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Complex Numbers A complex number is expressed in the standard form a + bi, where a and b are real numbers and i is defined by i^2 = -1 (that is, i is the square root of -1). For example, 3 + 2i is a complex number. The bi term is often referred to as an imaginary number (though this may be misleading, as it is no more "imaginary" than the symbolic abstractions we know as the "real" numbers). Thus, every complex number has a real part, a, and an imaginary part, bi. Complex numbers are often represented on a graph known as the "complex plane," where the horizontal axis represents the infinity of real numbers, and the vertical axis represents the infinity of imaginary numbers. Thus, each complex number has a unique representation on the complex plane: some closer to real; others, more imaginary. If a = b, the number is equal parts real and imaginary. Very simple transformations applied to numbers in the complex plane can lead to fractal structures of enormous intricacy and astonishing beauty. ![Merit Badge in Quill Award
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| Okay, here we go again. From Big Think: What if we’re alone? The philosophical paradox of a lifeless cosmos  If humanity lives in an otherwise barren Universe, we’ll have to forge philosophy that fills the void. As there isn't any credible, unambiguous evidence of extraterrestrial life, the headline seems to be arguing "paradox" from the widespread belief of "there just HAS to be other life out there," which in turn comes from the essentially misanthropic "we're not special" argument. The idea that we might have cosmic neighbors has captivated the human imagination for decades. It’s not just sci-fi enthusiasts who ponder the possibility of extraterrestrial intelligence (ETI) — the general public seems to lean strongly toward the belief that we’re not alone. And we're off to a bad start in the lede, which makes no distinction between "life" and "extraterrestrial intelligence." Now, one could argue that intelligence is a quality of all life, from bacteria to blue whales, but that's not what ETI usually connotes. ETI refers, in the public imagination at least (and the article is written for the public, not for scientists), to technology-wielders like ourselves. This is one reason I don't like to use "intelligence" in this context. Another is that there are demonstrably other intelligent species right here on Earth, and none of them seem inclined to make rockets or broadcast radio. So, for the purpose of my discussion here, I'll use ETL as shorthand for extraterrestrial life, and ETT for extraterrestrial technology. Both are entirely hypothetical, but, as I've noted dozens of times, ETL is far more likely than ETT (but it's almost certainly a prerequisite for it), and there's nothing about evolution that requires the development of a species that likes to send robots to neighboring planets. As for "the general public seems to lean strongly toward the belief that we're not alone," remember, "the general public" is pretty damn gullible when it comes to reports of strange lights in the sky and whatnot. Columbia University Professor David Kipping often finds that, when discussing astronomy and the potential for life elsewhere in the Universe, people almost universally insist, “Surely we can’t be the only ones!” Ah, yes. Argument from incredulity. It's a big universe. I mean, it's really, stupendously, mindbogglingly, incredibly big. So I understand the insistence that there's ETT out there somewhere. I actually share that belief. But just as you humans find it difficult to grasp just how horrendously huge the universe is, you also have little grasp of tiny probabilities. If you did, none of you would ever play the lottery. Oops. I meant "we" and "us," fellow humans. As we'll see, though, the chances of ETL alone are probably much higher, and it wouldn't surprise me a bit if we found some. Occam’s razor nudges us toward the notion that “life out there” is the easiest explanation — it just feels right. Snort. The interpretation of Occam's Razor is: "Entities must not be multiplied beyond necessity." It may "feel right," but throwing ETT or even ETL out there as a given is the literal definition of multiplying entities beyond necessity. That said, Occam's Razor isn't proof of anything. It's meant to guide hypotheses and research, not proclaim that the simplest explanation is definitely the answer. If you hear a funny noise from your car's engine, for example, you start by looking at the simplest possible explanation, maybe a loose belt. But if all the belts are tight, then maybe it's because your engine is fried: a much more complicated fix. The principle of mediocrity chimes in, reminding us that our little corner of existence is probably not all that unique. I can't argue against that bit. And the Copernican principle gives a knowing nudge, sweeping away humanity’s old, self-centered fantasy of cosmic importance. The difference between that and the idea of ETL is that we have concrete proof that Earth isn't the center of the universe, whereas there's no proof yet of ETL. To believe we’re alone in this vast, wild expanse feels not only improbable but strangely outdated, like clinging to some universal map where Earth is still at the center. Now, I have to admit that the idea of humanity's uniqueness is a religious tenet to many people, and I have no desire to argue one side or the other from a religious perspective. I want there to be ETT just to poke the eye of dogmatic religion, but that doesn't mean it's there. With the discovery of exoplanets, we’ve learned that our galaxy overflows with diversity: Billions of planets orbit stars in the so-called habitable zone, where conditions might support liquid water. Which is a fine argument in favor of ETL. Once, Earth’s oceans seemed unique; now, hidden seas on moons like Europa and Enceladus suggest that watery worlds may not be so rare after all. But let's not forget that Europa and Enceladus both orbit gas giants outside the Sun's habitable zone. Still, that only increases the chances for ETL to gain a foothold, right? Well, maybe. Venus and Mars are both in the Sun's habitable zone, and we still haven't found unambiguous evidence for life, extant or extinct, on either. As Professor David Kipping aptly points out, the data paints a tantalizing picture — just as compatible with a Universe brimming with life as it is with one where we stand solitary under the stars. To insist there must be life out there, he reminds us, is to trade evidence for optimism. And I think we all know where I stand with optimism. Still, it's absolutely important for us to make the search. It's also important to make the distinction between ETL and ETT as I do, because the signs we're looking for are different for each (though of course if we find ETT, it almost certainly implies ETL; just not the other way around). Even under ideal conditions, life doesn’t simply spring forth; no experiment has succeeded in replicating it. No, but that may be because the process takes longer than your usual lab experiments. What experiments have been done suggest that it's possible. Whether it's inevitable or not, or simply very difficult, we don't know, due to small sample size. Earth’s unique circumstances — a stabilizing Moon, plate tectonics, and precisely the right chemical mix — might be one in a trillion. If it's only one in a trillion, then the universe harbors quite a bit of life. But it might also be one in a googolplex. Or it could be one in 2. We don't know. As I've said before, once you've won the lottery, the odds of having won the lottery are 1 in 1, and prior probabilities make no difference except to remind you how lucky you got. Evolution adds yet another filter: While microbial life could be common, the leap to intelligent beings may require an almost comical series of accidents and near-catastrophes. There's that word again. Intelligence. It's too slippery in this context, and too easy to make self-contradictory jokes about (if you can type "there's no intelligent life on Earth either haha" the simple act of typing that and sending it over a worldwide network of computers, using old and new technology developed by humans, proves your statement wrong). We share the world with several other beings that can be labeled intelligent: crows, elephants, cats, octopodes, etc. And we don't have the capability to communicate with them beyond a very basic level, and, as I noted above, none of them are trying to send radio signals. So what we're looking for, again, is technology. Okay, I've gone on long enough, though the article continues for a while. I'll just point out one other thing that the article touches on later: it's remarkably hard to prove a negative. If we find ETL, that's enough to show that ETL exists. If we don't find it, though, that doesn't mean none exists. Like I said, it's a big universe, and the maps have always shown fantastic creatures in unexplored places. And sometimes, we search those places, and find fantastic creatures—just not the ones we were expecting. |
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The same can be said for 
Optimism is, after all, by its nature delusional; ‘realism’ or outright pessimism might seem more justifiable given the troubles of the present and the uncertainties of the future.
