joigus

And what about uniqueness?

Yes, you're on the right track, as this is a well-known property of combinatoric numbers. What you call "boundary conditions" are actually the pillars of what looks like calling for an induction proof. Induction on n and on k separately perhaps? A differential equation or similar would only apply for very big n, very big k, and very big n-k, so I don't think it would work. Approaching factorials by continuous variables is done when deriving the Stirling approximation, for example.

There are some funny --tho not hilarious-- moments in Seinfeld about that.

The thing I don't quite get is, why would you wanna do that, if it's more cumbersome and much, much less transparent? Dividing by factorials to correct for redundancy is the safest quickest way to master combinatorics. Except those in which repetition is allowed, and are driven by powers.

Sorry. You meant without division. I can't think of a way to do that that's as transparent as dividing by the factorials of whatever is irrelevant.

You've got 8 people. Count the number of ways you can make them sit in 8 chairs. It's 8! Now, every time you sit them, secretly, 8-3=5 of the chairs you declare irrelevant (not belonging to the committee), so you divide by 5! to discount those irrelevant reshufflings. They're not in the committee anyway. Now you're down to the number of ways in which you can select groups of 3 people from a group of 8 people and make them sit on 3 different chairs. The committee chairs. You've still overcounted, because it doesn't matter where they sit. So you divide by 3! and you're down to the combinatory number you need. It doesn't matter where they sit, and it doesn't matter where excluded people sit. It only matters who was selected for the committee.

Thanks. I'm not sure that's the way I'd phrase it though. I don't think proposals to reverse-pansperm --pardon my French-- other planets is really asking a well-posed scientific question, as it is an attempt to expand, command, and conquer --and also make profit. Some kind of interstellar Lewis-and-Clark effort. Or the Pizarro brothers. First, I think, we must really understand more deeply what life is. IMO, recent scientific developments have evidenced that we do not completely understand abiogenesis, the limits of life, minimum requisites for it, that we can merrily go round the galaxy planting our half-understood seeds out there. I resonate a lot more with what @Ken Fabian is saying in that I would love to see a lot more effort (and money) spent in sending probes under the seas of Enceladus, or the methane lakes of Titan, to mention just a couple of tantalising possibilities. There's also much to be learned on Earth that we don't understand. I have to say I'm biased on the whole thing. I always want more money spent on basic science, on really understanding what this is all about.

I'm aware of it. What Gerard 't Hooft is doing is trying to gradually build up towards models of deterministic variables that produce quantum mechanical dynamics in a simple way. More in particular the "logic of superpositions," but in such a way that there's a more satisfactory "ontological basis." But, he --and others-- declare that it's nowhere near quantum mechanics as to predictive and explanatory power. It's no substitute for quantum mechanics, let me tell you that. Here's a sample of some of his disclaimers, qualifications, and answers to criticism: https://physics.stackexchange.com/questions/600367/measurement-in-t-hooft-cellular-automation-interpretation-cai/601452#601452 https://physics.stackexchange.com/questions/32502/can-quantum-mechanics-really-be-the-same-as-underlying-deterministic-theory/34160#34160 https://physics.stackexchange.com/questions/30065/why-do-people-rule-out-local-hidden-variables/34073#34073 But this is not one of the topics I've studied more in detail, to tell you the truth. I don't think there's a significant number of experts, saying, "yeah, that's the ticket! How didn't we think of that before?"

Well, yes. If you think about it, it's arguably the case that even if the scenario that I painted was a possibility, we could afford it. After all there are something like the order of 1012 planets in our galaxy. Mind you, I said I don't have a strong opinion about it one way or the other. That's why. Some experiments may be worth the risk.

OK. I think it's by no means easy to tell what would be a good decision from what would be a really botched operation coming from too bold a move. Suppose there are microorganisms in that planet that we don't know about. Suppose these microorganisms live 5 km underground --as some chemolithoautotrophs we've discovered only as of the 1980's--. Suppose the microorganisms aren't based on the same catalogue of aminoacids that we, and our intestinal bacteria, and our parasites, are. Suppose they don't --even remotely-- have the same sequences of DNA in charge of the most basic functions --cellular respiration, fermentation, production of nucleic acid and proofreading of replication and translation of nucleic acid sequences, to name just a few that we share with many of the most primitive organisms on Earth. Suppose now that setting loose "our" bacteria or archaea from our "Earthly lineage" has the unfortunate result that the planet becomes neither ripe for our guys, nor for theirs --not anymore for them anyway. And as a result, a perfectly viable planet for life --only not our own kind-- ends up being forcibly ruined for their home residents, while we have failed in our attempt to set a foothold. Wouldn't that be a tragedy? We just don't know. I don't know. I'm just trying to plant the seeds of --hopefully-- constructively reasonable doubt. The picture of people thinking you can tell a planet's life content just from orbiting around it, honestly, gives me the chills. Don't get me wrong. I don't have a strong opinion about this one way or the other. Maybe it's worth the risk. I just tend to distrust iniciatives in which I see too much enthusiasm, while little focus on unforseen possibilities.

I see this as kind of the main problem (both ethical and practical.) How can we guarantee that world is really sterile? It's only very recently that we've discovered chemolithoautotrophs that live very deep underground (up to 5 km deep) and have life cycles nothing like anything we knew before. These things may be waiting for geological-scale times until their next breeding season. My point is: If we know so little about our own microbiota, what makes us think we're ready for colonising other planets with our archaea? Interesting TED Talks about chemolithoautotrophs: https://youtu.be/PbgB2TaYhio https://youtu.be/A2DzsgJSwcc The frontier between geology and biology is growing thinner and thinner.

Hey, you re-posted my post! Nah, never mind. You can re-post Modugno any time.

Camels, I suppose, must have been another form of currency. Pigs and eels were off the table, probably. Payment by IOU, the root of all evil. I think that's a good candidate for the original sin. And, just to keep on-topic by the skin of my teeth, yes, God is a Republican, and a fervent Catholic/Protestant (and so on), and a (fill in with your team's name) supporter, and is on our side in war no matter who we are, and writes straight with crooked lines, and... I almost forgot: And works in mysterious ways.

Another lesson to be had here is: Never pay your debts in goats.

LOL. What a bunch of old geezers we are. My dad loved these Italian songs, especially Domenico Modugno. For all old geezers out there, Domenico Modugno: Volare: https://youtu.be/t4IjJav7xbg Domenico Modugno: Ciao, ciao bambina: https://youtu.be/ygiHfNMwpdI You have no idea what a high price I've had to pay to scavenge for these oldies. I've been prompted that damned add selling funeral plans for British ex-pats two times in a row. They've fixed me with their cookies.

The latter actually represents two cats; one dead, the other alive.

Yes. Mmmm. Maybe you would have to do that. Which makes @Markus Hanke's accurate comment "in practice," if anything, even more compelling. You can always subdivide the cat into the cat's head, and the cat's rest of the body. With the head registering some crucial condition about "rest of the body," and conversely, \[ \left|\textrm{cat}\right\rangle =\frac{1}{\sqrt{2}}\left|\textrm{cat's head wrong}\right\rangle \left|\textrm{cat's body wrong}\right\rangle +\frac{1}{\sqrt{2}}\left|\textrm{cat's head OK}\right\rangle \left|\textrm{cat's body OK}\right\rangle \] With any interference terms of the kind \( \left|\textrm{cat's head wrong}\right\rangle \left|\textrm{cat's body OK}\right\rangle \) and \( \left|\textrm{cat's head OK}\right\rangle \left|\textrm{cat's body wrong}\right\rangle \) being erased due to decoherence. That goes to reflect in the quantum formalism that a cat's head cannot be alive of its own. Ultimately these states are macroscopic (made up of a huge amount of microscopic states consistent with statements like "the head knows the body is dead" and "the body knows the head is dead." Understanding "knows" as "has registered." IOW, microstates of the cat don't change the argument. Any "living states" of the cat's head immediately decohere with "dead states" of the rest of the body. And so on: (left part of the head vs right part...)

I agree. Unless you expand "context" to contemplate what your predecessors meant when they said that. Let's call it "tradition", if you want. Some idioms are still in use long after most speakers have forgotten the historical context. By the way, Wolfram Alpha didn't recognise your naval idiom: https://www.wolframalpha.com/input/?i=cold+enough+to+freeze+the+balls+off+a+brass+monkey It didn't recognise "cold enough to freeze your winnebago" either: https://www.wolframalpha.com/input/?i=cold+enough+to+freeze+your+winnebago Although it does recognise "winnebago": https://www.wolframalpha.com/input/?i=winnebago But I was able to check your naval idiom on the web: https://www.theguardian.com/notesandqueries/query/0,5753,-16960,00.html#:~:text=NOOKS AND CRANNIES-,The expression%3A "It is cold enough to freeze the balls,iron balls to fall out.

Interesting reflections, Studiot. Thank you. It seems that we need several levels of context to ascertain meaning to the point we think we understand what the other means. Another one is historical (idioms.) Aha! Nice examples. I suppose your first problem would be solved with a comma: "lane ends, merge left" versus, "lane ends merge, left" You can suggest that with your voice, but you can't with written language, unless you use punctuation! Interesting...

The only thing that looks "cohesive" to me on this thread are the cohesive attempts by members of the forums to have you explain --with a default-minimum maths, if possible-- how you can see a force in Schrödinger's equation; and what's more, how you can see any interactions that show up as "cohesive." The DeBroglie-Bohm model does not display cohesiveness either, BTW. The wave acts on the particle by means of the quantum potential in an equation that's formally a Hamilton-Jacobi formulation of dynamics; while the particle does not act on the wave. The wave goes its own way. That's the main reason why Einstein didn't like the model, BTW.

This would be the first level at which meaning could be hard to find. This is the aspect that interests me most. Another extreme example is --taking up @Genady on their suggestion of pragmatics: An intimate couple talking to each other can have a conversation like, --Really? --Nah There aren't many identifyable pieces of information there. Only they know what they're talking about. I suppose we interpret messages in some kind of optimisation strategy. There must be something like a critical time until you find the best match. An interesting example, perhaps. Consider the same couple who are very intimate. One of them, suddenly utters: "Every moon of every planet goes round and round" It's OK as to meaning, I suppose, but the other one would probably say, "what do you mean?" So meaning is --perhaps-- not exactly, or necessarily, about parsing a sentence and going, OK, I see no syntax mistakes, semantic mistakes, and so on.

I see your point. I didn't mean being serious when using it. I know real life speech is very much the way you depicted it in your @Genady depicted it in their example, which is a perfect example of pragmatics at work. And I wouldn't have it any other way. Otherwise daily conversation would be unsufferable. When I said "being serious" I meant it philosophically/scientifically. Edited: Sorry. It was @Genady who mentioned pragmatics, and gave the example.

Could you, please, elaborate? I'm prepared to accept that until we commit what we think to either paper, screen, or air --speech--, we're still not in the realm of meaning. It's only when there are at least two thinking agents that the question of meaning really arises. Is that anything like what you mean? There I go again. Thanks everybody for your contributions. I'll be reflecting on them ASAP. Interesting points here. Thank you. At first I translated, of course; but soon I realised that I'd better make it dynamic, emotionally involved, and intellectually involved, or else I would never acquire the language. Language needs to be a tool. Otherwise it's like learning lists, and logical trees, with no connection to any level of experience. And, as I've pointed out before in these forums, the brain is a very costly organ in energetic terms. Your brain is not going to commit. I do have a tendency to thinks maths in Spanish, but at some point I started forcing myself to do it in English. Now I can do it, although not as dexterously as in my mother tongue, of course. Maths in Italian must sound really charming.