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Sometimes There’s Math
But most often, there’s art! Like, almost always. • ‘main’ blog of @as-if-and-only-if, where you will more regularly find math
endofunktor
All numbers are imaginary. Some are useful.
So, @lipshits-continuous (I like your blog very much!) was bringing to my attention someone who was, let’s say, a little frustrated about having to learn “imaginary numbers”. You know. They’re not even real.
I don’t want to write about that person specifically, I don’t know them at all.
But.
My siblings in crisis.
There are no “real” numbers in this world. The act of counting is already an abstraction. If you have an apple and a pear, then counting them as two pieces of fruit is an act of forgetting all the properties that distinguish them and remembering only those that unite them. Same goes for two apples of course.
So the number 2 is already entirely imaginary. Made up. Nonsense.
This gets worse very quickly, by the way. You can’t have negative apples! Sure, you can have positive fractions of apples, but I can assure you: Two halves of an apple do rarely make one apple, except when they are the two halves of the same apple. And even then: Having them cut up makes it impossible to rejoin them. Think about that.
Now you know where this is going, because let’s talk about the so-called “"real”“ numbers. Nasty little things. Have a look at the irrational numbers. First, there are comparatively tame irrationals: Algebraic numbers. These arise as roots of integer polynomials. You know, like our favorite, the square root of 2, the positive zero of X^2 - 2.
Did you know that the Pythagoreans believed that everything was in integer relation to each other. (For example: the lengths of a rectangular triangle could be 3-4-5, meanging one cathetus is 4/5, the other 3/5 the length of the hypotenuse.) And that when someone found out that sqrt(2) was indeed not in integer relation to, say, 1, they had him murdered? That didn’t change the fact that they were wrong though. (This story is not true, of course, like all good stories. You could say: it is imaginary, but nonetheless an interesting tale to tell!)
And now, the non-algebraic, so called transcendent numbers. Like pi. We know (the abstract concept of) pi well enough to calculate the circumference of the observable universe up to the accuracy of a neutrino (or so they say, I’m not a physicist). We don’t have to know it any better. We could quit at however many digits we know and that would be pi. Perfectly rational. Because if we are looking for ”“"real”“” numbers only, why should we ever even concern ourself with those nasty things?
Did you know that transcendent numbers make the vast majority of real line? Of course you do! Rationals are countable, integer polynomials are countable and thus are their roots. The real line is famously uncountable, so must be the transcendent numbers.
So. Our so called “”“"real”“”“ numbers are mostly non-precisely calculable numbers. (Arbitrarily precise, but not precise.)
Do you know what kind of numbers are used in electrical engineering? Me neither because I’m not an electrical engineer. That’s right, complex numbers! Actual engineers have to actually work with the actually imaginary square roots of negative one.
Maybe Numbers aren’t a thing of this world.
Maybe Abstraction is a thing of our worldview.
And maybe we can learn to cherish abstract math the way we learn to cherish abstract art: Not always a true representation of reality, but a thing of interest and beauty in itself.