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# The pons asinorum of modern mathematics

 Posts: 2,736 Add as FriendChallenge to a DebateSend a Message 2/4/2016 5:26:18 PMPosted: 1 year agoThe pons asinorum of Greek mathematics was Proposition 5 of Book I of Euclid's Elements: The base angles of an isosceles triangle are equal. Its proof was believed to be a 'bridge': Anyone who can cross it is not a donkey, hence the name pons asinorum (yeah, 'asinorum' has to do with the word @ss). But times have changed and so have mathematics. What should the new pons asinorum be?I think it should be the first-order linear differential equation. It require you to apply nearly all knowledge of algebra, trignometry and calculus up to that point (I guess geometry is less relevant, but I don't like geometry...) It's not the easiest to learn, but also not hard once you get the hang of it.Here are some other possible pons asinorums (or whatever the correct Latin plural is) I thought of. What do you think?Quadratic equationsCentral limit theoremPolynomial equations and the Fundamental Theorem of AlgebraEquations of circlessin^2 a + cos^2 a = 1Taylor seriesDefinite integrals and the Fundamental Theorem of CalculusGeometric seriesProof of AM >= GM >= HMPythagorean theorem (any proof)Expoential and logarithmic equationsLogarithmic differentiationEpsilon-delta proofsVolume of a sphereInverse of a 3x3 matrixSimultaneous linear equations and Gaussian eliminationEuler's formula/De Moivre's TheoremPolynomial division and the remainder theoremI think it is well established that the only reason aliens come to earth is to slice up cows and examine inside peoples' bottoms. Unless you are a cow or suffer haemerrhoids I don't think there is anything to worry about from aliens. - keithprosser Don't be a stat cynic: http://www.debate.org... Response to conservative views on deforestation: http://www.debate.org... Topics I'd like to debate (not debating ATM): http://tinyurl.com...
 Posts: 6,033 Add as FriendChallenge to a DebateSend a Message 2/10/2016 9:39:25 AMPosted: 1 year agoAt 2/4/2016 5:26:18 PM, Diqiucun_Cunmin wrote:The pons asinorum of Greek mathematics was Proposition 5 of Book I of Euclid's Elements: The base angles of an isosceles triangle are equal. Its proof was believed to be a 'bridge': Anyone who can cross it is not a donkey. But times have changed and so have mathematics. What should the new pons asinorum be?DC, I wonder whether it's being able to explain why science:1) Relies on empiricism, yet continuously changes its model and methods;2) Is the continuing elimination of ignorance and error from knowledge, and not the conclusive proof of truth;3) Having derived from natural philosophy, and while still being stimulated by it and occasionally challenged by it, is no longer accountable to philosophy alone;4) While making extensive use of logic and math, is not created by either.In support for this contention, I find that members who understand this, can: understand a sketched rationale behind almost any accepted scientific result, recognise the difference between science and pseudoscience quite easily, and make a decent fist of understanding the philosophy of science and science history.Meanwhile, those who miss one or more of these seem to be missing critical knowledge in the discussion, critique and appreciation of science, while those who miss all of them seem to be missing the fundamental concepts of what science -- or knowledge -- are at all.I hope that may be useful.
 Posts: 2,736 Add as FriendChallenge to a DebateSend a Message 2/10/2016 6:02:55 PMPosted: 1 year agoAt 2/10/2016 9:39:25 AM, RuvDraba wrote:At 2/4/2016 5:26:18 PM, Diqiucun_Cunmin wrote:The pons asinorum of Greek mathematics was Proposition 5 of Book I of Euclid's Elements: The base angles of an isosceles triangle are equal. Its proof was believed to be a 'bridge': Anyone who can cross it is not a donkey. But times have changed and so have mathematics. What should the new pons asinorum be?DC, I wonder whether it's being able to explain why science:1) Relies on empiricism, yet continuously changes its model and methods;2) Is the continuing elimination of ignorance and error from knowledge, and not the conclusive proof of truth;3) Having derived from natural philosophy, and while still being stimulated by it and occasionally challenged by it, is no longer accountable to philosophy alone;4) While making extensive use of logic and math, is not created by either.In support for this contention, I find that members who understand this, can: understand a sketched rationale behind almost any accepted scientific result, recognise the difference between science and pseudoscience quite easily, and make a decent fist of understanding the philosophy of science and science history.Meanwhile, those who miss one or more of these seem to be missing critical knowledge in the discussion, critique and appreciation of science, while those who miss all of them seem to be missing the fundamental concepts of what science -- or knowledge -- are at all.I hope that may be useful.Hi RuvDraba :) I agree that the above may well be the pons asinorum of modern science, or indeed of a modern intellectual, but my OP was actually asking about the pons asinorum of modern maths. Which mathematical concept do you think could play the same role that Euclid's proof did in Ancient Greece?I think it is well established that the only reason aliens come to earth is to slice up cows and examine inside peoples' bottoms. Unless you are a cow or suffer haemerrhoids I don't think there is anything to worry about from aliens. - keithprosser Don't be a stat cynic: http://www.debate.org... Response to conservative views on deforestation: http://www.debate.org... Topics I'd like to debate (not debating ATM): http://tinyurl.com...
 Posts: 6,033 Add as FriendChallenge to a DebateSend a Message 2/10/2016 6:51:47 PMPosted: 1 year agoAt 2/10/2016 6:02:55 PM, Diqiucun_Cunmin wrote:At 2/10/2016 9:39:25 AM, RuvDraba wrote:At 2/4/2016 5:26:18 PM, Diqiucun_Cunmin wrote:The pons asinorum of Greek mathematics was Proposition 5 of Book I of Euclid's Elements: The base angles of an isosceles triangle are equal. Its proof was believed to be a 'bridge': Anyone who can cross it is not a donkey. But times have changed and so have mathematics. What should the new pons asinorum be?DC, I wonder whether it's being able to explain why science:1) Relies on empiricism, yet continuously changes its model and methods;2) Is the continuing elimination of ignorance and error from knowledge, and not the conclusive proof of truth;3) Having derived from natural philosophy, and while still being stimulated by it and occasionally challenged by it, is no longer accountable to philosophy alone;4) While making extensive use of logic and math, is not created by either.In support for this contention, I find that members who understand this, can: understand a sketched rationale behind almost any accepted scientific result, recognise the difference between science and pseudoscience quite easily, and make a decent fist of understanding the philosophy of science and science history.Meanwhile, those who miss one or more of these seem to be missing critical knowledge in the discussion, critique and appreciation of science, while those who miss all of them seem to be missing the fundamental concepts of what science -- or knowledge -- are at all.I hope that may be useful.Hi RuvDraba :) I agree that the above may well be the pons asinorum of modern science, or indeed of a modern intellectual, but my OP was actually asking about the pons asinorum of modern maths.You're right, DC! You did, and I overlooked that in part because I was having another conversation (in fact several) your question pinged off intuitively. Consequently, I got quite a lot from my own contribution, while contributing nothing to your topic. My apologies!Which mathematical concept do you think could play the same role that Euclid's proof did in Ancient Greece?I think the point of a pons asinorum is to set a bar below which it's not worth doing even remedial education. How tall must you be to go on the ride?In geometry, an isoceles triangle is the simplest regular polygon, and the key test under this proof is whether you can go from knowledge about lengths to inferences about angles. If you can, you've performed a fundamental step in geometry, using the simplest example. If you can't, there's no simpler example, so you may simply not be ready for the required abstraction.I think that in modern math, that abstractive role is occupied more by algebra than by the symmetries of triangles -- in fact, very few mathematicians now make inferences the way Pappus and Euclid used to. Visual intuitions aren't trusted so much as they once were, while the linguistic substitutions of algebra are still (mostly) unquestioned.Algebra also represents a major cognitive step in a math student, from the rote methods used in arithmetic to the design of what in informatics is now called patterns -- templated methods that you can compose, adapt and decompose. Algebra has relevance to computational procedures of all kinds, and ties to logic in a nice way too.So I think the pons asinorum needs to be a simple, irreducible algebraic problem with the same sort of ambiguity as the isoceles triangle problem, requiring the same cut-through intuitions, but like the triangle problem, having few moving parts.I think I'd pick a 3x3 linear algebra problem, DC (i.e, 3 equations with 3 unknowns) -- linear algebra because its impact on math is about as high as that of calculus, because it's part of the machinery generalising calculus, because it doesn't require any operation more sophisticated than basic arithmetic, because it draws on all the symmetries underpinning arithmetic, because it's key to all manner of logistical problems besetting business and daily life, and because linear algebra is the Sudoku of math. :D3x3 because it has just enough moving parts to bewilder, not so many that you can't see that there ought to be a way of solving it, nor so few that you can be immediately confident there's only one solution, and enough combinations that it's hard to guess by trial and error. :)As to what the actual problem should be, I'm not sure, but I think it should use small numbers and feature some hint as to the importance of the diagonal, like...1x + 2y + 3z = 33x + 1y + 2z = 22x + 3y + 1z = 1Solve for x, y, z.This one solves via reduced row echelon methods in eight steps, which is probably too many. The solution it produces isn't terribly exciting, and there might be a squint-and-stab approach that solves it without producing the necessary insights, so the problem might be improved by playing.Yet it's a taunting problem for anyone who hasn't solved something like this before, and I could imagine a dedicated twelve year-old spending some hours or even days on it (if you turned off their WiFi. :D), and like all good introductory problems, it ought to produce more questions than answers. :)I hope that may be more constructive than my last post. :D
 Posts: 6,033 Add as FriendChallenge to a DebateSend a Message 2/10/2016 7:24:21 PMPosted: 1 year agoAt 2/10/2016 6:51:47 PM, RuvDraba wrote:1x + 2y + 3z = 33x + 1y + 2z = 22x + 3y + 1z = 1Solve for x, y, z.It's a taunting problem for anyone who hasn't solved something like this before, and I could imagine a dedicated twelve year-old spending some hours or even days on it (if you turned off their WiFi. :D), and like all good introductory problems, it ought to produce more questions than answers. :)I should have added that in my country at least, four in five adults probably couldn't solve this problem, DC. Nevertheless, it's simpler than most household budgeting problems (where you might have two incomes, eight lines of credit and need to balance monthly repayments. :p)So if you want a pons asinorum numeracy benchmark that relates to real-world concerns, I think this one's probably not far off. :D
 Posts: 9,512 Add as FriendChallenge to a DebateSend a Message 2/10/2016 7:53:38 PMPosted: 1 year agoYou know, it's the damndest thing--until today, I had never heard the phrase "pons asinorum" before.