It’s true, we’re more likely to appear in a position other than one (not “any other position” specifically though), but your assumption that we are not the only ones is doing a lot of heavy lifting here.
The existence of life in the universe is clearly a non-0 probability event. Same for “intelligent” life. Although arguably almost all life on earth is intelligent so I’m assuming you mean human-level intelligence. Well, in which case we probably have a few contenders for at least getting fairly close if not matching given we have trouble defining intelligence (elephants, octopus, dolphin, some primates, crows etc). Let’s say life that can build technology and shape the world around them then.
If the universe is mathematically infinite, then it’s guaranteed we’re not the only ones because any low probability event will still happen infinitely many times in that case. So let’s say the universe is very very large but not mathematically infinite.
So then the question is a) what percentage of planets develop life b) what percentage of life develops a species capable of building technology c) what is the likelihood that species has access to resources on the planet to build that technology. We don’t know what a) is (just non-0) but using your equivalent argument for that of “we haven’t found any” I can counter by saying well, as far as we know b and c are 100% as the only planet we know to have life developed a intelligent technology-building species and the likelihood that that just happened by luck seems astronomically low (ie you’d need life to be extremely common and intelligent life not so but the odds seem against you if life is rare AND intelligent life is rarer still). I would argue that when life develops intelligent life is almost assured because pf evolutionary pressures. Now whether the right circumstances exist for them to evolve from basic tools to space ships, who knows. But that’s secondary.
So does life exist? Given the size of the universe it would be hard to bet against the argument that life has evolved somewhere out there in the universe. And given that it has, what are the odds that intelligent life came about? Again I’d say high because the same evolutionary pressures would exist to fill in niches that intelligence wins at.
So I don’t think it’s all that heavy a lift. Note that I’m not claiming they are space faring, have developed advanced mathematics, can communicate in any way that we’d understand each other or anything like that. I’m simply noting there’s multiple species even here on Earth who at a minimum come close to humans, which means the probability seems quite good unless our Earth is a magical place in the universe that is particularly special somehow at generating sentience. Thus the odds of there being intelligent sentient life out there seems to be the same as whether or not there’s any life out there. And since we know that’s non-0 and the universe is very very big, then it stands to reason there’s very likely intelligent life out there. Note even the observable universe itself is big enough that there’s likely other intelligent life within it. The total universe is much much larger than that and we’d never ever see evidence of life from there because we’ll never see anything from it because it’s forever outside of the expansion of the universe.
> If the universe is mathematically infinite, then it’s guaranteed we’re not the only ones because any low probability event will still happen infinitely many times in that case.
This isn't really true in any meaningful sense.
What is the probability of me tossing a coin immediately after writing this comment, and it landing heads? As I write now, it is 1/2. However, it's clear that only one outcome is possible. EDIT: it was tails. It didn't come up heads, however infinite the universe may be, it just didn't.
(Note: there is a non-zero probability that a similar being sits in front of his computer in a similar shirt, having a similar online discussion, who tosses a similar circular disk which comes up a different way, but that being is not me).
> the likelihood that that just happened by luck seems astronomically low
It seems that way, but as you are arguing elsewhere, unlikely things can and do happen in the universe.
> Note that I’m not claiming they are space faring
That's part of OPs premise, but I don't mind leaving it out of the discussion.
I actually like your formulation:
a. what percentage of planets develop life
b. what percentage of life develops a species capable of building technology
c. what is the likelihood that species has access to resources on the planet to build that technology
I'd argue with the 100% for b and c, because we have at least a reasonable belief based on observation that there is no other intelligent life in our solar system, and we have evidence that planets exist without those resources to build technology, but it's not the crux of my argument.
The crux of my argument is really a). We don't know what percentage of planets develop life. But as yet, we are unique.
On a personal level, by the way, I do actually suspect that we are not alone in the universe, but that's a gut feeling, and I don't believe there's maths to support it.
Side note while I think about it, actually we have more evidence than we think. Even on a planet clearly adapted to support life in massive varieties, it seems that all life on earth evolved originally from a single organism. We have millions of years of geological and paleontological evidence, and thousands of years of recorded history, and we have no evidence that this happened more than once.
It does certainly seem from our limited evidence that abiogenesis (or going from not alive to alive) is at best extremely uncommon, or at worst, an isolated anomaly.
