Overview
StackExchange's live-preview-with-$\rm\LaTeX$-support gives you the ability to see your mathematics properly typeset as you compose your post, but it has some drawbacks (as everyone who has edited anything substantial on the site knows). In particular, the fact that the entire question...
@anon: I finally came across a real inversive-geometryquestion today. I don't think my answer will get many votes since it requires knowing some inversive-geometry :-)
Does anyone care to explain math.stackexchange.com/a/145277/22144 to me? I feel like the issue is that it's inherently ignoring the idea of induction. A'' isn't ensured to be of all the same age.
Could you say that since A'' excludes a_1, the induction isn't possible. (You're metaphorically removing the first falling domino, so no dominos can begin to fall?)
Yes. I use understand that. That's what inspired the metaphor. The metaphor allows me to see the structure of the argument beyond words and ideas, if that makes sense.
@robjohn That was it! I feel like cursing now. I was thinking about $\exp$, but it is clearly the same idea, just you hit the nail in the head. Thanks for that.
if you allow the "principle of the excluded middle" and the existence of a well-ordered infinite set, then the well-ordering implies the principle of induction is true (since we cannot find a least counter-example)
so really, my question is: how do we know that the natural numbers exist (except by postulation)?
@DavidWheeler Sorry, are you considering the natural numbers to be part of the "construction" [I don't know any set theory or logic so I'll use scare quotes] of induction and hence I can't use the usual recursive definition of $\mathbf N$?
That is, $0 = \varnothing$ and then $n + 1 = n \cup \{n\}$?
You can't, formally speaking. But then again, if there's no finite set $\omega$, you don't need induction. You just repeat the inductive step enough times for each specific instance you need.
the thing is, if the set of ALL natural numbers isn't one (isn't a set), that still doesn't mean that the various statements "proved" by an inductive proof still aren't true for various natural numbers which i DO believe in
it seems to me, when people are first exposed to induction...that they are being asked to accept more than what is presented...that there is actually a "back-ground machinery" which is subtler than the naive way induction is presented. this seems "unfair" to me.
That is true, but only to the extent that in accepting formal mathematics you are probably signing a contract with ZFC. Ignoring the way we think of it today, induction is a very old technique that "intuitively" makes sense, and students can learn it in that naive framework.
well, i am thinking of what happened with euclidean geometry. i think non-standard set theories (or topoi, perhaps, i need to know more about those) should be explored in more detail
if one looks how mathematics is actually taught, it is taught "historically", the older concepts are taught first. it may take 2 decades or so before one is learning "current stuff". that seems inefficient.
in my experience, the utility of a concept is often dependent on how wide-spread it is. in a sense, we reinforce the utility of arithmetic simply by placing so much emphasis on teaching it. it becomes a self-fulfilling prophecy.
arguably, arithmetic "mod 2" is considerably easier to grasp then "ordinary arithmetic", and as far as i can see, very widely applicable. so what reason for the delay in teaching it?
a baby example: often students struggle with the fact, that in groups, ab = ba may not be true. but the idea of "commutativity" isn't one we are born with. we LEARN it. so students learing group theory for the first time have to UNLEARN it. how is that efficient?
@DavidWheeler And we’ve used them for so long precisely because they were found necessary. The claim that we find them necessary because we’ve used them for so long inverts the facts.
@DavidWheeler No. The main reason is everyday utility.
David, it's always easiest to start with the most relatable, concrete examples. Does one teach abstract topologies before open sets in R^n? Sometimes we need sacrifice optimal efficiency so that people can understand what's actually going on.
@IsaacSolomon Actually, that’s precisely how I learnt topology, and in general it’s my preferred approach to mathematical ideas. I know from years of experience teaching, however, that this is rather unusual.
@BrianMScott but how are we to gauge the utility of an unknown concept? it doesn't exist yet, but if it did, it's utility would be > 0. olders concepts always have more "utility" in the sense that they are applied more often, but that is often because people only use what they have.
@DavidWheeler I was talking about a comparison that you made between two known concepts, ordinary arithmetic and arithmetic mod $2$. Obviously we don’t teach unknown concepts, so they’re irrelevant to the question of what to teach.
well, arithmetic mod 2 has applications to "logical thinking" (something even used outside of mathematics), whereas most of the applications of "ordinary arithmetic" are internal to mathemematics, in some sense. understand, i'm being somewhat "tongue-in-cheeck", it's ludicrous to attempt to teach advanced math to kindergarteners.
@DavidWheeler I would say that the everyday applications of ordinary arithmetic are wholly external to mathematics, and that the application of arithmetic mod $2$ to logical thinking (as distinct from the abstract field of formal logic) is marginal at best.
this conversation began as a discussion of induction, in particular "why is it true?" it has, along the way, devolved to a discussion of pedagogical practices, which is a thornier subject
my point being, "infinite sets" are something we choose to make part of mathematics, but which are not intrinsically so. one could, for example, deal with the universe of hereditary finite sets. that is just one of many choices which can be made in deciding what is, and is not mathematics.
Even if some ‘we’ were to decide tomorrow to limit ‘our’selves to HF, the rest of set theory and the further complications of category theory would still be part of mathematics.
in my world, the sky is different colors at different times. at noon, on a cloudless day, its "blue" ( i could probably get a frequency range in angstroms, if you need that)
But then, I’ve never understood why people get all worked up over infinity. In my view the formal notion of induction, be it a consequence of ZF or an axiomatic assumption as in PA, is merely a formalization of very nearly as obvious as $2+2=4$.
i understand that if you take it as an axiom it's "true by default", and if you accept that there exists a well-ordered infinite set, it's true because there's no counter-example (assuming that the principle of the excluded middle is a given)
@JM My understanding is that it’s still in fairly common use, but the international committee that standardizes such things deprecates it, since the exponent isn’t a multiple of $3$.
@DavidWheeler At the intuitive level, utterly divorced from formal systems, I find it almost as obvious a logical principle as modus ponens.
@DavidWheeler Why shouldn’t you? They’re obvious intellectual constructs. As I said before, I simply don’t understand why people tie themselves up in knots over such matters.
inductively (yes, an ironic use of the word), if i believe in infinite sets, the existence of uncountable and inaccessible cardinals seems much more plausible
If I may ask a stupid question: how do you then consider a line segment to be continuous if you dispose of the notion that there are infinitely many points within the line segment?
@MattN Yes, and it has holes within holes. :) Here is a bigger picture.
@MattN Yes, that’s fine. If you now modify it by making each rational an isolated point, you have something similar to mine, though not quite homeomorphic.
Can anyone explain how dividing/multiplying $\theta$ in a polar coord affects the cartesian coordinate in the abstract case. i.e. if $r\cos(\theta) = x$, then what does $r\cos(\theta /2) = $?
@stariz77 If $x=r\cos\theta$, $x$ cycles from $r$ to $-r$ and back to $r$ with period $2\pi$. If $x=r\cos (\theta/2)$, it still cycles from $r$ to $-r$ and back to $r$, but it takes twice as long: the period is now $4\pi$, because $\theta/2$ changes only half as fast as $\theta$.
Right, ok yah I could visualize what was happening ok, it was just I'm used to quickly substituting $rcos(\theta)$ for x when needed without thinking what was happening, and drew a blank just then when $\theta$ was divided by 2 in the eq. Thanks.
true story: as i was waking up this morning, i found myself thinking about the uncountable particular point topology, although that's not what i called it.
@DavidWheeler It’s $U_5(105)$; $U_k(n)=\{x\in U(n):x\equiv 1\pmod k\}$, where $U(n)$ is the multiplicative group of units in the ring $\Bbb Z/n\Bbb Z$.
well, i proceed as follows: first, i only know how to add and multiply integers, so first i form the field of fractions of the integral domain Z. next, i note that Z is in fact a euclidean domain, so applying the euclidean algorithm to 3 and 18....
@MattN Every once in a while I get an upvote on a question that I barely remember answering. Which I suppose shows the site serving one of its purposes.
