a rational. Every such binary or decimal expression must denote a real number, because the set of finite truncations is a nonempty set bounded above by 1, hence has a least upper bound by the completeness of the real numbers. In other words the set {.1, .14, .141, .1415, ...} is bounded above by 1, hence has a least upper bound. This is the basic theory of the real numbers. Has nothing to do with the mathematical study of infinity. What is true is that in order to get the real numbers off the ground, we have to allow a "completed" infinity of the natural numbers {1, 2, 3, 4, 5, ...}. Of course nobody is making any claim that such a thing exists in the real world. No current physical theory includes infinite collections, but we don't know what physicists of the future will think. However when it comes to physical science, everyone uses standard math to model physical theories. And the standard math of the real numbers does include the assumption that there is an infinite set. So there's a bit of a philosophical puzzler. If math is based on an assumption (the existence of an infinite set) that's manifestly false about the real world, why does math work so well to describe the real world?

Infinitely many solutions with a surprising graph. https://www.wolframalpha.com/input/?i=x^y+%3D+y^x

That might be the heart of your misunderstanding. Physics [math]\neq[/math] math. Euclidean space is a continuum modeled by the real numbers. It's not known whether physical space is like that or not. You are right that electrons and photons don't flow into each other like the points on the real line. That's because the real line is not (as far as anyone knows) an accurate model of the real world. It's been generally understood that math describes logically consistent worlds, and not necessarily the physically true world, since the discovery of non-Euclidean geometry in the 1840's.

You are right. I looked up inclination last night and for some reason I thought it said the slope, but actually it said the angle. My mistake. Inclination is the angle.

A small quibble. OP said that a is the inclination, not the angle. The inclination is the slope of CX with respect to PC. In other words the angle is the arctan of a. OP did not provide a clarification so I assume this is what is meant. (In your pictures this would be more clear if you drew CX as having a positive angle with PC. In your pictures, the inclination is negative.)

By inclination do you mean the tangent of the angle XCP? If so you have side-angle-side and you can determine the third side PX. https://www.mathsisfun.com/algebra/trig-solving-sas-triangles.html In other words we know PC, and we know that CX = r, and we know the angle XCP as the arctangent of the inclination. Am I understanding inclination correctly? As the slope of the line with respect to CP, in other words imagining CP as the positive x-axis and then the inclination is the slope of CX?

I looked at your paper. You say: "So, I propose the following definition of continuity: A line is continuous between 2 points C and D if the space between them is zero." The problem is that the distance between any two real numbers is zero if and only if the two numbers are the same. This follows from the definition of the distance between real numbers, which is just the absolute value of their difference. You keep thinking real numbers are like bowling balls lined up in a row; and this false visualization is leading you into mathematical errors.

Uncountability is not sufficient. Suppose I take the real line and delete the point at 0. The resulting set is not connected and it is not what you would call "continuous," but it's uncountable. In fact the word "continuous" is wrong here because continuity applies to functions and not sets. However if you mean "no holes," plenty of uncountable sets have holes. A more striking example is the Cantor set, which is an uncountable set of measure zero. It's full of holes. https://en.wikipedia.org/wiki/Cantor_set ps -- I see you referenced the Cantor ternary set. So if you know this example, why does your exposition not deal with it? In other words you already know a striking counterexample to your idea. pps -- A few more idle thoughts. Bottom line you are confusing cardinal, order, and topological properties with each other. That's just not true. The standard construction of the real numbers within set theory shows that we do not need infinitesimals. There are no infinitesimals in the real numbers and they are not needed in math. It's true that there are nonstandard models containing infinitesimals but they don't add anything to the discussion and do not provide any more deductive power; so they are a distraction in these types of discussions. Of course that's not true. A 1-D line is made up of 0-D points. It is true that it is a philosophical mystery. But it's not a mathematical mystery! A better way to think of it is that a line is the path of a point through space. If you had a 0-D point moving through the plane, it would trace out a 1-D path. This was Newton's point of view. You haven't defined "continuity" of a point set. Until you do, your argument is not valid. Do you mean dense? Perhaps you mean complete, in the sense that every Cauchy sequence converges. In order to have an argument you have to say exactly what you mean by a continuous set. It would make it easier to understand what you're trying to say. The rationals have a dense linear order, but they're not complete because some Cauchy sequences don't converge. I have no idea what that means. Every real number has a decimal representation that is "fully determined" in the sense that all of its digits are "fixed." What do you mean determined? Every real number has a decimal expansion (or two). What does that mean to you? To the extent that you're trying to understand the nature of the mathematical continuum, that is a noble persuit. To the extent that you're here to deny Cantor's results, that's generally not a productive topic. Nobody doubts Cantor's results. He wasn't making any such claims at all. We agree that uncountability is not related to continuity, if by continuity you mean completeness. The Cantor set and for that matter the reals minus a point are uncountable but not complete. The real numbers with the discrete topology are a discrete set. With the usual topology they're a continuum. You're confusuing cardinality, order properties, and topological properties. It makes perfect sense. CH asks which Aleph is the cardinality of the reals. It has nothing to do with the topology on the reals. For example if we give the real numbers the discrete topology, then they are a discrete set. Yet their cardinality doesn't change, and it's sensible to ask what that cardinality is.

You can always fit finitely many values to a polynomial. https://en.wikipedia.org/wiki/Lagrange_polynomial

What do you make of the fact that Las Vegas bookmakers set specific odds on each and every sporting contest, and that the odds are very rarely 50-50?

https://aeon.co/essays/contagion-poison-trigger-books-have-always-been-dangerous

You're very welcome. Just a couple of years of grad school a long time ago and a lot of Internet surfing. The physicists in this thread certainly know far more differential geometry than I do.

Let me see if I can provide some context for MigL's remark. We have a mathematical object that we think of as some kind of geometrical space. For the moment forget Minkowski space and just think about the familiar Euclidean plane or perhaps Euclidean 3-space. Imagine rotating the standard Euclidean plane around the origin. It's clear that if you do one rotation then another, it's the same as if you'd combined them from the start. In other words the collection of rotations is closed under composition of rotations. Composition is associative (needs proof). The identity is the rotation through 0 degrees, which leaves the plane unchanged. If you rotate the plane you can just rotate it back to where you started, so we have inverses. Therefore the set of rotations of the plane forms a group under composition of rotations. In the case of the plane, the group is Abelian, after Niels Henrik Abel. In general, geometrical operations are not commutative. In Euclidean 3-space we can rotate around one of the standard axes or around some arbitrary line. There are more ways for commutativity to fail. If you have good 3D visualization (which I never did) you can see that 3D rotations do not in general commute. In the late 19th and early 20th centuries, mathematicians figured out that to study a geometric space, it was useful to study the groups of geometrical transformations that operated on a space. This is the general pattern. We have a space and we have various groups associated with geometrical transformations of that space. We study the groups to better understand the space. Minkowski space is (as I understand it) is the mathematical model of relativity theory. The Wiki page would take some time to work through, after which you'd know a lot of differential geometry and relativity. But basically it's just 4-dimensional spacetime with the funny metric that combines time with space to model modern relativity theory. (Apologies to the physicists for anything I've mangled here). And there are a number of interesting groups associated with various classes of geometrical transformations on it. For example in this page on Lie groups (Lie pronounced "Lee") we find MigL's example: An isometry is just a rigid motion, like a rotation or reflection or translation. Any transformation that preserves distances. So even though we may not know every detail of the Lorenz group; we can understand it as some group of rigid transformations of 4D spacetime. That's what we mean when we talk about groups in conjunction with geometry. We're considering collections of transformations that preserve some geometric property we care about. As long as the individual transformations are reversible, we'll have a group.

The poster in question was criticizing the behavior of another poster and used as an example the alleged bad behavior of some Americans. I don't think that's appropriate no matter what nationality was named. I expressed myself on the matter and I reiterate my displeasure with that comment. It has no place here. There is no language issue. Sensei said exactly what he meant. The question is why he said it. I await his response. Then why single out one particular country?