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 |    Not for the faint of art. |
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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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| My random number generator likes to have a laugh at my expense sometimes; today, it came up with another Quanta piece. This one, however, doesn't involve scary numbers. ‘Metaphysical Experiments’ Probe Our Hidden Assumptions About Reality  Experiments that test physics and philosophy as “a single whole” may be our only route to surefire knowledge about the universe. It does, however, discuss science and philosophy, as the headline warns. Metaphysics is the branch of philosophy that deals in the deep scaffolding of the world: the nature of space, time, causation and existence, the foundations of reality itself. Once something can be experimentally verified, though, it ceases to be metaphysics and becomes... physics. Or fact. Plenty of things we (for varying definitions of "we") know (for varying definitions of "know") now (for varying definitions of "now") were once in the realm of metaphysics. It’s generally considered untestable, since metaphysical assumptions underlie all our efforts to conduct tests and interpret results. And if it's untestable, it's not science. At least, not yet; the untestable can become testable. I've said before that philosophy guides science, while science informs philosophy. I stand by that assertion. What the article (and it's a fairly long one) focuses on is what to do when we don't even know that we're using philosophy to guide science. Intuitions we have about the way the world works rarely conflict with our everyday experience. Well, yeah, because intuition is largely based on everyday experience. But at the uncharted edges of experience — at high speeds and tiny scales — those intuitions cease to serve us, making it impossible for us to do science without confronting our philosophical assumptions head-on. Suddenly we find ourselves in a place where science and philosophy can no longer be neatly distinguished. A place, according to the physicist Eric Cavalcanti, called “experimental metaphysics.” My first reaction is to say that the phrase is misleading. However, sufficient explanation can suffice to un-mislead it. If this article isn't sufficient explanation, I don't know what would be. A book, maybe? For once, I don't see a book ad hidden in the article. You should definitely go to the link just to see the pictures of Cavalcanti, though. I think he and Orlando Bloom were separated at birth. In experimental metaphysics, the tools of science can be used to test our philosophical worldviews, which in turn can be used to better understand science. Okay, I don't have a problem with that. THE DIVIDING LINE between science and philosophy has never been clear. Often, it’s drawn along testability. Yeah, that's where I, an absolute amateur, generally draw it, at least in my head. I will note, though, that the system we call "science" developed out of what used to be called "natural philosophy." However, astronomy developed out of astrology, and chemistry out of alchemy, so I'm not sure that's a win for natural philosophy. As it turns out, though, the testability distinction doesn’t hold. Philosophers have long known that it’s impossible to prove a hypothesis. Which is why scientists test for whether something can be falsified, not proven. In 1906, though, the French physicist Pierre Duhem showed that falsifying a single hypothesis is impossible. Every piece of science is bound up in a tangled mesh of assumptions, he argued. Dang ol' French, messing things up for everyone. Take, for instance, the geometry of space-time. Immanuel Kant, the 18th-century philosopher, declared that the properties of space and time are not empirical questions. He thought not only that the geometry of space was necessarily Euclidean, meaning that a triangle’s interior angles add up to 180 degrees, but that this fact had to be “the basis of any future metaphysics.” And this is why we don't take the words of philosophers as absolute truth. Especially those of Kant. (I will resist the obvious pun.) The unit of empirical significance is a combination of science and philosophy. The thinker who saw this most clearly was the 20th-century Swiss mathematician Ferdinand Gonseth. You will note that he wasn't French. He probably spoke the language, though. For Gonseth, science and metaphysics are always in conversation with one another, with metaphysics providing the foundations on which science operates, science providing evidence that forces metaphysics to revise those foundations, and the two together adapting and changing like a living, breathing organism. As he said in a symposium he attended in Einstein’s honor, “Science and philosophy form a single whole.” I don't think I'd ever heard of this dude before, but that sounds awfully familiar. Like it's very close to my own ideas, reached independently, as noted above. Which means I have to be careful, lest my own confirmation bias kick in. The article goes on, like I said, for a while, with examples of the intersection of philosophy and science. Then, what to me is a pretty important example: Michele Besso, Einstein’s best friend and sounding board, was the only person Einstein credited with helping him come up with the theory of relativity. But Besso helped less with the physics than with the philosophy. Einstein had always been a realist, believing in a reality behind the scenes, independent of our observations, but Besso introduced him to the philosophical writings of Ernst Mach, who argued that a theory should only refer to measurable quantities. Mach, by way of Besso, encouraged Einstein to give up his metaphysical notions of absolute space, time and motion. The result was the special theory of relativity. Also not French. But the real point here is: in order to come up with his famous, and now well-supported, theory of relativity, Einstein had to shed some basic assumptions that he, maybe, didn't even realize he had. Once he did that, he made history and changed both science and philosophy completely. I won't quote more from the article, but I read the whole thing. I'll just say this: Einstein was famous for, among other things, popularizing the concept of the thought experiment, and the article leans heavily on that particular technique for figuring things out. And thought experiments, when based on known science, basically philosophy. I rest my case. |
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