These words should never be uttered by mathematicians

[Hercuflea]

I have to thank one of my undergraduate math professors for teaching me this invaluable lesson when I first began learning analysis. These are some words which are ubiquitous throughout mathematics and physics even Nobel laureates and top journals use them, and there is no excuse for using them other than laziness and bad style.

Math "bad words":
"Let" - ambiguous. It can mean "suppose", "choose", or "define" depending on the situation.
"Where" - lazy style. Always define your variables before using them.
"If...then..." and implication arrows: almost never used correctly. The only time it is safe to use these is in a heading to a theorem/lemma.

Use the following to make your proofs much more clear, precise, and readable:
Need to make a "for every ..." statement? "Suppose..."
Know something exists and want to use it? "Choose..."
Need to make a new definition? "Define..."

Miscellaneous helping words:
"such that": a qualifier, enables more precision.
"Since": helps make a justification for a passive statement
"using the fact that": like "since," but when you are using a piece of information to actually DO something.

The above 6 English words/phrases are just about all you need to write any proof in mathematics (besides mathematical symbolism, of course). Any other words added are just for making a proof look prettier.

 

[jedishrfu]

Removing fuzziness is word usage is good. It might make the proofs more like programming subject to correctness checks.

 

[Formagella]

honestly I don't think those words are so bad, but maybe it's because I studied in another language where very similar expressions are used and they're not ambigous at all.
Also those words have a direct translation to symbolic expressions like ∀⇔∃⇐⇒ without having to change anything up. These symbols are used instead of writing "for every" or whatever, hence when you read the theorem, it will be natural to say for every.

I agree on "where" though, at least in maths. Maths require precise language so it's good to define things first. Outside of maths, it doesn't always make sense though.

I don't agree with the "for every"-"suppose" exchange though.
If something is valid for every real x, it does not mean I'm supposing that x is real. x is whatever it is, if it's real, then the thing is valid, otherwise it's not, but x is free. Maybe some other thing is valid for complex x.

 

[Hercuflea]

honestly I don't think those words are so bad, but maybe it's because I studied in another language where very similar expressions are used and they're not ambigous at all.

I agree on "where" though, at least in maths. Maths require precise language so it's good to define things first. Outside of maths, it doesn't always make sense though.

What language did you study in? I'm sure most languages have words that mean "suppose", "choose" and "define" as these words have distinctly different meanings. The word "Let" is used in many mathematical contexts to mean anyone of the three, and it is often not explicitly stated which, and it is up to the reader to decode what the author means. I'm convinced in many cases the author him/herself doesn't know what they're trying to say either...

I don't have a problem with the use of symbols like ∀, ∃, ⇔ etc., but I do have a problem with authors using one word in different contexts that can take on different meanings...i.e. "let". In the case of the symbols, there is a one-one function from the set of symbols onto the set of words which the symbols represent, so there is no confusion (although the symbols are not to my personal taste, it isn't that much more difficult just to spell out the words if you're typing.)

In the case of ⇐⇒, these symbols are generally translated as "implies...". The word "imply" is also a lazy mathematician's word. It's essentially saying "I could describe this structure directly and in detail, but I'm just going to let you as the reader figure it out for yourself because I can't be bothered." It really makes a proof incomplete if you ask me.

Also those words have a direct translation to symbolic expressions like ∀⇔∃⇐⇒ without having to change anything up. These symbols are used instead of writing "for every" or whatever, hence when you read the theorem, it will be natural to say for every.

Quick clarification: I didn't mean that we should replace "for every" with "suppose." However, if you want to prove a statement about every member of a set S , you must begin by supposing x is a member of S. You can't just say "Let x in S" to start a proof. The reader then has to decide through intuition whether you are choosing a particular member x of S (choosing), or whether you are making a blanket statement about every member of S (supposing).

I don't agree with the "for every"-"suppose" exchange though.
If something is valid for every real x, it does not mean I'm supposing that x is real. x is whatever it is, if it's real, then the thing is valid, otherwise it's not, but x is free. Maybe some other thing is valid for complex x.

"Suppose" is a word you would use to actually begin a proof that something holds for every x in a set S. It doesn't lock you to a particular value of x, it is a statement about every member x of S.

Sure, something could be valid for members of a set of complex numbers, but that would be another proof?

 

[Formagella]

okay I see your point. I guess I've not been exposed to theorems in maths enough to notice the weird uses.

I studied in italian and we say "Sia x" to mean "let x be", or "Sia f(x) = x^2", same as "let f(x) be equal to x^2".

For me the usual use of that word is "Let x be \in S", which means x is a non-specified member of S. Honestly I've not seen many cases where it makes a difference to pick a specific member of the set or not, but I don't study mathematics.
Usually I let x be a member of S, and then come the conditions, like if x > 0 then, or if if x < 0 then. If I make a statement about x, it's about every member of S.

 

[Hercuflea]

okay I see your point. I guess I've not been exposed to theorems in maths enough to notice the weird uses.

I studied in italian and we say "Sia x" to mean "let x be", or "Sia f(x) = x^2", same as "let f(x) be equal to x^2".

Cool. I studied Spanish which is a similar language, and I'm pretty sure "suponer", "escoger" and "definar" can be interchanged with the English words. Like I said, I don't think it's a matter of native language, it's a fault in the fundamental teaching of mathematics.

For me the usual use of that word is "Let x be \in S", which means x is a non-specified member of S. Honestly I've not seen many cases where it makes a difference to pick a specific member of the set or not, but I don't study mathematics.
Usually I let x be a member of S, and then come the conditions, like if x > 0 then, or if if x < 0 then. If I make a statement about x, it's about every member of S.

In that case, you would be using the "suppose" meaning of "let". But what if I want to define δ= ε / 2 for every ε > 0? A lazy mathematician might say "for each ε > 0, let δ = ε / 2". This would be using "let" in an entirely different context and causes a lot of confusion, especially when the readers are new to the topic, which is generally the case in textbooks. A better, and equally easy way to say the same thing would be "For each ε > 0, we define δ = ε / 2". No difference in work input, but much more clear (a simple case).

Another example would be...say I know that w is not an upper bound of a set S. I can't say "let x > w and x in S". This is not precise. Is it saying something about every element of S that is greater than w or is it picking a single member of S greater than w? That's the distinction between suppose and choose. If you know w is not an upper bound of S, choose x such that w < x and x is in S.

Also, if...then... statements can never be used to actually prove anything. They may be valid statements, but simply saying something like "if x ∈ ℝ then x ∈ ℂ" doesn't actually prove that every real number is a complex number. That's why if...then... statements should only be headings to proofs, and not actually in the proof itself.

 

[Ben Niehoff]

I have a somewhat different opinion. I think "where" is useful if not overdone. Sometimes one wants to introduce the big picture first before getting into the details. But it would be bad style to do this all the time.

I have some other opinions about mathematical writing style as well:

I don't have a problem with the use of symbols like ∀, ∃, ⇔ etc., but I do have a problem with authors using one word in different contexts that can take on different meanings...i.e. "let".

I think one should never, ever use these symbols (nor other abbreviation symbols such as $\therefore$). It makes things very dense and confusing to read. Just write out what you want to say in natural language.

In fact, it's important in general to remember that you ARE writing something in natural language. Math is not a secret code for expressing mathematical truths; mathematical symbols should be used to clarify rather than to codify. Sometimes words have more clarity than symbols. Sometimes an equation written in symbols should ALSO be explained in words. And you should keep in mind that when you do use symbols, they are still part of a sentence in natural language, and have a grammatical function in that sentence.

There are, however, some other words that you should never say in a math paper:

"Obviously"
"Clearly"
etc.

In fact, it's best to leave out adverbs all together. If it is obvious, show how it is obvious. If you can't give a one-line proof that is elegant and crystal clear, then it is not "obvious". These kinds of words tend to insult the reader and puff up the author's ego ("What, it's not obvious to you? It's obvious to me!"), and more often than not are used when the author is unable to explain something clearly and hopes that the reader will just play along. Worse still, a major flaw in reasoning might be hidden behind an innocent "clearly".

 

[JorisL]

Ben Niehoff clearly has put a little more thought into this matter.

Usually I'm not to bothered with all this in part because I rarely read and or need pure mathematical results.
Some form of physics or a clear well understood application usually is sufficient for me to understand/agree with a statement.

 

[Hercuflea]

I have a somewhat different opinion. I think "where" is useful if not overdone. Sometimes one wants to introduce the big picture first before getting into the details. But it would be bad style to do this all the time.

This is probably true mostly for physics where many symbols are commonly standardized in the field. A physicist who had to define m=mass etc. in every equation/section would get quite redundant. However, I think where should be used very sparingly, especially in mathematics.

I have some other opinions about mathematical writing style as well:
I think one should never, ever use these symbols (nor other abbreviation symbols such as $\therefore$). It makes things very dense and confusing to read. Just write out what you want to say in natural language.

In fact, it's important in general to remember that you ARE writing something in natural language. Math is not a secret code for expressing mathematical truths; mathematical symbols should be used to clarify rather than to codify. Sometimes words have more clarity than symbols. Sometimes an equation written in symbols should ALSO be explained in words. And you should keep in mind that when you do use symbols, they are still part of a sentence in natural language, and have a grammatical function in that sentence.

I agree, personally it is not to my taste to ever use these symbols. However, I was just making the observation that symbols like these do not do anything to actually change the meaning of the proof, so technically it is safe to use them. There is a one-one function from the set of symbols to the set of English phrases represented by the above symbols, so they can't be misinterpreted if read correctly. It's just silly to do so. If one is typing, it takes infinitesimally more effort to make the proof much much clearer with words. However, words like "let" can change meanings depending on the context, and the reader can interpret them in a way that was not meant by the author, and thus should always be avoided.

Actually, it should be quite possible to write a perfectly valid mathematical research paper without ever using any symbolism at all, since every mathematical symbol can technically be read aloud in English. A paper like this would probably be as long as a sociology thesis, though!

There are, however, some other words that you should never say in a math paper:

"Obviously"
"Clearly"
etc.

In fact, it's best to leave out adverbs all together. If it is obvious, show how it is obvious. If you can't give a one-line proof that is elegant and crystal clear, then it is not "obvious". These kinds of words tend to insult the reader and puff up the author's ego ("What, it's not obvious to you? It's obvious to me!"), and more often than not are used when the author is unable to explain something clearly and hopes that the reader will just play along. Worse still, a major flaw in reasoning might be hidden behind an innocent "clearly".

I agree, these are almost always used to boost the author's ego, to intimidate the students, and/or because the author doesn't actually know how to write a clear proof, and thus shouldn't be teaching the material in the first place. However, "obviously" can be used between two people who mutually know that the other person can prove the statement on demand.

Ben Niehoff clearly has put a little more thought into this matter.

Usually I'm not to bothered with all this in part because I rarely read and or need pure mathematical results.
Some form of physics or a clear well understood application usually is sufficient for me to understand/agree with a statement.

It's true that many symbols in physics are commonly understood. But I have read many a physics textbook where the author will throw in completely obtuse symbols without ever defining their meaning. They also tend to drop terms in equations, sometimes without even mentioning it in the text. It doesn't matter which discipline you are in, if an author can't state clearly their reasoning, then they either don't know the material, they're lazy, or they're on some kind of ego trip. Either way, we should stay away from these books.

 

[nuuskur]

I hate it when in a textbook during a proof, there's written something like: "obviously, trivially, clearly [insert unknown fact here]". Being a first year analysis student, there is very little Trivial about it other than something along the lines of 1+1 = 2. The other textbook follows similar style, however, whenever there is something "trivial" it's indexed and I can look up the paragraph and see the explanation of what is so "trivial".
I have noticed that same bad style in my own writing, though. I, also, like to mention something is "obviously follows that...". I should aim toward my maths proofs being as neutral as possible without anything "obvious" being left out.

 

[JorisL]

[..] They also tend to drop terms in equations, sometimes without even mentioning it in the text. It doesn't matter which discipline you are in, if an author can't state clearly their reasoning, then they either don't know the material, they're lazy, or they're on some kind of ego trip. Either way, we should stay away from these books.

So you say we should use textbooks which literally show full derivations with highly detailed explanations?
Then a lot of students would never learn to understand this stuff. That's like high school books.
Do you think I learned anything about reasoning there? It's when I started to work through the books, filling in the gaps, using sidenotes, that I learned to really do/understand physics.