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When I want to write a paper, two problems bother me in my math writing!

First: what is the difference between be and is or are in math writing? For example, when I want to begin a theorem, I do not know if I should start with "Let $X$ be $Y$" or "Assume $X$ is $Y$". In some situations, it is really difficult to decide whether be is better or is.

Second: I do not know why but several times I used the expression as follows. For example, when I want to define Vandermonde matrix, I usually write as follows and I do not know which format is better or correct

  • The Vandermonde matrix is defined (as follows) or (as shown) or (in the following form) or (by) or (as below)

I would like to ask you to help me about these two difficult problems in my math writing.

Thanks for any suggestions.

Sahiba Arora
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Amin235
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    This is mostly a matter of taste. What you suggested sounds fine. You might enjoy reading this book: http://bookstore.ams.org/hwm – max_zorn Jan 06 '18 at 20:40
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    It is not really a difference between "be" and "is" but between the main verbs "let" and "assume". "Let" wants a direct object an an infinitive, "assume" want a subordinate clause with a finite verb. – hmakholm left over Monica Jan 06 '18 at 20:46
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    And there is no taste involved in the choice of which to use: let and assume always work as Henning describes. (And let and assume are not interchangeable: assume introduces an assumption, an hypothesis, while let introduces an object; the X in «assume X is Y» must have been introduced before that phrase. A good rule is that everything has to be introduced formally) – Mariano Suárez-Álvarez Jan 06 '18 at 20:48
  • As for your s cond point: why not say simply «the Vanderminde matrix is DRAW-THE-MATRIX»? Al.ost always nothing is gained from saying "is defined to be blah" instead of "is blah". – Mariano Suárez-Álvarez Jan 06 '18 at 20:50
  • Although some users voted that my question is off-topic, the @DavidK answer is excellent . Furthermore, I appreciate for your useful comments. – Amin235 Jan 07 '18 at 22:44

2 Answers2

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The use of "be" versus "is" is not a matter of mathematical style: it's basic English grammar.

"Let" is often followed by a clause with a verb in the present subjunctive, which is identical to the infinitive form of the verb. A mnemonic might be the title of the Beatles song, Let It Be.

The clause that follows "assume," however, is either a statement that could be a sentence on its own or is a phrase without a main verb at all. For example, "Assume it is true," which contains the statement, "It is true." Another example is, "Assume the contrary," which you can interpret by inserting "is true" at the end.

Regarding "as follows," this is very often perfectly OK to use. I would not use it to introduce something that was written in the form of a dictionary definition. "As shown below" works similarly. It's hard to give more specific advice without an example of the context.

David K
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  • In older books you can see " If x be 2...." instead of " If x is 2..." . – DanielWainfleet Jan 07 '18 at 03:01
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    The word "be" in "if he be awake" is in the present subjunctive mood; it's proper English, but that use of it sounds old-fashioned. (Looking this up, I realized that "let $x$ be $2$," while not sounding so old-fashioned, is also supposed to be in the present subjunctive mood, which just happens to be indistinguishable from the infinitive in modern English; I have modified the answer accordingly.) – David K Jan 07 '18 at 03:27
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    Very good. I didn't want to get into a discussion about the subjunctive. Most people who have English as a first language have no idea what it is. – DanielWainfleet Jan 07 '18 at 09:08
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    @DanielWainfleet My favorite web page title: "Long live the subjunctive!" – David C. Ullrich Jan 07 '18 at 13:56
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    The Canadian economist and humorous writer Stephen Leacock wrote an essay about a Japanese verb-mood that can be renderedi into English as "shouldn'ta oughta". I.e. " I think I should not have decided that I ought to do what I did", or "I shoudn'ta oughta done that." – DanielWainfleet Jan 07 '18 at 19:13
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"Assume" depends on the context, while "let" does not depend on the context. "let" describes a term or makes a statement that should be true. Use of "assume" is often to divide a compound statement in case of deletion or to prove an assertion.

Ex.: Let $x$ be a real number. Assume $x$ is positive.

Javi
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Ihab Megbil
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