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Complex Numbers
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.




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June 8, 2020 at 12:24am
June 8, 2020 at 12:24am
#985224
Last week, I posted an article and gave everyone a chance to win a Merit Badge. I'm considering making that a regular thing. Let's do it today, and see how it turns out.

Today's link is about language -- something that might be of interest to all writers. It's a few years old, but that's irrelevant to everything except maybe one of the examples the author provides.

https://slate.com/human-interest/2015/09/take-a-wittgenstein-class-he-explains-t...

The Limits of Language
Wittgenstein explains why we always misunderstand one another on the Internet.


The best class I took in college was on the philosopher Ludwig Wittgenstein. Until that point, I had avoided philosophy of language as simply being too esoteric and hermetic to be of use.

Um, you just described all of philosophy, for various philosophical definitions of "use."

Wittgenstein, who lived from 1889 to 1951, is most famous for a handful of oracular pronouncements: “The limits of language are the limits of my world.” “Whereof one cannot speak, thereof one must be silent.” “The human body is the best picture of the human soul.”

All three of which are prima facie bovine excrement. Yes, I, an utter nobody with an engineering degree, am calling a philosopher's pronouncements bullshit. This is my blog and I can do that.

(Since pretty much no one can agree on anything about Wittgenstein, I’m going to present things in the spirit of Pears’ interpretation, with the caveat that you could probably find a philosopher somewhere who would disagree with every following sentence.)

Well, that's a tautology if I've ever seen one. Anything one person can say, another can disagree with. I can say "the sky is blue" and some asshole will respond with "Well, where I am, it's gray."

To be perfectly fair, usually I'm the asshole pedant.

The article goes on to describe in great detail exactly what this author's interpretation of the periods of Wittgenstein's work represent.

The idea of words having relative meanings was not new, but Wittgenstein pioneered the controversial linguistic conception of meaning-as-use, or the idea that the meanings of words, relative or not, cannot be specified in isolation from the life practices in which they are used.

This bit makes all kinds of sense, though. Words are not, and cannot be, stand-alone parcels of meaning. As writers, I hope we're all aware of the relationship between denotation (or definition) and connotation, as well as context. And that's not even getting into words with multiple meanings, such as "run" or "sex."

I remember perusing a dictionary when I was a kid (I was a bored geek and this was before the internet) and marveling at the length of the entry for the word "run." Compared to that, "sex" was straightforward.

Unfortunately, this makes the study of language considerably more difficult, since examining the meanings of words now requires not just verbal definitions, but analyzing the whole “language-game” of situations and practices to which they are attached.

Any article on philosophy ultimately appeals to epistemology.

Here’s one example. The French equivalents for here and there are ici and là respectively. But if I point to a pen and say, “The pen is here,” the French equivalent is not “Le stylo est ici,” but “Le stylo est là.” In French, là is always used to refer to a specific place or position, while in English here or there can both work. This rule is so obscure I never learned it in French classes, but obviously all native speakers learn it because no one ever uses it differently. It could just as easily be the other way round, but it’s not. The situation is not arbitrary, but the way in which language carves up the interaction between mind and world varies in such a way that French speakers recognize certain practices as right or wrong in a different way than English speakers do.

I'm getting a lot of practical experience with this sort of thing as I continue to learn French. Learning a language creates a shift in one's mental processes and forces one to understand one's own language better in comparison.

This is one reason why Google Translate is notoriously unreliable, and why the most unbelievable science fiction on Star Trek is not warp drive, transporters, or phasers, but the Universal Translator.

Wittgenstein’s philosophy also accounts for the disastrous state of Internet discourse today. The shift to online communication, textual interactions separated from accompanying physical practices, has had a persistent and egregious warping effect on language, and one that most people don’t even understand. It has made linguistic practice more limited, more universal, and more ambiguous. More people interact with one another without even realizing they are following different rules for words’ usages. There is no time or space to clarify one’s self—especially on Twitter.

This is, quite obviously, still true five years after this article was written. Thing is, though, just like with everything else, we learn and we adapt. It's long been said that "tone doesn't translate to text," or words to that effect, but I think it does -- it just doesn't work the same way that it does in spoken language.

I mean, if I came home and said to my nonexistent spouse, "I had a rough day," she could tell by my body language, vocalization, and perhaps the way I threw myself onto the couch afterward, that it wasn't merely a rough day but a war zone. I type that in text or in my blog, though, and it probably comes across as a mere statement, devoid of nuance, and you might even be excused for thinking I'm engaging in either understatement or hyperbole (especially if you know I have a penchant for both).

Some people compensate for that with emoji (or, here, emoticons), but I try to use them sparingly, because I feel that as a writer, I should be working on expressing myself more clearly in writing. The best way to do that, in the above situation, is not to simply type "I had a rough day," and throw in a string of indecipherable emoji, but, in the grand tradition of show-don't-tell, to say exactly what it is that was rough about it. "My car died on the way to work, and while I was trying to fix it, some asshole stole my wallet." For example.

It's not that we lose nuance when typing. It's that a lot of people don't know how to express it.

And that, friends, is why I think "philosophy of language" does have use.

Now, the article I linked above is mostly about relating this philosophy to the problems of artificial intelligence, and it's interesting for those insights, but of course I view it through the lens of a writer, not a programmer. Just another example of how the meaning we take from something is not always the meaning the author intends.

*StarB* *StarB* *StarB*


And now for the chance to win a Merit Badge.

The linked article opens with an image that has the following caption:

“Describe the aroma of coffee—why can’t it be done? Do we lack the words? and for what are words lacking?—But how do we get the idea that such a description must after all be possible? Have you ever felt the lack of such a description? Have you tried to describe the aroma and not succeeded?” Wittgenstein, Philosophical Investigations.


Thing is, almost everyone knows what coffee smells like, even if we think it's a foul brew of sewage and nightmares from the depths of the Abyss. Describing coffee should not be the intractable problem that, say, describing the color "blue" to a person blind from birth would be.

So: describe the aroma of coffee in the comments here. I'll pick the one I like best and give the author a Merit Badge tomorrow. No, you don't have to agree with me; you just have to be thorough and/or funny. You have until midnight.


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