Mathematical pet peeves

[S.G. Janssens]

From another forum I once learned the English expression pet peeve.

More and more I realize that in fact I host a whole animal shelter of mathematical (in the broad sense) pet peeves and, maybe worse, sometimes I am causing them to others. In this thread I would like to hear about yours.

To start off, here are two of mine:

1. Writing things such as $f(x)$ when $f$ is meant, or the other way around, or even $f = f(x)$. (Here $f$ is supposed to be a function.)
2. Mandatory zero-based array indexing. (In a large number of programming languages.)

Note 1: I am not trying to pick on students that are "guilty" of these. As I said, I am sure I also cause similar pains to others.

Note 2: I do not intend to start a discussion about whether or not a particular thing should indeed be classified as a pet peeve. Rather, I am just curious about what kind of things make your toes curl.

 

[I like Serena]

1. Writing things such as $f(x)$ when $f$ is meant, or the other way around, or even $f = f(x)$. (Here $f$ is supposed to be a function.)

People have been pointing out to me that $f(x)$ can be seen as short for $x\mapsto f(x)$, since $x$ is unspecified.
That makes $f=f(x)$ short for $f = (x\mapsto f(x))$, which I think is correct isn't it?

But neh, it doesn't really do it for me either.
It still makes my toes curl.

2. Mandatory zero-based array indexing. (In a large number of programming languages.)

Just today, I was using Excel, and I was yet again annoyed by the fact that the row numbering is one off.
That is, I have a header at row 1, and the first value is at row 2.
That means that the last value is at row (n + 1) when I only have n values.
Really, the first row should be numbered 0 or something.
Ah well.

One of the first computer languages I learned was Pascal, which actually allows you to choose the lower bound for indexing.
However, it's conventional to start at 1.
And deviating from convention is a pain in the (oopsie).

As far as I know, it's in particular C/C++ that has 0 based indexing, and some later languages that inherited from it.
To be honest, I'm actually happy there's no choice.
And as C/C++ is a low level language, it kind of makes sense that a is literally the same as *(a + i), which requires 0-based indexing.
If it were not mandatory, as I see it, it would just increase the problems caused by off-by-one mistakes.
Still, if you really want to, you can start from 1 and ignore the value at 0.


Okay, enough of responding to your pet peeves.
Let me bring in one of my own pet peeves.
When doing a substitution, please, please, only do the substitution and nothing else.

 

[topsquark]

Written in the style I've seen on a number of exam sheets:

2x - 6 = 0 = 6/2 = x = 3

-Dan

 

[I like Serena]

Written in the style I've seen on a number of exam sheets:

2x - 6 = 0 = 6/2 = x = 3

-Dan

Yes, I recognize that.
And I've really been trying to teach students that $0 \ne 6/2$, although they tend to look at me and not take me seriously - they got the answer didn't they? Ah well, when I insist enough - each and every time - some of them try to make me feel better (and get rid of the hassle) and do as I ask.

 

[Opalg]

Written in the style I've seen on a number of exam sheets:

2x - 6 = 0 = 6/2 = x = 3

-Dan

All it needs is an arrowhead at one point: $2x - 6 = 0 \Rightarrow 6/2 = x = 3$. (Sun)

I believe that is probably how most students think of it when they write what looks like careless nonsense.

 

[S.G. Janssens]

$^w\log$ ("without loss of generality") is often used, but only sometimes true.

A complete proof using a $^w\log$-like statement first treats the special case (where generality is in fact lost) and then shows explicitly how the general case can be reduced to the special case. Everything else is just laziness. (Laziness, in turn, can be a virtue in mathematics, but in my opinion we should then write: "for simplicity" or something like that.)

 

[Evgeny.Makarov]

That makes $f=f(x)$ short for $f = (x\mapsto f(x))$, which I think is correct isn't it?

Up to η-conversion.

 

[S.G. Janssens]

Up to η-conversion.

Thank you, this is perhaps yet another reason to learn more about functional programming?

Could it be that I find the mixing of $f$ and $f(x)$ confusing (see post #1) because in ordinary mathematical text it is not always clear whether $x$ appears "free" or not?

 

[Evgeny.Makarov]

Could it be that I find the mixing of $f$ and $f(x)$ confusing (see post #1) because in ordinary mathematical text it is not always clear whether $x$ appears "free" or not?

Mixing $f$ and $f(x)$ is indeed confusing from the standpoint of types. If, for example, $f:\mathbb{R}\to\mathbb{R}$ and $x$ is a real number, which is denoted by $x:\mathbb{R}$ in typed lambda-calculus, then $f(x):\mathbb{R}$. That is, $f$ and $f(x)$ are entities of different type. In ordinary notation it may not be clear whether $f(x)$ means a function of a number. But when $f$ is an variable (not a meta-variable standing for some expression), then $x$ does not occur freely in $f$, so there is no doubt that $f$ and $\lambda x.\,f x$ (or $x\mapsto f(x)$) are η-convertible (basically the same thing).

 

[I like Serena]

$^w\log$ ("without loss of generality") is often used, but only sometimes true.

A complete proof using a $^w\log$-like statement first treats the special case (where generality is in fact lost) and then shows explicitly how the general case can be reduced to the special case. Everything else is just laziness. (Laziness, in turn, can be a virtue in mathematics, but in my opinion we should then write: "for simplicity" or something like that.)

Similarly it is trivial or it is obvious are often used when it is not trivial or obvious. I believe it is again out of laziness or actual lack of understanding. And it leads to frustration.

 

[MountEvariste]

My biggest pet peeve is when authors omit important parts of proofs, sometimes without even saying anything about the said omission, while at the same time finding the space to write out more simpler, less important parts. I genuinely believe people like Rudin intentionally committed this practice of writing out trivialities while omitting vital steps within proofs or relegating important results to exercises (I can at least see the utility of the last one... to an extent) so that they have their books enter the folklore of hard and deep mathematics books.

Another one is drowning the student with exercises; as in, hundreds per chapter, like Niven's book on number theory. How is the student supposed to know which exercises are there to consolidate the material? If you skip some exercises you may risk missing out on some excellent problem that would have helped you internalise some results you've just learned; but if you don't skip, it will take a very long time to finish a few chapters (especially since the problems are almost all non-trivial, and largely just novelties).

Don't even get me started on size of American multivariable calculus books!

 

[I like Serena]

Thank you, this is perhaps yet another reason to learn more about functional programming?

Could it be that I find the mixing of $f$ and $f(x)$ confusing (see post #1) because in ordinary mathematical text it is not always clear whether $x$ appears "free" or not?

Mixing $f$ and $f(x)$ is indeed confusing from the standpoint of types. If, for example, $f:\mathbb{R}\to\mathbb{R}$ and $x$ is a real number, which is denoted by $x:\mathbb{R}$ in typed lambda-calculus, then $f(x):\mathbb{R}$. That is, $f$ and $f(x)$ are entities of different type. In ordinary notation it may not be clear whether $f(x)$ means a function of a number. But when $f$ is an variable (not a meta-variable standing for some expression), then $x$ does not occur freely in $f$, so there is no doubt that $f$ and $\lambda x.\,f x$ (or $x\mapsto f(x)$) are η-convertible (basically the same thing).

I just realized that there is yet another notation.
What if we write $f=f(\cdot)$?
That is correct (i.e. equivalent) without ambiguity isn't it?
(Although we could wonder what the point is of writing something like that.)
That is, the confusing part is that $x$ is usually a real number (or equivalent), and it's not always clear or intuitive if it is free in the context or not.
However, a center dot ($\cdot$) in the position of an argument is always assumed to be a placeholder argument and as such free in the context.

 

[I like Serena]

Another one is drowning the student with exercises; as in, hundreds per chapter, like Niven's book on number theory. How is the student supposed to know which exercises are there to consolidate the material? If you skip some exercises you may risk missing out on some excellent problem that would have helped you internalise some results you've just learned; but if you don't skip, it will take a very long time to finish a few chapters (especially since the problems are almost all non-trivial, and largely just novelties).

Don't even get me started on size of American multivariable calculus books!

I found that there's usually a pattern to these exercises.
The first series are usually questions about the words, symbols, and definitions that were introduced in the chapter.
The second series are redoing the examples in the chapter with slightly different context and numbers.
The third series are questions that go beyond the examples and require a thorough understanding of what the chapter is about.
And the fourth series (often mixed with the third) are the starred questions for which a certain mathematical caliber is required, or otherwise an extreme amount of exercise including using different sources and asking help from professors and/or other experts (e.g here on MHB!).

If we feel comfortable with the definitions we can skip to the second series and so on.
Being able to redo the exercises is usually enough to pass an exam.
Doing sufficient problems of the third series guarantees a high grade.
The starred questions are for those seeking perfection and will usually have to be time boxed since time will tend to run out when the exam arrives.

 

[S.G. Janssens]

I just realized that there is yet another notation.
What if we write $f=f(\cdot)$?
That is correct (i.e. equivalent) without ambiguity isn't it?
(Although we could wonder what the point is of writing something like that.)

I think I get the point. (Wink)

That is, the confusing part is that $x$ is usually a real number, and it's not always clear or intuitive if it is free in the context or not.

Yes, exactly.

However, a center dot ($\cdot$) in the position of an argument is always assumed to be a placeholder argument and as such free in the context.

Yes, I agree. I see people do this sometimes even when there is just one argument, but personally I would only do it if there are other arguments in play - such as in https://mathhelpboards.com/differential-equations-17/compact-support-24292.html - where we would write $u(\cdot, t)$ to denote the one-parameter family of functions $x \mapsto u(x,t)$. When there is only one argument, I would indeed write $f$ instead of $f(\cdot)$.

 

[HOI]

Similarly it is trivial or it is obvious are often used when it is not trivial or obvious. I believe it is again out of laziness or actual lack of understanding. And it leads to frustration.

This reminds me of the math professor who, in the middle of a lecture, said "and now it is obvious that" and wrote a formula on the blackboard, then stood back and said "Why is that obvious?"

He then sat down at his desk, wrote furiously for about 15 minutes, then rose and said "Yes, it is obvious!"

 

[DrWahoo]

This reminds me of the math professor who, in the middle of a lecture, said "and now it is obvious that" and wrote a formula on the blackboard, then stood back and said "Why is that obvious?"

He then sat down at his desk, wrote furiously for about 15 minutes, then rose and said "Yes, it is obvious!"

I have to say I am guilty of this. Have been doing a better Job, but typically I only bring up trivial for the 0 case.

We are discussing tangent bundles, currently, and getting into a lot of Li Algebra and topological spaces that they are not use to seeing in an ODE class.

 

[Ackbach]

Similarly it is trivial or it is obvious are often used when it is not trivial or obvious. I believe it is again out of laziness or actual lack of understanding. And it leads to frustration.

I think some people, though by no means all, use this to intimidate the students. Essentially they're saying, "Look at how smart I am", when they should be working to clarify things for the students to show them how smart they are. If someone says that to me, and it's highly non-obvious to me, I have decided just to let that roll off my back now.

 

[Dethrone]

I think some people, though by no means all, use this to intimidate the students. Essentially they're saying, "Look at how smart I am", when they should be working to clarify things for the students to show them how smart they are. If someone says that to me, and it's highly non-obvious to me, I have decided just to let that roll off my back now.

I used to have a prof. that would say even 1 + 1 = 2 wasn't obvious to him... to motivate the axiom and properties of real numbers. (Rofl) He would later go on and state trivial limits, etc., basically anything 'obvious', was not obvious to him, to motivate the rigorous treatment of those concepts.