# counting to infinite **URL:** \ **Category:** Factual Questions\ **Created:** [January 6, 2004, 2:48am UTC](https://boards.straightdope.com/t/counting-to-infinite/222940 "2004-01-06T02:48:03Z")\ **Posts on this page:** 7\ **Page:** 3
**Author:** ![kitarak](https://avatars.discourse-cdn.com/v4/letter/k/898d66/32.png) [@kitarak](https://boards.straightdope.com/u/kitarak)\ **Post date:** [January 6, 2004, 9:24pm UTC](https://boards.straightdope.com/t/counting-to-infinite/222940/41 "2004-01-06T21:24:53Z")
> [@](#): > > \*Originally posted by ultrafilter \* > \*\*Number theory is probably a good one. AFAIK, it didn’t have any applications before the twentieth century. \*\* Ah. This is certainly true. I was thinking of more physicsy applications (what can I say? I’m a physics snob 🙂 ) , but this is a good general example. Thanks. ---
**Author:** ![Bytegeist](https://avatars.discourse-cdn.com/v4/letter/b/9dc877/32.png) [@Bytegeist](https://boards.straightdope.com/u/Bytegeist)\ **Post date:** [January 6, 2004, 10:02pm UTC](https://boards.straightdope.com/t/counting-to-infinite/222940/42 "2004-01-06T22:02:20Z")
> [@](#): > > _Originally posted by sadnil_ > **Bytegeist, you provided fantastic answers. … I can confidently say that I better understand mathematics fanatics. I never thought of math as an art.** Thanks for the kind words! I worry a little though about your comparing math to art, even if I’m the one who suggested the comparison. I’ll just add that although mathematicians and artists might typically have the same motive — a passion for truth that ignores practicality — mathematics is still supposed to be objective and impartial, whereas art is more about subjective truths, or even outright distortion for the sake of emotional effect. Not that that’s a bad thing, in its own context. > [@](#): > > \*Originally posted by kitarak \* > \*\*Depends on the mathematician. A lot of applied mathematicians do \… > \*\* Yes, my remark was too sweeping. Thanks for the correction. And thanks for a much better exposition of my point than I was able to manage. > [@](#): > > _Originally posted by Bytegeist_ > **… physicists often discover that mathematicians have already worked out the math that underlies their objects of study — sometimes centuries in advance, before any possible application was in sight.** > > \*Originally posted by kitarak \* > **Actually, could you give me an example of such subjects?** > > \*Originally posted by ultrafilter \* > **Number theory is probably a good one.** That’s the main one I had in mind: specifically, prime factorization as used in cryptography. And now, come to think of it, examples involving “centuries” might be hard to come by. Some others I had in mind for physics are: [ul] [li]Tensors: developed around 1890, but not applied in physics until Einstein’s general theory of relativity.[/li] [li]Quaternions: developed around 1850, but again not used until Einstein.[/li] [li]Non-euclidean geometry: first seriously investigated around the 18th Century, not thought useful until the 20th.[/li] [li]Polyhedra: not especially practical until the discovery of atomic theory and crystal structure.[/li][/ul] Maybe others can add to this list? ---
**Author:** ![ultrafilter](https://avatars.discourse-cdn.com/v4/letter/u/3d9bf3/32.png) [@ultrafilter](https://boards.straightdope.com/u/ultrafilter)\ **Post date:** [January 6, 2004, 10:11pm UTC](https://boards.straightdope.com/t/counting-to-infinite/222940/43 "2004-01-06T22:11:11Z")
I really can’t think of anything other than number theory. If you go back to 1850 or so, there were very few branches of mathematics, and most of them were still in their infancy, or at least nothing like they are today. E.T. Bell wrote of Poincare as being the last generalist, because he did significant work in algebra, analysis, arithmetic, and geometry. Apparently, Bell considered that list exhaustive of the major branches of mathematics, and he was in a position to make that judgment. ---
**Author:** ![The\_PHB](https://avatars.discourse-cdn.com/v4/letter/t/df788c/32.png) [@The\_PHB](https://boards.straightdope.com/u/The_PHB)\ **Post date:** [January 7, 2004, 9:21am UTC](https://boards.straightdope.com/t/counting-to-infinite/222940/44 "2004-01-07T09:21:09Z")
Mention of Poincare makes me wonder whether there’s an addition to **Bytegeist** ’s list. Poincare lived 1854-1912, so presumably most of his best work was around the turn of the century or a bit earlier. But AFAIK his work on dynamical systems didn’t see any real use until the modern interest in chaos theory kicked off, starting with Lorenz I guess in the early 1960’s. But I admit I’m a little out of my depth here. If any more enlightened member of the Teeming Millions is able to correct my understanding, I would appreciate it. ---
**Author:** ![Sergio](https://avatars.discourse-cdn.com/v4/letter/s/b77776/32.png) [@Sergio](https://boards.straightdope.com/u/Sergio)\ **Post date:** [January 7, 2004, 10:56am UTC](https://boards.straightdope.com/t/counting-to-infinite/222940/45 "2004-01-07T10:56:28Z")
> [@](#): > > \*Originally posted by Derleth \* > > Math types have names for these cardinalities. N has cardinality aleph-null. R has cardinality aleph-one. I don’t think too much about what might have cardinality aleph-two. The notion of cardinality aleph-aleph-null frightens me and makes me want to think about a nice, cozy padded room. It’s a long time (almost 40 years) since I studied number theory and, according to **Cabbage** , much have changed since. Anyway here is what I have learned: ℵ-0 is the cardinality of N; ℵ-1 = 2[sup]ℵ-0[/sup] is the cardinality of R (R[sup]2[/sup], R[sup]3[/sup]…); ℵ-2 = 2[sup]ℵ-1[/sup] is the cardinality of the set of all subsets in R (R[sup]2[/sup],…); Since we can associate the points in a plane with pairs in R[sup]2[/sup], ℵ-1 is the number of points in a plane and ℵ-2 is the number of curves in that plane. ---
**Author:** ![Wendell\_Wagner](https://avatars.discourse-cdn.com/v4/letter/w/8491ac/32.png) [@Wendell\_Wagner](https://boards.straightdope.com/u/Wendell_Wagner)\ **Post date:** [January 7, 2004, 11:03am UTC](https://boards.straightdope.com/t/counting-to-infinite/222940/46 "2004-01-07T11:03:31Z")
ultrafilter and Bytegeist and others have already answered some of sadnil’s questions, but there’s a couple of his statements that I’d like to address. sadnil writes: \> Anyway, I want to ask why you math people are so crazy about \> math. Sure, it’s exciting because it is absolute or whatever. \> Even if you don’t know the answer, there is always an answer. and later: \> So, if a math guy says “Math is interesting,” this sort of goes \> against the personal values that led them to enjoy math in the \> first place. You know, non-mathematicians often have weird ideas about what mathematicians are like. Speaking for myself, the notion of certainty was never part of what drew me to math. In any case, anyone who considered becoming a mathematician in the last half of the twentieth century should have know perfectly well that mathematics doesn’t offer certainty. Some theorems are undecidable, and there’s nothing one can do about that. What drew me to math was complexity. I enjoyed the fact that math showed me that the world was more complex than I had previously thought. But that’s why I enjoy lots of things. I’m interested in learning about many subjects, and math isn’t even the only one that I’ve got a degree in. It’s just the one that happens to be my paycheck. And I think that’s more common than you might suppose. There may be a few mathematicians out there who are interested in nothing but math and live just to do more math each day, but that’s pretty rare. Mathematicians, I must report, are actually rather boringly average people, with about the same range of interests as your neighbors. Furthermore, one of the things I’ve learned in my life is that I will never understand the world. The world is too complex and ambiguous for me or (I suspect) anyone else to understand. I’m not bothered by that and I haven’t noticed that other mathematicians are bothered by that. ---
**Author:** ![kitarak](https://avatars.discourse-cdn.com/v4/letter/k/898d66/32.png) [@kitarak](https://boards.straightdope.com/u/kitarak)\ **Post date:** [January 7, 2004, 3:08pm UTC](https://boards.straightdope.com/t/counting-to-infinite/222940/47 "2004-01-07T15:08:04Z")
> [@](#): > > \*Originally posted by Sérgio \* > \*\*It’s a long time (almost 40 years) since I studied number theory and, according to **Cabbage** , much have changed since. Anyway here is what I have learned: > ℵ-0 is the cardinality of N; > ℵ-1 = 2[sup]ℵ-0[/sup] is the cardinality of R (R[sup]2[/sup], R[sup]3[/sup]…); > ℵ-2 = 2[sup]ℵ-1[/sup] is the cardinality of the set of all subsets in R (R[sup]2[/sup],…); > Since we can associate the points in a plane with pairs in R[sup]2[/sup], ℵ-1 is the number of points in a plane and ℵ-2 is the number of curves in that plane. \*\* I’m afraid this is mostly wrong. As mentioned earlier in this thread the statement “ℵ-1 = 2[sup]ℵ-0[/sup]” is known as the continuum hypothesis, and it is an undecidable statement - one can’t deduce whether it is or true or false from the usual axioms os set theory. The statement that for any ordinal a, &alefsymsub[/sub] = 2[sup]ℵ-a[/sup] is known as the generalised continuum hypothesis and is similarily undecidable (even if we assume the continuum hypothesis) Secondly, ℵ-2 is not the number of curves in the plane even if we assume the continuum hypothesis. Because curves are defined as continuous functions there are a lot fewer than one might expect and the number of curves in the plane is in fact the same as the cardinality of the real numbers, usually denoted c. [Previous page](https://boards.straightdope.com/t/counting-to-infinite/222940.md?page=2)