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JA: The issue is moot, as I deleted the problematic word "unfortunately" three rounds ago. If you are referring to the "N o M a s, N o M a s", that's just a bit of kidding around, and at most Ad PseudoNom. Keep yer chin up. Jon Awbrey21:54, 4 May 2006 (UTC)[reply]
There are a large number of claims in many Wikipedia articles, many of them by HolyGanga, that people on the Indian subcontinent anticipated many (most?) European mathematical discoveries at a very early date. I am simply not competent to pass judgment on these claims, but it is, after all, the duty of the person making the claims to provide references. I've followed up some of these references, and what I've discovered is that scholars disagree greatly on the dates of certain manuscripts, sometimes by more than a thousand years! Whereas for Euclid and the later Greek discoveries we have many independent sources and a continuous tradition, the Indian mathematics seems not to have been part of a continuous tradition, but to be scattered in both place and time. Many sources are very difficult to date. Also, one book I read on the subject said that in Indian texts, good mathematics and the wildest superstition would often occur side by side in the same text. I finally decided that this was something only a person who read the various languages involved could possibly sort out. Rick Norwood14:54, 5 May 2006 (UTC)[reply]
You wrote: "Gödel's incompleteness theorems, proved in 1931, showed that many aspects of Hilbert's program had limited potential of succeeding: there exists true statements about, for example, the natural numbers, such that no finitary method exists to prove." But how did Gödel know these statements were true? Because he could prove them! The point was that given any fixed formal system F chosen in advance, he could construct a true statement S not provable from F. But then modify F by adding S as an axiom (which is justified because S is known to be true), and you have a new system F+S from which S is provable. So I think the change you made, in this form, is not correct.
You also wrote in an edit summary: 'Many mathemeticians wonder if the specific problem they are working on is "undecidable:" RH, PvNP, Birch/Swinnerton-Dyer, Navier-Stokes, etc.' I assume you mean this in the sense of independence. (I'm not sure how Navier-Stokes fits in, as it is not a conjecture.) The statement that the foundational crisis no longer assumes an important place in the awareness of mathematicians was definitely not meant to suggest that mathematicians now think all problems can be solved one way or another. It refers to the fact that mathematicians don't have the feeling they are on thin ice, and may any moment fall through a hole caused by a paradoxical inconsistency in the system, like they had in the 20s of last century. The issue referred to here is the perceived unreliability of mathematics, not completeness. Do you agree that indeed mathematicians are not particularly afraid of inconsistencies in ZFC + Law of Excluded Middle or whatever their belief system is, just like sailors are no longer afraid they'll drop over the edge of the world? If so, maybe your issue with the sentence can be resolved by rewording ir. --LambiamTalk00:40, 30 May 2006 (UTC)[reply]
Thanks for the edit; I think the text now leaves little room for misunderstanding the intention. I tweaked the wording a bit further. I'm not quite happy about the F+S thing yet. You are absolute right that there is always a "next" true but unprovable statement. A much simpler statement is that for any system there is a true but unprovable statement. In other words, no system is complete. However, as I tried to point out, the present text, taken at face value, is simply incorrect. I agree with the intention but not with how it is expressed in words.
I agree with you about your assessment of Gödel and Hilbert's program on Wikipedia. A lot of what there is on Wikipedia about the basic and foundational aspects of mathematics and logic is of an appalling quality. Gödel's results in particular seem to attract crackpots; just look at Talk:Gödel's incompleteness theorems/Arguments. But the article itself is also in bad shape. --LambiamTalk17:30, 30 May 2006 (UTC)[reply]
I've explained in a (hopefully) better and more extensive way why and how Gödel's theorems impacted on Hilbert's program. I've further removed the reference to the work by Matiyasevich since it is does not strengthen the "blow" Gödel dealt to Hilbert's program (in that respect it is not worse than G's first theorem, or Turing's halting problem), and the more so since the Grundlagenkrise, which is what the article is about, was raging half a century before M. obtained this stunning result. --LambiamTalk01:49, 31 May 2006 (UTC)[reply]
Thanks for showing me the way to truth, where I found a mention of "the discovery of ... statements whose truth or falsity is undecidable". Given the context the meaning must be "recursively undecidable". But no such statements have been discovered, nor will they ever be: see the last example in Nonconstructive proof. I've corrected it. By the way, the Foundational crisis article no longer exists as such; it has been swallowed up by Foundations of mathematics, but remains undigested. --LambiamTalk02:35, 1 June 2006 (UTC)[reply]
Hello, M a s. The section in question is somewhat controversial, and it has been discussed to an extent on the talk page of that article -- few advocate totally deleting it, many say it should be shortened, etc, etc -- I therefore think it should stay or be judiciously trimmed until such time as it there is further consensus Pablosecca10:46, 22 April 2007 (UTC)[reply]
About 5000 years ago an Egyptian architect at Saqarra came up with a method for describing the arc of a circle at least well enough that his contractor could use it to construct an arch. I know that the Egyptian Rhind papyrus of 1800BC gives the area of a circle as , where is the diameter of the circle, but thats much later.
17. Somers Clarke and R. EnglebachAncient Egyptian Construction and Architecture. Dover. 1990. ISBN0486264858.
A text found at Saquara dating to c 3000 BC or 5000 years BP has a picture of a curve and under the curve the dimensions given in fingers to the right of the circle as reconstructed to the right. It was presumed the horizontal spacing was based on a royal cubit but the results if based on an ordinary cubit are different. It appears the value for PI being used is 3 '8 '64 '1024. which at 3.141601563 is slightly better than the Rhind value.
The circumference of the circle is 1200 fingers and the diameter of the circle is 191 x 2 = 382
3 '8 '64 '1024 x 382 ~= 1200.0
The side of the square is 12 royal cubits and its area is 434 square feet.
The area of the circle is 191^2 x 3.141601563.
The algorithm suggests working with coordinates and numerical analysis to define a curve.
