This brings to mind something that has been bothering me for a long time: when should I accept that something is true? rgw points to nice list of definite integrals, each of which evaluates (it is claimed) to Catalan's constant, and then provides one of his own. Robert Baillie then casts some doubt. Are they true? The last one on the list uses C rather than G for Catalan's constant, which seems logical to me, but differs from all the others. Should I be more doubtful of that one? Somehow I am, but more because the calculation is longer. It is suggested that I should verify a calculation before accepting it. Well enough in principle, but I for one would struggle with any of those, pencil an paper, and be more likely to make a mistake than Mathematica. If I fail to corroborate, should I fail to accept the integral? Or should I hold all such statements in some "well, maybe it is true" category, and can I even control, by willing, the level of belief I attribute to something? My tendency would be to try another CAS, maybe maple. But if it failed to corroborate, I would likely say "well, thats CAS's for you, maybe one is right, maybe neither, and anyway they copy each other". Then, if I really cared about the integral I would try some numerical algorithm, and feel a "warm fuzzy" if the numeric value was near Catatlan. That's no proof .. but ultimately I am seeking belief that its true, acceptance .. and maybe that would happen and I would leave it there. How about FLT? I believed it 90% before it was proved. Now I believe it 99.9%. Shall I check it myself? In my case, and I suspect in the case of 99.9 percent of all people, that would be utterly impossible--I don't know enough and never will. How about the 4-color theorem. When I write computer programs they always have bugs. Did Appel and Haken's program have any bugs? When they finally got it to run clean, and those card-punch machines worked perfectly, were there still any hidden bugs? Can I really believe the proof is complete? How about Kepler's sphere-packing conjecture? I have nothing against Thomas Hales. But the problem is just too complicated. Didn't he have a lot of people helping in writing programs? Did they consider every case and cut no corners? Of course I believe the conjecture, but should I believe that it is proved? And the biggest fish of all, the classification of the finite simple groups, how can I know that it is nailed shut? How can anyone know, if the proof is 1000's of pages long, and its explication is a multi-volume project of books not yet complete? What is the status of this anyway? Probably I believe all of these things. But not without angst. Ultimately mathematics is an act of faith -- and more and more so as each day passes. I welcome opposing views and all comments. Jim Buddenhagen On Wed, Jun 11, 2008 at 6:37 AM, Robert Baillie <rjbaillie@frii.com> wrote:
the catalan identites are interesting, and it is likely that most of them are true.
however, the so-called "proofs" (i.e., mathematica output) are not really proofs. even the current version of mathematica has bugs. for example, version 6.0.2 says this series does not converge: Sum[((-1)^(n + 1) + Cos[n Pi/2])/n, {n, 1, Infinity}] even though the sum is Log[2]/2, and even though the two previous versions of mathematica gave the correct answer.
the bottom line is, don't accept output unless you can verify it through an independent calculation.
bob baillie ---
rwg@sdf.lonestar.org wrote:
from http://www.cs.cmu.edu/~adamchik/articles/catalan/catalan.htm In[1]:=Integrate[ArcSin[Sinh[t]], {t, 0, ArcSinh[1]}]
Out[1]=-Catalan + (1/4)*Pi*Log[3 + 2*Sqrt[2]]
(Not the trivial changevar of the ArcSinh[Sin[]].)