joigus

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, \[\frac{1}{5}={\color{red}0}\times\frac{1}{2}+{\color{red}0}\times\frac{1}{2^{2}}+{\color{red}1}\times\frac{1}{2^{3}}+{\color{red}1}\times\frac{1}{2^{4}}+{\color{red}0}\times\frac{1}{2^{5}}+{\color{red}0}\times\frac{1}{2^{6}}+\cdots\]

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

Test on hexadecimal multiplication table with matrix & colours. \[\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}10}\\ {\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}19} & 1e & 23 & 28 & 2d & 32 & 37 & 3c & 41 & 46 & 4b & 50\\ {\color{red}6} & 0 & 6 & c & 12 & 18 & 1e & {\color{purple}24} & 2a & 30 & 36 & 3c & 42 & 48 & 4e & 54 & 5a & 60\\ {\color{red}7} & 0 & 7 & e & 15 & 1c & 23 & 2a & {\color{purple}31} & 38 & 3f & 46 & 4d & 54 & 5b & 62 & 69 & 70\\ {\color{red}8} & 0 & 8 & g & 18 & 20 & 28 & 30 & 38 & {\color{purple}40} & 48 & 50 & 58 & 60 & 68 & 70 & 76 & 80\\ {\color{red}9} & 0 & 9 & 12 & 1b & 24 & 2d & 36 & 3f & 48 & {\color{purple}51} & 5a & 63 & 6c & 75 & 7e & 87 & 90\\ {\color{red}a} & 0 & a & 14 & 1e & 28 & 32 & 3c & 46 & 50 & 5a & {\color{purple}64} & 6e & 78 & 82 & 8c & 96 & a0\\ {\color{red}b} & 0 & b & 16 & 21 & 2c & 37 & 42 & 4d & 58 & 63 & 6e & {\color{purple}79} & 84 & 8f & 9a & a5 & b0\\ {\color{red}c} & 0 & c & 18 & 24 & 30 & 3c & 48 & 54 & 60 & 6c & 78 & 84 & {\color{purple}90} & 9c & a8 & b4 & c0\\ {\color{red}d} & 0 & d & 1a & 27 & 34 & 41 & 4e & 5b & 68 & 75 & 82 & 8f & 9c & {\color{purple}a9} & b6 & c3 & d0\\ {\color{red}e} & 0 & e & 1c & 2a & 38 & 46 & 54 & 62 & 70 & 7e & 8c & 9a & a8 & b6 & {\color{purple}c4} & d2 & e0\\ {\color{red}f} & 0 & f & 1e & 2d & 3c & 4b & 5a & 69 & 76 & 87 & 96 & a5 & b4 & c3 & d2 & {\color{purple}e1} & f0\\ {\color{red}10} & 0 & 10 & 20 & 30 & 40 & 50 & 60 & 70 & 80 & 90 & a0 & b0 & c0 & d0 & e0 & f0 & {\color{purple}100} \end{array}\] \[\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}\]

\[7={\color{red}1}\times2^{2}+{\color{red}1}\times2^{1}+{\color{red}1}\times2^{0}={\color{red}111}\] \[7={\color{red}1}\times2^{2}+{\color{red}1}\times2^{1}+{\color{red}1}\times2^{0}={\color{red}1}{\color{red}1}{\color{red}1}\]

Because one country can be worked into one unit of awareness and political action. If that country happens to have 1.5 billion people --or thereabouts-- in it and a concerted action can be taken so that they all --or most-- do their part of the deal, the situation will improve considerably. It's not like the ice of Greenland is gonna say: "Wait a minute, don't melt just yet; that CO2 is Chinese!" China is not only very highly populated. It's very densely populated as well, when compared to, eg., Russia and Canada.

I've read with utmost interest all your comments, and I have to say that, to me, trying to judge the matter of suicide under the scope of ethics is hopeless, pretty much like looking at a compass needle on one of Earth's magnetic poles. The needle goes round and round in circles. Just declaring some action right or wrong in the abstract doesn't do it for me, while the regular dilemma punishment vs rehabilitation doesn't even begin to get a handle on the problem. Ethics, I would say, is about mending your ways or making you responsible for something you did. These concepts don't seem to apply here at all. I tend to think all concepts have a limit of applicability. To me suicide is not within the realm of ethics. It may be for those around, but not for the person who commits suicide.

I haven't said I agree. I don't. When I posed this question to my teacher back then it was precisely because he was making an argument that suicide is a sin, and it's wrong --he was a Catholic priest. That's why I jumped "then it should be punishable by law --both human and divine-- for those people who aren't successful, wouldn't it?" He said, "that's an interesting question, but we don't have time for it." That's why I would try to bring it to the context of action and reaction. Right or wrong are pretty much just tags to me.

I think this is a very interesting question. This is far from my area of expertise, but what I can say from my own thinking is that it'd better be phrased in a way that has a practical content, like, e.g., 1) Should a failed attempt of suicide be punishable by law? 2) Should a successful suicide be subject to investigation, with similar intent than in other police investigations, with the purpose of bringing to account those responsible for the situation that put the person on the brink? Some people may have suicidal tendencies without much help from others; other people may be just pushed to them by abuse or extreme injustice. It is possible to conceive situations in which the differences can be discerned, and action be taken. When I was 16 I posed question number 1) to a philosophy teacher who unfortunately dismissed it on the grounds that we didn't have time to deal with it. Ever since this question popped up in my mind I've thought about it, but I haven't quite made up my mind about it.

Thank you. I'll add some more from Wikipedia, for completion. https://en.wikipedia.org/wiki/Meteora

The tepuis (tepuyes in Spanish) from Venezuela, Brazil, Guyana and Colombia. Karst topography is awesome almost beyond words or concepts. But not beyond belief, because it's there. https://en.wikipedia.org/wiki/Tepui Because the first continents had no plant cover to protect them from erosion, for eons upon eons sediments formed over vast regions, which later became exposed to more selective wearing down, sculpting canyons, plateaus, grottos, and seemingly bottomless chasms. That's what a blind, unconstrained by intention, relentless force can do. No thinking is necessary, if given enough time. Gigantic pillars carved out of the depositions of a long-lost world, where once big dinosaurs roamed, and tiny mammals scurried around, waiting for their moment to arrive, these monuments are silent, patient witnesses to the existence of Gondwana. No human-made temple is remotely comparable to this. No religious feeling can echo in our minds what the first people coming from the Bering Strait must have felt when they first saw this more than 15'000 years ago. Picture from: https://hananpacha1.wordpress.com/2017/07/07/tepuy/ (In Spanish.)

This is a very good question. IMO, the only reason to withhold something that you think to be true, after reflection and examination of evidence, is not because it may cause anger or demoralization, but because of the danger of this piece of knowledge being revealed. I remember this point to have come up before in my life, and I've compared the anger or demoralization that you mention to the cauterization or sterilization of an injury: Pain or annoyance are different from harm. I'm a firm believer that people are better off if they are able to rule out assumptions that are not worth considering. seriously. As to topics about which I haven't made up my mind yet, I prefer to stay quiet, as you say, and let others talk until I find my position, if at all. What's your position?

I'm not 100 % sure of what's bothering you, but it's useful to distinguish in these integrals the field point \( x \) and the source point \( x' \). Now, the sources \( \boldsymbol{J} \) do not extend to spatial infinity. Nor do the fields \( \boldsymbol{A} \) (at the field points: the points at which the field is calculated, \( x \) ). So, \[\int\frac{\partial}{\partial x_{i}}\left[f\left(x\right)g\left(x-x'\right)\right]=\left.fg\right|_{\textrm{boundary}}\rightarrow0\] The boundary is spatial infinity. I hope that helps.

@MigL: I meant 'that's quite common among men my age.' Never a bad choice of preposition was as open to misinterpretation as this once.