John Keil's preface to Euclid | Sociology and Pure Mathematics

John Keil's preface to Euclid | Sociology and Pure Mathematics | N J Wildberger

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@rexwhitehead8346 3 жыл бұрын

When did the use of Euclid begin to decline? When I was 10 or 11 my grandmother, then in her 60s mentioned learning "euclid" at school. When I asked "whats that?" she mentioned triangles and circles and we concluded that it was what I knew as geometry. She, like most Australian children of the time, would have had six years of primary education and two of secondary before leaving school at 14. So I guess that, in Australia at least, "euclid", the subject if not the book, was being taught as late as the early 1900s. My own Greek hero is Eudoxus, and some believe that it was mainly his work that Euclid was expounding in the "Elements".

@Stelu0407 4 жыл бұрын

Great Video! Interesting to hear Keil's critique of the unneeded complexity (in Keil's opinion at least) that was coming into mathematics, and his view of Barrow, that the tendency for brevity in one's mathematical expressions may tend to one's works falling into obscurity! I think this captures something really interesting that is taking place in mathematics around this time, as it transforms from a corpus of problems in computation and a canon of knowledge, to a far more expansive and specialist program. Keil raises an interesting point that becomes ever more pertinent, especially in the 200 years of so following this work's publication: why is it that mathematics is increasingly spoken about in such a complicated way? Perhaps that is simply due to the inherent nature of the subject matter? Or is there something else in play here? Not sure that these are math questions per se, which makes this video series all the more interesting. I can't help but take a liking to Keil, with his appreciation that Euclid, (as opposed to those Keil takes aim at), presented a mathematics that was learnable. After the endless hours coming to terms with things like Type Theory, Category Theory, or the twists and turns of Modern Logic, it would be nice, frankly, if it was all a bit more learnable, all a little simpler, because when you finally get to the bottom of some of the topics in modern mathematics, it seems they could certainly could be. And in that sense Keil is breath of fresh of air. Keep it up - Looking ford to the next one!

@njwildberger 4 жыл бұрын

@Jamie Gabriel, Thanks for the interesting comment. I think you are onto something here.

@christopherellis2663 4 жыл бұрын

The long s was used in all Places except the Final, which was always the short s. Capitalisation of Nouns was standard Practice.

@thomaskember4628 4 жыл бұрын

I didn't know before that English nouns were ever written with the first letter in upper case. In my attempts to read German, I have found this practice quite useful. It is good to know that a word is definitely a noun; this helps to sort out German's complicated grammar.

@bobby-3x5x7mod8is1 4 жыл бұрын

Norman, I love your deference to the great thinkers before you. I find it amazing that even when you disagree vehemently you tone down your deportment and your demeanor becomes subdued honoring them sincerely with all your person. When you read, your use of inflection and impassioned body motions clearly indicate the important points. I am a logic hobbyist and never understood numbers. But my study of the Bible turned me on to group theory through the ancient Augustine of Hippo. I came to find out that a geometer, Felix Klein, formalized some basic Bible truth that Augustine verbally stated 1400 years prior [Trinity] and Klein started a whole programme of classification that you have improved and extended. You are a master teacher and I post videos [uhg-39] like this video in the hope that my family and friends will, once in their useless modern lives, engage thoughts that are enriching, beautiful, and motivating. If I was religious I would praise you. But alas, I am a Christian so all praise be to the God and father of my lord and savior, the man, Jesus of Nazareth. Norman, I needed to hear you say, [uhg-39] "This is probably the most important Mathematics video that you are likely to see."

@XylyXylyX 3 жыл бұрын

This was a lovely read, thank you Norman!! I spent considerable time reading critical material written as late as 1935 regarding the Special Theory of Relativity. Einstein’s critics sounded VERY similar to Kiel (even the style!), praising Newton. They made vague assaults on the theory, implied arrogance on the part of Newton’s critics, suggested that simplicity had an authority all its own, and intimated that the motivations of the supporters and contributors to relativity theory were suspicious. The only thing missing from Keil’s writing compared to the critics of Einstein is anti-Semitism. What I would love to see is some literal rebuke of these later authors: where exactly do they think the modern imperfections were introduced? Also, I would like to know if Kiel is worried about *bad* new texts, that are actually easy targets. Einstein’s critics did attempt specific challenges to Einstein’s logic but those challenges are themselves unintelligible, confusing, and almost impossible to digest. It is hard to find the exact point where the error is, I gave up many times.

@rexwhitehead8346 3 жыл бұрын

Silly, wasn't it? I wonder if any of those "critics" ever knew that there is almost nothing in Special Relativity that was not known and experimentally tested before Einstein first set pen to paper. The central discovery of Einstein's paper "On the Electrodynamics of Moving Bodies", the Lorentz Transformation, had been discovered already by err... Lorentz, the relativistic mass increase had been measured, around 1900, by Kauffman, the electromagnetic significance of the group of Lorentz Transformations, which was called the Poincare Group, after err... Poincare, was clear. Einstein's genius was to derive all these hard won discoveries from scratch in a short paper with almost no mathematics and which almost anybody could digest. The other discovery in that paper was that the frequency of light and its energy transformed under Lorentz transformations in exactly the same way as one another, which he already knew, or suspected, from Planck's work, and that played the central role in Einstein's other Annus Mirabilis paper on the Photoelectric Effect. The third great 1905 paper, on Brownian Motion and the existence of molecules, was I believe, also connected to the other two but in a much more roundabout way.

@rickshafer6688 3 жыл бұрын

Kiel was correct in calling out the accusations of critics of Euclid as projections of their own not understanding , but his critique of Burroughs and Clavious are nitpicking. One thing I do agree with him is the simplest diagram contains explanation for algebra. So Graphs are useful.

@99bits46 4 жыл бұрын

Sir, What is modern equivalent of Euclid Elements?

@alexroch6058 4 жыл бұрын

Probably Newton's principia

@99bits46 4 жыл бұрын

@@alexroch6058 Lol that is another incomprehensible book, I need more of a book in modern terms with same content

@alexroch6058 4 жыл бұрын

@@99bits46 im sure you can find translated versions somewhere

@fakegandhi5577 2 жыл бұрын

A more modern example would be principia mathematica by bertand russe this is THE book of constructing mathematics from first principles and axioms. This is more difficult to read in my opinion and is more symbolic than geometric in its reasoning.

@mimzim7141 4 жыл бұрын

Around 14:00 seemed to me he was saying Euclid had a definition for proportions for both rationals and irrationals and those people wanted to assume it only for rationals and then demonstrate it for irrationals.

@hyperduality2838 4 жыл бұрын

@diegocolomes 4 жыл бұрын

Maybe a comment on Oliver Byrne's color edition of the first six books of Euclid can be made in these series👍

@mokranemokrane1941 4 жыл бұрын

Pure pleasure to listen.

@keyblade134679 4 жыл бұрын

hi professor Wildberger, if I want to continue pursue a phd in mathematics but want to avoid all that set theoretic stuff, would you recommend I focus more on applied math than pure math?

@njwildberger 4 жыл бұрын

It really depends on what you find most interesting and accessible.

@rexwhitehead8346 3 жыл бұрын

Once upon a time I decided to read Newton's original statements of the three laws to my class of first year physics students, in Newton's original Latin, from, moreover, Lord Kelvin's personal copy of the Principia, complete with his pencilled annotations. "Corpus omne perseverare in statu .... etc, etc." I got one, and only one, question at the end of the lecture: "Why was it in Latin?"I never bothered again.

@robinbrowne5419 4 жыл бұрын

Happy New Year Professor Wildberger. I have been watching your videos for a few years and I have been fascinated by your objections to the use of infinity. At first I resisted, saying "What do you mean. Of course there is infinity." But now I am not so sure. It leads to a lot of contradictions. For instance: We say that pi has an infinite number of digits. But then we say that infinity is not a number. So, how can we say that there are an infinite number of digits? This is likely semantics, but it does illustrate one of the problems of using infinity without really thinking about the implications. Anyway, Happy New Year in 2021.000000000000... :-)

@hyperduality2838 4 жыл бұрын

Rational is dual to irrational -- mathematics. Noumenal (rational, analytic) is dual to phenomenal (empirical, synthetic) -- Immanuel Kant. The word "irrational" has a different meaning in philosophy. Concepts are dual to percept -- the mind duality of Immanuel Kant. The intellectual mind/soul (concepts) is dual to the sensory mind/soul (percepts) -- the mind duality of Thomas Aquinas. According to Kant & Aquinas the mind/soul is dual! Mind is dual to matter -- Descartes. Matter duality: Waves are dual to particles -- quantum duality. Symmetric wave functions (Bosons) are dual to anti-symmetric wavefunctions (Fermions). Active matter (life) is dual to passive matter (atoms, forces). Mind duality is dual to matter duality. "Always two there are" -- Yoda. Platonic forms (definitions, deduction) are dual to to particulates (induction) -- Plato. Deductive inference is dual to induction inference. Being is dual to non-being creates becoming -- Plato. Absolute truth is dual to relative truth, the relation of ideas is dual to the matter of facts -- Hume's fork. Homology is dual to co-homology. Elliptic curves are dual to modular forms. Syntropy (prediction) is dual to increasing entropy -- the 4th law of thermodynamics!

@michielkarskens2284 4 жыл бұрын

Euclid's Elements also seems to have served a specific socialogical/ political purpose. I believe it was commissioned by Ptolemy Soter I to oppose and replace the Pythagorean foundations of mathematics, which had been dominant throughout the Hellenic world. According to the Pythagoreans (pure) mathematics flowed from arithmetic and geometry only came second. In contrast Euclid structured the Elements to start with geometry in Books I-VI. Not a coincidence either Euclid choose to divide the material over the first SIX books...

@njwildberger 4 жыл бұрын

@Michiel Karskens That is certainly an interesting conjecture. Is there any support for such an idea?

@michielkarskens2284 4 жыл бұрын

@@njwildberger Since you ask. Yes, there is....:

@michielkarskens2284 4 жыл бұрын

I have spent the last two years testing this hypothesis, and the evidence is quite overwhelming (bundled in a book: store.bookbaby.com/book/spiral-mechanics , I will gladly send you a copy so you may judge for yourself). Central is the passage of Plato’s Theaetetus 147d --148c, in which the boy Theaetetus divides all the (natural) numbers in two classes: square numbers and oblong (rectangle) numbers. Squares and rectangles.., it is clear (also from your youtube lectures on the history of mathematics) where (exactly) Plato/ the Pythagoreans would have gotten this idear from ;-). Historically, the passage has been used to assign 'incommensurability/ irrational numbers' to Theodorus and Theaetetus. However, overlooked has been the fact Plato clearly places prime numbers in the class of 'oblong numbers', and thus prime numbers can be (and evidently were) represented TWO dimensional as rectangles at the Academy of Athens. Aristotle, having studied for twenty years at Plato's Academy, will have been familiar with Plato's two-dimensional representation of (prime) numbers, but he nevertheless maintained (contrary to the facts) prime numbers are "in one dimension only" (Metaphysica Book Δ 13, 1020 b 4). Euclid's definition VII.2 (1 is not a number) and VII.11 (a prime number has only 1 divisor) follow Aristotle verbatim. A prime number cannot be and is never a plane number (definition VII.16) in the Elements. Euclid represents numbers as one-dimensional lengths throughout the Elements, see for example his proof of Proposition VII.2. A plane number is, By Definition, a composite number in Euclid's book (VII.16 ), in direct contrast to (and contradiction of) Theaetetus' representation of a prime number by a rectangle with a side 1 and the other side the prime number in question (in accordance with/ based on the fact a prime number has two divisors). Why does Euclid follow Aristotle to the letter? Because he was ordered to do so by Ptolemy Soter I. Ptolemy - before becoming Soter I, founding the library of Alexandria and commissioning the Elements- was tutored at the Court of Macedonia by the same man as Alexander the Great: Aristotle of Stagira (see your Math Foundations 251). Regarding arithmetic as the first of the sciences; there is widespread agreement among scholars (both ancient and modern) the Pythagoreans believed arithmetic was the source of all the sciences (geometry, astronomy, music, etc.). Nicomachus of Gervasa is an example of the older scholars (2nd century AD), apart from both Plato and Aristotle as sources in antiquity as well as the fragments of Philolaus. Wilbur Knorr is a contemporary source: "A point we must emphasize is that the Pythagorean mathematics was fundamentally arithmetical." (page 132, The Evolution of the Euclidean Elements). Finally, it has been established (beyond any reasonable doubt) Pythagoras spent time in Babylon. He was taken from Egypt to Babylon in 525 BC, when the Persian king Cambyses conquered Egypt. Would it be possible for Pythagoras to have spent any amount of time in Babylon and not to have learned (of) the standard Babylonian identity: length x width= ((length + width)/2)^2 - ((length - width)/2)^2 ?? (You mention this identity in your latest lecture on Babylonian mathematics) In arithmetic. What happens when the width is set equal to 1, and the length is a square number? And what happens when the length is an oblong number? In geometry. Take a flat surface (horizontal line), put the smaller square (of the half difference) on this line and place the larger square (of the half sum) diagonally against the smaller square... The length in between the two squares will “put on of itself” the square root of the product of length and width. There exists no better pedagogy to explain (in-)commensurable length, the Pythagorean theorem, and quadratic equations. All in/ from One elegant identity. In your lecture on Babylonian mathematics you state you do not believe the Babylonians used this identity for doing basic multiplication. You are right, they used it for much, much more.. From the passage in Theaetetus it is evident Plato (who was a Pythagorean according to everybody except Aristotle) considered 1 a number and believed a prime number has two divisors. In the Elements a number and a prime number are defined as the exact opposite in definitions VII.2 and VII.11. Interestingly Euclid also defines an odd number exactly the opposite of the Pythagorean definition (of an odd number) given by Nicomachus of Gervasa in his Introduction to Arithmetic… Practical use of the Babylonian identity is made impossible BY DEFINITION by Euclid. The Babylonian identity is (almost) nowhere to be found in the Elements. Incommensurable length which is so saliently explained and easily derived using this identity is treated by Euclid in the - notoriously difficult- Book X. The geo-arithmetic or geo-algebra of Euclid in Book II is very difficult. The Babylonian identity is hidden (Fubar) in Book II, the Definitions (II.1 and II.2) and Propositions (II.5-II.10 among others) of the Elements. F***ed up beyond all recognition, pardon my French, because it is all explained from (cutting) an one-dimensional (number) line. Proposition II.5 “If a straight line is cut into equal and unequal segments, then the rectangle contained by the unequal segments of the whole together with the square on the straight line between the points of section equals the square on the half.” Euclid should be commended for having found the most unintelligible means of stating and proving that every product of unequal factors (rectangle) is a difference of two squares (aka a gnomon)! Starting from the arithmetic of a two-dimensional number line as Plato has Theaetetus explain, the material (mathematics) can be understood by everybody and requires no more than 7th grade arithmetic and simple cut and paste proofs. The Elements was commissioned and composed for suppression of the Babylonian foundations, which Pythagoras had introduced into the Hellenic world.

@michielkarskens2284 4 жыл бұрын

P.S. To the list (of uses for the Babylonian identity) of multiplication, explaining incommensurable length/ irrational numbers, the Pythagorean theorem, and quadratic equations, may be added: Exact Trigonometry ‘B.E.S.T.’ thanks to you and Daniel Mansfield. To be fair, Joran Friberg’s igi - igibi problems also fit exactly. It seems to me the entire identity is what is broken off from Plimpton 322. You and Daniel sum up (argue) the righthand side of the identity (is missing): the half difference (base) and the half sum (diagonal). Joran Friberg also lists the left-hand side of this identity in his “Amazing traces of Babylonian origin in Greek Mathematics”(page 89, 2007): the length is the ‘igi’ and the width is the ‘igi.bi’ resulting in the ‘takilti siliptim’ (what you and Daniel call diagonal) and the ‘takilti sag’ (your and Daniel’s base). The point is not Either igi/ igi.bi problems (Friberg), Or exact ratio based trigonometry (you and Mansfield), but All in(/from) One: Arithmetic (left-side) And geometry (right-side), And exact trigonometry, And igi / igi.bi problems, And Pythagorean triples, And quadratic equations. Donald Knuth was right on the mark when he stated: “The calculations described in Babylonian tablets are not merely the solutions to specific individual problems: they actually are general procedures for solving a whole class of problems. The numbers shown are merely included as an aid to exposition, in order to clarify the general method.” The identity explains 90% of the mathematics I was taught in the first two years of high school. Using the identity you can easily teach this 90% to a ten-year-old in one afternoon.

@ChrisTietjen_00 4 жыл бұрын

T-hank you.

@whig01 4 жыл бұрын

It is a worthy mission to purge irrationality from mathematics. There remain incommensurables, of course, and approximations of projections.

@hyperduality2838 4 жыл бұрын

@whig01 4 жыл бұрын

@@hyperduality2838 I'm well aware of the terminological meaning, but irrational numbers do not exist, only approximations can be made, or gestures, in one dimension, however in matrix form any algebraic number can be precisely expressed if within range of what can be expressed.

@ppkrex 4 жыл бұрын

1723 is the same year Anderson's Masonic Constitution was published it's tries to moralize the recorded mention of building and geometry through western literature.

@firstnamegklsodascb4277 4 жыл бұрын

Being a mathematician was so easy back in Euclid's time. Nothing had been discovered yet

@palmtoptigeri9797 4 жыл бұрын

On the contrary

@fakegandhi5577 2 жыл бұрын

We have all the tools and education to become a mathematician. Back then i would think you would have to reason about ideas that were never thought of using logic that was never presented to you