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I have noticed that many authors tend to use "let" instead of "for all". For example, they write something like this:

Let $n$ be an even natural number. Then also $n^2$ is even.

I wonder, why they use "let" instead of "for all", also in cases where the "for all"-version sounds quite good:

For all even natural numbers $n$, $n^2$ is even.

Note that "let" has a slightly different meaning than "for all":

The statement "Let $n$ be an even natural number. Then also $n^2$ is even" translated into the logic calculus would be something like: $\mathrm{even}(n)\vdash \mathrm{even}(n\cdot n)$ (this means that "$\mathrm{even}(n\cdot n)$" is true when we are supposing that "$\mathrm{even}(n)$" holds). On the other hand, the statement "For all even natural numbers $n$, $n^2$ ist even" can be translated into a single formula $\forall n.\ \mathrm{even}(n)\implies \mathrm{even}(n\cdot n)$. EDIT: In the formalization of the examples the quantifier $\forall$ ranges only over natural numbers, so this is the type of "object" we are considering.

I think that in most cases the second version ("For all ...") is meant, but the authors however use "let".

Here is my question:

Why do so many authors write their statements in the form "Let [Variable] be a [Type]. Then ...", even when they actually mean "for all" and even when the version written "for all ..." sounds quite good?

Here a example where this causes confusion:

Theorem: Let $G$ be a planar graph, and let $V$ be the number of vertices, $E$ the number of edges and $F$ the number of faces. Then $V-E+F = 2$.

Proof: by induction on the number of edges $E$.

Why is this confusing? Because a proof by induction gives us a "for all"-statement.

Maybe I take the formalization of proofs too serious and exact. In this case: Sorry for the question.

wejtjqwie
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    Do you have an example where this actually causes confusion? – Jason Starr Dec 07 '15 at 15:10
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    I don't know anything about this, but there is a brief discussion from the technical point of view at https://en.wikipedia.org/wiki/Deduction_theorem – Maurício Collares Dec 07 '15 at 15:16
  • How exactly does one parse "For all even natural numbers $n$[...]"? It may be common to say this, but is it really a correct construction? –  Dec 07 '15 at 15:18
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    I know the differences from the technical point of view. In particular, I studied Gentzens calculus of natural deduction. – wejtjqwie Dec 07 '15 at 15:20
  • quid: How do you mean your question? Haven't I given a formalisation of the "for all"-statement in my post? – wejtjqwie Dec 07 '15 at 15:22
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    When you try to prove a universal statement of the form $\forall x P(x)$, you let $x$ be some arbitrary object and prove $P(x)$, from which you can derive $\forall x P(x)$. So, I think, this usage of the word has something to do with the following: https://en.wikipedia.org/wiki/Universal_generalization – Burak Dec 07 '15 at 15:23
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    Generally, I try to have my "then"s paired with "if"s. So that instead of "Let $n$ be an even natural number. Then $n^2$ is also even", I would write "If $n$ is an even natural number, then $n^2$ is also even". In this light, it is a bit clearer why this is equivalent to "For all even natural numbers $n$, $n^2$ is even", which is a way of abbreviating "For all $n$ (if $n$ is an even natural number, then $n^2$ is an even natural number)". – Arturo Magidin Dec 07 '15 at 15:24
  • Burak: Yes, the formulations "Let n be an even natural number. Then also n2 is even" and "For all even natural numbers n, n2 ist even" are equivalent, this says the "universal generalization" – wejtjqwie Dec 07 '15 at 15:24
  • As to your final question: remember that many authors don't write very well. – Arturo Magidin Dec 07 '15 at 15:24
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    I am just not sure if the sentence-fragment "For all even natural numbers $n$" is really grammatically correct (but I am honestly not sure). It seems like some kind of jargon to me. The specific issue I see is the combination of plural and the $n$. –  Dec 07 '15 at 15:27
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    @quid: In a sense it is a bit of jargon, as there is an elided "which we denote by $n$". It's very common, though. – Arturo Magidin Dec 07 '15 at 15:29
  • @wejtjqwie: What I'm saying is that if you were to translate proofs in English to formal proofs, the part where you "let" some object be something would be the part where you introduce a new variable. The universal statement is only proved after you prove the particular instance of the formula for the variable you introduced. So, in order to obtain "for all even n, n^2 is even", you already have to use universal generalization. – Burak Dec 07 '15 at 15:29
  • @Burak: I know, I know. – wejtjqwie Dec 07 '15 at 15:34
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    For me, I think there is kind of a subconscious assumption that the "scope" of the words "for all $n$" is only the rest of the current sentence. If I want to say more than one sentence of things that hold for all even $n$, I'll say "Let $n$ be even". – Nate Eldredge Dec 07 '15 at 15:57
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    Do you see the difference between the question here, and http://math.stackexchange.com/questions/1556708/difference-between-let-and-for-all? – Asaf Karagila Dec 07 '15 at 16:21
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    I have voted to close this as not a research question. I can't really see anyone getting confused by the Theorem given as an example. – Lucia Dec 07 '15 at 16:39
  • @Lucia: I think nobody does research in 'mathematical-writing'. So questions in this section can't be research-questions. But maybe it is interesting for professional mathematicians to discuss when to use "let" and when to use "for all" in their papers. – wejtjqwie Dec 07 '15 at 16:48
  • Somewhere I read an article or post about how type theory more closely matches mathematics as practiced, where one of the examples was that math theorems usually come in three parts: a setup declaring what the objects in the statement are, an assumption about those objects, and a conclusion. Can't remember where though. – Noah Snyder Dec 07 '15 at 16:53
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    There is another issue in your 2nd example, where you have "...$n$, $n^2$ is even." It is very bad style to use mathematical notation this way, separated only by punctuation. Such usage can often cause unnecessary confusion and ambiguity; the reason is that it often isn't initially clear whether one is to parse the thing $n, n^2$ as a single mathematical expression, or as two expressions separated by punctuation. One solves the problem by insisting that words appear between mathematical expressions. In this case, I would write: For every even natural number $n$, the number $n^2$ is also even. – Joel David Hamkins Dec 07 '15 at 17:31
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    Another distinction is that a statement of the form "Let $n$ be even. Then $P(n)$ is true" implicitly suggests that the statement is already going to be useful in applications for a single $n$, whereas a statement of the form "For all even $n$, $P(n)$ is true" implicitly suggests that the main importance of the statement comes from the fact that it holds uniformly for all $n$ in the domain of universal quantification (in this case, the even numbers). – Terry Tao Dec 07 '15 at 17:33
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    For instance, your example of Euler's formula is already very interesting when applied to a single graph $G$, and in many applications one just wants to apply Euler's formula to a specific graph of interest. On the other hand, the importance of a uniform bound such as "For all $x \in X$, $|f(x)| \leq M$" often lies in its uniformity; it's not so much that there is one or two interesting values of $x$ that one has in mind to apply the bound to, but one needs to know that the bound is uniform across all values of $x$ (e.g. in order to control the integral of $f$). – Terry Tao Dec 07 '15 at 17:35
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    This also helps explain the apparent dissonance between the statement you provided of Euler's formula (which is aimed towards applications), and the proof of that formula. In the inductive proof, it now becomes important that the formula is true uniformly for all graphs with a certain number of edges $E$, including those graphs that were not initially of interest to one's application. So the proof is indeed more naturally aligned to a "for all" formulation. However, it is usually better to state theorems to be oriented towards their applications, rather than towards their proofs. – Terry Tao Dec 07 '15 at 17:40

3 Answers3

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The authors do that for two reasons first, to give the reader a breather, second, because they want do do more with the notation than just finish this one sentence. After "Let $n$ ne a natrual number" there is a pause. A pause in which time the notation sinks in, so that people transfer it from their ultrashort memory to their short memory so that it can then be used for various purposes. In particular, the notation then has a longer half-life than the notation in "For every even natural number $n$, the number $n^2$ ist even." In this last sentence, the meaning of $n$ being a natural number, is erased with the period. Not so in the previous case, where it can be used on.

  • Interesting answer, I think there is definitively some truth in it. But the answer does not satisfy me: I think there are many examples where someone writes a "Let"-Theorem where the analogue "For all"-statement is readable. In my example, "For all even natural numbers n, n^2 ist even" is perfectly readable. Besides, changing the MEANING of a statement is not justified by psychological facts, I think. Keep in mind that the "Let"-version is fundamentally different than "For all". – wejtjqwie Dec 07 '15 at 15:42
  • Why has this answer so many likes? My question is not answered. My post contains many more aspects that aren't clear only when reading this answer. – wejtjqwie Dec 07 '15 at 16:16
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    The answer has been upvoted because it answers your question 'I wonder, why they use "let" instead of "for all" ' – Lee Mosher Dec 07 '15 at 16:27
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The "Let ... Then ... " statement is an abuse of language which is also grammatically incorrect. See http://www.math.illinois.edu/~dwest/grammar.html#letthen.

user2944440
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    This does not answer my question at all. – wejtjqwie Dec 07 '15 at 15:45
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    Why is it grammatical incorrect? The page you link to continues to say 'Used at the beginning of a sentence, the English word "Then" is temporal, as in "Then we left." Since the implicative sense of "then" is so common in mathematics, the temporal sense should rarely be used, to avoid confusion. Usually the temporal "then" at the beginning of a sentence can be changed to "Now" or "Next" with less confusion and essentially the same (and more accurate) meaning, especially in a proof.' This sounds like an opinion to me, not like the assertion of something being incorrect. –  Dec 07 '15 at 15:45
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    As in the debate on 'associate to' and 'associate with', I would be careful to just designate common ways of speaking or writing as grammatically incorrect. It is not even entirely clear to me what it means; probably that it conflicts with the usage of English in common manuals. But like a dialect, mathematical English does not need to be identical with "standard English". – Lennart Meier Dec 07 '15 at 15:48
  • @quid - the page also says, of the Let/Then construction, "The second sentence is not a sentence, since the implicative sense of "then" plays the role of a conjunction". I'm not sure why you glossed over that part, since it's exactly the explanation you're asking for.

    Note that temporal "then" is not being used as a conjunction, but as an adverb, so the observation that starting a sentence with temporal "then" is common English usage isn't an argument in favor of the correctness of starting a sentence with implicative "then".

     – Gregory J. Puleo Dec 07 '15 at 15:56
  • @GregoryJ.Puleo I glossed over it as no actual sentence is specified. The assumption there seems to be that every author using a let/then construction wishes to express a condition/conclusion. Then it is argued that this is incorrect. However, the premise is not valid. (Added: It may be the case for the example in OP, but I understood the answer's scope to be broader.) –  Dec 07 '15 at 16:07
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    Checking the nearest dictionary, I find that (as I expected) one of the meanings of "then" is "in that case." Are you claiming that this meaning is not appropriate at the beginning of a sentence? – Andreas Blass Dec 07 '15 at 16:55
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Language, written and spoken, is a flexible beast. English is particularly flexible. There are often many grammatically correct ways to say the same thing. Good, careful writing requires the ability to use this flexibility and to rein it in as appropriate, but this takes a lot of time and requires lots and lots of self-editing. Quick, not-so-careful writing exploits this flexibility, depending on the flexibility of the language to avoid loss of information. Both the careful and the quick ways of writing are useful and important in professional mathematical writing.

For example, consider your two sentences:

Let $n$ be an even natural number. Then also $n^2$ is even.

For all even natural numbers $n$, $n^2$ ist even.

You, and I, and others in this thread, and probably most other experienced mathematical readers, understand the meanings of these two sentences, and we probably all get the same mathematical information from reading them. We probably even instantaneously spell-check "ist" and get "is".

Now it may be that some logical parser tranlates these two sentences into different statements of some symbolic calculus.

But, your parser and my brain might be different. And I, writing the first sentence, might be trying to convey something different than I, writing the second sentence. I might have some didactic reason for writing it one way rather than the other, despite the more "efficient" or "correct" or "machine readable" advantage the other has over the one.

For example, I might have various reasons for expressing a universal quantifier in the fashion of "Let". If you will indulge me, here is one thing I might be trying to convey:

Let $n$ be an even natural number. Any one at all. Like, even one with a gazillion digits. I'm not just talking about $2$ or $4$ or $6$ here!!! No matter WHAT even natural number $n$ we take, also $n^2$ is even.

I'm not trying to be silly here, I'm just trying to point out that conveying a mathematical idea in a human fashion (as opposed to a machine fashion) sometimes requires different modes of expression.

Community
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Lee Mosher
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  • My english is not so good. It would be helpfull if you could summarize your answer in a conclusion or so. Could you give a SHORT answer to the following question: Which reasons are their to use a "Let"-construction in a theorem, when trying to convey a "For all"-statement? – wejtjqwie Dec 07 '15 at 16:33
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    Your post contains many more aspects that aren't clear only when reading such a short answer. – Lee Mosher Dec 07 '15 at 16:38
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    Some time ago, I wrote a piece on my blog that concurs almost exactly with what Lee is saying here: https://terrytao.wordpress.com/advice-on-writing-papers/take-advantage-of-the-english-language/ – Terry Tao Dec 07 '15 at 17:15
  • Standard etiquette is that answering and voting to close should be treated as mutually exclusive. –  Dec 07 '15 at 18:19