If Q is not a written source, the Q hypothesis falls apart. The point is to explain what Matthew and Luke (but not Mark) have in common, much of which is word-for-word the same. Whole sentences between them that are verbatim in the Greek point to literary dependence, not oral tradition.
On the Farrer hypothesis, Luke used Matthew as a source, accounting for their shared material. Mark Goodacre may be the current leading academic proponent of the Farrer hypothesis. His article ‘Fatigue in the Synoptics’ (https://www.markgoodacre.org/Q/fatigue.htm) is a short read that makes part of the case.
Setting aside the truth value of any religious claim — I find ‘Jesus **** Christ’ to be in gratingly poor taste. Imagine going around saying ‘Prophet **** Muhammed’ or even ‘Martin **** Luther King.’ It’s gauche.
> the set of such roots is actually closed under multiplication, addition, and subtraction, and there is even an analogue of prime factorization if you squint
I did a maths undergrad, but I don’t think I ever studied algebraic integers. That’s something I shall have to remedy now, thanks!
If you took abstract algebra (which presumably you did as a math major), you certainly encountered these at least in the exercises as groups of the form ax + b where x is some irrational number (or imaginary) and a and b are integers are a staple of chapter 1–2 proofs. Gaussian integers (ai + b) are a special case that are loads of fun to play with it. They are not unique factorization domains like the integers (e.g., 5 can be expressed as both 1∙5 and (1 - 2i)² where 1, 5 and 1 - 2i are all irreducible).
Nit: while it is not generally the case that rings of algebraic integers must be unique factorization domains, it is the case for Gaussian integers! In your example, 5 is uniquely factorizable up to units as (1-2i)(1+2i).
Indeed, the integers have the same limitation -- factorization is unique only up to units. 1 = -1 * -1
In elementary mathematics, people wave away "-1" by saying silly things like "positive integers", before Gaussian integers arrive and force us to figure out precisely what we are trying to say without silly ideas from analysis like "ordering". :-)
> The V is gradually being dropped in Spain and in the last few decades the process seems to have accelerated
I own a Spanish textbook printed (in English) in 1958 that says about «usted»: ‘It is the universal respectful address of society, and the only one the foreigner is ever likely to employ or hear addressed to him, unless he marries a native or forms intimate friendships.’ How times have changed!
This is what I was taught as well when taking Spanish 30 years ago. Fast forward 20 years and me visiting Mexico City for the first time, and I got some really strange looks using usted
The web is, in theory, an open venue, and somebody publishing on the web is not unlike somebody performing in the street. It is not your duty, as the consumer, to ensure the producer’s income—particularly not at the expense of your privacy. The producer has something to say, and you the consumer are willing to hear it: that may just as well be the extent of your relationship. How, or even whether, the producer monetizes this state of affairs is not the consumer’s responsibility, though some consumers (who can) may choose to patronize the producer.
