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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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| I just found this one yesterday, added it to the queue as usual, and behold, its number came up. This is either random chance, or the Universe trying to tell me something. (It's the former.) How to read philosophy  The first thing to remember is that the great philosophers were only human. Then you can start disagreeing with them "The great philosophers were only human" except for Nietzsche (of whom there is a sketch at the top of the article). No one with a mustache like that could possibly be only human. You might say he was superhuman. An Übermensch, if you will (please don't). It might seem daunting to read philosophy. No more difficult than crossing Antarctica in a sweatshirt. Giants of thinking with names like Hegel, Plato, Marx, Nietzsche and Kierkegaard loom over us with imperious glares, asking if we are sure we are worthy. Yeah, no, they all had to take a shit sometimes. Some of them were real jerks. Here’s Arthur Schopenhauer on his fellow German philosopher Georg Wilhelm Friedrich Hegel, for instance: ‘a flat-headed, insipid, nauseating, illiterate charlatan, who reached the pinnacle of audacity in scribbling together and dishing up the craziest mystifying nonsense.’ Wow, if only they were alive today, I might actually sign up for Twatter just to watch them fight there. That is, why read philosophy in the first place? The chief goal is, simply, the improvement of your own soul. I'm going to go ahead and assume that they're using "soul" metaphorically. No one should read philosophy just to sound smart, or intimidate others, or have impressive books on the shelf. Not that those aren't fun things to do. One should read philosophy because one wants a better mind, a better spirit, and a better life. (Or, at least, one wants a better understanding of why none of these things are possible, or why none of them matter; philosophy leaves no possibility unexplored.) And also so you can fully appreciate things like this. (That link will take you to the latest installment of Existential Comics. You can peruse earlier comics or do what I do and just hit "Random.")Reading philosophy gives us richer perspectives, casts us into deep wonder, and helps us grapple with the biggest questions a human can ask. It is a heady call to action – and one that can only be met by diving in to the works themselves. So: how does one read philosophy? Um... one word at a time? To this day, there are bookstores with sections titled ‘Philosophy’ that include titles like The Seven Secrets to a Happier Life, or Get Your Sh*t Together, or Living With Your Heart Wide Open. These are self-help books, and some of them may actually prove helpful in giving you a better perspective to overcome or live with the obstacles in your path. There's a word to describe œuvre like this: copium. Rather than helping with the Big Picture, it zooms in on details, trying to help people cope instead of encouraging thought. And the only person a self-help book really helps is the author, provided they sell enough copies. Everybody needs a helpful nudge from time to time, and probably every self-help book has helped someone somewhere. Every newspaper horoscope has probably helped someone somewhere. That doesn't make it worthwhile; the stopped clock analogy applies. Philosophy typically raises less personal questions, such as whether time is real, or whether humans can exempt themselves from laws of nature, or if a person is just what their brain does, or whether we have moral obligations to strangers. And sometimes it gazes so deeply into its own navel that the navel starts staring back. (Sorry again, Nietzsche.) But more generally, philosophical problems are the ones that unavoidably come along with being conscious. If you can think, you have problems: this is why there is philosophy. No, this is why there is C-4. I don't remember where it was; it was probably on a bumper sticker: "There are very few personal problems that cannot be solved with a suitable application of high explosives." But how does one get started? What books are good to begin with? Philosophy isn’t like mathematics, where people generally agree that you need to start at one place and take a sequence of steps that steadily build upon one another. Mathematics has its own branches, and every attempt to unify them thus far has, as far as I know, utterly failed. Including attempts by philosophers. Especially attempts by philosophers. (Lots of mathematicians were also philosophers, like Russell and Descartes.) ...but perhaps the most important point to bear in mind is that there’s no single or best way to begin. Start anywhere, and follow your interest wherever it goes. Yeah, I don't know (that phrase, incidentally, is what I consider the most important in all of science and philosophy). Personally, I'd start with logic. Without a good background in logic, including a study of the fallacies we're all subject to, it would be more difficult to read other philosophy critically, to identify any flaws in reason or logic as a philosopher lays out an argument. For example, I remember a graffito I saw on a bathroom stall in the philosophy department of the university I attended: God is love. Love is blind. Ray Charles is blind. ∴ Ray Charles is God. The logical fallacy of which I leave as an exercise for the reader (I hate when math books do this). Muddying the waters there (as philosophers love to do), there's Kant's Critique of Pure Reason. So there's that. (Like Existential Comics, I'm strenuously avoiding Kant/Can't puns.) Dogs must think that we slip into comas when we are reading books because we hardly move at all. Assumes facts not in evidence: do dogs think, and if so, do they have any concept of "coma?" It has been said that there are ultimately two replies to any philosophical claim: ‘Oh yeah?’ and ‘So what?!’ That is right. They forgot "Oh hell no" and "Snort." You should be thinking of counterexamples to the general claims that are made, or other possible explanations, or asking whether the philosopher is handling similar cases consistently. You mean like I do here? Thinking of counterexamples, I mean. I'm anything but consistent. "Do I contradict myself? Very well then I contradict myself. I am large; I contain multitudes." Whitman, by the way. Poet, not philosopher. Though that line is blurry as hell. In all likelihood, the philosopher you are reading is not an idiot... Snort. Let us not be book snobs: Plato himself thought that real philosophy takes place only in live conversation, and a written text is at best only an imitation of the real thing. (And yes, he wrote that down in a text.) As you can see, self-contradiction is baked into the very sourdough of philosophy. One nice thing about a discipline as old as philosophy is that textbooks don’t exactly go out of date, and sometimes some old introductory textbook from 1970 can provide a nice overview of a tangled set of questions. Unless you're taking philosophy classes at a modern college, where the 18th edition differs from the 17th edition by three punctuation marks in the Foreword, so you have to spend $250 on the 18th instead of buying the 17th used for a buck and a half. Anyway. I've banged on long enough. There's way more at the article, if you're interested. And if you're not, well, can't say I blame you. As usual, I can't end an entry about philosophy without embedding this video: |
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