Last time I opened with basic exponentiation and raised it to the idea of complex exponents (which may, or may not, have been surprising to you). I also began exploring the ubiquitous exp function, which enables the complex math needed to deal with such exponents.
The exp(x) function, which is the same as ex, appears widely throughout physics. The complex version, exp(ix), is especially common in wave-based physics (such as optics, sound, and quantum mechanics). It’s instrumental in the Fourier transform.
Which in turn is as instrumental to mathematicians and physicists as a hammer is to carpenters and pianos.
For the last two weeks I’ve written a number of posts contrasting physical systems with numeric systems.
(The latter are, of course, also physical, but see many previous posts for details on significant differences. Essentially, the latter involve largely arbitrary maps between real world magnitude values and internal numeric representations of those values.)
I’ve focused on the nature of causality in those two kinds of systems, but part of the program is about clearly distinguishing the two in response to views that conflate them.
Too Weird For Words!
I started with the idea of physical determinism and what it implies about free will and the future. Then I touched on chaos theory, which is sometimes raised as a possible way around determinism (short answer: nope). In the first article I drew a distinction between “classical” mechanics and quantum mechanics because only at the quantum level is there any sign of randomness in reality.
It turns out that the quantum world is decidedly weird, and while we have math and models that seem to describe it extremely well, it can honestly be said that no one actually understands it. This time I’ll tell you about some of that weirdness and how it may (or may not) apply to the world as we know it.
The key question here is whether our brains make use of quantum effects.