wtf

That, if a straight line falling on two straight lines make the interior angles on the same side less than two right angles, the two straight lines, if produced indefinitely, meet on that side on which the angles are less than two right angles.

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.

In an axiomatic formulation of Euclidean geometry, such as that of Hilbert (Euclid's original axioms contained various flaws which have been corrected by modern mathematicians),[10] a line is stated to have certain properties which relate it to other lines and points. For example, for any two distinct points, there is a unique line containing them, and any two distinct lines intersect in at most one 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.

Self awareness, feelings etc are complex emergent processes of simpler molecular processes. It's the same with electronics; one reaches some, as yet unkinown, level of complexity and artificial intelligence and biological intelligence becomes indistinguishable from an operating and output point of view.

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.

It is a difficult problem. What are you thinking by asking "why"?

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.

What I do here is to propose new ideas. But I'm unable to construct mathematical building.

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.

The question I want to ask is "will every row of this matrix have an infinite number of digits?"

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 ...

If they do, then will a number like pi/4 be in one of those rows?

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.

The proper length of a lightpath is infinite since it is accomplished in zero proper time.

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.

Maybe but what is the 'proper length' of a lightwave according to Einstein's relativity?

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

If you draw a line that is the path of a point through space, then between points there is not continuity. See the electron that passes from one point to the following, it will not be able to forego the point in between.

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.

I don't suppose your visits overlapped.

 

I think formal study is difficult in India and the OP is trying his best with the English language, so I took it to mean to angle

 

I'm glad it was a small quibble since inclinometers (I have several) measure angle.

 

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

 

They were much used in the survey of India.

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.

Note that for acute angles ...( a<90)

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.

The real line is made of real numbers which are points. Points are discrete objects, but lines are continuous objects. How does continuity arise out of discreteness when points make line? The idea of uncountability solves this problem.

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.

Remark: The construction of geometrically continuous line proves that the controversial infinitesimal number really exist, otherwise, continuity cannot arise.

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.

One-dimensional geometrically continuous line is constructed only with one-dimensional objects.

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.

How did Georg Cantor link uncountability to continuity? In fact, he constructed the continuum ℝ in two steps: 1) ℝ is uncountable; 2) Uncountability of ℝ creates continuity for the real line.

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.

However, does the limit of an and bn really exist? A limit is a real number which must be fully determined, that is, all the digits from 1st to th are fixed,

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?

In addition to this flaw which is explained in «On Cantor's first proof of uncountability», Georg Cantors later proofs, the power-set argument and the diagonal argument, contain also flaws, which are explained in «On the uncountability of the power set of ℕ» and «Hidden assumption of the diagonal argument». So, all 3 proofs that Georg Cantor provided fail and uncountability possibly does not exist.

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.

About the second step Georg Cantor did nothing but simply claim that ℝ is a continuum; probably he assumed that uncountability really created continuity. But it is shown above that uncountability is not related to continuity. So, uncountability has lost its utility and becomes useless except for itself.

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.

Continuum hypothesis

The continuum hypothesis states that there is no set whose cardinality is strictly between that of the integers which is a discrete set and the real numbers which is a continuum. The idea behind this hypothesis is that there cannot be set that is discrete and continuous at the same time.

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.

Anyway, ℝ is not a continuum and the continuum hypothesis makes no longer sense.

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

A friend of mine correctly predicted who would win a football game (college I think so no tie) the other day. He was congratulated by my other friend but another dismissed his prediction as 50/50. An argument ensued as obviously football games themselves are not (necessarily) 50/50 like heads vs tails on a balanced coin is 50/50. My question is: Does my friend who dismissed my other friend's prediction have a point in any way? He did acknowledge that there were odds in play but insisted that "it's still 50/50." Not sure what "it" is then.

 

Also, I know that if you guess on a multiple choice test (4 choices to a question), you have a 25/75 of getting it right. Is that like what's going on here? Since there are only 2 answers to who will win the game, it's 50/50 of being correct, right?

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?

Socrates said that most people couldnt handle written text on their own. He feared that for many especially the uneducated reading could trigger confusion and moral disorientation unless the reader was counselled by someone with wisdom. In Platos dialogue, the Phaedrus, written in 360 BCE, Socrates warned that reliance on the written word would weaken individuals memory, and remove from them the responsibility of remembering. Socrates used the Greek word pharmakon -- drug -- as a metaphor for writing, conveying the paradox that reading could be a cure but most likely a poison. Scaremongers would repeat his warning that the text was analogous to a toxic substance for centuries to come.

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

Thanks very much for that wtf.

You're very welcome.

Although his profile does not indicate much wtf posts as though he is or was a professional mathematician so it was good to see my words in post 3 echoed and much extended, in a particularly understandable and helpful way.

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.

I thought I had accepted MigL's correction in post#2 where he said: "Minkowsky space-time ( flat, ignoring gravity effects ) is not a group."

...

I understand now that the elements in the Minkowski space are (acc MigL) "certain transformational operations ( isometries IIRC ) "

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:

The Lorentz group is a 6-dimensional Lie group of linear isometries of the Minkowski space.

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.

Did the poster really mean

 

US citizens, usually poorly educated ones?

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.

bearing in mind that every country has well educated and poorly educated citizens.

Then why single out one particular country?