Irrational numbers can never be real magnitudes.
There, I said it. Irrational numbers cannot exist as magnitudes of length or weight or whatever in the world of mobile substance. My stock example: you can never forge a sword with a blade whose length is √2 feet.
(Aside: that’s a gimmick in a to-be-written short story of mine–a magical Sword of Impossibility whose blade really is √2 feet in length and instantly annihilates whatever it cuts.)
Back to my math heresy. Why do I think this?
Well, here’s one intuitive appeal:
1.4 feet < 1.41 feet < 1.414 feet < 1.4142 feet < …
If the decimal is infinitely expanding, so is the magnitude. In reality, whatever your magnitude is, you will locate it with precision on a tape-measure (in the case of length). You do not need an infinite number of “zoom-ins” to gain the precision necessary to mark off its length. It will be a close approximation of √2 feet but never an infinitely expanding decimal.
For this thought experiment, we should set aside problems like Planck-length and decay and the fact that we are swirling clouds of electrons. Even with an unreal level of control over scale, the length we are dealing with will always, always, always be a rational number. Always.
That does seem to imply that there can never be two truly identical lengths, else we could construct a square angle from them and the hypotenuse would be √2 feet. Or perhaps the problem is rather that two such lengths could never enter into the point-precise arrangement necessary to generate a perfect square angle. I’ll bite the bullet on that consequence.
Flip to circles: no circle in the universe has a circumference which is a multiple of the infinite decimal 3.14159265… If you “unrolled” that circle and measured it, you would get a finite result. Or go in reverse: coil a finite length into a circle and you will not get a diameter which is a fraction of π.
This is not true of rationals, by the way. Or at least so I claim. Rationals can all happily exist as magnitudes in the real world. Just not irrationals.
Is this a specific problem inside the wider measurement problem? The nature of numbers and mathematicals?
I take this problem to indicate the power of our intellect to transcend the material-physical. Also my power to irritate my math-teaching colleagues.