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Notation as a tool of thought

Paper · 1980 · Kenneth E. Iverson

Summary

Iverson's 1979 Turing Award Lecture asks whether one language can be both a programming language and a medium for mathematical reasoning. Conventional mathematical notation is compact and suggestive, but its meaning shifts with topic, author and context. Most programming languages are general-purpose, unambiguous and runnable, yet mathematicians seldom use them to think. Iverson argues that APL, which began as a notation for writing and teaching before it was implemented, combines both sets of strengths. He introduces APL gradually, in the course of working problems rather than up front. Early examples include counting atoms in close-packed crystal layers and deriving triangular numbers with sum reduction and sum scan. Later sections cover polynomials, conversions between representations of functions such as permutations and directed graphs, and formal proofs of identities used earlier. The lecture closes by comparing APL with standard notation and discussing efficiency. The work matters because it presents programming notation as a way to explore and prove ideas, not only a way to instruct machines.

Key points

  • The paper names five qualities of a good notation: it should express problem constructs easily, be suggestive, subordinate detail, be economical, and be amenable to formal proof. Executability and universality are added on top of these.
  • Worked examples show how the notation suggests general results:
    • the sum of ⍳N equals ((N+1)×N)÷2;
    • the power function relates to times in the same way that times relates to plus;
    • figurate numbers and factorials arise from repeated scans;
    • a single duality pattern covers De Morgan's laws and multiplication via logarithms.
  • The paper has four stages: notation characteristics (§1), polynomials (§2), transformations between function representations (§3), and identities with formal proofs (§4). A concluding comparison with mathematical notation follows.
  • Appendix A summarizes all APL notation used, with fuller coverage in APL Language (IBM, pp. 24–60). Appendix B gives an APL compiler that converts direct-definition functions into canonical, executable form, illustrated with a Fibonacci function.
  • On efficiency, Iverson warns that counting only arithmetic operations ignores housekeeping costs such as indexing. He also notes that languages and hardware have shaped each other in a circular way.

Our summary of the paper, in our own words — not its text. The work belongs to its authors; read it at the links above.

 Institutions

🔬IBM Research - Thomas J. Watson Research CenterYorktown Heights · United States57 works · 49,627 citations

 Fields, topics and concepts

📚 Shelves 3The field of science this work belongs to (OpenAlex fields).Computer Science · 19,503 worksArtificial Intelligence · 18,805 worksComputer Science Applications · 184 works

🧭 Topics 1Research subjects and languages the work is about.Reasoning · 1,948 works

Cite

Cite as Kenneth E. Iverson. 1980. Notation as a tool of thought. Communications of the ACM 23(8):444–465 (Turing Award Lecture 1979).

The paper itself, not this page. BibTeX from the library's record of the work.

@inproceedings{iverson1980notation,
  title = {{Notation as a tool of thought}},
  author = {Kenneth E. Iverson},
  booktitle = {Communications of the ACM 23(8):444–465 (Turing Award Lecture 1979)},
  year = {1980},
  doi = {10.1145/358896.358899},
  url = {https://doi.org/10.1145/358896.358899}
}

References

22 references, 0 of them with a card of ours. ↗ leads to the work, «find» searches for one we have no id for yet.

  1. Boole, G. An Investigation of the Laws of Thought, Dover Publications, N.Y., 1951. Originally published in 1954 by Walton and Maberly, London and by MacMillan and Co., Cambridge. Also available in Volume II of the Collected Logical Works of George Boole, Open Court Publishing Co., La Salle, Illinois… find
  2. Cajori, F. A History of Mathematical Notations, Volume If, Open Court Publishing Co., La Salle, Illinois, 1929. find
  3. Falkoff, A.D., and Iverson, K.E. The Evolution of APL, Proceedings of a Conference on the History of Programming Languages, ACM SIGPLAN, 1978. find
  4. APL Language, Form No. GC26-3847-4, IBM Corporation. find
  5. Iverson, K.E. Elementary Analysis, APL Press, Pleasantville, N.Y., 1976. find
  6. Iverson, K.E. Algebra: an algorithmic treatment, APL Press, Pleasantville, N.Y., 1972. find
  7. Kerner, I.O. Ein Gesamtschrittverfahren zur Berechnung der Nullstellen von Polynomen, Numerische M a t h e m a t i k , Vol. 8, 1966, pp. 290-294. find
  8. Beckenbach, E.F., ed. A p p l i e d Combinatorial M a t h e m a t - ics, John Wiley and Sons, New York, N.Y., 1964. find
  9. Tarjan, R.E, Testing Flow Graph Reducibility, J o u r n a l of C o m p u t e r a n d Systems Sciences, Vol. 9 No. 3, Dec. 1974. find
  10. Spence, R. Resistive Circuit Theory, APL Press, Pleasantville, N.Y., 1972. find
  11. Iverson, K.E. A P r o g r a m m i n g Language, John Wiley and Sons, New York, N.Y., 1962. find
  12. Iverson, K.E. A n I n t r o d u c t i o n to A P L f o r Scientists a n d Engineers, APL Press, Pleasantville, N.Y. find
  13. Orth, D.L. Calculus in a new key, APL Press, Pleasantville, N.Y., 1976. find
  14. Apostol, T.M. M a t h e m a t i c a l Analysis, Addison Wesley Publishing Co., Reading, Mass., 1957. find
  15. Jolley, L.B.W. S u m m a t i o n o f Series, Dover Publications, N.Y. i find
  16. Oldham, K.B., and Spanier, J. The F r a c t i o n a l Calculus, Academic Press, N.Y., 1974. find
  17. A P L Quote Quad, Vol. 9, No. 4, June 1979, ACM STAPL. find
  18. Iverson, K.E., Operators a n d Functions, IBM Research Report RC 7091, 1978. find
  19. McDonnell, E.E., A Notation for the GCD and LCM Functions, A P L 75, P r o c e e d i n g s o f an A P L Conference, ACM, 1975. find
  20. Iverson, K.E., Operators, A C M T r a n s a c t i o n s on Prog r a m m i n g L a n g u a g e s A n d Systems, October 1979. find
  21. Blaauw, G.A., D i g i t a l S y s t e m I m p l e m e n t a t i o n , Prentice-Hall, Englewood Cliffs, N.J., 1976. find
  22. McIn{yre, D.B., The Architectural Elegance of Crystals Made Clear by APL, A n A P L Users M e e t i n g , I.P. Sharp Associates, Toronto, Canada, 1978. Communications of the ACM August 1980 Volume 23 Number 8 find

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