You’d never get Kessler syndrome at Starlink altitudes.

Starlink satellites orbit at around 550km, and get dragged by the little bit of atmosphere that is at that altitude. Each collision might make more debris, but the conservation of momentum means that any debris that gets kicked to a lower orbit will probably burn up on the atmosphere while any debris that gets kicked to a higher altitude will be smaller mass and therefore cause less damage on the next collision after that.

Collisions can still happen, but the runaway conditions where debris begets debris won’t happen at those orbital velocities and altitude.

At a certain point my pattern recognition skills reached their limit to where learning each new concept was still the same, but I had a lot more trouble organically seeing and identifying when a particular technique was useful for a particular type of problem. Which is something that happens to a lot of different people at different stages of their math education, just happens to different people at different points.

And maybe I could’ve stuck with it, or used it enough to where I eventually got it easily the way I had done with all math topics before that, but I ended up steering the rest of my engineering education into topics that weren’t as heavy on that type of math. More programming and logic and microcontrollers, less electromagnetics and radio signals.

For me, the leap to multivariate calculus gave me a lot of trouble.

Differential equations was doable but no longer fun for me, either.

It was a combination of multivariate calculus, linear algebra, and differential equations where I just wasn’t having fun with math anymore. Those subjects represented the end of pure math education for me, and later engineering classes requiring knowledge in those were also not a ton of fun. Went from a self-described math and science guy to just…not describing myself in that way anymore.

The anti vaxx movement was a conspiracy by big pharma to get people to stop taking the cheapest way to prevent disease, so that they can profit on the expensive ways to treat those easily preventable diseases.

Hello fellow American. This you should vote me. I leave power. Good. Thank you, thank you. If you vote me, I’m hot. What? Taxes, they’ll be lower, son. The Democratic vote for me is right thing to do Philadelphia, so do.

asking to have the banana

Yeah that’s just a quirk of the English language in that “ask” means both inquiring, trying to learn information from a response, and request, a communication to another that the “asker” wants something.

I’m pointing out the fallacious reasoning behind your view that with enough chances, it is inevitable that every possible outcome occurs at least once. That does not necessarily follow, simply because it is possible to generate events of infinitesimal probability, simply because n! grows much faster than x^n. That’s just plain math.

Turning to whether the rare earth hypothesis itself is correct or not, I don’t actually have a strong view on this. I just know that you can’t reason your way into disproving the rare earth hypothesis simply by saying “the earth is possible and therefore common, because everything that is possible is inevitably common.”

I don’t see how else this could be anything but probabilistic. Unless you’re saying every star is the same size as the sun and every star has an earth-like planet orbiting it in he habitable zone, the probability of those things is obviously less than 100%. We can already observe counterexamples that proves those aren’t 100%.

So if you want to argue that there’s no way the probability is less than 1 in 10^21, fine. Then we’re having the conversation about the actual probabilities. But my whole point, since my first comment in this thread, is that it is not enough to say “I think there are 10^21 planets so life is inevitable.” That’s not sufficient to support that conclusion.

Debate whether a large moon, plate tectonics, a magnetic field, an atmosphere, an ozone layer, a Jupiter-like neighbor, a G-type star, and what ratios of specific elements need to be present on a planet to qualify. I’ll leave the actual estimates of those probabilities to others. But each of these factors has a non-100% chance of happening on any given planet, and it becomes a question of whether the probabilities stack in a way that overcomes the sheer number of stars and planets there are. And that’s the thing I’m sure about, that you simply can’t ignore the factorial expansion of those factors because you think that there are enough planets in the universe to make that irrelevant.

So a planet isn’t just 50% likely to form with rocky bias withín the frost line, it is certain to do so.

No, you’re skipping a step. For any n number of chances, the likelihood of something with probability p happening at least once is 1 - (1 - p)^n . You may think that with high enough n that it doesn’t matter what p is, because the exponential increase from n overwhelms the math to where the whole term basically converges onto 1, but my point is that there are combinatorics where the exponential increase in n is still dwarfed by the effect of the factorial increase in 1/p.

The probability of a rocky planet to form within a habitable zone is about 20% for any given star, according to your earlier link. How many will have a moon like ours? How many other life-sustaining characteristics will it have? If your argument is that the probability is 100% for every star, well, that’s just wrong. If your argument is that it is inevitable in that the probability approaches 100% if you look at enough stars, then you’re ignoring the entire point I’ve been making here, that you would have to show that the probability p is large enough that one would expect the overall probability to be found in at least some of the n stars viewed.

The fact that something has happened nearly every time we see a chance of it happening very much does make it a high probability event, cf. Bayesian inference.

No, my deck of cards counterexample directly disproves this conjecture of yours. And you can’t talk about Bayes theorem while simultaneously saying that this isn’t a discussion about probability.

And you also can’t talk about natural laws without probability, either, as quantum mechanics itself is probability distributions.

So I’ll continue to point out that the vastness of space might mean that the n is in the order of 10^21, but I can simultaneously recognize that 10^21 is a mind bogglingly large number while still not being large enough.

The math I’m talking about still works with weighted probabilities or conditional probabilities. The underlying factorial math expands the number of possibilities way faster than the number of “tries” can increase the likelihood of at least one hit.

The point is: the fact that something has already happened is not proof that it is a high probability event. The deck of cards hypothetical is merely an example of that phenomenon. Applying different weights (e.g., ignoring the suits of cards) doesn’t change that basic mathematical phenomenon, both only re-weights the probabilities to be bigger. But lining up a bunch of probabilities in a row still multiplies them in a way that results in a infinitesimal probability.

If there are only billions of earth-like planets in our galaxy, and only trillions of galaxies, that’s still only 10^21 chances at life. Yes, that’s an unfathomably large number for the human brain to process, but it’s also nowhere near the numbers that can be generated through factorial expansion, so if the probability of life arising is something like 10^30 on any of those planets, the expected number of life bearing planets would be pretty much zero.

If we live on a habitable planet then it’s logical to make the assumption that habitable planets are common.

That’s what I take issue with. I don’t think that follows.

If I have a random deck of cards, I can’t assume that the deck order is common. Or, if I flip a coin 20 times I can’t assume that the specific heads/tails order that results is commonly encountered, either. Just because it actually happened doesn’t mean that the a priori probability of it happening was likely.

The Copernican Principle is assuming that all decks of cards or all flipped coins follow the same rules. I’m not disagreeing with that premise, but I’m showing that no matter how many decks or coins you use, the probability of any specific result may be infinitesimal even with as many decks as there are planets in the universe.

Showing me good reason to believe that earth sized planets have a 20% chance of showing up in habitable zones still doesn’t answer the other questions I have about plate tectonics, elemental composition, magnetic fields, large moons, etc. Stacking dozens of variables with conditional probabilities can still produce numbers so small that even every star in the universe representing a “try” might not lead to a high probability result.

I’m not disagreeing with you on any of the physics of solar system formation, just disagreeing with your interpretation it means that habitable planets are high probability.

When clouds of dust and gas settle into spherical planets, what makes them rocky? What makes them have magnetic fields, atmospheres, water? What makes it so that the planet in the habitable zone hits those conditions.

The tendency of certain things to develop isn’t a lockstep correlation of 1 between these factors.

We can believe that stars are common. And so are planets. But what combination of factors is required for life, and does that combination start leveraging the math of combinatorics in a way that even billions of planets in each of trillions of galaxies wouldn’t be enough to make it likely that there are other planets that can give rise to life as we know it.

My point isn’t actually about cosmological physics. It’s a point I’m making about the math about probabilities being counterintuitive, in a way that “the vastness of the universe” doesn’t actually mean that life is inevitable. It might still be, but it doesn’t necessarily follow.

We know little about solar system formation, but sufficient to say it’s not a card deck shuffle,

Well it’s different in several factors competing in different directions, and it’s not clear to me what the overall aggregate direction is.

The fundamental force of gravity is going to drive a lot of disparate starting points to collapse into similar results.

But in the end, we’re still talking about the probabilistic chances that certain lumpiness in the distribution of mass from supernovas or whatever forms the matter of solar systems, and how each solar system’s spinning disk coalesces into planets with their own elemental composition and orbits and rotations and moons and internal rotation and energy that might make for magnetic fields, plate tectonics, etc.

If the probabilities of those may still have some independence from one another, then even if there are lots of stars like ours and maybe even lots of planets that are earth sized, and lots of planets with the oxygen to make water or carbon to make organic chemistry or the iron to make a magnetic field, we might still recognize that the correlations between these not-fully-independent variables still require stacking probabilities on probabilities at a factorial rate.

While the number of opportunities for those conditions to hit might go up at an exponential rate, if the probabilities are small enough and there are enough necessary factors for life stacking on each other, it’s entirely possible that the exponential expansion of more solar systems than we could fathom is still too small to make for an appreciable probability of the conditions of life.

I don’t know what the probabilities actually are. But I can see how the math of the combinatorics can totally dwarf the math of the vastness of the universe, such that the overall probability remains infinitesimal.

But in the vastness of space, it was practically guaranteed to happen somewhere.

Do we know this for sure?

When we thoroughly shuffle a deck of 52 cards, we’re almost certainly creating a new deck order that has never been seen before and will likely never be seen again in a random shuffle.

The number 52! is 8 x 10^67, so large that we can make the equivalent of a billion (1 x 10^9 ) shuffles per second per person on earth (8 x 10^9 ), so that in any given millennium (3.15 x 10^10 seconds) we’ve covered a percentage so small it’s got 36 leading zeros after the decimal point for the percentage, or 38 leading zeroes for the ratio itself.

My impression is that factorial expansion for probabilities moves up much faster than the vastness of space itself, but I don’t know how to calculate the probabilities of each of these priors.

I like to call it lost wages.

Mammals evolve into ant eaters.

The real advice is to realize that every job has components that are not fun.

There are professional athletes who still love to play their sport, and intend to retire into coaching, but hate dealing with marketing and promos and media availability. Lots hate the travel. Some don’t like some of their teammates or coaches.

I know doctors who hate dealing with the paperwork, and programmers who hate dealing with documentation or testing, and lawyers who hate tracking their timesheets. But each of these are part of the job. The question is whether the entire bundled package deal is a pretty good job or not for yourself.

You’re getting the labels mixed up.

As a labeling requirement under U.S. law, anything labeled “American Cheese” must be pasteurized process cheese made from some combination of cheddar, colby, washed curd cheese, or granular cheese, which the law also defines pretty strictly. It must be made from these cheeses, heated and emulsified with an emulsifying salt (usually sodium citrate).

American cheese is allowed to have some optional ingredients and still be labeled American Cheese:

• Food safe acid (as long as pH stays above 5.3)
• Cream or milkfat, such that this added fat can account for up to 5% of the weight of the finished product.
• Water (but the total moisture content of the resulting product must still be within the other limits in the regulation)
• Salt
• Artificial coloring
• Spices or flavoring that do not simulate the flavors of cheeses
• Mold inhibitors from sorbate up to 0.2%, or from proprionate up to 0.3%
• Lechitin, if sold in slices

You can add milk, cream, buttermilk, whey, or certain other dairy products up to 49% of the finished product, but then you’d have to call it “Pasteurized American Process Cheese Food” instead of just American Cheese.

American cheese is made from almost entirely cheese ingredients. The individual slices being sold at the store, though, vary by brand on whether they’re even trying to be American Cheese (or whether they’re some kind of lesser “cheese food” or even lesser “cheese spread” or even lesser “cheese product”)

Regular Kraft singles aren’t American Cheese. Look at the label. They’re “cheese product.” Even the Deli Deluxe line has taken a hit in quality in recent years, even if they are labeled Cheese.

Go with other brands that actually put together a decent tasting American Cheese, and check the label to make sure it’s made with 100% cheese instead of 51% cheese (or less).

It’s even funny the way the parent comment described it: a girl town right next to the normal town.

India basically introduced toilets in a single generation.

According to this article, in 1993, 70.3% of the Indian population did not have access to toilets. By 2021, the number dropped to 17.8%. So literally more than half the population of India got access to toilets within 30 years.