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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.
February 24, 2020 at 12:12am February 24, 2020 at 12:12am
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Language.
It's ubiquitous -- you're using it right now -- and yet its origins are as mysterious as that of life, the Universe, and everything. Of course, that doesn't stop us from speculating.
https://www.theatlantic.com/science/archive/2018/06/toolmaking-language-brain/56...
A Sneaky Theory of Where Language Came From
It might have hijacked our early ancestors’ brains.
I’ve come to see Bovaird, who teaches wilderness-survival skills in western Massachusetts, because I want to better understand the latest theories on the emergence of language—particularly a new body of research arguing that if not for our hominin ancestors’ hard-earned ability to produce complex tools, language as we know it might not have evolved at all.
I'm just going to throw out here the observation that when I think of "wilderness," one place that doesn't come to mind is western Massachusetts. But, okay. That's irrelevant to the discussion. I just thought it was funny.
It's well-known by now that humans aren't the only species to use tools, or for that matter communicate with each other. And yet, as far as I've been able to determine, we're the only extant species that uses tools to make tools. This nesting of concepts is, in my view, essential for language as we know it today.
Oren Kolodny, a biologist at Stanford University, puts the question in more scientific terms: “What kind of evolutionary pressures could have given rise to this really weird and surprising phenomenon that is so critical to the essence of being human?”
And here we start to steer into my least favorite subject: evolutionary psychology.
Stout found that his students’ white matter—or the neural connectivity in their brains—increased as they gained competence in flintknapping. His research suggests that producing complex tools spurred an increase in brain size and other aspects of hominin evolution, including—perhaps—the emergence of language.
Yeah, put like that, we're also steering perilously close to Lamarckian evolution  -- unless it's just badly phrased.
When hominins like Homo ergaster and Homo erectus taught their close relatives how to make complex tools, they worked their way into an ever more specialized cultural niche, with evolutionary advantage going to those individuals who were not only adept at making and using complex tools, but who were also able—at the same time—to communicate in more and more sophisticated ways.
Which puts us back on secure Darwinian footing.
Rudimentary language, which evolved in the context of toolmaking and teaching, was ultimately able to break away from its immediate contexts—this is the hijacking part—eventually employing those original cognitive pathways for its own unique purposes.
I think I see what they're getting at here. It's the development of the metaphorical aspect of language, an idea I find fascinating. Basically, at some point, we developed the capacity to relate something to something else that's superficially unrelated. For instance, you might count your sheep and find that there are 13 of them. And you might count the number of trees in your immediate vicinity and also get the number 13. The cognitive leap is to make 13 a separate concept - and suddenly, there can be 13 of almost anything that's countable. Or, if numbers somehow instill a feeling of deep dread, one might notice that the caudal appendage of a cat is similar to the caudal appendage of a dog, and rather than having separate words for the two anatomical protuberances, one might allow the word "tail" to do double duty - which can then be extended to describe what sticks out of the hind end of almost any animal that has one.
These are concepts we take for granted, but they represent massive cognitive leaps.
The article goes on to describe some pushback that this hypothesis has gotten, which is good -- though the article is clearly biased to the hypothesis. I'm not sure this particular article proves or disproves anything. As I always say when I'm not quite sure whether I like a beer or not, "further research is necessary."
Scientifically, the question is, okay, they've made the hypothesis of the connection between complex tool-making and language. What predictions does this make, and how can we test the predictions? Until more study is done, it remains just an idea. It's entirely possible, for instance, that other forces led to both the development of complex tools and the development of complex language; correlation doesn't necessarily prove causation, as is the case in every discipline.
Still, interesting enough for me to read and share the idea. After all, that's what language is for. |
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