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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Writer_Mike

about 11 hours ago
In response to "Paris, I Guess"

Elisa: Middle Aged Stik

- I thought it very interesting, to say nothing of extraordinarily coincidental, that your video mentions a lock-adorned bridge in Paris. I just finished reading Spymaster by Brad Thor, in which the Russian mastermind takes strolls from Kaliningrad's mainland to Kant Island. The bridge he uses is also festooned with locks placed by couples declaring their undying love for one another.

1 Like

QPdoll

a day ago
In response to "Landmarks"

I hope you're enjoying every minute in France! What an opportunity.

2 Like

Wordsmitty ✍️

a day ago
In response to "Landmarks"

Hey! Why are the signs at that Mac's in English if you're in France?

With all that walking, I hope you stayed hydrated.

2 Like

Annette in France

a day ago
In response to "Paris, I Guess"

Bienvenue en France.

1 Like

WakeUpAndLive️~Happiness

a day ago
In response to "Paris, I Guess"

last one today:https://parisjetaime.com/eng/event/olivier-giraud-how-to-become-a-parisian-in-one-ho ur-e047

1 Like

WakeUpAndLive️~Happiness

a day ago
In response to "Paris, I Guess"

Could not resist sending the link: https://www.atlasobscura.com/things-to-do/paris-france

1 Like

Elisa: Middle Aged Stik

Sep 21, 2024 at 7:34am
In response to "Paris, I Guess"

Your remarks on Paris looking like many cities reminds me of this video (which I promise is short). It's a sassy look at Paris, and I'm here for it.

https://youtu.be/ByUPFYxy5vw?si=lT57RpTDntAhFdBs

2 Like

WakeUpAndLive️~Happiness

Sep 21, 2024 at 6:46am
In response to "Paris, I Guess"

if you have two hours to spare tomorrow? A beer bike tour in Paris. Check it out, still available. https://parisbeerbike.com/

3 Like

WakeUpAndLive️~Happiness

Sep 21, 2024 at 6:16am
In response to "Paris, I Guess"

Je suis sure, vous avez aimez la ville!!!!

2 Like

Aiva Raine

Sep 20, 2024 at 10:13pm
In response to "Get Baked"

FYI- basting them with olive oil before placing them directly on the oven rack- NO tinfoil- will allow the skin to crisp up a little as well and add a nice flavor to the skin. Recommend Idaho baking potatoes as well for best results.

February 27, 2023 at 12:51pm

#1045623

What You Will

Ugh. I would get this weighty topic on a Monday.

Why the Classical Argument Against Free Will Is a Failure
Despite bold philosophical and scientific claims, there’s still no good reason to doubt the existence of free will.

There's no good reason not to, either. But I'll get to that.

My problem with this article is it uses the strawman fallacy. But if you subscribe, as I do, to the notion that we don't have free will, the author couldn't have written otherwise. Not then. Now, they could, once the strawman is pointed out. But they won't; I've found that some people are invested in the idea of free will, and get upset if you present evidence to the contrary.

In the last several years, a number of prominent scientists have claimed that we have good scientific reason to believe that there’s no such thing as free will — that free will is an illusion.

I've talked a lot about illusion in here. In short, let's not confuse "illusion" with emergent or bulk properties. For example, consciousness is an emergent property of our bodies, but it's, by definition (cognito ergo sum) real. But some things that people think are real are, necessarily, illusions; the obvious example of that is, well, look up any number of optical illusions online; until they're explained, they certainly seem real.

If this were true, it would be less than splendid.

Assertion without evidence.

And it would be surprising, too, because it really seems like we have free will. It seems that what we do from moment to moment is determined by conscious decisions that we freely make.

"Seems" is the key word here.

We need to look very closely at the arguments that these scientists are putting forward to determine whether they really give us good reason to abandon our belief in free will. But before we do that, it would behoove us to have a look at a much older argument against free will — an argument that’s been around for centuries.

The older argument against free will is based on the assumption that determinism is true.

And that's the strawman here. I'm sure there are scientists who still accept determinism, but once quantum theory began to take shape about a century ago, all that went right out the window (the article notes this later). I won't copy their version of the strawman here, but it's there at the link if you want to read it.

There’s a big problem with the classical argument against free will. It just assumes that determinism is true.

On this, we can agree. But again, no one's promoting the classical argument anymore. Or, at least, very few are.

The idea behind the argument seems to be that determinism is just a commonsense truism.

Which is another reason why common sense is neither. Another person's "common sense" might lead them to conclude that we do have free will. After all, as the article noted up there, it certainly seems as if we do.

Quantum mechanics has several strange and interesting features, but the one that’s relevant to free will is that this new theory contains laws that are probabilistic rather than deterministic.

True.

We can understand what this means very easily.

False. No one understands quantum physics.

But if this is right, then it means that at least some physical events aren’t deterministically caused by prior events. It means that some physical events just happen.

This is the crux of the issue, though. These "physical events that just happen" include the quantum effects in your brain. It's not hard to understand that the result of this can be some truly random firing of neurons, which in turn affects our thoughts and/or actions. Despite metric shit-tons of ink wasted on "you can control your quantum reality" bullshit pseudoscience books, it's really the other way around.

And so we don’t know whether all physical events are completely caused by prior events. In other words, we don’t know whether determinism or indeterminism is true.

And as regards free will or lack thereof, it doesn't matter. Any quantum effects, inside or outside the brain, that we observe or are otherwise affected by, inform our perceptions, which in turn inform our decisions.

But now notice that if we don’t know whether determinism is true or false, then this completely undermines the classical argument against free will.

Yes, but only the classical argument.

Now, to be fair, the article goes on to present that as an argument, along with a more philosophical argument. I take no issue with they way they're both described. But there is no good reason to devote most of the article to debunking determinism, as that has already been done, and thoroughly.

My own view is that neither of these new-and-improved arguments succeeds in showing that we don’t have free will. But it takes a lot of work to undermine these two arguments. In order to undermine the scientific argument, we need to explain why the relevant psychological and neuroscientific studies don’t in fact show that we don’t have free will. And in order to undermine the philosophical argument, we need to explain how a decision could be the product of someone’s free will — how the outcome of the decision could be under the given person’s control — even if the decision wasn’t caused by anything.

Yes. "It takes a lot of work." Because they're good arguments. I followed the same arguments this author (who is a professor, but let's not add "argument from authority" to our list of sins here; he's a philosophy professor) describes and came to a different conclusion.

Well, I guess in the end, we can agree on one thing: More study is necessary.

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