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[]].)

On 6/11/08, Stephen Gray <stevebg@roadrunner.com> wrote:

... The most complex proofs must be accepted provisionally, since all advanced science and math is accepted on the basis of approval by experts. As much as I dislike it, even mathematics is therefore subject to authority. ...

[A touch of subject-line creep poking its head up here, I fear ...] It's not even that simple (is it ever?). Without having to think about it, I can immediately recall three separate occasions on which I have identified clear, serious logical fallacies in a purported chain of reasoning, only to find my objections dismissed out-of-hand by other (competent, experienced) mathematicians. This phenomenon is by no means uncommon, and occurs --- occasionaly with serious repercussions for the refuter --- in many other apparently objective, scientific disciplines --- but it is disconcerting to discover how often it arises in a subject lacking any obvious reliance on experimental interpretation. The first conclusion I draw from this phenomenon is that, while the fallible human cannot always rely on his own critical powers, neither should he rely on current consensus, often driven as much by emotional and social factors as by intellectual consideration. The second is that psychology plays a larger part in our acceptance of a proof than we might like to think. In particular, as soon as a generally accepted notion is challenged, cognitive dissonance kicks in; the mind --- composure threatened by a looming contradiction which may not even have been recognised at a conscious level --- starts casting around frantically for an "explanation" of the observation which does not involve having to accept that one's current comfortable mental model of the world may have to be abandoned. To put it another way, once you accept that a proof is correct, you invest emotional security in it, which inevitably compromises your future capacity to review it critically. There really seems no way around this: if you keep on chipping away at the foundations, your tower will never reach the first floor. At some stage you have to make a qualitative decision to trust the current construction and proceed farther. But watch your back --- or maybe it should be your feet! Fred Lunnon

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[]].)

rcs>Cosmetic change:

I'd restate the pi term as log(1+sqrt2) pi /2. --Rich

Maybe Mma 7.0 with have CosmeticForm.-) Actually, it usually takes the lazy approach of brutally canonicalizing. NoninvasiveForm would be ArcSinh[1] Pi/2 . WFL> [...]but it is disconcerting to discover how often [fatuous arrogance] arises

in a subject lacking any obvious reliance on experimental interpretation.

Can anyone supply a non-experimental proof or refutation of inf n + 1 ==== ==== \ \ j > > j (- 1) log(j) binomial(n, j - 1) = 0 ? / / ==== ==== n = 0 j = 1 Failing that, let's experiment with it and quarrel over the results! Mike Stay> What's the definition of q-deformed pi, pi_q? Just take that infinite product following (d37) as the definition--it's a q-extension of Wallis's product. (-> pi as q -> 1). rwg>> [Re the display labeled (d183)]

...

near the end of www.tweedledum.com/rwg/idents.htm, suppose a,b>0. How can the 4th equand be negative, but not the 5th?

Mike Stay>If a,b > 0 then W(-e^{-a} b) is negative, so the entire expression is

positive. The only questionable equand in the sequence is, as you noticed, the fourth. But the trick there is that the k = -1 term in the sum outweighs the nonnegative terms.

Spot on, as usual. OK, cruel demystifier (or anyone else puzzle-prone), why does n (n + 1) --------- n 2 (sqrt(2) + 1) pi (- 1) sec(-----------------) 4 4 limit ------------------------------------- = -- ? n -> inf n pi (sqrt(2) + 1) --rwg

That's the q-pochhammer symbol: (c1224) BLOCK([FANCY_DISPLAY : FALSE],QPOCH(A,B,C,Q^2,N),STRINGPOCH(%%) = MAKEPROD(%%)) n - 1 /===\ 2 | | 2 i 2 i 2 i (d1224) (a,b,c;q ) = | | (1 - a q ) (1 - b q ) (1 - c q ) n | | i = 0 I should also re-mention that there's a polynomial relating pi_q, pi_q^2, and pi_q^4 that Gene Salamin turned into a quadratic algorithm for pi. --- On Thu, 6/12/08, Mike Stay <metaweta@gmail.com> wrote:

From: Mike Stay <metaweta@gmail.com> Subject: Re: [math-fun] Missing Catalan identity To: "math-fun" <math-fun@mailman.xmission.com> Date: Thursday, June 12, 2008, 10:41 AM On Thu, Jun 12, 2008 at 4:15 AM, <rwg@sdf.lonestar.org> wrote:

...

Mike Stay> What's the definition of q-deformed pi, pi_q?

Just take that infinite product following (d37) as the definition--it's a q-extension of Wallis's product. (-> pi as q -> 1).

Can you remind me how the semicolon notation works? -- Mike Stay - metaweta@gmail.com http://math.ucr.edu/~mike http://reperiendi.wordpress.com