Your questions
1. I had that feeling when I was at school time ago, mnyah mnyah, it seemed to me that the important thing was to find the numbers x for which e.g. x^2 - 4x would add up to -3, there seemed to be this mania for expressing it as finding the x for which (x^2 - 3x + 3) amounted to 0. I avoided expressing it that way whenever possible, I tolerated the other way since it was obviously equivalent so it didn't seem all that important and I never asked about this habit; and there weren't the places with nice people like here
in those days either mnyah mnyah.
Now I realize that it was because it points you in the direction of a solution, and that this and maybe at least half the
useful things in maths depend on the exceptional properties of the number 0! The key property that 0 X anything = 0. It is the only number with the property n X anything = n. So then all algebraic equation solving is about
factorising expressions like (x^2 - 3x + 3), to (x – 1)(x – 3) because if any of the separate factors is 0 so is the whole expression by the above property.
Even if we can find solutions by other methods like guessing, trial and error or numerical approximation we would miss things. E.g. we would not have the simple theorem that the number of roots equals the highest degree in x. We would find instead only two solutions to x^3 -5x^2 + 7x = -3, (i). Then you would have a complicated discourse about exceptions I suppose. Instead if you see it as (x – 3)(x – 1)^2 = 0, you see it as three factors always, two of which happen to be the same.
Then e.g. in physics you have conservation laws, which say that despite this and that feature of the situation changing there are some things like total mass, or energy etc. which don’t, their changes add up to nothing. So it gives a feeling of finality somehow to have that nothing on one side of the equation. Then maybe it is used beyond where it needs to be, and not always used, you could say the change in this equals the balancing change in that, but it is not a bad habit.
2 You have the same sort of exceptions if you do not allow complex numbers and the same resolution maintaining the theorem if you do. When you first meet complex numbers when solving quadratic equations, they may seem an irrelevant nuisance and you wave them away; I think the ancient Islamic mathematicians did that while recognising their existence. But then in the fifteenth century Italian mathematicians found that if they had the crazy idea of believing this complex roots meant something and you could work with them like the familiar numbers, then that enabled them to solve
any cubic equation like (i). Moreover you
cannot do it algebraically
without complex numbers even when the solutions are all ornery real numbers.
It was pure formalism without any concrete representation (such as now will be given you) of complex numbers, a leap into abstraction, therefore I think perhaps the most important step in the development of mathematics as we know it. The usefulness and indispensability of complex numbers (e.g. in quantum mechanics) has only ever increased since then.
3 About the Riemann hypothesis there are now popular books like “Dr. Riemann’s zeros” and “Music of the Primes” that ‘explain’ it to laypersons. Well at least they give you an idea of what it is all about and the academic gossip and sociology around those who work on it.
A thing that strikes me though reading these books is what a different thing almost maths of these people is from that of what we’re used to. I mean we are able to go through a chapter or a book and then master some methods with which we can handle a certain area of applications, maybe there are things we wouldn’t have thought of unaided, though some even of these are obvious with hindsight. But what these other guys do seems in a different world, like needing a team that takes a year to check through a proof and you wonder how anyone can possibly see the way to arrive at such a proof in the first place.
I would welcome some comments from professionals on this impression of mine, that maths, at least some of the cutting edge, has become qualitatively a different kind of enterprise from the sort of thing we common users know as maths? Maybe this is not the thread for it.
