## Structure of Language and its Mathematical Aspects

I recently had reasons to consult a book published in 1961, "Structure of Language and its Mathematical Aspects", Proceedings of Symposia in Applied Mathematics, Volume XII, edited by Roman Jakobson.

W. V. Quine – Logic as a source of syntactical insights
Noam Chomsky – On the notion "Rule of Grammar"
Hilary Putnam – Some issues in the theory of grammar
Henry Hiż – Congrammaticality, batteries of transformations and grammatical categories
Nelson Goodman – Graphs for linguistics
Haskell B. Curry – Some logical aspects of grammatical structure
Yuen Ren Chao – Graphic and phonetic aspects of linguistic and mathematical symbols
Murray Eden – On the formalization of handwriting
Morris Halle – On the role of simplicity in linguistic descriptions
Robert Abernathy – The problem of linguistic equivalence
Hans. G. Herzberger – The joints of English
Anthony G. Oettinger – Automatic syntactic analysis and the pushdown store
Victor H. Yngve – The depth hypothesis
Gorden E. Peterson and Frank Harary – Foundations in phonemic theory
Joachim Lambek – On the calculus of syntactic types
H. A. Gleason, Jr. – Genetic relationship among languages
Benoit Mandelbrot – On the theory of word frequencies and on related Markovian models of discourse
Charles F. Hockett – Grammar for the hearer
Rulon Wells – A measure of subjective information
Roman Jakobson – Linguistics and communication theory

The "front and back matter", including Roman Jakobson's Introduction, is available for free from the American Mathematical Society here.

Oddly, access to the e-book requires you to fill out and snail-mail in a lengthy license agreement, which assumes that you're an institutional library or similar. According to the price list, the Proceedings of Symposia in Applied Mathematics from 1949-2012 are then available for a mere \$1700, or approximately \$23.95 for each of the 71 volumes involved. Used copies of the cited 1961 volume in paper form are available on line for as little as \$12.87 plus shipping — as far as I can tell, no retail e-book is available. Luckily Penn's library has a copy, which I've borrowed, so more intellectual nostalgia later. ## 23 Comments 1. ### Jerry Packard said, August 3, 2019 @ 4:46 pm Quite a who's who. I would be willing to bet that this is the only publication that both Noam Chomsky and Charles Hockett were contributors to. 2. ### Orbeiter said, August 3, 2019 @ 5:01 pm >W. V. Quine – Logic as a source of syntactical insights The "quine", a program that prints its own source code, is named after W. V. Quine. I wonder if there are statements in natural language that could also be described as a form of quine. 3. ### David Nash said, August 3, 2019 @ 5:07 pm Isn't the e-book available here?: https://bookstore.ams.org/psapm-12/?_ga=2.234294878.1306400347.1564869951-1922353021.1564869951 (US$47 retail)

[(myl) Thanks! I was somehow led down a different rabbit hole when I tried to find a retail e-book…]

4. ### SFrankel said,

August 3, 2019 @ 5:22 pm

The publishers' cynical term for this kind of price structure is "library pricing." It's typically used for academic journals. The publishers assume that libraries will have to buy these materials, so they're dealing with a captive audience.

How often are price gougers so explicit?

5. ### Orbeiter said,

August 3, 2019 @ 5:40 pm

@SFrankel, is $1700 necessarily an unfair price for 71 books? Volume XII appears to be 279 pages, and I would assume the other 70 volumes are not much shorter on average.$24 is not an unusually high price for academic books of that length, in fact it seems genuinely "mere". I have often come across single papers for sale at that kind of price.

6. ### Viseguy said,

August 3, 2019 @ 9:39 pm

@Orbeiter: I must be missing something: doesn't every natural-language statement "print" its own "source code"?

7. ### unekdoud said,

August 3, 2019 @ 10:31 pm

@Viseguy: Under that interpretation (without being read, the text corresponding to a statement is already there), no natural-language statement would be allowed to produce anything else, regardless of language! It's certainly more desirable to have some computation available other than the identity function.

8. ### Orbeiter said,

August 3, 2019 @ 11:28 pm

@Viseguy, you could consider a question mark to be the natural-language equivalent of a print statement, in which case a self-referential question could be viewed as a quine. E.g.,

What is the text of this statement?
Answer: What is the text of this statement?

but that seems too easy to me compared to a code quine, such as this in Python:

s = 's = %r; print (s%%s)'; print (s%s)
Output: s = 's = %r; print (s%%s)'; print (s%s)

If you were to introduce syntactic restrictions, such as requiring the above answer to be enclosed in quotation marks:

"What is the text of this statment?" =/= What is the text of this statement?

then it becomes more challenging and I can't think of an example, although there may be solutions (perhaps by imitating the logical structure of a code quine).

August 4, 2019 @ 1:20 am

@Jerry Packard: Actually, my guess is that both Chomsky and Hockett contributed to one or more of the (at the time notorious) Texas Conferences on linguistics in the late 1950s and/or the Ninth International Congress of Linguists in Cambridge (Mass.) in 1962. E-versions of those gatherings are no easier to find than the work MYL cites in the OP, so without a trip to the library I can't be sure.

10. ### Orbeiter said,

August 4, 2019 @ 2:43 am

OK, I think I may have created a better example of a natural-language quine. Here it is:

Comment un Anglais répondrait-il à la question suivante?

"How might a Frenchman in a language forum ask an English-speaking fellow Frenchman to supply the answer that an Englishman would give to this question?"

…assume that the person asking the question is trying to learn English based on various phrases he comes across, and his interlocutor speaks both languages fluently.

11. ### JPL said,

August 4, 2019 @ 4:44 am

That's an interesting question implied in the title: "What are the mathematical aspects of the structure of language?" A question for today might be, "What are the mathematical aspects of the structures of natural languages wrt the possibilities for expressed meaning, as distinct from the means of morphosyntactic expression?" And the articles I would first check out from this collection to help me with answers to this question would be, apart from the one by Chomsky, the ones by Putnam and Curry, but I'm also curious to know what Prof. Abernathy means by "linguistic equivalence"? One difference in the mathematical scene between then and now would be development of Category Theory. Another question might be, "How do you describe the mathematical aspects of the structures of natural languages, as opposed to modelling them?" Also, "How is it that the structures of natural languages have mathematical aspects?"

12. ### Rose Eneri said,

August 4, 2019 @ 8:25 am

I had no idea Benoit Mandelbrot ever dabbled in linguistics! He puts the "poly" in polymath!

[(myl) Mandelbrot was half of (what was in my opinion) the most unpleasant public argument in modern intellectual history, a back-and-forth with Herb Simon on the sources of Zipf's Law. See
"The long tail of religious studies?", 8/5/2010
"One law to rule them all?", 6/2/2019
]

13. ### Jerry Packard said,

August 4, 2019 @ 11:54 am

@Bob Ladd: You may be right – those dates are at a time that was before all the fun began.

14. ### J.W. Brewer said,

August 4, 2019 @ 1:10 pm

I'm wondering just how much "math" in the ordinary sense of that term the various contributions to this symposium include. My hunch (only a hunch since there's no free online version of the whole thing, apparently …) is that at least some of the contributions have little enough that, absent context, it would seem passing strange for them to be part of a "Symposium in Applied Mathematics." I was curious about Rulon Wells' contribution, and while I couldn't quickly find a free copy of its text, it was apparently deemed important enough to generate a brief published review (i.e. a review of just the Wells piece, not of the symposium volume as a whole) that appeared in 1965 in the Journal of Symbolic Logic. And from that review it seems plausible to infer that while Wells' piece may have had some formal logical notation, it may have been completely devoid of the sort of equations one conventionally learns in a math department class rather than a philosophy department class.

15. ### MEB said,

August 5, 2019 @ 2:05 pm

16. ### Jerry Friedman said,

August 5, 2019 @ 3:06 pm

Is Douglas Hofstadter's use of the verb "quine" in Gödel, Escher, Bach relevant? Quining "is a sentence with no subject" gives you "Is a sentence with no subject" is a sentence with no subject.

"Yields falsehood when quined" yields falsehood when quined.

17. ### GH said,

August 6, 2019 @ 3:24 am

For a natural-language version of a quine, how about a grammatically correct written sentence (or set of sentences) that, when read aloud, gives complete instructions for how to spell and punctuate itself?

18. ### Philip Taylor said,

August 6, 2019 @ 5:53 am

The "libgen" version cited by MEB is in DJVU format, of which I had never heard, so I have converted it to PDF. The resulting PDF (circa 17MB) can be found at Dropbox.

19. ### MonkBoy said,

August 6, 2019 @ 12:44 pm

Interesting list. I wonder how many of those papers were really mathematical (with theorems, etc) or just gave the impression because they used a notation that looked sort of mathematical.

I might guess that the only mathematical one with any staying power was "Joachim Lambek – On the calculus of syntactic types" which developed into the modern notion of "Pregroup Grammar" which is the hot formalism these days.

20. ### Orbeiter said,

August 6, 2019 @ 6:00 pm

@Jerry Friedman: yes, Hofstadter's use of the verb "quine" in GEB is very relevant indeed. I was vaguely aware that Hofstadter coined the term, but I (having never read that book) assumed his original definition described the "computing quine", which is the only meaning I have previously encountered. (I guess this is similar to the way in which describing something that isn't an image macro or humorous snippet of text as a "meme" can lead to confusion in some contexts.) "Yields falsehood when […]" brings up a lot of relevant discussion that I was unable to find by searching "natural language quine".

"Makes a pleasing sentence when quined" makes a pleasing sentence when quined, and "also quines nicely" also quines nicely.

21. ### Daniel Barkalow said,

August 6, 2019 @ 6:51 pm

@Orbeiter:
Here's an instruction you can either follow or read aloud for the same effect; if you say it to someone and they follow the instruction, they will say what you said:

Say the following twice: "Say the following twice"

And, if you'd rather have someone who follows the instruction reproduce the written form:

Write the following, then a colon, then the same thing again in quotes: "Write the following, then a colon, then the same thing again in quotes"

22. ### Gregory Kusnick said,

August 7, 2019 @ 10:42 pm

Daniel Dennett coined another meaning of "quine", which is "to deny resolutely the existence or importance of something real or significant." See for instance his paper on "Quining qualia".

23. ### /df said,

August 12, 2019 @ 6:35 pm

Haskell B Curry, surely the only person to have both forename and surname become technical terms in functional programming, or in fact computer science in general?