Wikipedia:Reference desk/Archives/Mathematics/2018 June 30

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June 30

Facebook page likes

My apologies, it's been about 20 years since I took algebra and I was wondering if someone would be kind enough to do the math for me? I run a Facebook page that has 1,500 likes and is growing at 1.8% per week. My competitor's Facebook page has 2,900 likes and is growing at 0.2% per week. Assuming the same growth rate, how long will it take my Facebook page to surpass my competitor's Facebook in likes? A Quest For Knowledge (talk) 12:45, 30 June 2018 (UTC)

This is a Geometric progression, whose sum can be found with the formula \frac{a(1-r^n)}{1-r}, and you want to find the smallest $n$ where your progression is larger than their progression. You have $a=1500$ and $r=1.018$ and your competitor has $a=2900$ and $r=1.002$. Plugging these numbers in to a calculator gives an answer of 76 weeks (75 weeks and 1 day to be precise). Iffy★Chat -- 14:48, 30 June 2018 (UTC)

Um, actually, what you want here is just the progression/sequence, not the series/sum. There's no need for the summation formula. You want the solution to $a_{1}(1+r_{1})^{n}=a_{2}(1+r_{2})^{n}$, meaning $n={\frac {\log(a_{1}/a_{2})}{\log((1+r_{2})/(1+r_{1}))}}$. In our case we get $n=41.6$. -- Meni Rosenfeld (talk) 20:24, 30 June 2018 (UTC)

If you assume continuous growth you get $n={\frac {\ln(a_{2}/a_{1})}{r_{1}-r_{2}}}$, slightly simpler. I get 41.2 weeks with this which is probably close enough, and the like count would be 3150. One thing to keep in mind with this kind of problem is that logistic growth is often a more accurate model than exponential growth long term in real life. --RDBury (talk) 21:08, 30 June 2018 (UTC)

Yep, you're both right, I edited my comment for accuracy. Iffy★Chat -- 12:16, 1 July 2018 (UTC)

Resolved

 – Thanks! I appreciate everyone's help. A Quest For Knowledge (talk) 09:37, 3 July 2018 (UTC)

Defining identity by Second-order propositional logic

Is the following proposition used somewhere as an axiom (for defining identity)?

$\forall (x,y){\biggl (}(x=y)\leftrightarrow \forall \phi {\bigl (}\phi [x]\leftrightarrow \phi [y]{\bigr )}{\biggr )}$

HOTmag (talk) 20:46, 30 June 2018 (UTC)

See identity of indiscernibles. --Trovatore (talk) 20:32, 2 July 2018 (UTC)

Which to me seems incompatible with the catholic belief in Transubstantiation. ;-) Dmcq (talk) 21:34, 2 July 2018 (UTC)

I know you're sort of joking, but really it doesn't seem to me to have much to do with transubstantiation. The unblessed wafer and the transubstantiated host do differ in at least one property, namely that of having been blessed, so they don't have to be identical. --Trovatore (talk) 22:32, 2 July 2018 (UTC)

Thankxs to your reference, I now know that the principle I've asked about is called Leibniz's Law. Anyway, it seems that you accept this principle, don't you? As far as I'm concerned, I accept it. Further, I think it's also non-trivial, because it's sensitive to the language involved. For example, an axiom like: $\exists (x,y)(x\neq y)$, with respect to a language containing no constants/predicates (other than the identity sign), is illegitimate, as it contradicts Leibniz's Law; However, when allowing constants/predicates (besides the identity sign), an axiom like: $\exists (x,y)(P(x)\land \neg P(y))$, that entails the previous proposition: $\exists (x,y)(x\neq y)$, doesn't contradict Leibniz's Law, so it's legitimate. HOTmag (talk) 08:20, 3 July 2018 (UTC)

I think much depends on which system of logic you're talking about and what the universe of discourse is. There's no reason in pure logic that the universe can't be empty; though you'd get some strange results such as $\forall (x)P(x)$ does not imply $\exists (x)P(x)$. You can also have systems of logic where '=' is not defined, see First-order logic#First-order logic without equality. Generally first order logic has axioms for =. There's no reason that the formula above can't be taken as the definition of = in second order logic, so it doesn't need to be an axiom. --RDBury (talk) 00:10, 4 July 2018 (UTC)

"There's no reason in pure logic that the universe can't be empty". of course, that's an old well known fact.

"strange results such as $\forall (x)P(x)$ does not imply $\exists (x)P(x)$". Not strange in an empty universe.

"You can also have systems of logic where '=' is not defined". of course, that's an old well known fact, but I can't see how this has anything to do with my question, that tries to define identity in systems other than the systems you are talking about.

"There's no reason that the formula above can't be taken as the definition of = in second order logic". Our article identity of indiscernibles, provides some reasons (I don't accept) against the formula above, whereas I'd explained (in my previous response you've responded to) why the formula above can be taken as the definition of = in second order logic. HOTmag (talk) 08:30, 4 July 2018 (UTC)

Actually, standard first-order logic does not permit an empty universe. The assertion $\exists x(x=x)$ is provable without using any axioms at all (except for the logical axioms); it is considered a logical truth.

There are variants of first-order logic, called free logic, that permit an empty universe. As far as I can tell, there is no real advantage to this weakening, but yes, you can do it. --Trovatore (talk) 09:19, 4 July 2018 (UTC)

Of course, that's an old well known fact, first discovered in the Fifties. HOTmag (talk) 11:54, 4 July 2018 (UTC)

I wasn't aware that first-order logic does not permit an empty universe, but certainly such an assumption is not needed in set theory since the existence of ∅ is an axiom. As for the business about "well known facts", that's pretty much what we're supposed to be providing here, so if you're already familiar with all the literature in on the subject then this is not the place to post your question. --RDBury (talk) 17:08, 4 July 2018 (UTC)

"I wasn't aware that first-order logic does not permit an empty universe". Almost. Actually, only standard first-order logic does not permit an empty universe; However, non-standard first order logic, known as "free logic", does.

"if you're already familiar with all the literature in on the subject then this is not the place to post your question". I have always been familiar with free logic, as well as with systems that do not define identity, but I didn't know whether the formula I was asking about in this thread had ever been used, nor did I know that it was called Leibniz's Law, and that's why I asked (Eventually I got an answer in this thread). HOTmag (talk) 18:09, 4 July 2018 (UTC)

"Eventually" meaning "in the very first response, 2 days after you posted it"? Oy vey. --2601:142:3:F83A:BC7C:4FF3:BC84:995E (talk) 16:27, 5 July 2018 (UTC)

Yep ! HOTmag (talk) 18:12, 5 July 2018 (UTC)

"Normally distributed and uncorrelated does not imply independent" in practice

The examples in our article Normally distributed and uncorrelated does not imply independent are contrived ones that don't seem to me similar to any joint distributions that could occur by accident (i.e. without someone cherrypicking the definition of a variable -- if not invoking an otherwise-useless synthetic variable -- to create an example). Are there any real-world examples of data sets where the null hypotheses of normally-distributed and uncorrelated variables cannot be rejected, but the stronger null hypothesisn of independence between the variables can, where the variable definitions served some applied purpose other than demonstrating the possibility of dependence without correlation? NeonMerlin 23:40, 30 June 2018 (UTC)

• Not to answer the question, but: I AfD'd the article. Tigraan^{Click here to contact me} 09:33, 2 July 2018 (UTC)