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I'm currently assisting a basic course where the students have to write some proofs. Most of them use terminology like "Let $x$ be a random integer", instead of "Let $x$ be an arbitrary integer" or plainly "Let $x$ be an integer". Am I doing the right thing in correcting them?

My point of view is that a random integer is an integer chosen by some kind of random distribution. In that interpretation, there's some truth in saying that "Let $x$ be a random integer, then $x$ is not equal to 25", since the probability might be zero for that to happen, and in that sense the randomly chosen integer is not going to be 25. Meanwhile the sentence "Let $x$ be an integer, then $x$ is not equal to 25" is blatantly false, since we can take $x=25$ deterministically (but definitely not randomly).

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    $\begingroup$ I don't know why this received a downvote. $\endgroup$ Nov 20 '17 at 17:05
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    $\begingroup$ Yes, you should correct them. Arbitrary means an implicit quantifier in a way that random doesn't. $\endgroup$ Nov 20 '17 at 17:09
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    $\begingroup$ You are right that they are wrong. Whether you are right in correcting them is a completely different question, and depends, among other things, on the students' age / cognitive level. Although if they are writing proofs, they are probably far enough along that you should correct them. $\endgroup$
    – Arthur
    Nov 20 '17 at 17:10
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    $\begingroup$ @Arthur, I'd say no matter the age, it is the right thing to do to point that there is difference in meaning. What depends on age/cognitive level is how deeply one should go in explaining the difference and how insisting one should be that the students use those phrases correctly. $\endgroup$
    – Ennar
    Nov 20 '17 at 17:14
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    $\begingroup$ It isn't true that the probability of a randomly chosen integer being equal to 25 is 0. It depends on the actual distribution which is used to give a meaning to "randomly chosen integer". Since here is no uniform distribution on the set of integers, there is no default meaning for that phrase. $\endgroup$ Nov 20 '17 at 21:51
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I originally wrote a more ambivalent response, but thinking about it further I've changed my mind.

It's clear that the phrase "let $x$ be a random integer" is mathematically . . . bad. What is at question is whether:

  • it is misleading to the student,

  • it is worth correcting,

  • and as a bonus, whether it is worth penalizing when repeated.

I think the answer to (3) is no (unless one is in a class dealing with probability), and the answer to (2) is yes, since if nothing else explaining why the phrase is wrong lets you preemptively address some of the usual confusions around quantifiers (e.g. we're allowed to pick a number that happens to be a counterexample "out of a hat").

I think the answer to (1) (and here's where I've changed my mind) is "yes" - or rather, it is "yes" enough that we should treat it as "yes." I think this is a case where poor use of language early on could set the student up for more confusion down the road, even if they are not being confused by the phrase at the moment. (And this is generally an argument for helping students with language use in mathematics.)

That said, I still think the answer to (3) is no (again, unless the class is dealing with probability).

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In common parlance, random and arbitrary are often used interchangeably. A quick check of on-line dictionaries confirms that the semantic overlap is well established in spite of the different origins of the two words.

The fledgling proof-writers need to be made aware that this is not the case in math, with random being used when probabilities are involved. On the other hand, "Let $x$ be an arbitrary integer; then $P(x)$ holds" translates $\forall x \in \mathbb{Z} \,.\, P(x)$ into English.

Next, it would probably help the aforementioned fledglings if they were shown why the distinction is useful. One practical reason is simplicity. If one deals with an arbitrary integer $x$, all that is assumed is that $x \in \mathbb{Z}$. Could $x = 25$ be true? Of course! Could $x = 25$ be false? Certainly!

If, however, $x$ is a randomly chosen integer, not much may be said without knowing the distribution from which $x$ was drawn. The probability of $x = 25$ may be greater than $0$ if the distribution is not uniform (as it must be if the sample space is countable). Besides, as you may well know, zero probability doesn't mean impossible. By avoiding the use of random all these issues are sidestepped.

In more advanced courses, students will be able to appreciate more reasons for keeping random and arbitrary, as well as probabilistic and nondeterministic, distinct. But the example above should be enough to get them started. At any rate, in framing my feedback to students at their first attempts with proofs, I'd assume that they had the right concept in mind, but didn't pick the correct mathematical term to express it.

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    $\begingroup$ "zero probability obviously doesn't mean impossible". It's not so obvious to me, could you elaborate? Do you mean "probability asymptotically approaches zero" (integers are infinite, so the probability of choosing zero is infinitesimal)? $\endgroup$
    – Barmar
    Nov 20 '17 at 22:13
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    $\begingroup$ > if the distribution is not uniform. and it's always not uniform $\endgroup$
    – RiaD
    Nov 20 '17 at 22:21
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    $\begingroup$ @Barmar, choose random number (uniformly) in interval (0; 1). what was the initial probability of the number you have chosen? $\endgroup$
    – RiaD
    Nov 20 '17 at 22:32
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    $\begingroup$ Infinitessimal, I think, but not zero. $\endgroup$
    – Barmar
    Nov 20 '17 at 22:54
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    $\begingroup$ @Barmar That probability is zero. Probabilities are real numbers. "The probability of X is infinitesimal" means nothing. $\endgroup$ Nov 20 '17 at 23:21
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Actual dictionary definitions: Doing some quick dictionary searching for "arbitrary" gives the definition: "based on random choice or personal whim, rather than any reason or system." The definition given for "random" is "made, done, happening, or chosen without method or conscious decision." In fact, "random" is listed as a synonym for "arbitrary" on an online dictionary. Therefore the interchanging of the two terms is completely understandable.

Technical vs natural language: While it is true that "arbitrary" vs "random" have very different technical meanings in mathematics, they are nearly interchangeable in natural language. It is important to distinguish between natural and technical language usage/meaning.

I would explain this distinction between natural and technical language to my students. That is something that is students of any discipline should be aware of. There may be a risk of muddying the waters though since mastering the actual mathematics at hand may or may not be helped by this discussion of language.

Are you doing the "right" thing in correcting them? Offering a student relevant correct information is always the "right" thing to do. However, it may not always be the right thing to do if there is sufficient risk of it causing more confusion.

Language is a huge problem in mathematics. It's not something that is taught well in my opinion, in terms of how to actually speak mathematics. At least, if the way my students talk is any indication, there is generally a huge gap in being able to do mathematics and being able to explain it verbally in a coherent fashion using technical terminology correctly.

The physical process of choosing a number: Now let's consider the actual physical process of a human choosing an arbitrary number (in the technical sense here). It might be the case that such a physical process could be modeled using a random variable.

So, inasmuch as it is a real process of coming up with an actual example of an arbitrary number, it might actually be a type of random number in the probabilistic modeling sense. Of course, in the actual mathematical context where the number is to be used, it is just an arbitrary number, e.g. to be plugged into an equation.

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