Elements
Hard to say if it was "widely read", but the Elements was certainly read by Lincoln in the mid-1800s. This text book of the Elements was first published in 1888, updated in 1900, and with reprints every year or two until (at least) 1915 http://openlibrary.org/books/OL24198679M/Text-book_of_Euclid.... I'll agree with thesis that it wasn't uncommon about 100 years ago as a standard text, and agree that that was about when it stopped being common. If you read the preface to the 1888 version, you'll see: "From first to last we have kept in mind the undoubted fact that a very small proportion…
(Yes, this is part II. I got interested in the topic.) I empathize with the difficulty of defining 'educated person.' As you rightly point out, there are different threads in what an "education" means. http://www.educationengland.org.uk/history/chapter03.html goes into the three main groups - 'public educators', 'industrial trainers', and 'old humanists' - which participated in that debate, ending "The curriculum which evolved during the 19th century was 'a compromise between all three groups, but with the industrial trainers predominant' (Williams 1961:142). This was 'damaging both to…
I see the split too. I'll add that each camp is frustrated and feels the other is missing the point and would make information security worse if its worldview won. You can do some empirical analysis. Someone downthread linked to a paper claiming to being able to reach a few million vulnerable devices over IPv6 and not IPv4. This kind of analysis isn't dispositive, though, because there are all sorts of second-order effects and underlying philosophical differences. Facts seldom change minds when you can build multiple competing true stories around these facts. I'll call one camp the…
I've meant to compile this list, so you just inspired me. I wish the Durants had had the stamina for one more volume. I don't know much about the European revolutions of 1848 and would have like to read about that time from them. I already had read a lot of ancient Greek literature in translation as well as The Pentateuch, all the historical books of the Old Testament and the Gospels. Western Civilization springs from intellectual roots in Athens and Jerusalem, so any survey has to include that heritage. My degree is in Comparative Literature, so I put a lot of stock in origninal source…
> In step 1 you haven't assumed that you know all the primes up to p_n, and then in step 2, you assume that you do have them. If you don't make step 1 more precise, saying that you know all the primes up to p_n, then when you multiply them all together and add 1, step 3 fails. Why? We know that the largest prime is p_n, and the numbers from 1 to p_n are finite. Each of them might or might not be prime. It's perfectly legitimate to say "multiply together all the primes between 1 and p_n" without knowing which numbers in particular those happen to be. Since I don't know which numbers I…
Here's the proof I learned: 1. Assume there is no prime number larger than p_n. 2. Compute the product of all the prime numbers less than or equal to p_n, plus one. Call this number c. 3. By construction, no prime number less than or equal to p_n divides c. 4. But all integers greater than one have one or more prime factors. (This is not proven.) 5. Therefore, since c has no prime factors less than or equal to p_n, it must have one which is not less than or equal to p_n, or in other words c must have a prime factor greater than p_n, which contradicts step (1). Viewing…
Ah, my favorite example of a pons asinorum: http://mathworld.wolfram.com/PonsAsinorum.html > An elementary theorem in geometry whose name means "asses' bridge," perhaps in reference to the fact that fools would be unable to pass this point in their geometric studies. The theorem states that the angles at the base of an isosceles triangle (defined as a triangle with two legs of equal length) are equal and appears as the fifth proposition in Book I of Euclid's Elements. Generalizing the concept a bit, a pons asinorum is a critical concept in any field, such that if you don't grasp it you'll…
Actually, this example makes Euclid even more of a logician. Recall that Euclid stated his Fifth Postulate (where his postulates really mean axioms). Geometers for thousands of years afterward, guided by intuition, attempted to prove that in fact the Fifth Postulate could be proved from the other axioms. But by the 19th century, it was actually clear that Euclid was correct to include the postulate; had he not included the postulate, his geometry would not convey what he wanted to convey. Euclid in this way was fundamentally a logician! He realized that he needed the Fifth Postulate to prove…
oldest "math text" certainly not. Wikipedia says "Elements is the oldest extant large-scale deductive treatment of mathematics." There is a category for which we can count it as "the oldest" The only older "important publications in math" Wikiepedia has are * "Moscow Mathematical Papyrus" from 1850 BC. This is referred to as a manuscript rather than a "text" though. * "Baudhayana Sulba Sutra" from 8th century BC. Which by all accounts seems to be actually a mass produced text with large reach. But also seems like these general don't contain proofs. and are more like a reference book than…
There are a ton of extremely Hard problems to solve there that we are not likely going to solve. One: English is terribly non-prescriptive. Explaining an algorithm is incredibly laborious in spoken language and can contain many ambiguous errors. Try reading Euclid’s Elements. Or really any pre-algebra text and reproduce its results. Fortunately there’s a solution to that. Formal languages. Now LLMs can somewhat bridge that gap due to how frequently we write about code. But it’s a non-deterministic process and hallucinations are by design. There’s no escaping the fact that an LLM is making…