Are you just looking for an easy way to determine whether expressions written using ordinary arithmetic operations and ordinary constants like e and π come out transcendental or not? This would indeed be a breakthrough.
BoForHire> I am running out of time presently, but, This! Directly above. I am aware that I did not specify a particular operation in my proposed definition of an “algebra of transcendentals”. Moreover, I deliberately “low-balled” the requirement for there to be an algebra of transcendentals by specifying a minimal algebra (not a technical term here) – that is, one operator and not even an inverse. I was trying to point out that it is actually quite easy to create an “algebra” of transcendentals, in the sense of modern algebra, in the sense that transcendentals admit to some structure. We both agree here. I also concur that, for an “algebra” over transcendentals to be useful, it would need to have at least one operator and its inverse (so that we could have a “walk” up and down a number line) with suitable properties of closure, associativity, commutativity, identity, etc. I am also quite willing to agree that it might not be an interesting “algebra” unless it were at least a ring. Or, we could define what we mean by “interesting” by what it can do. That is, we want the “span” of axioms which allows us to pick out transcendentals (for example).
I still don’t consider any of this to make π “mysterious”. This seems a kind of mysticism which infects pop mathematics which I am not fond of. There are things we understand very well about π. There are also things we don’t understand so well which invoke π. The same is true of 2 (is the Collatz Conjecture (involving division by 2) true? Is the Goldbach Conjecture (involving multiples of 2, as well as 2 primes) true? Is sqrt(2) a normal number in any or all bases?) and everything else.
BoForHire> I hear you. I have reached a point of contentment, for my part. I believe that we understand one another. What you don’t like in amateurs (pop mathematics) I find interesting and salutory. A professional mathematician, armed with confidence and years of successful proof-making experiences, sees a subfield of mathematics that is at a stumbling block, “stuck” as it were, merely as a challenge to the profession. But an amateur, for whom even the very things which you did earlier (running out trivial sets of transcendental numbers from your head) seem un-intuitive and difficult, thinks of the very same thing as “mysterious”. Do you see, friend, that it is a semantic issue, to be sure, but it is one that matters…?! If we (as a society) want amateurs to be interested in mathematics - whether to inspire another generation of professionals or of more amateur to buy books and pursue the field from a distance, or whatever – then is it not incumbent upon us to indulge their view that things which professionals cannot yet solve are, in fact, “mysterious”? Unlike professional thinkers, amateur thinkers like mystery. There is a romance in it, a feeling of touching on something great and deep. PI not mysterious…? People write books for amateurs on just the history and nature of this number, and these books sell like hot cakes! Mystery to a mathematician is just another spur to rational investigation, but to those who feel overwhelmed by the field, it is like being born into a different universe. I cannot say more, I am at a loss to explain feelings that I myself have had. For my part, I am a wayfarer midway, so to speak, between these worlds, with some skill but with life circumstances that do not admit time to pursue mathematical insight with passion, indeed, with obsession.
So at the end of the day, I must accept my own advice. It is a matter of perspective. To you, it is not mysterious. You have more clarity and see transcendental numbers and their theory (and especially one of its most celebrated members) as just another challenge, because you can envision a day when some bright young person will construct something that clears up the confusion. And, on that same day, many amateurs (likely, myself included) will stand below and be dazzled, wishing that we might contribute in some small way.
You all are the rock stars, we the audience; you are the sports stars, we the fans. Do not spurn, I adjure you, those who are beneath you in ability, just because they fail to utter a clear question. Find the question inside the question. Be a teacher. Accept the difference in perspective, then instead of dismissing, do what stars do…
…dazzle.
Make numbers shine for us.
Yours Truly,
-BFH