Attempt to formally generalize log, exp functions to 3,4,5..(n,m) log exp - Printable Version +- Tetration Forum ( https://math.eretrandre.org/tetrationforum)+-- Forum: Tetration and Related Topics ( https://math.eretrandre.org/tetrationforum/forumdisplay.php?fid=1)+--- Forum: Mathematical and General Discussion ( https://math.eretrandre.org/tetrationforum/forumdisplay.php?fid=3)+--- Thread: Attempt to formally generalize log, exp functions to 3,4,5..(n,m) log exp ( /showthread.php?tid=175) |

RE: Attempt to formally generalize log, exp functions to 3,4,5..(n,m) log exp - bo198214 - 06/03/2008
Ivars Wrote:No, for all real points . Such a function can not exist. If then there must exist a such that for all , because otherwise - if in each left neighborhood of there is an with - there is a sequence with which means that does not exist or is 1. But if in whole left neighborhood of then for . RE: Attempt to formally generalize log, exp functions to 3,4,5..(n,m) log exp - Ivars - 06/03/2008
So we have to skip reals. At least formally it seems possible to define such function on hyperreals or superreals (second is more likely as they are discontinued by definition, with gaps) , but as I do not know enough to make such definition I will study a little from both Conway orginal book and Hyperreals, and Cantors ordinals/cardinals because these things does not work without them. It seems that definition of functions on these numbers is not a very popular topic, most works try to link them to real functions as soon as possible, but I have seen brief mentioning of transcendental functions of surreals in the net, as well as surcomplex numbers so far made by simple adding of I to surreals. Here is excerpt of one link I have started to read: Nicolau C Saldanha on surreal functions Quote:Unfortunately, I know of no written reference to this material. I have not any opinion about this, yet. Ivars RE: Attempt to formally generalize log, exp functions to 3,4,5..(n,m) log exp - bo198214 - 06/03/2008
Sorry, but I dont see any meaning for hyper operations in defining a fucked up function that is so crude that you can not define them on the reals. That may be an interestic topic for itself but then rather goes in the direction of Cantor sets or general topology. Good luck. RE: Attempt to formally generalize log, exp functions to 3,4,5..(n,m) log exp - Ivars - 06/03/2008
bo198214 Wrote:Sorry, but I dont see any meaning for hyper operations in defining a fucked up function that is so crude that you can not define them on the reals. Well I found a popular citation that expresses my thoughts rather well and perhaps is more authoritative. From my point of view, given the fast divergent nature of hyperoperations, there can not be a better place to use smaller infinitesimals than used in normal calculus than hyperoperations. Quote:According to Kruskal, these problems could disappear if Ivars RE: Attempt to formally generalize log, exp functions to 3,4,5..(n,m) log exp - andydude - 06/04/2008
Ivars Wrote:Kruskal Wrote:According to Kruskal, these problems could disappear if You need to learn the difference between divergent and false. Andrew Robbins RE: Attempt to formally generalize log, exp functions to 3,4,5..(n,m) log exp - Ivars - 06/04/2008
bo198214 Wrote:Sorry, but I dont see any meaning for hyper operations in defining a fucked up function that is so crude that you can not define them on the reals. andydude Wrote:You need to learn the difference between divergent and false. The other point is who cares about functions on reals if they are so fucked up in principle that they can not deal with simple reduction of tetration to simpler analysis. Disctinction between true and false in tetration and above can not be determined by the properties of function on reals. But of course this is just uneducated intuitive opinion. Though I do not see a problem to define a function like f(xL,x)= 0, f(x, xR)=1. xL, Where xL, xR are Left and Right surreal numbers between any 2 real values of x.Function is just a unique relationship between 2 sets, be it reals and surreals or what ever. Once defined, it exists. Next step is to study if it has any reasonable properties and does it fit the purpose of understanding hyperoperations. Ivars RE: Attempt to formally generalize log, exp functions to 3,4,5..(n,m) log exp - bo198214 - 06/04/2008
Ivars Wrote:The other point is who cares about functions on reals if they are so fucked up in principle that they can not deal with simple reduction of tetration to simpler analysis. I dont know what you mean, the tetrational functions we considered so far are quite well behaved, continuous or even analytic. There is no complicated analysis needed, as far as I can see. For your function you need complicated analysis, and I still dont see the connection/meaning that it could have for hyperoperations. I know Ivars you are great in speculations, but perhaps then mathematics is not the right discipline for you. In mathematics there is the possibility (and the necessity) to verify things, to verify ideas. If you dont like verifying (and this impression is quite strong) then go to philosphy where nobody can disprove you for sure. The gap between what you know and what you want to be true is just too huge. RE: Attempt to formally generalize log, exp functions to 3,4,5..(n,m) log exp - Ivars - 06/04/2008
bo198214 Wrote:I know Ivars you are great in speculations, but perhaps then mathematics is not the right discipline for you. In mathematics there is the possibility (and the necessity) to verify things, to verify ideas. If you dont like verifying (and this impression is quite strong) then go to philosphy where nobody can disprove you for sure. Unfortunately I absolutely am sure that they both converge ultimately ( mathematics and philosophy). So there is no chance to prefer one over another, though the gap you mention is present in both, and my aim is to reduce it as much and as fast as possible using these speculations as motivators. Excuse me for heating up much above the level of clarity of the problem statement I presented. Hyperoperations is the place where new things will be discovered, because they are required by nature to operate, and I want to be part of it, even if only as active observer, if not able to be a contributor really. Ivars RE: Attempt to formally generalize log, exp functions to 3,4,5..(n,m) log exp - Ivars - 12/04/2008
Hi, After some pause, I have found a solution why generalized logarithms will work, despite the proofs by bo, Andy that real number line does not support such extension. I can give only a general confirmation, without details yet. Briefly, because they will take values from INSIDE real number line , where the organization of Real number line will be changed by each level generalized logarithm. From the representation of numbers as ordered lengths of Euclidean straight lines to much more complex ways to organize the Numbers. Each of these INTEGER level number organization ways will be linked to each other via transformation groups of respective projective spaces to spinor spaces (e.g. h(CP1)=Spin(2)). NON-INTEGER levels will be presented by a non-integer iteration point in process a continuous transformation ( like tetration) that maps Real number line with one organization to Real number line with different ordering. Thus, also non-integer operations and dimensions, iterations will obtain their geometric meaning and intuitive understanding will be possible. Here is a link to more detailed first ideas about how tetration transforms +RP1, -RP1, CP1 etc. with comments from Tony Smith about group structure of such transformations. http://math.eretrandre.org/tetrationforum/showthread.php?tid=216 Plus some ideas from Tony Smith about this transformation ( still working on it): Quote:CP1 = SU(2) / U(1) = S3 / S1 = S2 by the Hopf fibration S1 -> S3 -> S2 = 2-sphere Ivars RE: Attempt to formally generalize log, exp functions to 3,4,5..(n,m) log exp - BenStandeven - 05/12/2009
(06/04/2008, 07:20 AM)Ivars Wrote:bo198214 Wrote:Sorry, but I dont see any meaning for hyper operations in defining a fucked up function that is so crude that you can not define them on the reals. Let 0 = {|}, 1 = {0|}, -1 = {|0}, as usual. Then {-1 | 1} = 0. So 1 = f({|0}, 0) = f(-1, 0) = f(-1, {-1 | 1}) = 0. The function is not well-defined. (It would be well-defined on combinatorial games, except that it is of course not defined on all of them.) Also, if f(x^y) = f(x) f(y) on the surreals, then either f is identically 1, f is identically zero, or there is some y with 1^y not equal to 1, or some x with x^1 not equal to x. This follows by bo198214's proof. I daresay all the standard definitions of exponentiation on the surreals have the latter two properties, so if you still want this setup, you'll need a new definition of exponentiation. I would recommend making it commutative, since surreal multiplication is. |