While we are in this gossip mode about my hero Beethoven, just in case you did not know a fact about my hero Newton, well, he was a complete nut case outside of physics, and a nasty character too. This is an important lesson, that one can admire a person's works despite one's revulsion for other aspects of their behavior.
I cannot recall who said it, but it was something like, "The role of proof is to sanction the conquests of intuition". Oliver Heaviside, British electrical engineer and physicist, was the inventor of the Laplace Transform. Now, if ever there was an injustice in the annals of mathematics, it is the fact that the Laplace Transform is not called the Heaviside Transform. True, Heaviside did not define his operational mathematics the way you saw in the link you just hit, but what the heck, it had the same effect. The problem, so it seems, was that Heaviside did not (perhaps could not) prove the validity of his (effectively) transform calculus that reduced differential equations to algebraic ones. When mathematicians protested his lack of rigor, Heaviside responded, "Should I refuse a good dinner simply because I do not understand the process of digestion?" Touche! The Heaviside Transform was eventually validated about 20 years after he introduced the idea.
One of the greatest physicists who ever lived was Michael Faraday. He knew virtually no mathematics, yet his physical intuitions were so powerful that all the knowledge encapsulated in Maxwell's Equations did no more than summarize in mathematical form the laws of electromagnetics discovered by Faraday. (A confession! These equations were my first love, remembered with greater nostialgia than the girl whom I first dated.) Of Maxwell, the greatest mathematical physicist of the nineteenth century, one can say of his role in electromagnetic theory, vis-a-vis Faraday's, was that (apologies to Jesus) "he came not to destroy but to fulfill".
Having said all that, a beautiful proof is a joy forever! But the notion of what constitutes a proof may not be as innocent as it looks, as explained in that page by Alexander Bogomolny. This guy maintains a very " informative and mathematically entertaining site in which the one I just cited is but a page. The famous Chinese Remainder Theorem was reported by the 13th century mathematician Chin Chiu-Shao at a time when the notion of "proof" in China was quite different from ours today. Western (and therefore ALL modern) mathematics inherited its notion of proof from Euclid. So, in a sense, when we do mathematics today, we are all Greeks!
Have you ever wondered what the world would be like if the Nazis had won World War II? It is not so hard to imagine that Hitler could have invaded Britain if Churchill was not such a stubborn PM. Or, what the state of Christianity would be like if the Reformation had failed and the Vatican Councils succeeded in bribing the rebellious German princes into betraying the nascent Lutheran protest?
Well, wonder no longer! Help is at hand! There are thousands, nay, millions like us dreamers -- so many that there is a lucrative market for novels and novellas speculating on such dreams. Ian Montgomerie's Alternative Histories webpage has all the interesting links to alternative timelines. You can feast on trilogies that delineate an alternative path if the American south had won the Civil War. If your Catholic soul sometimes wishes to undo the Protestant success, your waiting is over! Keith Roberts' Pavane is a collection of episodes or fragments of what happened after the Spanish Armada succeeded. This was my first excursion into a variant sci-fi soon after my exhaustion of the available Asimov, Clarke, Niven, etc. classics.
Incidentally a scholarly yet surprisingly engaging historical account of The Defeat of the Spanish Armada by Garrett Mattingly is probably in your local library. It is a great read! I did not know until I read it that in fact the Spanish Navy was virtually doomed from the start with its dinosaur-like galleons that were slow and relied on (would you believe it!?) oarsmen! Contrast them to the small but agile ships commanded by (practising) pirates like "Admiral" Francis Drake, and the outcome of the Armada was really a foregone conclusion.
One cannot just write any story and expect it to be received as an alternative history. It has to have strong asects of what I would call credible counterfactuality. A counterfactual is a philosophical concept that is at the essence of good alternative time-lines. Suppose you want to dream about what the world would look like if, say, the Spanish Armada had succeeded. It is not a good idea to also hypothesize that at the same time the Mongol hordes swept all over continental Europe. A counterfactual, being a hypothesized alternative to what actually happened, should not simultaneously change other things that had also happened. Otherwise, it stops being a dream but beocmes a nightmare. The notion of a counterfactual is intimately tied to that of causation. To this day there is no universally accepted formal account of counterfactuals.
A really sexy string quartet, the Bond Girls have stimulated appreciation for classical string music like no others before them. In some ways this is the inevitable sequel to the selling of the more sedate yet just as subliminally erotic Vanessa Mae. And Mae herself is a manifestation of an unfolding dialectic, with an almost Hegelian inevitability, that seemed to have started with Anne-Sophie Mutter, the thinking man's sex symbol. Mutter is married to Andre Previn.