Ken Iverson @ 100

[Roger|Dyalog]

On 2020-12-17 a Zoom meeting, organized by Stephen Taylor and Jake Jacobs, was held to commemorate the centenary of Ken Iverson's birth. A few follow-up thoughts.

The Circle Function

Both Morten Kromberg and Rebecca Kelly (kx) spoke about the circle function APL. Herewith an anecdote about the function, from Eugene McDonnell Quotations and Anecdotes [Hui 2009].

The second thought determined the values of the controlling parameters, by recalling that the sine and the tangent functions were odd functions, as were the hyperbolic sine and hyperbolic tangent. This suggested that odd numbers be used to designate them. The values 1, 3, 5, and 7 seemed appropriate. (An odd function is one for which F-x is equal to -Fx . Signum is an odd function, for example; ×-5 is equal to -×5.)

Actually, 1 and 3 were chosen first, more or less by accident, for the sine and tangent, along with 2 for the cosine function, by listing the functions in the order in which they were taught me in high school, and then the observation was made about sine and tangent being odd functions. The hyperbolic functions simply fell into place afterwards.

— Eugene McDonnell, The Story of ○ [McDonnell 1977]

Circle also stars in another anecdote:

Reasons for Liking ⍟

• It's kind of cute, possessing a radial symmetry.
• It denotes a function for which conventional mathematical notation [Abramowitz and Stegun 1964, §4.1] does not have a good symbol:
  ⍟y ←→ ln y or log y
  x⍟y ←→ logx y
• It alludes to 0=1+*○0j1, the most beautiful equation in all of mathematics [Hui 2010], relating in one short phrase the fundamental quantities 0, 1, e, π, and 0j1 and the basic operations plus, times, and exponentiation.
• It is a visual pun — the symbol looks like the cross section of a felled tree, i.e. a log [McDonnell 1977].

— Roger Hui, My Favorite APL Symbol [Hui 2013]

The connections, like Cleveland, border on the eerie.

Four Jeans

Rob Hodgkinson was the program chairman for the APL 88 conference in Sydney, Australia. On his arrival in Australia, Ken Iverson told Rob that his party included his wife Jean, and Eugene McDonnell and his wife Jeanne, observing that he brought three "jeans" to Australia.

Actually, it was four jeans, because Ken himself was Kenneth Eugene Iverson.

Forever Young

Joey Tuttle related a comment made by Stephen Wolfram at the APL 89 Conference in NYC, that he saw a lot of grey hair and that it was not a good thing. Joey was encouraged to see young faces in the Zoom meeting. Whereupon Morten showed a slide of photos of recent Dyalog APL problem solving competitors. Not a grey hair in sight. It is a good thing.

In a related thread, an attendee said that he knew kdb but doesn't know APL, and asked how long it takes to learn APL. I replied that the Dyalog APL problem solving competitors learn APL on the fly, taking at most a few weeks, and do quite well at it.

Forever

When I was doing grad studies at the University of Toronto some professors commented that to get your degree you have to prove a theorem. (I think they were at least semi-serious.) I am thinking what is the analog for a mathematician to make his or her mark, and I think one way is to devise a function and/or notation which are then used thereafter. On this basis, I think Ken is well on his way.

What are some candidates? (A non-exhaustive list.)

• + - × ÷ etc. working on entire arrays
• ⍟ for logarithm (both the monadic and dyadic cases). See above and My Favorite APL Symbol [Hui 2013].
• * for exp(x) and exponentiation.
• ⌊ (floor) and ⌈ (ceiling). See The Art of Computer Programming, volume 1 [Knuth 1968].
• Propositions (=, ≤, >, etc.) returning a 0 or 1 result. See Two Notes on Notation [Knuth 1992].
• The function denoted by j. in J, especially its dyad: x plus 0j1 times y. To quote [Hui 2020a], "it's not everyday you come across a function whose peers are - and ÷".

Why didn't I say this during the Zoom meeting? A clear case of staircase wit.

Google Doodle

My only disappointment for the day was that Google did not come through with a Google Doodle. I knew that Ken had strong competition, because by some accounts 2020-12-17 was the 250-th anniversary of Beethoven, and if the Google Doodle on December 17 was for Beethoven I would understand. (Beethoven was baptized on 1770-12-17; he himself agreed that he was born on the 16-th.) But Beethoven was not granted a Google Doodle, neither on December 16 nor on December 17. Therefore, there is no excuse.

My Presentation

Orginally, I was not going to do anything more for Ken's centenary, thinking that I have said all I wanted to say in APL Since 1978 [Hui & Kromberg 2020]. But when Stephen Taylor told me he was organizing a meeting and invited me to present, I felt that I can not turn it down.

The text of my presentation can be found here [Hui 2020b].


References

• Abramowitz, M., and I.A. Stegun, Handbook of Mathematical Functions, National Bureau of Standards, 1964.
• Dyalog, APL Problem Solving Competition, 2010-2020.
• Hui, R.K.W., editor, Eugene McDonnell Quotations and Anecdotes, 2009.
• Hui, R.K.W., Euler's Identity, J Wiki Essay, 2010.
• Hui, R.K.W., My Favorite APL Symbol, Vector, volume 26, number 1, 2013-09.
• Hui, R.K.W., j., APL Chat Forum Post, 2020-04-09 (a).
• Hui, R.K.W., Ken Iverson @ 100, 2020-12-17 (b).
• Hui, R.K.W., and M.J. Kromberg, APL Since 1978, Proceedings of the ACM on Programming Languages, volume 4, number HOPL, 2020-06.
• Knuth, D.E., The Art of Computer Programming, volume 1, Addison-Wesley, 1968.
• Knuth, D.E., Two Notes on Notation, American Mathematical Monthly, volume 99, number 5, 1992-05-01.
• McDonnell, E.E., The Story of ○, APL Quote Quad, volume 8, number 2, 1977-12.

[Bob Armstrong]

I thought Paul Penfield at MIT was also involved in the discussions of the definitions of the circle fns .

I posted a comment that I was catercorner from a dinner napkin discussion between Ken an Wolfram at APL89 where I felt Wolfram was missing the importance of what Ken was saying about operations on boolean vectors .
As was obvious to anybody familiar with Gérard Langlet work .

Also , it happens that yesterday SV-FIG had their monthly Zoom , already up on YT . A part was a Forth challenge to find all the palindromic numbers of 5 digits or less : https://youtu.be/BOnqhOrEwxc . ( I was wearing my APL97 T ) .

I think I saw Morten present a 5 token algorithm which itself was a palindrome , but my attention had wondered ( exacerbated by having a 4k screen ) . I'll wait for the YT to see the reality .
I also definitely want to capture that so diverse it's PC picture of Dyalog winners . And clue the SV group in about the competitions and the nature of the APL level questions . The Forth tend to be very much more machine level . ( 2 of the talks were presenting new little thumb sized processor boards with Forth in flash , etc . )