Timeline andydude Long Time Fellow Posts: 509 Threads: 44 Joined: Aug 2007 05/01/2009, 08:17 PM I made this timeline to put hyperopertions research in perspective, so let me know if I got anything wrong, or missed anything. I'll try to remember to add Maurer and Kneser in a later post. All of these are chronological. 1852 E. M. Lemeray writes Sur les fonctions iteratives er sur une nouvelle fonction (Association Francaise pour l'Avancement des Sciences, Congres Bordeaux 2) in which he calls tetration the "fourth natural algorithm" (according to Knoebel). 1898 H. Schubert writes Grundlagen der Arithmetik (Encyklopadie der Mathematischen Wissenschaften mit Einschluss ihrer Anwendungen) in which he discusses tetration, and calls this the "operation of the fourth kind" (according to Knoebel). 1915 A.A. Bennett writes Note on an Operation of the Third Grade (available here) where he mentions the idea of hyperoperations (specifically commutative hyperoperations). 1928 Wilhelm Ackermann writes Zum Hilbertschen Aufbau der reellen Zahlen (available here) in which he defined the hierarchy $ \phi(a, b, n) = \left\{ \begin{tabular}{ll} a + b &\text{if } n = 0 \\ 0 &\text{if } n = 1, z = 0 \\ 1 &\text{if } n = 2, z = 0 \\ a &\text{if } n > 2, z = 0 \\ \phi(a, \phi(a, b - 1, n), n - 1) &\text{if } n > 2, z > 0 \end{tabular}$ While at first glance this looks identical to hyperoperations, it is in fact not. $ \begin{tabular}{rl} \phi(a, b, 0) & = a + b \\ \phi(a, b, 1) & = a b \\ \phi(a, b, 2) & = a^b \\ \phi(a, b, 3) & = a \uparrow\uparrow (1 + b) \\ \phi(a, 0, 4) & = a \\ \phi(a, 1, 4) & = a \uparrow\uparrow (1 + a) \\ \phi(a, 2, 4) & = a \uparrow\uparrow (1 + a \uparrow\uparrow (1 + a)) \\ \phi(a, 3, 4) & = a \uparrow\uparrow (1 + a \uparrow\uparrow (1 + a \uparrow\uparrow (1 + a))) \\ \end{tabular}$ As you can see, $\phi(a, b, 4)$ is different from pentation in a nontrivial way. 1935 Rozsa Peter writes one of these (not sure which one, got this from Robert Munafo here). This is where the modern Ackermann function was formed, and where the base was assumed to be 2 (which then made the Ackermann function a bivariate function). However, Robert Munafo says that Peter's initial conditions make the "zeration" equal $2b + 1$, which would make this function produce very different values from the Ackermann function today. 1947 Reuben Louis Goodstein writes Transfinite Ordinals in Recursive Number Theory (available here) in which he defines the hierarchy $G(k, a, n) = a {\uparrow}^{k-2} n$. Apparently, this is the first time that addition was assigned $k=1$. Together with the counting of exponents (as opposed to counting operators), this makes it equivalent to hyperoperations as we know them today. Goodstein also coins the terminology tetration, pentation, etc. 1948 Raphael M. Robinson writes Primitive recursive functions (available here) in which he references Peter's work. Robert Munafo says that it is here that the "zeration" becomes $b + 1$ making the Ackermann function we know today. This means that the 2-argument Ackermann function would more correctly be called the Ackermann-Peter-Robinson function (which is quite a mouthful). 1976 Donald E. Knuth writes Coping with Finiteness (available here), in which he coins the up-arrow notation we use all the time. 1987 Nick Bromer writes Superexponentiation (available here), which is probably the source of the super-prefix (used for super-logarithm). He defines an arrow notation similar to Knuth's arrow, but offset by one such that $a \uparrow b = {}^{b}a$ and $a \downarrow b = a^{a^{b-1}}$. Bromer references Knuth, so he was certainly aware of his notation, but apparently he dropped an arrow somewhere... 1993 Markus Müller writes Reigenalgebra (available here) in which he defines both up-arrow and down-arrow notations exactly like Bromer's, but offset from Knuth's arrow notation. 1994 K. K. Nambiar writes Ackermann functions and transfinite ordinals (available here) defines yet another notation for hyperoperations. 1999 Marc Wirz writes Characterizing the Grzegorczyk hierarchy by safe recursion (available here). 2000 Stephen R. Wassell writes Superexponentiation and Fixed Points of Exponential and Logarithmic Functions (available here) which continues Bromer's terminology. Campagnolo and Moore and Costa write An analog characterization of the Grzegorczyk hierarchy (available here, later published in 2002) in which they consider various algorithmic complexity classes, which involve iterated exponentials. They use the notation $\exp^{[n]}(x)$ (base e) and $2^{[n]}(x)$ (base 2) for iterated exponentials. I believe this is the earliest reference to what has evolved into $\exp_b^n(x)$ notation. 2001 Harvey M. Friedman writes Long Finite Sequences (available here) Yunhi Cho and Kyunghwan Park write Inverse Functions of $y = x^{1/x}$ (available where?) uses MacDonnell's hyperpower terminology. Andrew Robbins bo198214 Administrator Posts: 1,389 Threads: 90 Joined: Aug 2007 05/01/2009, 09:40 PM G. Königs: Recherches sur les intégrales de certaines equations functionelles Schroeder, 1884 P. Lévy: Functions à croissants régulière et itération d'ordre fractionnaire, 1928 I think they found the regular iteration limit formulas for the Schröder and Abel functions in the non-parabolic and parabolic case, respectively. BenStandeven Junior Fellow Posts: 27 Threads: 3 Joined: Apr 2009 05/02/2009, 09:57 PM Here's a few more: 1777: Leonhard Euler: De formulis exponentialibus replicatis, Acta Academiae Scientarum Imperialis Petropolitinae, No. 1. (177. He also wrote a second paper, IIRC. I don't remember its name. 1927: Gabriel Sudan: Bull. Math. Soc. Roumaine Sci. 30 (1927), 11 - 30; Jbuch 53, 171. About a precursor to the Ackermann function. 1953, Andrzej Grzegorczyk: Some classes of recursive functions, Rozprawy Matematyczne, 4, 1953, 3-45. at http://matwbn.icm.edu.pl/ksiazki/rm/rm04/rm0401.pdf : Original definition of the Grzegorczyk hierarchy. Ivan Newbie Posts: 1 Threads: 0 Joined: Jul 2020 Yesterday, 05:00 PM When Hilbert (1926, but in a footnote, it says “Vortrag, gehalten am 4. Juni 1925”) reported about Ackermann’s result, he used a slightly different function from the one that Ackermann used in his 1928 paper. Hilbert’s $\phi_n(a,b)$ is actually the same as $\mathrm{hyper}n(a,b)$. (He started at $n=1$.) Hermann Schubert (1899) already considered the general concept of hyperoperations (implied by “and so on”) and addition as a direct operation of first order (“Stufe”), anticipating Hilbert. Here is what he wrote (translated from German): Quote:Addition and subtraction are called basic arithmetic operations of first order, multiplication and division of second order. Furthermore, addition and multiplication are called direct, subtraction and division indirect basic arithmetic operations. […] In the same way as multiplication emerges from addition, exponentiation from multiplication, one could derive from exponentiation as the direct operation of third order a direct operation of fourth order, from it one of fifth order etc. But already the definition of a direct operation of fourth order is, although logically justified, unimportant to the progress of mathematics, because the commutative law already loses its validity at the third order. « Next Oldest | Next Newest »