this post was submitted on 25 Nov 2024
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Showerthoughts
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A "Showerthought" is a simple term used to describe the thoughts that pop into your head while you're doing everyday things like taking a shower, driving, or just daydreaming. A showerthought should offer a unique perspective on an ordinary part of life.
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Why? What does it mean for something to be real?
I believe pure mathematics isn't concerned with its correspondence with reality.
I recall hearing a quote from the guy that coined the term "imaginary number", and how he regretted that term because it conveys a conceptualization of fiction. IIRC, he would rather that they would have been called "orthogonal numbers" (in a different plane) and said that they were far more real that people tend to hold in their mind. I think he said "they are as real as negative numbers" along the same lines of one not being able to hold a negative quantity of apples, for example.
The stray shower thought (beyond simply juxtaposing the discordant terms of 'imaginary' and 'real') was that infinity by contrast is a much weaker and fantastic concept. It destroys meaningful operations it comes into contact with, and requires invisible and growing workarounds to maintain (e.g. "countably" infinite vs "uncountably" infinite) which smells of fantasy, philosophically speaking.
So Descartes coined the term specifically as a dig because he didn't see any geometric possibilty to the concept. The concept seems to have roots going back to ancient Egypt, but the modern inquiry goes to the Renaissance. I think Gauss wanted to call them laterals.
This isn't always true. The convergent series comes to mind, where an infinite summation can be resolved to a finite number.
Furthermore, it is meant to highlight the fact that people gleefully embrace the concept of infinity, but try their hardest to avoid and depreciate the concept of imaginary numbers. It would appear to me that the bias ought to be reversed.
I mean, complex numbers are important for quantum mechanics. In that sense, they are closer to reality as they are used to describe the underlying blocks of reality to our current best understanding
You don't even have to go into quantum mechanics. I vaguely recall using a real/imaginary plane with a rotating vector to do something about electricity in first year engineering?
Don't worry I'm not actually an electrical engineer.
But my point is that there are applications for imaginary numbers with very practical engineering applications. Foundational, even.
It's a usefull technique to model the symmetry between magnetic and electrical power.