Yet there aren’t any 37-page threads where people argue stubbornly that what they learned in elementary school definitively proves that ladybugs are mammals, and zoologists who disagree are just educated stupid and evil eggheads who don’t understand the real world.
Also, I think that Weierstrass’ \delta-\epsilon approach to limits and analysis in general is being given short shrift as honstandard analysis becomes more widely known and discussed. There’s obviously nothing sketchy about hyperreals, but the former generalizes more naturally to other spaces (particularly metric spaces or topological spaces in general). The main appeal of the latter in teaching, irrespective of its role in actual math, is that it appeals to people’s intuitive idea of what they expect infinitesimals to be— which, as the 0.999… = 1 thread demonstrates, is not something that really deserves to be encouraged.
Delta-epsilontics is great (and not actually that far from Robinson-style nonstandard analysis; just framed a little differently), but I’ll note that hyperreals generalize just fine to metric spaces or topological spaces in general: e.g., one common approach to hyperreals is as functions from N to R modulo a free ultrafilter, and in exactly the same way, we can consider hyper-elements of an arbitrary space T as functions from N to T modulo a free ultrafilter.
Indeed, I personally think of the definition of continuity in point-set topology in this way: When is f continuous at x? When we are assured that each hyper-element y which is infinitesimally close to x has f(y) infinitesimally close to f(x). Identifying “infinitesimally close to” with “in all the same open sets as”, and actually considering nontrivial hyper-elements, this reproduces the standard definition.
That’s a reasonable way to think of it… But the definition is even more clever. It’s kind of “legalistic.” It’s a challenge, so to speak, from the believer to the skeptic.
“Go ahead. Name a very small positive number. I can squeeze in a solution to the function that is within that distance of the limit.” i.e., if you say “1 millionth,” I can show a solution that is actually within “1 two-millionth” of the limit. No matter how small the number you offer, I can show a solution within that tolerance…and better.
It’s a kind of proof by exhaustion – by which I mean fatigue! You can keep naming small numbers as long as you want, and I can always construct an answer within that distance. Since you can never actually stymie me, in due course, you give up.
This is the “legal argument” answer to the .999…farrago. Go ahead: name a very small number. 1 millionth? I win by putting down seven nines. Try again. 1 billionth? I win by putting down ten nines. I can always win.
I honestly do not get why this is not convincing. It’s the definition of the “epsilon delta” proofs that underlie calculus.
As far as I can see it, the problem is psychological. After being unsuccessful in naming a bunch of really small numbers, the skeptic just shouts, “It ain’t! You’re fibbing! Prove what you say!” and then goes “blahblahblah” when you say anything.
(Like the vaguely amusing scene in Guardians of the Galaxy.)
I am not joking when I talk about how innumerate I am: liberal arts student and all those excuses. I recognize that the gang of idiots in that thread know more math words and symbols than I. So why is it that their abject wrongness is even obvious to me? Why is it that I can look at .99999…, see how the ellipsis says it goes on forever, and conclude that yes, that’s pretty much 1, and move on with my life? Worse, what could be more blindingly obvious than .3333… = 1/3? They seem too smart to be retarded and too committed to be trolls. Are they insane?
You know that you’re not a mathematician, and so you’re totally fine accepting that 0.999… = 1. You agree with the statements and arguments you understand; for the others, you can at least trust that mathematicians do know what they’re talking about and note that all sources agree on the matter, and move on. And there’s nothing wrong with not knowing something. I know nothing at all about, say, biology, but I don’t go on threads about biology and medicine and insist that, say, I’ve disproved germ theory.
Crackpottery is an entirely different business. It’s like the difference between being an idiot and being a fucking idiot, but that’s a bit hard on people who are genuinely trying to learn. Crackpots are people who are totally convinced that they’re brilliant geniuses who would revolutionize the state of math or science if only the orthodoxy were wise enough to listen. They have some pet theory (0.999 != 1, Einstein was wrong about special relativity, quantum mechanics means that anything can happen at any time for no reason, etc.) that they can’t resist expounding upon at every opportunity, and they summarily dismiss any criticism of it.
Take the recent fuckwit Oh noes so many 7s!!111 or the little-mourned Conspiracy Tide, for instance. They’re mathematically literate, yet they’re irrationally convinced that they’ve figured out some clever point of high-school math that mathematicians have not only missed for centuries, but are unable to understand even now. What makes them really crackpots, though, is that they’re just unable to comprehend the idea that they might be wrong. Both posters I mentioned stopped trying to respond to criticisms of their remarks early on, for example, and just parrot back their initial claim over and over again. If they deign to respond to specific arguments at all, it’s just to rant about how mathematicians and scientists don’t understand the real world, how the scientific orthodoxy is suppressing their brilliance, how we’re all too short-sighted to think as creatively as them (see schooner26’s recent dumbassery about how he can’t be limited by something as trivial as definitions), etc.
The same thing happens all the time with special relativity and quantum mechanics, though it’s fortunately a bit rarer in math. We had some fuckwit post here a while ago about how some ridiculous setup with rotating mirrors totally disproved special relativity. Or, at least it would if scientists would get out of their ivory towers and understand the real world. This particular poster started screaming when the board predictably asked him to use math to clarify his ramblings. See, he couldn’t handle middle-school level algebra, but he could totally revolutionize the world of special relativity because he was just that inherently awesome. (And he checked out literally a hundred books on relativity out of the library. Holy fucking shit!)
There are some common traits I’ve noticed among crackpots:
They suck at math, and they don’t think science should have anything to do with math.
They have backgrounds in engineering, or at least pretensions to backgrounds in engineering. I don’t mean to say anything derogatory about engineers or engineering, and certainly 99.999% of engineers are clever and not crackpots. There’s a sort of mentality, though, that thinks that everything works according to classical, Newtonian physics, and that quantum mechanics, relativity, or even real numbers should act like things in the real world, whatever that means.
They’re utterly incapable of realizing that they’re wrong. It’s not just that they’re mistaken, but that there is no possible way in which they can be wrong. They don’t even bother responding to counterarguments; they just rant about how they’re smarter than everyone else (despite clearly failing at it) or how everyone else is brainwashed into accepting the prevailing orthodoxy.
They consider themselves to be experts in the field in question despite (or in fact because of) being amateurs. You don’t really need to study quantum mechanics, relativity, analysis, etc. to understand it. Besides, study would just be an indoctrination. The latter fellow I mentioned, for example, repeatedly insisted that he was a physicist, despite having no background in physics or conducting any work in physics or writing any papers in physics.
And so on. In short: No, they’re fucktarded. You’re not; you’re just not a mathematician, which doesn’t mean that you’re stupid or a crackpot.
(And sorry for the long post. ‘In short’, indeed.)
Are you sure about that? As a matter of fact, for much of the past 37 years my job title included the word “Engineer,” though I found it pretentious and mostly called myself a “Glorified Drafter” or “CAD Guy.” And I must admit I’ve found few conspiracy theories I didn’t love dearly. Doesn’t mean I believed them; just means my head is easily turned by well-crafted crackpottery,
Since even now my errors in Algebra often turn on transposed or missing numbers and when totalling a list of numbers I do it two or more times and settle on the total I reached twice as being probably correct, I’ve begun to suspect some undiagnosed dyscalculia (they didn’t have that condition when I was a kid) might have limited me, and it makes me sad.