June 2020 - mathfun@mailman.xmission.com

Re: [math-fun] [EXTERNAL] Re: Much ado about nothing (was Re: Proof that 0^0 = 1)
by Dan Asimov 15 Jun '20

15 Jun '20

Among statisticians there is total agreement on how to define median for a finite multiset of numbers: Order the numbers, repeating them according to multiplicity. If there's a unique middle number, then that's it. If there are two middle numbers, then it's the halfway point between them. —Dan William Cordwell wrote: ----- If we have a finite set of discrete real numbers, {x_i}, then we can ask: What values of a,b,c minimize, respectively \sum_i | xi - a |^2 \sum_i | xi - b |^1 \sum_i | xi - c |^0 a = mean b = median c = mode, if we define 0^0 = 0 (or 0.0^0 = 0) I find this useful when discussing mean, median, and mode with middle-school students, as a way of looking at the values in a more abstract way. I note that there does not seem to be agreement on how to define median. -----

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Re: [math-fun] Quantifiers
by Henry Baker 15 Jun '20

15 Jun '20

As a failed linguist (I briefly considered doing my PhD thesis on natural language), my conclusion about natural language ambiguity is: A speaker constructs an utterance for a specific audience with an assumed shared understanding. The speaker then maximizes ambiguity in order to minimize the size/length of the utterance, all consistent with the information he/she is attempting to transmit. Of course, the most skilled speakers can construct utterances that can simultaneously have different meanings for different audiences ('dog whistles'); these people are called 'politicians'. Logicians attempt to do the opposite: to construct utterances that have a universal, unambiguous meaning, regardless of size/length. This is why trying to construct unambiguous utterances makes one sound extremely pedantic -- because one has to make the obvious, obviously obvious. W.r.t. pigeon attacks: the speaker assumes that the audience already knows about pigeons and pigeon attacks; the only thing new is 'New York' and the *rate* of attacks in New York. As a truly pedantic engineer, I'm amazed that all of these pigeons have been somehow synchronized with some sort of giant clock to periodically and synchronously attack at precisely the 30-second mark. "In New York City, someone sees the sun rise every 24 hours". At 05:28 AM 6/15/2020, James Propp wrote: >Math gives us one way to dissect the ambiguity of sentences like "In New York City, someone is attacked by a pigeon every thirty seconds" (is it always the same person? is it always the same pigeon?) by way of quantifiers. > >Does linguistics have its own way of talking about the different interpretations of such a sentence? > >Jim

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[math-fun] Much ado about nothing (was Re: Proof that 0^0 = 1)
by Keith F. Lynch 15 Jun '20

15 Jun '20

As a computer guy, I consider real numbers and integers to be two very different things. The English language shares this distinction, in that real numbers go with "less" and integers go with "fewer." I'll use 0 to mean the integer zero and 0.0 to mean the real zero. I would say that 0^0 = 1, and 0.0^0.0 is undefined. If anyone disagrees, I'd like to know why. I'd also like to know if anyone has an opinion on 0^0.0 or 0.0^0. And if sticking a minus sign in anywhere would change anything. Thanks.

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Re: [math-fun] Another horribly confusing Quanta article
by Dan Asimov 14 Jun '20

14 Jun '20

I agree that it's not a good article. I'd say very few Quanta articles on math seem good to me — they're so dumbed down that they convey very little information. But for that reason I'd never look to a Quanta article to learn much about a field of math. —Dan ----- If you are trying to learn representation theory, the Quanta article looks to be useless, especially because it only references other Quanta articles (probably with similar problems). -----

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[math-fun] Draft of June 17 blog post
by James Propp 14 Jun '20

14 Jun '20

Here's a link to the current draft of my June essay. Comments of any sort are welcome (even "You should probably stay in the pure-math lane, Jim" if you think my attempts at relevance only expose my own ignorance), as befits an essay on the theme of negative feedback... http://mathenchant.org/062-draft2.pdf Jim Propp

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Re: [math-fun] Draft of June 17 blog post
by Dan Asimov 14 Jun '20

14 Jun '20

That would be Norbert Wiener. My father thought I should make sure to get to know him when I arrived at college, but unfortunately he died about six months before my freshman year began. —Dan ----- with regard to wolves and deer, you could tell the story of the deliberate introduction of myxomatosis to control the rabbit population in Australia… I recently re-read Weiner’s Cybernetics, and found it for the most part to have aged quite well. -----

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Re: [math-fun] Another horribly confusing Quanta article
by Dan Asimov 13 Jun '20

13 Jun '20

Bruce Berndt has a series of books I think proving as much as possible from "Ramanujan's Notebooks" (which are their titles: I, II, III). Here is a summary article he wrote about these books: <https://faculty.math.illinois.edu/~berndt/articles/aachen.pdf>. —Dan Cris Moore wrote: ----- This is a beautiful article, but it doesn’t explain where any of these identities (Ramanujan and successors for \pi) come from. I know this might be a heavy lift, but are there any of these that can be explained on an elementary level, even if non-rigorously? -----

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[math-fun] Proof that 0^0 = 1
by Andres Valloud 13 Jun '20

13 Jun '20

I hear 0^0 should be defined as 1 for convenience, but perhaps there are proofs this has to be so (the *value* zero raised to the *value* 0, not the indeterminate limits of the form 0^0). What's your favorite proof that 0^0 = 1? Here's one. 0^0 = (1 - 1)^0 = \sum_{k=0}^0 \binom{0}{k} 1^{0-k} (-1)^k = \binom{0}{0} * 1 * 1 = \frac{0!}{0! \cdot 0!} * 1 = 1 * 1 = 1 Andres.

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Re: [math-fun] Proof that 0^0 = 1
by Dan Asimov 13 Jun '20

13 Jun '20

Neil Sloane wrote: ----- Let's not start that discussion again. 0^0 is undefined. End of story. ----- Agreed. Without meaning to extend the discussion, I'd say that for certain applications, especially in combinatorics like counting the number of functions between two finite sets, there might be some value in defining 0^0 = 1 for that purpose. For situations that are more gooey than crunchy*, not so much (e.g., the value of x^y as (x,y) approaches (0,0) along some curve in the half-plane x > 0). So there's no point in trying to say what its only correct value ought to be. —Dan ——————— * Back around 1970 my grad-student friends and I used to classify various fields of mathematics as either gooey (continuous) or crunchy (discrete). Mathematicians, too.

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Re: [math-fun] Another horribly confusing Quanta article
by Dan Asimov 12 Jun '20

12 Jun '20

One magazine that's been around since at least 1978 is the "Mathematical Intelligencer", published by Springer, which has good articles that more or less assume a decent math background but manage to be almost universally readable. All of them are written by actual mathematicians, though there are also articles about math history or culture that may be written by other experts. It's rarely seen on magazine racks, but can be ordered directly from the publisher. But I would very much like to start a new math publication that runs articles on math, and on philosophy of physics and of math. —Dan ----- > Is there *any* popular science publication that has good math articles? Excellent opportunity to fill the niche: there's no competition. So, what is the audience that should be targeted? What kind of articles will this audience read? What do they need? What are their interests? Why should they read this hypothetical publication? -----

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