---
title: "Notation as a tool of thought — paper by Iverson (1980)"
description: "He introduces APL gradually, in the course of working problems rather than up front. 22 references, cited by 5 papers here."
authors: ["Kenneth E. Iverson"]
year: 1980
venue: "Communications of the ACM 23(8):444–465 (Turing Award Lecture 1979)"
source: "https://www.sigapl.org/Articles/"
licence: "All rights reserved — free to read at the source"
summary_licence: "Elamur Cosmic License"
canonical: "https://research.elamur.ai/free/en/papers/iverson-1980-notation-as-tool-thought/"
---
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# Notation as a tool of thought

**Authors:** Kenneth E. Iverson · **Year:** 1980 · **Venue:** [Communications of the ACM 23(8):444–465 (Turing Award Lecture 1979)](https://md.research.elamur.ai/free/en/venues/communications-of-the-acm/) ([Communications of the ACM 1980](https://md.research.elamur.ai/free/en/venues/communications-of-the-acm/1980/)) · **Publisher:** [Association for Computing Machinery](https://md.research.elamur.ai/free/en/publishers/association-for-computing-machinery/) · **Licence:** All rights reserved — free to read at the source

**Links:** [Publisher's page](https://doi.org/10.1145/358896.358899) · [PDF](https://www.sigapl.org/Articles/Notation%20as%20a%20Tool%20of%20Thought_a1979-iverson.pdf) · [OpenAlex](https://openalex.org/W1987727095)

**Citations:** 189 citations (OpenAlex, 2026-10-14) · [5](https://md.research.elamur.ai/free/en/papers/iverson-1980-notation-as-tool-thought/cited-by/) in our library, of the 149,439 papers whose bibliographies we read

## 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 Center](https://md.research.elamur.ai/free/en/labs/ibm-research-thomas-j-watson-research-center/) — Research institute, Yorktown Heights, United States · 57 works · 49,627 citations

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🧭 Topics (1): [🧮 Reasoning](https://md.research.elamur.ai/free/en/topics/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.

```bibtex
@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}
}
```

- [← previous Learning and reasoning by analogy](https://md.research.elamur.ai/free/en/papers/winston-1980-learning-reasoning-analogy/)
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- [↑ back to papers of 1980](https://md.research.elamur.ai/free/en/papers/1980/)

## 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](https://scholar.google.com/scholar?q=Boole%2C+G%2E+An+Investigation+of+the+Laws+of+Thought%2C+Dover+Publications%2C+N%2EY%2E%2C+1951%2E+Originally+published+in+1954+by+Walton+and+Maberly%2C+London+and+by+MacMillan+and+Co%2E%2C+Cambridge%2E+Also+available+in+Vol)
2. Cajori, F. A History of Mathematical Notations, Volume If, Open Court Publishing Co., La Salle, Illinois, 1929. [find](https://scholar.google.com/scholar?q=Cajori%2C+F%2E+A+History+of+Mathematical+Notations%2C+Volume+If%2C+Open+Court+Publishing+Co%2E%2C+La+Salle%2C+Illinois%2C+1929%2E)
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](https://scholar.google.com/scholar?q=Falkoff%2C+A%2ED%2E%2C+and+Iverson%2C+K%2EE%2E+The+Evolution+of+APL%2C+Proceedings+of+a+Conference+on+the+History+of+Programming+Languages%2C+ACM+SIGPLAN%2C+1978%2E)
4. APL Language, Form No. GC26-3847-4, IBM Corporation. [find](https://scholar.google.com/scholar?q=APL+Language%2C+Form+No%2E+GC26%2D3847%2D4%2C+IBM+Corporation%2E)
5. Iverson, K.E. Elementary Analysis, APL Press, Pleasantville, N.Y., 1976. [find](https://scholar.google.com/scholar?q=Iverson%2C+K%2EE%2E+Elementary+Analysis%2C+APL+Press%2C+Pleasantville%2C+N%2EY%2E%2C+1976%2E)
6. Iverson, K.E. Algebra: an algorithmic treatment, APL Press, Pleasantville, N.Y., 1972. [find](https://scholar.google.com/scholar?q=Iverson%2C+K%2EE%2E+Algebra%3A+an+algorithmic+treatment%2C+APL+Press%2C+Pleasantville%2C+N%2EY%2E%2C+1972%2E)
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](https://scholar.google.com/scholar?q=Kerner%2C+I%2EO%2E+Ein+Gesamtschrittverfahren+zur+Berechnung+der+Nullstellen+von+Polynomen%2C+Numerische+M+a+t+h+e+m+a+t+i+k+%2C+Vol%2E+8%2C+1966%2C+pp%2E+290%2D294%2E)
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](https://scholar.google.com/scholar?q=Beckenbach%2C+E%2EF%2E%2C+ed%2E+A+p+p+l+i+e+d+Combinatorial+M+a+t+h+e+m+a+t+%2D+ics%2C+John+Wiley+and+Sons%2C+New+York%2C+N%2EY%2E%2C+1964%2E)
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](https://scholar.google.com/scholar?q=Tarjan%2C+R%2EE%2C+Testing+Flow+Graph+Reducibility%2C+J+o+u+r+n+a+l+of+C+o+m+p+u+t+e+r+a+n+d+Systems+Sciences%2C+Vol%2E+9+No%2E+3%2C+Dec%2E+1974%2E)
10. Spence, R. Resistive Circuit Theory, APL Press, Pleasantville, N.Y., 1972. [find](https://scholar.google.com/scholar?q=Spence%2C+R%2E+Resistive+Circuit+Theory%2C+APL+Press%2C+Pleasantville%2C+N%2EY%2E%2C+1972%2E)
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](https://scholar.google.com/scholar?q=Iverson%2C+K%2EE%2E+A+P+r+o+g+r+a+m+m+i+n+g+Language%2C+John+Wiley+and+Sons%2C+New+York%2C+N%2EY%2E%2C+1962%2E)
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](https://scholar.google.com/scholar?q=Iverson%2C+K%2EE%2E+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%2C+APL+Press%2C+Pleasantville%2C+N%2EY%2E)
13. Orth, D.L. Calculus in a new key, APL Press, Pleasantville, N.Y., 1976. [find](https://scholar.google.com/scholar?q=Orth%2C+D%2EL%2E+Calculus+in+a+new+key%2C+APL+Press%2C+Pleasantville%2C+N%2EY%2E%2C+1976%2E)
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](https://scholar.google.com/scholar?q=Apostol%2C+T%2EM%2E+M+a+t+h+e+m+a+t+i+c+a+l+Analysis%2C+Addison+Wesley+Publishing+Co%2E%2C+Reading%2C+Mass%2E%2C+1957%2E)
15. Jolley, L.B.W. S u m m a t i o n o f Series, Dover Publications, N.Y. i [find](https://scholar.google.com/scholar?q=Jolley%2C+L%2EB%2EW%2E+S+u+m+m+a+t+i+o+n+o+f+Series%2C+Dover+Publications%2C+N%2EY%2E+i)
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](https://scholar.google.com/scholar?q=Oldham%2C+K%2EB%2E%2C+and+Spanier%2C+J%2E+The+F+r+a+c+t+i+o+n+a+l+Calculus%2C+Academic+Press%2C+N%2EY%2E%2C+1974%2E)
17. A P L Quote Quad, Vol. 9, No. 4, June 1979, ACM STAPL. [find](https://scholar.google.com/scholar?q=A+P+L+Quote+Quad%2C+Vol%2E+9%2C+No%2E+4%2C+June+1979%2C+ACM+STAPL%2E)
18. Iverson, K.E., Operators a n d Functions, IBM Research Report RC 7091, 1978. [find](https://scholar.google.com/scholar?q=Iverson%2C+K%2EE%2E%2C+Operators+a+n+d+Functions%2C+IBM+Research+Report+RC+7091%2C+1978%2E)
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](https://scholar.google.com/scholar?q=McDonnell%2C+E%2EE%2E%2C+A+Notation+for+the+GCD+and+LCM+Functions%2C+A+P+L+75%2C+P+r+o+c+e+e+d+i+n+g+s+o+f+an+A+P+L+Conference%2C+ACM%2C+1975%2E)
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](https://scholar.google.com/scholar?q=Iverson%2C+K%2EE%2E%2C+Operators%2C+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%2C+October+1979%2E)
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](https://scholar.google.com/scholar?q=Blaauw%2C+G%2EA%2E%2C+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+%2C+Prentice%2DHall%2C+Englewood+Cliffs%2C+N%2EJ%2E%2C+1976%2E)
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](https://scholar.google.com/scholar?q=McIn%7Byre%2C+D%2EB%2E%2C+The+Architectural+Elegance+of+Crystals+Made+Clear+by+APL%2C+A+n+A+P+L+Users+M+e+e+t+i+n+g+%2C+I%2EP%2E+Sharp+Associates%2C+Toronto%2C+Canada%2C+1978%2E+Communications+of+the+ACM+August+1980+Volume+23+)

*Updated 3 November 2026*