Wikipedia:Reference desk/Archives/Mathematics/2016 September 1

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September 1

Great circles, rhumb lines, isoazimuth

At rhumb line and Great-circle distance, we see a sentence like: "A great circle arc is, together with the rhumb line and the isoazimuthal, one of the three lines that can be drawn between any two points on the earth's surface." The language is vague but suggests some uniqueness theorem - that these three types of arcs are unique with respect to some properties. Is there such a characterization? Staecker (talk) 12:08, 1 September 2016 (UTC)

As far as I can see from a quick scan of the relevant articles, these types of arcs do have unique properties that are defined by two points. A great circle is well known and defined, and is unique unless the two points are diametric opposites. A Rhumb line is a wacky curve that joins 2 points such that every crossing of a longitude meridian is a defined angle - the path varies depending upon the angle chosen - it is a sort of spiral on a sphere, but is (I think) unique. For an isoazimuthal choose one point as the centre and the other as being elsewhere - the isoazimuthal is the circle that goes through the second point - again unique unless you swap the point roles over. -- SGBailey (talk) 15:32, 1 September 2016 (UTC)

A rhumb line (or loxodrome) might be "wacky" from a pure-math POV, but historically it's been fairly important, because it's easy to follow on a ship. You just keep your heading constant. The distance you traverse will be greater than if you followed a great circle, but in lots of cases it's not that much greater. --Trovatore (talk) 00:03, 3 September 2016 (UTC)

Yes, the historical and navigational perspective might be worth a mention. I've been bold and simply stripped out the statements in the article leads in the form they were given. —Quondum 00:17, 3 September 2016 (UTC)

Yes it's true that any two points are joined by a unique great circle, a unique rhumb, and a unique isoazimuth. But I could imagine several other types of "unique" arcs connecting any two points on the sphere- like between any two points, take the great circle line connecting them, but make it wobble a bit. If you prescribe the wobbliness specifically you'll get another scheme for producing arcs between points which is not one of the 3 listed above (great circle, rhumb, isoazimuth). Since I could wobble however I want, I can describe infinitely many "arcing schemes" connecting pairs of points- most of these would be useless. I'm wondering if there is some special property that only the 3 listed schemes have. Staecker (talk) 22:59, 1 September 2016 (UTC)

The property of having a name? (This is tongue in cheek, but it may explain what the person who wrote that sentence was thinking.) --JBL (talk) 00:13, 2 September 2016 (UTC)

I think the characterization that gives the appearance of elevating those three curves into a distinct class is unfortunate. A great circle is unique given two points on a sphere, but the other two require an identified third point (the "north pole") to be unique. So a rhumb line and isoasimuthal are "less unique" than a great circle: they need three points to be specified, not two. The statements in the articles could be toned down to reflect this. —Quondum 22:30, 2 September 2016 (UTC)

Good point there. Staecker (talk) 23:03, 2 September 2016 (UTC)

Looking at it more, those statements should simply be removed. There are (OR alert) exactly two circles on a sphere through a pair of non-antipodal, non-incident points that are uniquely determined under any characterization: a great circle and a smallest circle. Characterizing a rhumb line and isoazimuthal as special curves in the sense of being the only other curves that are uniquely determined is a big stretch. For example, the circle through any two points that also intersects the north pole is at least as special as these two. And I'm sure one could come up with zillions of curves through two points that are uniquely determined given a third point and a characterization, with a bit of creativity. —Quondum 23:44, 2 September 2016 (UTC)

Paris–Harrington theorem

I would like to add the following paragraph to our article Paris–Harrington theorem. Please correct whatever needs a correction.

Actually, the Strengthened Finite Ramsey Theorem (as opposed to the regular Finite Ramsey Theorem), is not expressible in Peano Language (because of the condition that the number of elements of Y is at least the smallest element of Y), so no wonder this theorem is unprovable in PA. In fact: the statement - whose unprovability in PA is claimed by Paris-Harringtion Theorem, was not the Strengthened Finite Ramsey Theorem itself, but rather was a much more complex one - whose phrasing in Peano Language is extremely long - similarly to that of some statements which are both expressible in Peano Language and encoded in Goedel numbers, as follows: The Strengthened Finite Ramsey Theorem can be encoded in a Goedel number - n - about which there is an arithmetical statement φ(n), which is expressible in Peano Language, and whose meaning in the Goedel deciphering method is - that the statement encoded in n is unprovable in PA. Paris-Harringtion theorem states, that even though φ(n) is both provable outside PA and is expressible in Peano Language - it's still unprovable in PA.

HOTmag (talk) 15:34, 1 September 2016 (UTC)

Hi HOTmag, you seem to be quite worried about this general issue. But just because you don't immediately see a direct translation into the first-order language of arithmetic doesn't mean there isn't one. This stuff comes down to things that feel more like "programming" than they do like "mathematics" — a collection of fairly standard, fairly tedious tricks that mostly no one really bothers to remember. I would work a little harder on finding a way of stating it in the language of arithmetic, before you conclude it can't be done. --Trovatore (talk) 22:27, 1 September 2016 (UTC)

@HOTmag: I'm not qualified to comment on the issue but I just wanted to point out that this isn't the place to have the discussion. Even if everyone here said "Yes, absolutely you are correct!" that wouldn't be very relevant because the people you need to collaborate with are the other editors working on that page who quite likely aren't following the discussion here. The place to have this discussion is on the talk page for the article: Talk:Paris–Harrington_theorem This forum is for people who have general questions on topics not about editing articles. I see you have posted on the talk page and some other user disputes your assertion that the theorem is not statable in Peano arithmetic. I think the best thing is to go back to the talk page and to discuss further with that editor to reach a consensus. --MadScientistX11 (talk) 02:35, 2 September 2016 (UTC)

Let's not get too formalistic here. Yes, procedurally, proposed changes to an article go on the article talk page. But that's probably a lightly patrolled page, and posting it here serves a couple of purposes; first, it may draw in people who know something about the issue, and second, it can be interpreted as a question (can you phrase the theorem in the language of arithmetic?) to which we can give an answer here (yes, you can). --Trovatore (talk) 02:39, 2 September 2016 (UTC)

Yes: I've really thought that - even though the Strengthened Finite Ramsey Theorem can be encoded in a Goedel number - still that theorem isn't expressible in Arithmetic Language (because of the condition that "the number of elements of Y is at least the smallest element of Y"), but if you think I'm wrong, then I would be very eager to see how Arithmetic Language can express the idea that "the number of elements of a given set is at least the smallest element of that set". anyway, I would really be grateful if you could find a way of stating the Strengthened Finite Ramsey Theorem in Arithmetic Language, and I think all of that would contribute a lot to the reliability of our article. HOTmag (talk) 06:48, 2 September 2016 (UTC)

So the central thing you need here, I think, is to see that finite sets (or finite sequences; doesn't make much difference) are expressible in the language of arithmetic. That is, there should be a way of coding up any finite set/sequence as a natural number, in such a way that the questions you want to ask about the set or sequence are definable in arithmetic.

For example, you might try to come up with a coding that allows any natural number to be decoded into a finite set, and find formulas φ and ψ such that, for any m and n, φ(m,n) holds if and only if n is an element of the finite set coded by m, and ψ(m,n) holds if and only if n is the cardinality of the set coded by m.

Now you've broken the question up into subproblems, possibly easier to solve. The subproblems are: (1) What's the coding? (2) How can you see that every finite set can be coded (possibly with repetition; don't waste time trying to make the codes unique)? (3) What are φ and ψ? (4) Given that you have this, how do you express the strong finite Ramsey theorem?

Hint: Use the Chinese remainder theorem. --Trovatore (talk) 07:03, 2 September 2016 (UTC)

Thanks. HOTmag (talk) 19:27, 3 September 2016 (UTC)