Natural logs and products of no primes

The e-mail you get after you write an article about number theory is very interesting.  For one thing, you’re reminded of phrasings which have one meaning among mathematicians, but a slightly different one outside the tribe.

The majority of the e-mail I’ve gotten about the bounded gaps piece concerns two questions of this kind:  I’ll answer them both here, in case other readers are following the link from Slate to the blog.

Q:  You say that the number of primes less than X is about X/log(X), but don’t you mean X/ln(X)?

A:  When mathematicians say “log” we mean the natural log, the thing which in some other contexts (e.g. Google’s search bar calculator) is denoted “ln.”  But mathematicians never say “ln.”  (To be honest, we kind of think the base-10 logarithm should be called “lu.”)

Q:  You say that every positive number is the product of primes, but this is not true for prime numbers themselves, which can’t be expressed as products.

A:  A prime number is indeed the product of prime numbers!  It is the product of just one prime number, itself.

What about 1?  It’s the product of zero prime numbers.


10 thoughts on “Natural logs and products of no primes

  1. Terence Tao says:

    With regards to the second question, the general issue here is that mathematicians tend to include the degenerate and trivial cases by default when interpreting a mathematical concept (because this gives much better closure properties with respect to mathematical deduction), whereas non-mathematicians tend to exclude them (because this makes for more informative and efficient communication, provided that nobody involved is overly pedantic or trying to be a wise guy). For instance, “X or Y” is usually interpreted by default as the exclusive or by the public, but the inclusive or by mathematicians. And “if X, then Y” is interpreted by default as material implication by mathematicians, but an indicative implication by non-mathematicians.

    I encountered this sort of issue in my own posts on Goldbach type problems, for instance it was disputed that 7 was the sum of three primes (because to a non-mathematician, the phrase “three primes” defaults to “three distinct primes” rather than “three primes, not necessarily distinct”).

  2. JSE says:

    Reminiscent of the old joke that a mathematician is a person who, when asked, “Do you want the soup or the salad?” responds “Yes.”

  3. Jason Starr says:

    Jordan, were you still a grad student when the Clay Millenium Prizes were announced in the press? There was a great story from those days (which I cannot personally confirm): the phone rang in one of the grad student alcoves. The grad student answers to find a layperson excited about his progress on one of the prize problems — the Riemann Hypothesis. The layperson has just one question for the grad student: what is that funny little squiggle?

  4. KCd says:

    Jordan, you write that mathematicians don’t ever say “ln”. I imagine you meant that they don’t ever write “ln”. My experience in Russia was that the default notation for natural logarithm there is ln rather than log. I couldn’t believe it at first. C’est la vie. Also, it might be the case that applied mathematicians do write ln instead of log. Euler wrote the natural logarithm just as a single letter l.

  5. JSE says:

    I’m suddenly remembering a talk I saw Serre give, where at one point he said “I introduce this notation only in order to object to it.”

  6. Yiftach says:

    Jordan, let me disagree with you, I believe that in areas close to CS the base will usually be $2$ rather than the natural one. Also if you are thinking about $p$-groups or pro-$p$ groups you might think about base $p$ (if you are at all concerned about it).

  7. Cultural differences are a bit of a headache if you work in a field where people come from different disciplines. From my experience, electrical engineers tend to, but certainly not always, exclude degenerate/trivial cases while theoretical computer scientists and theoretical physicists are like mathematicians or at least they don’t complain.

    Usually I write the way the target audience would. The thing is that choosing one style doesn’t mean your editor picks all referees from the particular community you have in mind; you write a paper mainly aiming at fellow mathematicians, and one referee gets confused by a trivial case and go, “how come the empty set is an example?” while the other referees are all happy with it.

    But perhaps the most difficult of all is to make your co-authors who are not familiar with culture outside mathematics understand that the writing style you chose is the way to go when the target audience includes non-mathematicians. You may be able to convince them, but you’ll be explaining the exact same thing again when writing another paper with them: “Those convinced against their will are of the same opinion still.” — Dale Carnegie

    It’s good to be reminded of what’s important in one culture and what’s different in another culture though.

  8. Just to be pedantic about the “log” vs “ln” question… I’ll go on and say that both are acceptable within mathematics :)

    This is an example of a paper by respected authors where “ln” makes an appearance

    Click to access S0002-9939-1983-0699419-3.pdf

    and this is a recent preprint on combinatorics where they both are used *in the same paper* :))

    Click to access 1305.5518.pdf

    I’d say that though “log” is predominant, it is very common to find “ln” within mathematical physics. Many mathematicians won’t acknowledge the latter as a part of mathematics… but that would leave poor the mathematical physicists homeless, because physicist sure won’t take them :)

    I like to use “ln” because that’s just what my teacher at school used. I think it’s best to be aware of both: it’s redundant, but it’s also a nice reminder of the fact that notation is just notation (or is it? :) )

  9. d.n. says:

    Agreed with Yiftach. In the subfield of mathematics known as Theoretical Computer Science, I think most people write log to mean log_2, ln to mean log_e. There is also some trend to write lg to mean log_2, reserving “log” for the-thing-you-write-inside-O(.)’s-because-it-doesn’t-matter.

  10. Nick says:

    “I’m suddenly remembering a talk I saw Serre give, where at one point he said ‘I introduce this notation only in order to object to it.’ ”

    And this reminds me of the story about a trick played on Serge Lang because of his hatred of ugly/cumbersome notation: (end of pg. 546 – top of pg. 547)

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