Complex Numbers (Book)

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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.

http://www.ddewey.net/mandelbrot/

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June 5, 2022 at 12:01am

#1033371

Natural Philosophy

Today's article is about four years old, and has been languishing in my queue for a while (though not four years) until the Random Number Generator deigned to notice it. But despite its advanced age, I don't think the subject matter has an expiration date.

Physics Needs Philosophy / Philosophy Needs Physics
Philosophy has always played an essential role in the development of science, physics in particular, and is likely to continue to do so

I've said it before: science informs philosophy, and philosophy guides science. For example, we know that we can get a lot of good answers doing testing on unsuspecting humans (because some assholes have done so), but it takes philosophy to realize that doing so is usually Bad. Science is great for understanding the universe, but on its own it's ethically and morally neutral.

A certain current anti-philosophical ideology has had damaging effects on the fertility of science. The recent momentous steps taken by experimental physics are all rebuttals of today's freely speculative attitude in theoretical physics. Empirical results such as the detection of the Higgs particle and gravitational waves, and the failure to detect super-symmetry where many expected it, question the validity of philosophical assumptions common among theoretical physicists, inviting us to engage in a clearer philosophical reflection on scientific method.

I have to admit I've been largely ignorant of any such fray. I'm merely a spectator here.

Against Philosophy is the title of a chapter of a book by one of the great physicists of the last generation: Steven Weinberg. Weinberg argues eloquently that philosophy is more damaging than helpful for physics—it is often a straightjacket that physicists have to free themselves from.

Anyone else see the irony here? That taking a position against philosophy is itself a philosophical position? It's a bit like the old "This sentence is a lie" false paradox.

Here are a few examples of this influence, from astronomy and physics. Ancient astronomy—that is, everything we know about the Earth being round, its size, the size of the moon and the sun, the distances to the moon and the sun, the motion of the planets in the sky and the basis from which modern astronomy and modern physics have emerged—is a direct descendent of philosophy.

Physics (and science in general, though the philosophical argument could be made that all science is, at its core, physics) is really quite good at answering "how" questions. "Why" questions have been more elusive, though, and are generally the domain of philosophy, in my view. And religion, perhaps, but I consider religion a subset of philosophy.

We're curious, but to get meaningful answers, we have to ask the right questions. "Why are we here?" is vague and contentious, but "How did we get here?" can be answered through science.

Why this influence? Because philosophy provides methods leading to novel perspectives and critical thinking. Philosophers have tools and skills that physics needs, but do not belong to the physicists training: conceptual analysis, attention to ambiguity, accuracy of expression, the ability to detect gaps in standard arguments, to devise radically new perspectives, to spot conceptual weak points, and to seek out alternative conceptual explanations.

Here, I might have cause to quibble. Science is all about "attention to ambiguity" and "accuracy of expression." Sure, they borrowed those traits from philosophy -- which is much, much older than the modern scientific method -- but from my understanding and experience, they're absolutely taught as part of a science curriculum.

Whether that's the case for advanced physics, I have no idea; like I said, I'm mostly just a spectator here.

Here is a second argument due to Aristotle: Those who deny the utility of philosophy, are doing philosophy.

See? Even dead Greeks agree with me there.

The article provides many examples of how, as I put it, philosophy informed science over the centuries. And even how science affected philosophy in turn. Really, I'd call it a symbiotic arrangement. If your philosophy isn't grounded in objective fact, it's mere word games. (They can be fun, too, of course.) And if you don't have a clear methodology for doing science, then all you're doing is groping around in the dark (which, under the right circumstances, can also be fun, or so I've heard).

Just the other day, I saw a study that examined -- well, let me show you the blog post

discussing it.

If you can't be arsed to click on that link, or even if you can, just to be clear, that study was about "do guys living under a neoliberal capitalist realism hellscape like big boobs?"

Philosophical question: WHY??

Scientific question: How in the hell did they manage to score a grant for that?

Speaking of which, going back to the original article:

I think that this methodological philosophy has given rise to much useless theoretical work in physics and many useless experimental investments.

Apparently, not just physics. But to be honest, the only branch of science I trust less than evolutionary psychology is nutritional science.

Anyway, the article has a lot more information than the little snippets I'm pasting and, fair warning, sometimes goes over my head. But I think I get the gist (though that may be my confirmation bias talking).

Here is one last argument from Aristotle: More in need of philosophy are the sciences where perplexities are greater.

You can tell when something is truly profound when it twists grammar around. See also: Yoda.

(For all I know, that might be a literal translation from Aristotle's language. I know next to nothing about ancient Greek.)

That's really all I have to say about it, but if any of this shit interests you even a little bit like it interests me, the article is absolutely worth a look.

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