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Knowing that the words in a written language can be represented by combinations of symbols (e.g. letters of an alphabet), I would be interested to learn what kinds of structural restrictions there are to individual words. Specifically, I have two questions:

1) In principle, can a written language contain infinite words?

2) In principle, can a language contain some words that are represented only by a non-linear structure of letters? For example, can the following conglomerate of symbols be considered as a "word", if we assume that mathematics is a language?

\int_{a}^{b}{e^{x}}dx

  • This looks to be of interest. amazon.com/… – user4894 Jan 24 '17 at 6:25
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    About the use of an infinite sequence of symbols to represent a word in a language : how to write/utter a phrase if the first word will never end ? – Mauro ALLEGRANZA Jan 24 '17 at 7:17
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    I would suggest that vocal inflection is part of any language. If you wanted to actually accurately represent an inarticulate whine, in full detail, it might require infinitely many decorations, and many of them would be diacritical or graphic elaborations, and therefore not linearly placed. Language is not naturally written, so the question kind of comes from an odd place. Of course, in recording expression, at some point, we 'round off' real sounds to an available arrangement of symbols, but the nature of the approximation is not fixed. – jobermark Jan 24 '17 at 10:48
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    Yeah, musical notation seems like a better example. But as long as we're talking LaTeX, I'd remind you that there's even LaTeX markup for that, e.g., stackoverflow.com/questions/648429/typesetting-music-in-latex Of course, your original "infinite" wouldn't happen either way. – John Forkosh Jan 25 '17 at 6:38
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    89.898989(89) is not an infinite string of symbols: it is clearly finite. And about "the constant e is actually represented by the infinite sequence of digits: 2.71828 ....", it is the other way round: teh number e (its name is clearly a finite string of symbols) can be calculated with a sequence of digits taht starts with 2.71828. – Mauro ALLEGRANZA Jan 25 '17 at 8:56
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I recently came across term Morpheme

In linguistics, a morpheme is the smallest grammatical unit in a language. In other words, it is the smallest meaningful unit of a language. The field of study dedicated to morphemes is called morphology. A morpheme is not identical to a word, and the principal difference between the two is that a morpheme may or may not stand alone, whereas a word, by definition, is freestanding. When it stands by itself, it is considered as a root because it has a meaning of its own (e.g. the morpheme cat) and when it depends on another morpheme to express an idea, it is an affix because it has a grammatical function (e.g. the –s in cats to indicate that it is plural).[1] Every word comprises one or more morphemes.

Also, relating to the concept of a word, it's interesting to note the term in a computer science context where word size is considered.

In computing, a word is the natural unit of data used by a particular processor design. A word is a fixed-sized piece of data handled as a unit by the instruction set or the hardware of the processor. The number of bits in a word (the word size, word width, or word length) is an important characteristic of any specific processor design or computer architecture.

  • Thank you, Ron. I think, this is a good addition to this topic. – Noviff Feb 1 '17 at 17:18
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In response to (2),

"The definite integral, from x=a to x=b, of Euler's number multiplied by itself x-number of times" is not a sentence, but rather is a predicate term when combined with a copula. Therein, it can be a word as a propositional term that is a referent of the aforementioned predicate term.

It is a word to the extent that a predicate without a copula can be a word. It is a predicate because it refers to the properties that are in the extension of that mathematical function. To equate the function to something is to create a sentence wherein "equals" is the predicate's copula. For example, "The definite integral, from x=a to x=b, of Euler's number multiplied by itself x-number of times equals the number Z", in first-order logic, has the following form (wherein P is the predicate term "is equal to the number Z" and wherein Q is the predicate term "is equal to the definite integral, from x=a to x=b, of Euler's number multiplied by itself x-number of times":

∃y:Py∧Qy.

Whether or not the categorical proposition is true is another matter entirely. It's possible that ∃y:Py∧Qy is false such that ∄y:Py∧Qy possibly is true.

  • I agree that this integral is not a sentence. My question is "Can we consider this integral as a word in the mathematical language?" And from your answer I conclude that "it can be a word as a propositional term ..." – Noviff Feb 3 '17 at 18:24
  • I have made an edit. Please let me know if I can be more clear. – אהרן רובין Feb 3 '17 at 21:38
  • (+1) Thank you for your clarification. – Noviff Feb 4 '17 at 1:58
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1) In principle, can a written language contain infinite words?

Most languages can generate expressions that have an infinite number of words. The clinching argument for that, for me, was this post by Allan C. Wechsler.

(2) if and only if there exist utterances of infinite length.
This inference is false. A simple counterexample: Although there are
only a finite number of digits (= language elements) and although no
decimal numeral (= utterance) is of infinite length, there are
undoutedly an infinite number of decimal numerals.

Our innate grammars allow for such phrases to exist. Also consider this list of longest words across languages, where many of the languages allow for arbitrary compounding. The longest example there is a 431-letter-long word in Sanskrit that was written sometime in the Middle Ages. Or consider the English example of the scientific name of Titin, which consists of 189,819 letters. The rules of physics and chemistry may prevent us from creating infinitely large molecules, but the rules of IUPAC nomenclature do not prevent us from creating names for infinitely long molecules. Whether we're using Sanskrit's sandhi rules or IUPAC nomenclature, the grammars do not prevent us from forming infinitely long words.

In principle, can a language contain some words that are represented only by a non-linear structure of letters?

Most Indic scripts are not as linear as Latin scripts. Devanagari features compounds and conjuncts where some letters modify other letters from left, right, above, below, or within. But why is that relevant? Indian languages can be written in a linear form as well, commonly with IAST or ITRANS. The same goes for mathematical expressions. Your two-dimensional example is often written in a linear way in LaTeX or Mathematica.

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Unless you really dislike Kanji, and logorams in general, hoeroglyphs, and ideograms, I have no idea why.

You could I suppose argue that ideogarms are letters, but I am fairly sure these characters are not an alphabet.

Written signs in other writing systems are best called syllabograms (which denote a syllable) or logograms (which denote a word or phrase).


Spoken words are made up of units of sound called phonemes, and written words of symbols called graphemes, such as the letters of the English alphabet.

Clearly the symbols are graphemes:

There are additional graphemic components used in writing, such as punctuation marks, mathematical symbols, word dividers such as the space, and other typographic symbols.


As to your questions:

  1. All languages are construtions, and any infintely long word would be impossible to write, and that language could not be written. Unless you allow a grapheme which says e.g. that the preceeding letters had to be uttered an infinte number of times. That would be a planned language

  2. I can only really guess what you mean by "non-linear structures of letters", but e.g. in Chinese there are 214 radicals to the Kangxi Dictionary, and

Radicals may appear in any position in a character. For example, the radical 女 appears on the left side in the characters 姐, 媽, 她, 好 and 姓, but it appears at the bottom in 妾.

I would be interested to see a language, aside from mathematics, which uses an alphabet similarly.

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This seems like a very strange question in terms of languages as they actually are.

In principle, may a language contain some words that can be represented only by infinite sequences of letters

No. Science after all looks at evidence; and the evidence appropriate here are natural languages; there are no languages with 'infinite sequences of letters'; and this can easily be deduced from a principle - who would have the time to write out such a sequence?

Grammars are represented by finite-state-automata; there is the Chomskian hierarchy of formal grammars of types 0-3.

The type of grammar that would fit in such a language as you're suggesting is type-0; but ths is because it allows anything, they're equivalent to a Turing machine, so more like a computer than a actual, living grammar; although I should not that even actual computers have finite resources so there is no such thing as an actual turing machine with an infinite tape.

  • Thank you for your answer. In your response, you stated that "there are no languages with 'infinite sequences of letters'". How about the language of mathematics? In this language, the mathematical constant e is actually represented by the infinite sequence of digits: 2.71828 ...., and each digit is nothing but a letter in the numerical alphabet. I agree with you that the language of mathematics is not a formal language in Chomsky's hierarchy and it's statements can not be processed by Turing machine; but, it seems a valid language for me. – Noviff Jan 24 '17 at 18:16
  • @Noviff: you got it backwards. the infinite seq 2.71828... is represented by the symbol e, not the other way around. lots of natural language words "represent" infinities without themselves being infinite. e.g "God", "love", "infinity". – user20153 Jan 24 '17 at 20:21
  • @mobileink, Good point. However, I think, symbols and letters are two different things. The purpose of the numerical alphabet is to represent any numbers (including transcendental ones) by digits, and this alphabet does not contain a digit e. For this reason, a symbol e is meaningless if it does not have a digital interpretation. – Noviff Jan 24 '17 at 21:10
  • Re e, pi, etc, they're computable reals, whereby there exist relatively short computer programs that output them, digit-by-digit (granted, they take a long time to run to completion:). So e can be represented by the sequence of symbols comprising any such corresponding program. However, there also exist uncomputable reals, which comprise the vast majority of reals (indeed, the computable reals are measure zero). And it would take an "infinite sequence of symbols" to fully represent any of them. But in an epsilon-delta sense, a finite sequence can denote them to any accuracy – John Forkosh Jan 24 '17 at 23:47
  • @noviff: lke the man says, you've got it backwards; its the letter e that represents the number. The numerical representation you are referring to is always approximate as you can never write down the infinite expansion; whereas, the letter e that represents it, is always exact. – Mozibur Ullah Jan 31 '17 at 8:59
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Linear structures of words, sentences, paragraphs, and even books in natural languages could be explained by the property of a human brain to process symbols sequentially, one at a time. Interesting, but sounds and images we process differently, and that is why we can hear at once all musical instruments in the orchestra and we are able to see visual objects at once. For this reason, musical accords are not just sequences of notes and pictorial elements on drawings are not arranged linearly.

Another case is programming languages. Used for human/computer interactions, these languages are designed to present the same information in two different forms: as source codes for humans and as machine codes for computers. While source codes are linear structures - they look almost as texts in natural languages; machine codes are not linear structures - computers do not process machine codes sequentially, in a predefined order.

Furthermore, the language of mathematics is probably the most difficult case. For example, no doubt, the mathematical formula that represents the law of gravity contains some information; however, I am not certain who/what are originators and primary consumers of this information: are they physical objects them-self, a gravitational field around these objects, or it is just my imagination proposes this law? Moreover, it is not even clear how much information does this formula contain: are there few bytes of data that just enough for writing this formula or an infinite amount of information that defines all possible gravitational trajectories in our Universe.

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