Youtube comments of B. Xoit (@b43xoit).

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  102. Piss Bucket Rocket Scientist Before people had calculators, and they had to do arithmetic by either algorism or abacism, multiplying large numbers was more time consuming than was convenient. This mattered, because there were circumstances when people needed those products, at least approximately. Logs were invented to ease that situation. When you have the logs of two numbers, you can add those logs, then look up the antilog of that sum, and you have approximately (often close enough for practical purposes) the product of the original two numbers that you wanted to multiply. A slide rule is an analog calculator that is based on logs. It is marked in a log scale. When you work the slide rule, you are adding distances. When you read off the answer, the scale converts the distances to their antilogs, so when you add the logs, you are multiplying the numbers that you can read off. So that explains the initial purpose why logs mattered. One of the convenient properties of logs is that the logs of multiplicative inverses are additive inverses. So how does this work, mathematically? A logarithm is a solution of an equation b^x = q. In this equation, "^" means "raised to the power", b is some fixed number, called the base, q is the number you want the log of, and x is its log. Any nonzero value for b will suffice to give the logs the useful properties I described above. For example, say we want to multiply some number c by another number d. Suppose we can look up log c and log d in a table of logs. Once we add those logs, we are going to look up the antilog of the sum by reading the table the opposite way. The answer we obtain that way is b ^ (log c + log d). By the properties of powers, this is (b ^ (log c))(b ^ (log d)). Consider just the first factor in that expression, b ^ (log c). Going back to my equation at the start of this paragraph, we can substitute log c for x and substitute c for q, because I said that the log has to be a solution to that equation. So we have b ^ (log c) = c. Doing the parallel thing to the other factor, tells us that our result (b ^ (log c))(b ^ (log d)) is indeed equal to cd, which is what we wanted. It is conventional to use one of three values for b, the base. There are circumstances where people like to use 10, which makes the logs decimal logs. In a few circumstances, they use 2, which leads to binary logs. However, the best value to use in many circumstances is the mathematical constant known as e, appx. 2.71828, the base of the natural logs. The most conventional viewpoint says that log functions map dimensionless numbers to dimensionless numbers, and that the decimal log, the binary log, and the natural log are just three different functions (although as I mentioned, any of them could be used to multiply). When I mention "centineper", I am taking an alternative viewpoint that says we can treat log as one function that if applied to a pure number, produces a dimensioned result. If we take the binary log, the result is in bits. If we take the decimal log, the result is in decades. If we take the natural log, the result is in "nepers" or "nats", both words having the same meaning. One way to look at is that a neper is equal to one (like we can take a viewpoint that a radian is equal to one), and that a bit is some pure number or that many nepers, and a decade is some pure number or that many nepers. In this viewpoint, a centineper is 1/100 neper. So, to get the log of a number in centineper, you take its natural log, and multiply by 100. I find this unit convenient for things like interest rates, because for small interest rates, the log in centinepers is close to the conventional additive rate in %. So for example, 1% interest is very close to 1 centineper interest, and 5% interest is not outrageously far from 5 centineper interest. Consider 5% interest for example. If I borrow $100 from you, how much do I have to pay back? The answer is $105. What matters to me is the ratio of how much I have to pay back, to how much I borrowed. That ratio is 1.05. What is the natural log of that ratio? My calculator says 0.04879016416943205. 100 times that is about 4.9, not outrageously different from 5. Conversely, if the loan rate is exactly 5 centinepers, (I'd like to write 5 % neper for that), then in conventional terms it is a 5.13% interest rate approximately. Suppose I borrow $100 from you at what we conventionally denote as 5% per annum compound interest. Suppose it is two years before I can pay you back. How much do I owe you at the end of the two years? One way to look at is to suppose that after one year, because I can't afford to pay you back on the loan, but for some reason I must, I'm going to borrow $105 from you to pay back the loan. So I use the $105 to pay back the original loan, but now I'm in debt to you for $105 and am going to pay 5% interest on that. So at the end of the second year, I owe you $105 times 1.05, or $110.25. If we had naively forgotten to compound the interest, we might have reasoned that since the interest is 5% per year, and two years went by, we should be able to multiply, and get 10% as the interest rate for the two years, which would mean you get cheated out of 25 cents. This bugs me, because the usual meaning of the word per implies you can multiply. If for example you are an exterminator and you charge $5 per raccoon removed and I want to hire you to get rid of three of the critters that are nesting in my attic, I expect to pay you $15, because of the usual meaning of per as it appears in "five dollars per raccoon removed". But this meaning did not carry over into the convention to talk about compound interest with an expression such as "5% per year". If on the other hand, we had agreed to a loan rate of 5 centinepers per year, then the two-year rate would indeed be 10 centinepers as expected. Here's another example. Suppose I bake bread and I charge so much per pound of it. But because you are a very regular customer, I give you a 10 centineper discount. But suppose you are resellling the bread in your 'hood and your markup is 10 centineper. Then your customers are paying you my regular price for the bread.
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  980.  @isobar5857  A clear case of "events" is meetings. Say you and I meet in New York. That's one event. You get on a 'plane with your atomic clock and I stay put with mine. You fly around the world about four times. You land in New York and have another meeting there between the two of us. Your clock can register a different amount of time since our prior meeting than mine registers. I think this meets your definition of one of us traveling into the other's future. So, the "same events" are our two meetings, and your clock and mine register different time durations (lengths of time) between those same two events. As to your question about the terms "past" and "present". At a given time in your experience with your clock, that time is your present. The times of the events you can, in principle, remember accurately, or read about in your journals, are your past. You might be able to predict something about your future, but prediction is not as accurate as memory, in general. Another way to think about distinguishing past from future is that in a closed thermodynamic system, entropy increases toward the future. By "closed", I mean nothing is passing into or out of it that would take away entropy. So, this brings up a question I have been wondering about for the last 24h or so, and I don't know the answer. Is it possible to create a temporary local condition where entropy is decreasing over time? I'm envisioning a box with heat insulation around it, and in the middle is a little ball, and it's mounted on a stalk, and there are pipes running through the stalk that allow us using external equipment to refrigerate the shit out of the little ball. We plan a sequence where the box and the gas in it and the ball start out at room temperature, and we let that sit for a while, then we turn on the refrigerator and cool the ball for a while, and then end the experiment. So for a time, the flow of energy via radiation and heat conduction go from the walls of the box toward the ball, and from the gas toward the ball. This is time-reversed from a model where there is a hot ball and energy flows out from it. And this time-reversed picture is like the world you and I actually live in. We live in a space some distance from a hot ball, our Sun, and radiation flows out from that. Now in our world, it's possible to build and run a computer, and have it record events, and make decisions about what it will do in the future. But I'm pretty sure that in the cold-ball experiment I describe, it won't be possible to insert a computer in the box and have it record events from the future (what we as experimenters standing outside the box would regard as the "future") and make decisions about what it is going to do in the past. So that confuses me, because my understanding is that the past and future are distinguished from one another by a gradient of entropy over time (it's increases into the future), but it sort of seems as though the experimental design I describe creates a local and temporary pocket in spacetime where the entropy gradient is reversed. The radiation goes inward toward the little ball, which is the reverse of what we experience with the Sun, where the radiation radiates outward, so it sounds like reversed time, but I don't believe a computer or a person or other information-processing machine or being placed in the box is going to experience reversed time. So the question still confuses me.
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  1374. Another example of the uncertainty principle is applied in astronomy. Astronomers are interested sometimes in the "apparent diameter" of a star as seen from Earth. This is the angle it would subtend if you could make out the edges of it and measure the angle. If the actual diameter is somehow known or inferred, then from that and the apparent diameter, you can get the distance, or the distance is somehow known and you can measure the apparent diameter, you can get the actual diameter. But when you look at a small or distant star with a regular optical telescope, you can't measure the apparent diameter (it isn't really apparent), because it always just lights up one pixel on your detector. So, according to what I read (sorry, source amnesia), here is what astronomers do. Stars can be microwave sources and dish antennae can pick up this radiation. Since a photon of microwaves coming from the direction of a star is overwhelmingly likely to have originated from the star, by the time that photon gets to the radiotelescope on Earth, its momentum is determined to within the apparent diameter of the star. For a star with a small apparent diameter, the momentum of a photon from it to Earth is very precisely known. Since the momentum is well determined, the position isn't. This makes the photons big and sloppy. By varying the distance between two dish antennae (one of them is mounted on rails so it can be trundled along), they can do interferometry and see out to what distance the photons are interfering with themselves. From that, they use the uncertainty inequality to infer the apparent diameter of the star. For me, a fascinating question in connection with this measurement technique is how do they do the interferometry. I get the impression that there's a receiver at each antenna and the bringing together of information from them happens as classical information transmission, not quantum. Does that work for interferometry? How? Another interesting dimension of the fact that this technique works is that it seems like a time reversal in a way. I'm used to hearing about experiments where a photon hits a screen and is detected to have a location, for example the double-slit experiments. From the setup of those experiments, QM cannot predict that all the photons put through the experiment will land on the screen at a precise spot. So, that's uncertainty about the destination and destiny of the photon. But in the star-apparent-diameter-measuring exercise, there is uncertainty in the origin of the photon. We think it's somewhere on the face of the star, because everything is very hot there and photons are boiling off every which way, but for a given photon, its origin isn't precisely determined. It's as though the photon doesn't know the past from the future.
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  1514. Given that grammar arises at least once in the lightcone prior to and including ourselves, someone must have been first to have reached that ability. There is no logical reason to conclude that we are not the first. So I have established the possibility; let's proceed to the question of the probability. As at least a couple of other commenters allude to, there are arguments floating around to the effect that the arising of human-like capability as it happened in the only case we know about, i. e. our own case, depended on a long string of improbable and rare events. Evidently most of the planetary systems found are characterized by the planets within them having close to the same size within a given system, but ours is exceptional and has multiple sizes. Saturn and Jupiter seem to have changed places. Both of these gas-giants, located in high orbits of Sun, which I gather is exceptional in that most systems have gas giants near their star, clear a lot of inflying crap that would otherwise likely bombard our planet and kill most of us off so often that we wouldn't get enough breathing time to build up grammar. Then there's the chemical difficulty of going from a few amino acids here and there to a complex self-reproducing molecule or system of molecules in association with each other (something like RNA and rhibosomes). Who can be sure how rare or common that is? Then there is the transition from prokaryotic life to eukaryotic. The concentration of energy characterizing only the latter gives us enough power to think and communicate and model. Then there is the protection against cosmic rays provided by Earth's magnetic field, which in turn comes from convection within the planet. So, it seems to me that a big pile of reasons exist to doubt whether we shall ever see any sign of intelligence or life beyond the one case we know about, which the anthropic principle requires that to exist, so looking at it as a telling example amounts to the biggest cognitive bias of all time and of all theoretical possibility with respect to what degree of cognitive bias could ever occur.
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  1644. $ dc 4 10000^p 398027684033796659235430720619120245370477278049242593871342686565238\ 635974930057042676009749975595510836461137504912702831400376935319143\ 621753470415827025981215282426893498224826615977707595539466961019588\ 699726772279731941315198182787264034852821200164566127930390710398182\ 979935327718016873784821349516406114982916691867361875370024545872140\ 793827277482562824192439237801588697814168520338650090909697535966525\ 032757049430286459482977357373598020450589927318365663076719136934132\ 593126761906696003770385305284570331119691001526584347722012386381881\ 779425549210851696458253943578557699072154639655630793883941961378971\ 846841113804188730258903839103669626086974468150655710480841592465655\ 211805257863007811676888839555017536731758113448656752514158601444051\ 645154665514388431619042396106716755762338728183461369854648923972904\ 427556158821823778729193111453445844216979095435045778144571378954652\ 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320493601682133709318222647489080531644015321391157387178232154126828\ 007760313716872242209614200967522180475716199973689467714010404673961\ 454146466045855232217196687665143147612199151921277432309700460321430\ 381533385245877431330533479476152339364503436322919665631042328740463\ 612565842560411947020174006507893396276103834436233140915025391014386\ 119201176462659556388343058600326710618903683746516577021214276933289\ 179021059956925949717956040857979165914170970056212869933593589268626\ 151996676594370800885093048230687152803213254735594741799076039453057\ 272319884322341883241036382617598401889439130301876975498681736174215\ 711287053447013711596004574803562701388246822510391522419061320663740\ 921321754344166744899588160649291823535983386025904942040724581017615\ 968429577015808090360968544059204594200069304612417366398776831532265\ 596224715750301792207725607932534543693758772262010387360435567635232\ 718343420679693057360004073679493008945813961012439574397373178636054\ 628207647520675194420244271036343729318858430871461978866964772362057\ 290577326080664463129657590249859748544101333842092713653096656066266\ 827446079145590196644643417403723220085696202719321533233027169599734\ 928971588850348415000070034027025298183104148343980297663148971586607\ 903771717880683175436445585810610546882073571556162324659351310326560\ 804448974229349743425637164834242799991427145050899469511954834774847\ 172360693568437689147399455672090773686782511054291185172381917008889\ 957645311339950993044779783607140593766508017935992581357858306525303\ 783231752425242008347844867988333025417249944092118578113687403158162\ 707075154006053416374075765162668533127078605316562826337193606242535\ 290683224423660462222408680300498714149607265550441220738075941633988\ 435051594487256802874182264814425923111193188280632013127802897889605\ 338783089532740877202304122498193625454768343775535498872821099981620\ 497070810489137457106892573248498734243717184800822956334469415666818\ 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  2140. I want to think that if you see a star, one particle is relating an event on the surface of the star to another event in your retina. But maybe my intuition about this is not well supported by the evidence. I'm not sure whether this is relevant, but there is a radio interferometric technique for measuring the apparent or angular diameter of distant stars. The "apparent" diameter is not apparent in an ordinary telescope-and-camera arrangement, because the star is going to light up just one pixel in the detector regardless. Astronomers are interested in apparent diameter because if they think they know the real diameter of the star, they can infer the distance, or vice versa. The technique uses two radio dishes and one of them is on railroad tracks and so its distance to the fixed dish can be adjusted. The signals from radio receivers attached to the dishes are somehow combined to detect the presence or absence of interference. When there is interference, this indicates that the photons are big enough to cover both antennae and interfere with themselves. Even though the photon lands at the detection site including both dishes, the origin of the photon is spatially sufficiently non-determined that it could be anywhere on the disk of the star as seen from Earth. The narrower the apparent diameter of the star, the more determined the momentum of the photon is, and so by the uncertainty principle, the less determined is its position. So, the fact that this technique is practical seems to me to counter the argument you mention from Huygens Optics. Something about a photon where the light wave is interacting with detectors on Earth is saying something about the origin of the "same" photon from the hot surface of the distant star.
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