Comment by drob518
4 hours ago
Some of these seem forced. For instance, does Chapernowne's number (number 7 on the list, 0.12345678910111213141516171819202122232425...) occur in nature, or was it just manufactured in a mathematical laboratory somewhere?
It is indeed manufactured specifically to show the existence of "normal" numbers, which are, loosely, numbers where every finite sequence of digits is equally likely to appear. This property is both ubiquitous (almost every number is normal in a specific sense) and difficult to prove for numbers not specifically cooked up to be so.
Okay, fair. It just seemed to me to have pretty limited utility.
It's fame comes from the simplicity of its construction rather than its utility elsewhere in mathematics.
For example, Graham's number is pretty famous but it's more of a historical artifact rather than a foundational building block. Other examples of non-foundational fame would be the famous integers 42, 69, and 420.
All the transcendental numbers are "manufactured in a mathematical laboratory somewhere".
In fact we can tighten that to all irrational numbers are manufactured in a mathematical laboratory somewhere. You'll never come across a number in reality that you can prove is irrational.
That's not necessarily because all numbers in reality "really are" rational. It is because you can't get the infinite precision necessary to have a number "in hand" that is irrational. Even if you had a quadrillion digits of precision on some number in [0, 1] in the real universe you'd still not be able to prove that it isn't simply that number over a quadrillion no matter how much it may seem to resemble some other interesting irrational/transcendental/normal/whatever number. A quadrillion digits of precision is still a flat 0% of what you'd need to have a provably irrational number "in hand".
It appears quantum phenomena are accurately described using mathematics involving trig functions. As such we do encounters numbers in reality that involve transcendental numbers, right?
You don’t need quantum mechanics. Trigonometric functions are everywhere in classical mechanics. Gaussians, exponential, and logs are everywhere in statistical physics. You cannot do much if you don’t use transcendental numbers. Hell, you just need a circle to come across pi. It’s rational numbers that are special.
They're accurately modeled. Just as Newtownian phenomena are accurately modeled, until they aren't. Reality is not necessarily reflective of any model.
Yes, it occurs in the nature of the mathematician's mind.