joigus

The word "point" in itself does not tell you what it is. A point on the real line: \( x \in \mathbb{R} \) A point on the real plane \( \left(x,y\right) \in \mathbb{R}^2 \) ... etc. Edited: A point is a locus, location, place in a set. When you say "point" normally you imply some kind of position (distance-->geometry, topology...). When you say "element" or "member" you normally imply just set theory. There's context missing. And as Ahmet suggested, "light cones", "faster than light", "hyperplanes"... That has nothing to do with your drawing or the concept of points. The impression I get is, again, you're trying to connect too much in one simple concept. Points don't need light in order to be defined.

(My emphasis.) You've given no answer to any of my points. I've provided you with references and reasons why many of the things you hold as true about the past simply cannot be correct. Then you engage in an argument about bricks by using 16/17th-century language. The fact that you desperately try to attack the man, "you're a cynic", while fleeing from the argument tells me I must be doing something right. People always do that when they're logically cornered. "Debating is always cynical" is the bit that I've decided to leave uncommented because it needs no further comments from me. I don't know what to say. You might as well say "reason is always cynical". Maybe you simply don't know what "cynical" means.

A message from the Bronze Age from an invisible being, compiled by people from the Iron Age, written in English from the 16th century. Not very illuminating to me, I'm sorry. If I want to be understood, I use 21st-century English. That's why safety warnings, for example, are not written in 16th-century language: Being understood could be a matter of life and death in that case.

I stand by every word here. There are many sources of possible mistakes. A very common one is forgetting that \( g^{\mu \nu} \) is the inverse of \( g_{\mu \nu} \). It is always the best idea to go over the calculation again, instead of believing you've found a shortcut to the Nobel Prize. It's a natural rite of passage. You do it in the abstract, with indices; you do it with polar coordinates on the plane, you do it with hyperbolic polar coordinates. You convince yourself that it's correct. You turn on and off the contravariant to covariant "switch". It still works... Oh my. It must be true. That's the path.

Muons orbiting nuclei? Lifetime of a muon is 2.2 ms. That's a mighty ephemeral magma.

Physical four velocities are not made of arbitrary numbers. As Markus said, in just a bit more explicit notation and rephrasing what he said, although I think it was clear enough, \[ \left( v_t, v_x, v_y, v_z \right ) \] but constrained to, \[ \left( \frac{c}{\sqrt{1-v^2/c^2}}, \frac{v_x}{\sqrt{1-v^2/c^2}}, \frac{v_y}{\sqrt{1-v^2/c^2}}, \frac{v_z}{\sqrt{1-v^2/c^2}} \right) \] Same with 4-momenta. 4-momentum is the product of \( u^{\mu} \) --the 4-velocity-- times the mass. So it doesn't do to fill arbitrary numbers in the slots, so to speak.

\[ \left( \frac{1}{\sqrt{1-v^2/c^2}}, \frac{v_x}{\sqrt{1-v^2/c^2}}, \frac{v_y}{\sqrt{1-v^2/c^2}}, \frac{v_z}{\sqrt{1-v^2/c^2}} \right) \]

"Par excellence" after a noun meaning "a very good example of something": https://www.oxfordlearnersdictionaries.com/definition/english/par-excellence_1?q=par+excellence No excellence implied in the literal sense.

I haven't had time to review in detail. The numeric relation seems correct to me. As to the claimed contradiction, I haven't got around to it yet. Contradiction with what exactly? Don't have time to read Markus' comments. Maybe later.

There is no conflict in the operation of taking covariant derivatives of any tensor, at any order of derivation. Again, can you explain here, instead of linking to a document, please?

OK, the fact that you mention Adam and his immediate offspring as factual is enough for me to know this discussion is not leading anywhere useful. Some biblical myths are inclusions from Babylon. Ezra re-edited the Torah, because it had been lost after Nebuchadnezzar II destroyed the Second Temple. The myth of the flood from the Epic of Gilgamesh is very recognizable. Another one is the story of a man whose wife cannot conceive, so that they arrange that it is the slave who is going to play that role --Abraham--. The latter story is foreshadowed in the Nuzi tablets over and over, and over again. Also in Mari --Mesopotamia. It is more than likely that they picked it up by the rivers of Babylon, because they came to know it was a common Babylonian story. Also you say all deities in the Bible stand for God. We already know this cannot be true -beyond any reasonable doubt. There are also inscriptions speaking of Yahweh and his Ashera (his wife). The existence of a pantheon of gods is very clear in the archaeological record. Baal is not, as you seem to suggest, another representation of Yahweh, but the bull god that appears in many places of the Middle East and features prominently in Exodus, different in name and in the statuettes --in the human form, El, sitting and serene; while Baal, aggressive, in smiting position, and using his strength. It's Baal-Zebub, the Lord of the Flies, that in Christian iconography became to be known as Satan. Yahweh, in the Sinaitic depictions, looks nothing like El. Why would he? They are different gods from different regions. Are you going to believe what a book which was copied again and again, recompiled hundreds of years later after its partial destruction, probably recited at some points; or are you going to believe the fragments of script that are dug from the ground and tell us what the Canaanites of that time probably believed? Faith-based religion is not like a message passed down in its pristine form generation after generation; it is more like a game of Chinese whispers played throughout the centuries in which you never know what the message is going to become. That much we already know beyond any reasonable doubt. Understanding the process, rather than the details, makes it very easy to see how you can throw in a new element and make it part of the broth, keeping some words but changing the meaning, etc. Like your kefir. It is the lack of logical strictures which allows to do that. This appears not to be true: https://weareisrael.org/spiritual-seed-2/male-child/betulah-vs-almah/ But, when Eliezer recalls his story to Rebekah’s father (Bethuel), he calls Rebekah a young woman (עלמה, al-mah’), a sexually mature woman at the prime age for work, because he was not privy to her actual sexual status. Helenization of Roman Jews started in the 4th century BCE, but found strong resistance that culminated in the Maccabean revolt during the Seleucid rule. So they were not completely Helenized, especially considering the Maccabeans were successful, unlike the rebels of Masada. It was after the diaspora that most transcriptions of Torah appear only in Greek, at least in Europe. But the sect of Qumran still copied the Bible in Hebrew, and I'm sure the rebels of Masada also did so, at the time of the Jewish revolt. Some of the parchments have been found to appear to have been dropped on the ground of the caves in Qumran as the legions came to arrest them. Those texts range from old copies to contemporary copies --at the time. They are all in Hebrew.

You've got to admit that Christopher Hitchens was spot on when he said "with religious types you never know what you're gonna hear next."

@iNow. Fair enough. "Beyond a reasonable doubt" actually sounds more like me.

Archaeological evidence proves to the contrary. 900 BCE Jerusalem was a small town, as proven by Israel Finkelstein. The perimeter is well established by the old tombs and the dating of the pottery is ambiguous at best. The gates of Hazor, Megiddo, and Gezer, rebuilt by Solomon according to the Bible, are not Solomonic at all. They are probably Iron Age. In the time of David the kingdoms of Israel and Judah were two independent kingdoms. Monumental architecture of the time corresponds to king Omri of Israel, not to Judah. Masonry signatures testify to that. Just after king Josiah accesses the throne of Judah, local temples disappear and Jerusalem stands alone. Careful studies of the archaeology of the countryside have proven that the population wasn't enough for an army. There is no trace of scribal activity or pottery bearing the seal of the king in the Davidic kingdom. We know there was a chieftain named David because of the Tel-Dan stele.But his presence was very thin on the ground. We know Omri was the big king of the time because of the Mesha stele. Also the only to boast monumental architecture in Samaria, as well as abundant excavated luxury goods. LOL. It hasn't for me. Yes, I was referring to the archaeology of the written hearsay. Of course material culture and inconsistencies between the Bible and what's been unearthed are most important.

It is ironic the way in which the persecutors par excellence like to play mentally with the idea of being persecuted. No, not censored. You are just ignorant. "Virgin" in "virgin Mary" (parthenos, in the Septuagint) is a mistranslation from Hebrew almah ("young woman".) As the Dead Sea Scrolls have shown, there is no trace of the Hebrew word bethulah ("virgin") referring to Mary. So for all we know from science (archaeology), Mary was not a virgin, but just a "young woman." https://en.wikipedia.org/wiki/Septuagint#Christian_use El and YHWH (probably "Yahweh", archaic Semitic scripts had no vowels) were different deities. One came from Canaan, the other from the outskirts of the Sinai desert. One was a god of the Canaanite hills, the other from the desert. There were many Canaanite deities, like Ashera, or Baal. It was king Josiah who fused together El and Yahweh, decreed a unified place of worship in Jerusalem, and substituted all of them for the common name adonai as a conveniently ambiguous placeholder for "god". It is not Yahweh the name for "the lord" in Hebrew, it is adonai. All for political reasons well understood in terms of the decline of the Assyrian Empire and the political situation that resulted --need for unification of two kingdoms. Some of these points are debatable, and different scholars hold different views, but what seems to emerge clearly from the ground is that the Israelites and Judahites were not de facto monotheists until after the Babilonian takeover. There were many deities among the Nabateans too --the precursors of Islam, which is the reason why Ibn S'ad, Ibn Ishaq ol Al-Tabari mention them in the so-called Satanic Verses, but they were conveniently whitewashed by later traditions. There are still death penalties for those who dare talk about it. Whitewashing, abrogation... very common in faith-based religions. As I imagined, nothing whatsoever on this thread having to do with "the abstract vs not the abstract." You could at least learn a bit about where your book comes from.

Can you summarize it here, please?

No, no, no. First come numbers. Then comes topology (neighbourhoods in a set defined by the relation \( \subseteq \) "contained in") Then comes geometry (defined by distance, a number assigned to pairs of "points": \( d\left(x,y\right) \)) From metric (distance) come angles, defined as ratios of distances, as @Sensei has told you. Topologies are possible to define even when there is no notion of a metric. Numbers don't have geometry built in them. You need numbers first. How else could you define the distance, which is a positive number? Topology is more primitive. You only need a notion of inclusion, open and closed sets, etc. Closed set: contains its boundary Open set: does not contain its boundary Edit: Dimension you can define with vectors (tangent space) or with analysis (number of real variables necessary to describe your set analitically). And so on... What a wasted effort!

Because you couldn't be farther off the mark. That's not what bases are about. I and others have been telling you until we're blue in the mouth. You're using the oldest trick of the game, which is non-sequitur. It's as if someone tells you, "Mountains arise from mechanical tensions and thermal processes in the Earth's interior" and you say, "Then why are elephants winged creatures?" 1st) Elephants are not winged creatures (a false premise embedded in a question is called a sophism) 2nd) The question does not follow from the previous statement at all (that's called a non-sequitur) If you think for a moment most users here don't see right away what you're trying to do, you're quite wrong. You're not discussing in good faith. It's not about disagreement. It's about you not being intellectually honest. You're free to keep playing your game for as long as you want, but you're just calling for action from the mods and very justified annoyance from other users. Have a good day.

Although nothing would amuse me more than the picture of you being preyed upon by legal counsellors, I'd advice you to think it twice. In a previous post you bitterly complained about not being offered a job, as some kind of reward for your brilliant thinking. Set your priorities right, is all I can say. I don't wish you any wrong, in spite of your misled smugness and total disregard of the efforts of many users trying to help you to the best of their --our-- abilities. The bullying that you mention is about a post by @iNow on another thread that didn't even mention you. I almost forgot: numbers are not geometrically motivated. They come first. You can study their properties with topology --a basis of neighbourhoods-- or with geometry --distance, metric--. If you have n-tuples of numbers, then you can introduce angles, also from the metric.

That's probably because you're ready to ignore all answers and keep diverting into new questions. Case in point. Is that a question, or word origami?

I didn't write any computer code. That was all by hand. They way it was done before computers arrived, other that Leibniz's calculating machine.

Sorry, I wrote 1/2 in binary. As I was presenting them in decimal, it should be, \[\frac{1}{2}=0.5\]

CuriosOne, I've never seen anyone who understands so little and claims to understand so much at the same time. I'm very nearly done with you too.

The evaluation or products --or fractions, for that matter-- does not depend on the base. Take, eg., \( 7\times3=21 \). In binary, the numbers \( 7 \) and \( 3 \) are written, \[7={\color{red}1}\times2^{2}+{\color{red}1}\times2^{1}+{\color{red}1}\times2^{0}={\color{red}1{\color{red}1{\color{red}1}}}\] \[3={\color{red}0}\times2^{2}+{\color{red}1}\times2^{1}+{\color{red}1}\times2^{0}={\color{red}0{\color{red}1{\color{red}1}}}\] You can even reproduce the algorithm for multiplication that you learnt at school, you only have to remember that, in binary, \( 1+1 \) gives zero, and carries \( 1 \). Then, \[\begin{array}{cccccc} & & & {\color{red}0} & {\color{red}1} & {\color{red}1}\\ & & \times & {\color{red}1} & {\color{red}1} & {\color{red}1}\\ & & & 0 & 1 & 1\\ & & 0 & 1 & 1\\ & 0 & 1 & 1\\ & {\color{green}1} & {\color{green}0} & {\color{green}1} & {\color{green}0} & {\color{green}1} \end{array}\] Input numbers are in red, intermediate calculations are in black, and output is in green. Sure enough, it gives you \( 10101 \) which, in binary, is \( 21 \), \[{\color{red}1}\times2^{4}+{\color{red}0}\times2^{3}+{\color{red}1}\times2^{2}+{\color{red}0}\times2^{1}+{\color{red}1}\times2^{0}=16+4+1=21\] Floating-point numbers are floating-point numbers in any base. For example, \( \frac{1}{2} \) is \( 0.5 \) in decimal. In binary, eg, the only peculiarity is that they are expanded in terms of, \[\frac{1}{2}=0.1\] \[\frac{1}{2^{2}}=0.25\] \[\frac{1}{2^{3}}=0.125\] etc. Here's a trick with which you can convince yourself that decimal numbers in base 10 are decimal numbers in base 2 too: https://indepth.dev/posts/1019/the-simple-math-behind-decimal-binary-conversion-algorithms By successively multiplying by \( 2 \) and extracting the integer part as a sum of ones you can in principle get the whole series of floating-point digits (zeroes and ones). You must distinguish digits --(\( 0 \) and \( 1 \) in binary, \( 0 \), \( 1 \), \( 2 \), \( 3 \), \( 4 \), \( 7 \), \( 8 \), \( 9 \), in decimal, or \( 0 \), \( 1 \), \( 2 \), \( 3 \), \( 4 \), \( 7 \), \( 8 \), \( 9 \), \( a \), \( b \), \( c \), \( d \), \( e \), \( f \) in hexadecimal--. from the base powers -- \( 1=2^0 \), \( 2=2^1 \) etc. in binary; \( 1=10^0 \), \(10=10^1 \), etc., in decimal, and so on. You can use the same trick for hexadecimal base with the help of the page I gave you and the multiplication table, \[\begin{array}{cccccccccccccccccc} {\color{teal}\times} & {\color{red}0} & {\color{red}1} & {\color{red}2} & {\color{red}3} & {\color{red}4} & {\color{red}5} & {\color{red}6} & {\color{red}7} & {\color{red}8} & {\color{red}9} & {\color{red}a} & {\color{red}b} & {\color{red}c} & {\color{red}d} & {\color{red}e} & {\color{red}f} & {\color{red}1{\color{red}0}}\\ {\color{red}0} & {\color{purple}0} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ {\color{red}1} & 0 & {\color{purple}1} & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 & a & b & c & d & e & f & 10\\ {\color{red}2} & 0 & 2 & {\color{purple}4} & 6 & 8 & a & c & e & g & 12 & 14 & 16 & 18 & 1a & 1c & 1e & 20\\ {\color{red}3} & 0 & 3 & 6 & {\color{purple}9} & c & f & 12 & 15 & 18 & 1b & 1e & 21 & 24 & 27 & 2a & 2d & 30\\ {\color{red}4} & 0 & 4 & 8 & c & {\color{purple}g} & 14 & 18 & 1c & 20 & 24 & 28 & 2c & 30 & 34 & 38 & 3c & 40\\ {\color{red}5} & 0 & 5 & a & f & 14 & {\color{purple}1{\color{purple}9}} & 1e & 23 & 28 & 2d & 32 & 37 & 3c & 41 & 46 & 4b & 50\\ {\color{red}6} & 0 & 6 & c & 12 & 18 & 1e & {\color{purple}2{\color{purple}4}} & 2a & 30 & 36 & 3c & 42 & 48 & 4e & 54 & 5a & 60\\ {\color{red}7} & 0 & 7 & e & 15 & 1c & 23 & 2a & {\color{purple}3{\color{purple}1}} & 38 & 3f & 46 & 4d & 54 & 5b & 62 & 69 & 70\\ {\color{red}8} & 0 & 8 & g & 18 & 20 & 28 & 30 & 38 & {\color{purple}4{\color{purple}0}} & 48 & 50 & 58 & 60 & 68 & 70 & 76 & 80\\ {\color{red}9} & 0 & 9 & 12 & 1b & 24 & 2d & 36 & 3f & 48 & {\color{purple}5{\color{purple}1}} & 5a & 63 & 6c & 75 & 7e & 87 & 90\\ {\color{red}a} & 0 & a & 14 & 1e & 28 & 32 & 3c & 46 & 50 & 5a & {\color{purple}6{\color{purple}4}} & 6e & 78 & 82 & 8c & 96 & a0\\ {\color{red}b} & 0 & b & 16 & 21 & 2c & 37 & 42 & 4d & 58 & 63 & 6e & {\color{purple}7{\color{purple}9}} & 84 & 8f & 9a & a5 & b0\\ {\color{red}c} & 0 & c & 18 & 24 & 30 & 3c & 48 & 54 & 60 & 6c & 78 & 84 & {\color{purple}9{\color{purple}0}} & 9c & a8 & b4 & c0\\ {\color{red}d} & 0 & d & 1a & 27 & 34 & 41 & 4e & 5b & 68 & 75 & 82 & 8f & 9c & {\color{purple}a{\color{purple}9}} & b6 & c3 & d0\\ {\color{red}e} & 0 & e & 1c & 2a & 38 & 46 & 54 & 62 & 70 & 7e & 8c & 9a & a8 & b6 & {\color{purple}c{\color{purple}4}} & d2 & e0\\ {\color{red}f} & 0 & f & 1e & 2d & 3c & 4b & 5a & 69 & 76 & 87 & 96 & a5 & b4 & c3 & d2 & {\color{purple}e{\color{purple}1}} & f0\\ {\color{red}1{\color{red}0}} & 0 & 10 & 20 & 30 & 40 & 50 & 60 & 70 & 80 & 90 & a0 & b0 & c0 & d0 & e0 & f0 & {\color{purple}1{\color{purple}0{\color{purple}0}}} \end{array}\] In any base you must use as many digits as your base (always a positive integer different from one). In hexadecimal you can only use \(0,1,2,3,4,5,6,7,8,9,a,b,c,d,e,f\). Some properties of real numbers are counter-intuitive, and you seem to have lots of problems with them. For example: two different real numbers can never "touch each other" (be just next to each other with no other real number in between). This property you are grappling with is not one of those counter-intuitive properties. Swansont, MigL, and Studiot are doing a great job of explaining. I've tried to add auxiliary explanations that you're free to ignore if you find they don't help you. And, as @studiot said, be careful to distinguish pure numbers from physical quantities. Physical scalars are a different thing. They carry units. So they are subject to transformation laws. Only ratios of scalars are pure numbers. As a final exercise, let's write \( \frac{1}{5} \) in binary: \[\frac{1}{5}=0.2\] \[2\times0.2={\color{red}0}+{\color{red}0}.4\] \[2\times0.4={\color{red}0}+{\color{red}0}.8\] \[2\times0.8={\color{red}1}+0.6\] \[2\times1.6={\color{red}1}+{\color{red}1}+0.2\] etc. You keep going. The numbers in red are the binary digits of your fractional number. You get, \[\frac{1}{5}=0.001100110011...\:\textrm{(base two)}\] Which means, \[\frac{1}{5}={\color{red}0}\times\frac{1}{2}+{\color{red}0}\times\frac{1}{2^{2}}+{\color{red}1}\times\frac{1}{2^{3}}+{\color{red}1}\times\frac{1}{2^{4}}+{\color{red}0}\times\frac{1}{2^{5}}+{\color{red}0}\times\frac{1}{2^{6}}+\cdots\]

\[\begin{array}{cccccc} & & & {\color{red}0} & {\color{red}1} & {\color{red}1}\\ & & \times & {\color{red}1} & {\color{red}1} & {\color{red}1}\\ & & & 0 & 1 & 1\\ & & 0 & 1 & 1\\ & 0 & 1 & 1\\ & {\color{green}1} & {\color{green}0} & {\color{green}1} & {\color{green}0} & {\color{green}1} \end{array}\]