The Sordid Past of the Cubic Formula
quantamagazine.org
quantamagazine.org
One such interesting tidbit is the notion of a different mathematical culture at the time, which valued “duels” — exchanges of mathematical puzzles. Whoever solves the most, wins. This practice (we read) incentivized keeping some clever problem solutions secret, as ammo for a future contest.
Of course you can write them as a limit of such operations, but that's true for all numbers.
Most operations for defining real numbers are computable. The reason for this is that, in practice, continuous functions and functionals are the same thing as computable functions and functionals. Continuous functions which aren't computable are generally pathological curiosities. One consequence of this heuristic is that since the Riemann integral is a continuous functional, one might guess that it is also computable. Indeed it is (but the proof isn't obvious). This then implies the computability of the constant pi because pi is equal to the integral
1
⌠
⎮ ________
⎮ ╱ 2
⎮ 2⋅╲╱ 1 - x dx
⌡
-1
Isn’t that a tautology or at least selection bias? “Operation” seems to imply we can carry it out, which means the number can be computed.
Even if we give “operation” a wider meaning to include those that can’t be carried out, are we deceiving ourselves that most of them are computable because most of those we have thought of in the history of mathematics are?
Or do we really know something about the cardinalities of those sets?
- Probability that a bit flip will introduce a bug - Number of reasonable ways to solve a problem in a given language for some measure of reasonableness (e.g., as a heuristic when examining whether two similar programs are similar out of necessity or if there might have been influence from one to the other or via some joint hidden variable) - (generally, most statistics over a space of programs) - Optimal super-optimization with jumps allowed - Minimal boilerplate complexity to describe an algorithm in a language - (generally, most questions about optimizing some target over a space of programs) - ...We don't care about all of them, but a vast array of questions about classes of programs are simply not answerable in any finite period of time.
You might be able to answer the above for sufficiently small and well-behaved inputs (e.g., in restricted languages whose constraints enable certain sorts of static analyses), but in general you cannot, and it happens often enough to definitely be a problem for real-world programs.
For what it's worth the numbers in question are computable.
It's the probability of a randomly generated program halting. One can imagine it being of interest, at least among theorists.
That said, I struggle to think of other useful non-computable numbers, save contrived examples.
We know from school that there is a solution for any equation of degree 1 (linear) with integer coefficients using plain arithmetic, and degree 2 using radicals and arithmetic. The linked article mentions that the same holds for degrees 3 and 4.
What Abel proved for degree 5 and Galois for any degree >= 5, is that for some equations of these degrees there's no expression involving radicals and arithmetic (*) that is a solution.
To reiterate, Abel and Galois' results are about the very existence of a specific form of a solution, not "findability".
(*) More technically: any finite formula involving composition of radicals, arithmetic operations and natural numbers.
The chapters are usually short, maybe 8-12 pages, with about half that being lots of exercises to help cement your understanding of the material. This greatly helps with self-study. And it's a Dover edition so is inexpensive.
[1] https://www.amazon.com/Book-Abstract-Algebra-Second-Mathemat...
On the other hand, for a say fourth degree equation, it just says that some solution expressible in terms of higher roots exists, but not how to find it. To find the expression, one needs to study the "generic" equation.