The Gods help me!
First I thought I'd stay up all night, now I believe I'll start my new job on Monday with black circles around my eyes.
The Gods help me!
First I thought I'd stay up all night, now I believe I'll start my new job on Monday with black circles around my eyes.
I'm not sure what you mean "classical mathematics". Complex numbers were introduced in the 16th century and for strictly mathematical reasons. Chemistry as we understand it now did not exist then. Saying that complex numbers are outside classical mathematics is at least controversial, unless you mean mathematics of classical antiquity, but this is not how the term is usually understood.
Cardano devised them some time around 1600, yes, but he said the square root of -1, i, was "as subtle as it would be useless" and they were called imaginary because they were considered an impossibility. I did not say they didn't exist then, I said they were considered impossible and purely theoretical. They turned out to be useful in chemistry many years later. Classical mathematics I simply meant as the mathematics of the period covering classical physics (from Galileo to quantum mechanics and relativity), as we use the term classical music for the same period.
Yes, they were. Similarly, zero and negative numbers were considered either impossible or useless. All three inventions were major breakthroughs in the history of mathematics. It's actually quite interesting how many great mathematical concepts were ridiculed at the moment of their birth. Today, non-Euclidean geometries have a strong position despite the contempt the idea had to deal with in its beginnings. Cantor's "actual infinities" have become an organic part of modern mathematics. Complex numbers, once "impossible", are now considered extremely useful and very beautiful, and half of modern mathematics would be impossible or very difficult without them. Infinitesimal calculus, despite George Berkeley's essentially correct criticism, is now a part of rigorous mathematics and its value is impossible to overestimate.
Now you seem to be agreeing with me, but you may have missed my point: a purely theoretical concept, nay, a theoretically impossible concept, was later seen to have empirical relevance in atomic structure. Quantum mechanical solutions to wave equations for orbitals show real orbitals for solutions in i. Electrons can occupy such orbitals. To me, that's pretty interesting. Also, please pay attention to what I say before you misinterpret me and argue with things I am not saying.
Usually, by "classical mathematics", we mean mathematics that is based on standard logic and standard set theory. The history of mathematics isn't so much tied to the history of physics as to make the natural divisions into periods the same in both. While the discoveries of 20th-century physicists had an enormous influence on many branches of mathematics, if not all, the common perception isn't that it constituted a transition from a classical period in mathematics to some other period. "Modern mathematics" is considered to have started around the beginning of the 19th century with the work of Carl Friedrich Gauss.
Yes, it is interesting and yes I was agreeing with you after you explained what you meant. I thought I didn't need to explicitly say I agreed with you, since it was clear. I have no idea where I misinterpreted you and where I did not pay attention.
Anyway, complex numbers are not "theoretically impossible", unless you specify what it means. The theory of complex numbers is consistent if only the standard set theory is consistent, which is widely believed to be true. This means that there exists a model of complex numbers, and, actually, the main problem in constructing such a model is with constructing real numbers first. As I said though, I'm not really sure what "theoretically impossible" means to you. Do you mean some kind of philosophical theory? I'm completely unprepared to defend then.