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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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We'll end July with an article that's mostly for clarification, not for me to rag on. Now, the author has some bias, as always, but it happens to be a bias that I share -- only in her case, it's backed up by facts and education.
Last month in book club we were talking about an adorable newborn baby we had just met over Zoom.
Normally, I'd Stop Reading Right There because everything about that sentence is just plain wrong. But I was able to move past the anecdote and get to the important stuff.
Someone commented that the baby looks like their father. Another person piped up with an evolutionary explanation they had heard before — newborn babies look more like their fathers so that the father will know they’re his and will help raise and protect them.
You know, assuming I were the type of person to do book club meetings over Zoom (meetings that are interrupted by people bragging that they successfully fucked), which I'm not, I wouldn't make uninformed statements about evolutionary effects, however plausible they might sound, in front of an evolutionary biologist. That's just begging for a smackdown.
Some of us started wondering how something like that would even work. We agreed that this kind of “paternal effect” was possible and moved on. Later I looked it up only to learn that this popular hypothesis had been debunked nearly ten years ago.
Just in case you were thinking that this sounds reasonable.
Here's the important part; the rest is commentary:
Any discussion of a scientific hypothesis to explain a pattern should ask first, "is the pattern real?" and second, "could anything else lead to the same pattern?" There is a fine line between “it’s possible” and “it’s probable” and the consequences of mistaking a possibility for the truth can be far-reaching.
Evolution is often treated as synonymous with natural selection, and natural selection is often boiled down to a single phrase: survival of the fittest. This oversimplification is easy to apply to everyday life — but should we? If we don’t carefully manage our assumptions and biases, we are in danger of the same sort of thinking that leads to Social Darwinism, in which Darwin's theory of natural selection is applied to justify social inequities between different groups of people. To prevent that, let’s tease apart the concepts of evolution, natural selection, and fitness.
The article goes on to do so, and I obviously won't be quoting all of it. But I felt it was important to emphasize this thesis.
Some evolution is random, but not all.
It can be important to choose one's words carefully. One of the persistent refrains from the anti-evolution crowd is something like "human beings didn't appear at random!" Thing is, no, we didn't, and neither did cockroaches or oak trees. Random mutations, as the article notes, can be detrimental, neutral, or beneficial to survival and reproduction. It's like if you keep rolling a pair of ordinary six-sided dice, but you only keep the results of 10, 11, or 12. Random shit happens, but the higher number are selected for (in that case, through human decision, so it's an imperfect analogy).
In evolutionary biology, the word "fitness" is the capacity of an organism to survive and reproduce in a given environment.
Another thing that some people misunderstand: we're not "more evolved" than mice, for example. Mice are just as suited for their natural environment (which is not mazes and laboratories) as we are for ours (which might include mazes and laboratories).
Also, evolution doesn't have a "goal." It's a result of mutations and environmental factors, and has no way of looking ahead and going, "There's going to be climate change soon, so we need to adapt to that eventuality."
In social species like humans, this contribution can be indirect — for example, helping to raise your sibling's kids increases your own fitness.
This is another important point to note. People (ironically some of the same people who don't "believe" in evolution) have used the idea that certain people don't reproduce to argue that, for example, being gay is "unnatural." That is arrant nonsense.
Armchair evolutionary psychologists love to reinforce gender stereotypes, touting that men are built for fighting and hunting and not caregiving, or women are less promiscuous and more inclined to prefer monogamy than men.
And I've been railing against that kind of bullshit for years.
Confirmation bias kicks in: people who want these statements to be true will believe them to be true without looking for good evidence. And of course, this danger goes beyond gender stereotypes to racial and ethnic stereotypes.
You know what's worse, in my estimation, than someone going "it's this way because God made it this way?" Its misusing science to marginalize individuals or entire groups.
Moving past evolutionary biology background facts, the author finishes with opinion (one based on said facts), but like I said, it's one that I share:
The existence of natural selection in nature does not justify its application to social policy. The synergy of intelligence, compassion, and social structure in humans has created a unique opportunity to prioritize equity and even the fitness playing field within our species. Some of our greatest achievements as a species have increased access to resources and raised each other’s capacity to survive and reproduce. These innovations are cultural adaptations rather than genetic adaptations — they are passed down through culture rather than DNA.
Some people have internalized the idea that competition is the driving force in evolution. It's certainly a factor. But among humans, cooperation, not competition, is the defining attribute. |
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