It clearly is being done, though, even if only implicitly; there’s not really any difference between “When I write infinity on the right hand side of an equals sign like so, what I mean is this, and note how if A = infinity and B = whatever, then A + B = so and so, and if A = infinity and B = whatever, then A * B = such and such, and…” and in fact having defined an abstract entity in some abstract arithmetic game to call “infinity”. Whether they’re comfortable admitting it or not, anyone who has ever said something like “Well, the limit of f at this point is infinity, and the limit of g at this point is -3, so the limit of f * g = infinity * -3 = -infinity” has been working perfectly competently with the system of affinely extended real numbers.
That is: Sure, you can think of limits as just a mathematical fiction on top of whatever, and whenever one discusses the reified limit, what one really means can be explicated in prior terms. But all mathematical entities are mathematical fictions; rationals are just pairs of integers with certain rules for manipulating them, integers are just pairs of naturals with certain rules for manipulating them, whatever. The entities used in describing limits are no less “actual things that you can talk about…” than the entities used in describing ratios or the entities used in describing differences; they’re all equally just abstract pieces in abstract games.
Not, mind you, that you need even bring up limits, as such, to think about the affinely extended reals; if one thinks of reals as Dedekind cuts on the rationals or anything along those lines, then the positive and negative infinity come along for free, right up until one explicitly bars them. [Of course, no one in high school discusses any kind of formal account of what the real numbers “are”, nor necessarily should they]
(Similarly, when one first ponders the rationals as ratios of integers, their projective extension comes along for free, right up until one explicitly bars it; these aren’t esoteric concepts that the average person will never run into. They’re right up in the thick of everything else we teach in introductory math, but we’ve just standardized on quietly ignoring these particular threads. Which is alright, but let’s not kid ourselves that it’s anything more intrinsic than curricular tradition)
I know a very ingenious computer scientist who spoke of “The square root of NOT.” In circuitry, two invertors in series do nothing:
True (resp. False) → invertor → invertor → True (resp. False)
but two of his SRON (square root of NOT) would invert:
True (resp. False) → SRON → SRON → False (resp. True)
Totally puzzled, I went about my own business. A few years later, believe it or not, I read that, building on this concept, he had developed an actual counting circuit with fewer transistors than ever hitherto achieved! (Hint: The true/false to 1/0 mapping reverses each clock cycle, IIRC.)
For what it’s worth, in quantum computing, one of the most important gates is indeed a square root of NOT.
The guy I’m thinking of (maybe you know him?) carried three polarizing filters in his pocket. He brought them out to spin in his fingers depending on how conversation was going…
Heh, I doubt I know him, but who is he, by any chance?
[As for the quantum sqrt(NOT), if anyone is interested, it actually can be thought of as just 45 degree rotation (in two-dimensional complex vector space, whose associated projective space represents qubit superpositions). Applied twice, you get 90 degree rotation, which sends the horizontal onto the vertical and the vertical onto the horizontal; this is what counts as NOT, and thus 45 degree rotation is sqrt(NOT). The particularly interesting thing is that 45 degree rotation in itself acts as a kind of randomizer (sending the orthogonally aligned axes onto the diagonals perfectly between them), so we have an instance where the interference in quantum computing allows “double-randomization” to cancel out into something not at all random]
A question for the mathematicians, hopefully not too far off-topic.
Is there an extension of the quaternions that’s analogous to what the Riemann sphere is to complex numbers? How about octonions?
Yes, at least for the quaternions, you can do the exact same thing; construct the projective space of one-dimensional subspaces (equivalently, ratios a:b, where a and b are not both zero, modulo ratio equivalence), with the obvious notions of arithmetic, continuity, etc.
This probably works just as well for the octonions, but I haven’t thought through the problem of associativity. (I mean, you can do the same construction, obviously, but I don’t know if you would want to)
Missing words restored in bold.
Also, according to Baez, the associativity is a problem for the above definition of the projective line, but there is still a useful way to do it for the octonions.
The non-associativity is a problem, I mean. You know what I mean. (Hopefully)
I do (I think), and thanks for the answers.
Actually there is. A coin is just a cylinder where d >> l, where d = diameter, and l = lenght. The definition would therefore be just a designation of the two ends of the cylinder as opposite “sides”. 
Still, to summarize the thread for the OP, your question is not really answerable, because mathematicians have a number of different definitions of infinity, each applicable to a separate subset of mathematics. Each is useful in it’s subset, but is incompatible with the others. “Infinity” is no one “thing” that can be examined in isolation from the formal mathematical system to which it belongs. Nor can it really be compared between 2 different formal systems. Sorry… 