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The pons asinorum of modern mathematics

Diqiucun_Cunmin
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2/4/2016 5:26:18 PM
Posted: 10 months ago
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, 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 equations
Central limit theorem
Polynomial equations and the Fundamental Theorem of Algebra
Equations of circles
sin^2 a + cos^2 a = 1
Taylor series
Definite integrals and the Fundamental Theorem of Calculus
Geometric series
Proof of AM >= GM >= HM
Pythagorean theorem (any proof)
Expoential and logarithmic equations
Logarithmic differentiation
Epsilon-delta proofs
Volume of a sphere
Inverse of a 3x3 matrix
Simultaneous linear equations and Gaussian elimination
Euler's formula/De Moivre's Theorem
Polynomial division and the remainder theorem
The thing is, I hate relativism. I hate relativism more than I hate everything else, excepting, maybe, fibreglass powerboats... What it overlooks, to put it briefly and crudely, is the fixed structure of human nature. - Jerry Fodor

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RuvDraba
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2/10/2016 9:39:25 AM
Posted: 9 months ago
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.
Diqiucun_Cunmin
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2/10/2016 6:02:55 PM
Posted: 9 months ago
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. Which mathematical concept do you think could play the same role that Euclid's proof did in Ancient Greece?
The thing is, I hate relativism. I hate relativism more than I hate everything else, excepting, maybe, fibreglass powerboats... What it overlooks, to put it briefly and crudely, is the fixed structure of human nature. - Jerry Fodor

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...
RuvDraba
Posts: 6,033
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2/10/2016 6:51:47 PM
Posted: 9 months ago
At 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. :D

3x3 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 = 3
3x + 1y + 2z = 2
2x + 3y + 1z = 1


Solve 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
RuvDraba
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2/10/2016 7:24:21 PM
Posted: 9 months ago
At 2/10/2016 6:51:47 PM, RuvDraba wrote:
1x + 2y + 3z = 3
3x + 1y + 2z = 2
2x + 3y + 1z = 1


Solve 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
Diqiucun_Cunmin
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2/11/2016 2:49:13 PM
Posted: 9 months ago
At 2/10/2016 6:51:47 PM, RuvDraba wrote:
At 2/10/2016 6:02:55 PM, Diqiucun_Cunmin wrote:
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!
Haha, no worries. I read your discussions from time to time - I think I've read your posts on and off since you joined - and I understand what you mean. (Pity that UndeniableReality isn't on DDO any more, btw. :( )
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 don't think there's no simpler example - Euclid's proof involved three pairs of congruent triangles, and I think there are proofs that are easier than that. So I think there's still a sort of complexity there - the same sort that I find first-order linear ODEs. I understand your point though - the pons asinorum should also signify some kind of transition to 'higher' maths.
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.
Very true. The only geometric proof I remember doing that isn't directly related to shapes is that of the AM >= GM >= HM inequality.
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.
I agree. Actually, the introduction of factorisation was the time I struggled most in mathematics - I overcame it eventually by lots of drilling! (Though by that time it was three years after I was first introduced to algebra.)
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. :D
Haha, I agree with all those, particularly the last line. :)
3x3 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 = 3
3x + 1y + 2z = 2
2x + 3y + 1z = 1


Solve 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 would make the coefficients a tad more complicated than that, personally. I think a 12-year-old can solve this fairly easily using the substitution and elimination methods that they've already learnt - I've done the problem both the Gaussian way and the algebraic way, and I think the former isn't significantly faster :P Or maybe an additional unknown could be introduced - that would make the problem much harder to solve using elementary algebra.

But I agree that if given necessary instruction, simultaneous equations could be an instructive introduction to the world of linear algebra and 'higher' mathematics in general (in fact, it was one of the candidates I mentioned in the OP XD).
I hope that may be more constructive than my last post. :D
It was very insightful :)
The thing is, I hate relativism. I hate relativism more than I hate everything else, excepting, maybe, fibreglass powerboats... What it overlooks, to put it briefly and crudely, is the fixed structure of human nature. - Jerry Fodor

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...
Diqiucun_Cunmin
Posts: 2,710
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2/11/2016 2:51:59 PM
Posted: 9 months ago
At 2/10/2016 7:24:21 PM, RuvDraba wrote:
At 2/10/2016 6:51:47 PM, RuvDraba wrote:
1x + 2y + 3z = 3
3x + 1y + 2z = 2
2x + 3y + 1z = 1


Solve 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)
And you want to minimise tax :P A knowledge of optimisation and linear programming will probably go a long way toward saving money in real life as well, though I'm not sure how many people remember these from school to apply them to daily life. XD

I think most adults can still do simultaneous equations using the substitution/elimination methods if they're pressed to do it, though the majority probably won't recall the reduced row echelon form (or the row echelon form for that matter).
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
Yep :)
The thing is, I hate relativism. I hate relativism more than I hate everything else, excepting, maybe, fibreglass powerboats... What it overlooks, to put it briefly and crudely, is the fixed structure of human nature. - Jerry Fodor

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...
Diqiucun_Cunmin
Posts: 2,710
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2/11/2016 2:52:55 PM
Posted: 9 months ago
At 2/10/2016 7:53:38 PM, Cody_Franklin wrote:
You know, it's the damndest thing--until today, I had never heard the phrase "pons asinorum" before.

Hi Cody! Always great to see you around and I hope you're online more :P You're always insightful, and your posts are always thought-provoking.

And wow, there are words you don't know? :O
The thing is, I hate relativism. I hate relativism more than I hate everything else, excepting, maybe, fibreglass powerboats... What it overlooks, to put it briefly and crudely, is the fixed structure of human nature. - Jerry Fodor

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...
RuvDraba
Posts: 6,033
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2/11/2016 9:33:51 PM
Posted: 9 months ago
At 2/11/2016 2:49:13 PM, Diqiucun_Cunmin wrote:
At 2/10/2016 6:51:47 PM, RuvDraba wrote:
At 2/10/2016 6:02:55 PM, Diqiucun_Cunmin wrote:
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!
Haha, no worries. I read your discussions from time to time - I think I've read your posts on and off since you joined - and I understand what you mean. (Pity that UndeniableReality isn't on DDO any more, btw. :( )
I miss him too, but he's a busy boy, completing a PhD and working in a start-up when last I heard. :)

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 don't think there's no simpler example - Euclid's proof involved three pairs of congruent triangles, and I think there are proofs that are easier than that.
Yes, Pappus has a fun one that uses only one triangle viewed two different ways. [http://www.themathpage.com...] It has sometimes been called an 'Irish bull' proof -- because the triangle is in two places at once. :D [https://en.wikipedia.org...] By way of explanation, the obscure term 'Irish bull' normally applies to witty but contradictory aphorisms like:
He'll regret it till his dying day, if ever he lives that long, or
Always go to other people's funerals, otherwise they won't come to yours

In the TV animation series Archer, the spy Archer has a cyborg nemesis called Barry -- who often engages in dialogue with himself as 'other Barry'. I think Pappus' proof is like that -- an isoceles triangle talking to the other isoceles triangle with the same sides and angles as itself. :D

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. :D
Haha, I agree with all those, particularly the last line. :)

Well, Gauss went a bit up-market from my idea. He felt that the best pons asinorum in modern math was the Euler identity:

e ^ (i * pi) + 1 = 0

Aside from being beautifully simple, it connects logarithms, complex analysis, geometry and number theory. Gauss' view was that if you could see right away why it was true you'd be a great mathematician. If you couldn't, you kinda shouldn't bother. :D

A simple argument for why it's true comes form Euler's formula in complex analysis:

e ^ (i * x) = cos x + i sin x.

When x = pi, the result drops out easily. But you need to know complex analysis, or it'll still be mysterious. However, there are number-theoretic and geometric explanations too on the Wikipedia page [https://en.wikipedia.org...'s_identity]

Yet I still like the linear algebra example simply because just about anyone with high school math can attempt it, yet (I suspect) few can complete it despite its apparent simplicity.

3x3 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 = 3
3x + 1y + 2z = 2
2x + 3y + 1z = 1


Solve for x, y, z.

I would make the coefficients a tad more complicated than that, personally. I think a 12-year-old can solve this fairly easily using the substitution and elimination methods that they've already learnt.
It's possible that a twelve year-old might do a better job than most adults on this, DC, but if I handed this problem to my clients (most of them white-collar professionals in senior business roles) and demanded they solve it or I'll fire-bomb their Volvo, I reckon most would be hunting and pecking at guessed numbers. :p

But I agree that if given necessary instruction, simultaneous equations could be an instructive introduction to the world of linear algebra and 'higher' mathematics in general (in fact, it was one of the candidates I mentioned in the OP XD).
Yes -- I realised after I pushed 'send' that you had already listed a matrix inversion I hadn't acknowledged -- so let me do so now. :D

I hope that may be more constructive than my last post. :D
It was very insightful :)
Thank you for a fun question -- and congrats on getting your thread stickied (however briefly) as an exemplary Science topic. :D
keithprosser
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2/11/2016 10:00:19 PM
Posted: 9 months ago
But times have changed and so have mathematics. What should the new pons asinorum be?

Whether times and maths have changed I will not consider, but human minds haven't changed. The point of the original PA is that it is a test for an untutored mind, for 'mathematical virgins'. I think that anything requiring even minimal exposure to mathematical concepts can't be a PA, ruling out advanced algebra and calculus.

I think its still true that a mind capable of appreciating Euclid's 5th can learn much of mathematics (ancient and modern), and a mind that can't get it is probably best off hoping for a football scholarship. And of course footballers are paid more than mathematicians, so who is the a$$ anyway :( ?
Diqiucun_Cunmin
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2/12/2016 5:00:43 AM
Posted: 9 months ago
At 2/11/2016 9:33:51 PM, RuvDraba wrote:
At 2/11/2016 2:49:13 PM, Diqiucun_Cunmin wrote:
At 2/10/2016 6:51:47 PM, RuvDraba wrote:
At 2/10/2016 6:02:55 PM, Diqiucun_Cunmin wrote:
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!
Haha, no worries. I read your discussions from time to time - I think I've read your posts on and off since you joined - and I understand what you mean. (Pity that UndeniableReality isn't on DDO any more, btw. :( )
I miss him too, but he's a busy boy, completing a PhD and working in a start-up when last I heard. :)
Wow, he's working in a startup as well :O No wonder why he's busy. I'd like to do that too - hopefully, after I declare CS as my second major, I'll be able to find internship positions in tech startups...
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 don't think there's no simpler example - Euclid's proof involved three pairs of congruent triangles, and I think there are proofs that are easier than that.
Yes, Pappus has a fun one that uses only one triangle viewed two different ways. [http://www.themathpage.com...] It has sometimes been called an 'Irish bull' proof -- because the triangle is in two places at once. :D [https://en.wikipedia.org...]
Haha, I think this proof is actually easier than Euclid's...
By way of explanation, the obscure term 'Irish bull' normally applies to witty but contradictory aphorisms like:
He'll regret it till his dying day, if ever he lives that long, or
Always go to other people's funerals, otherwise they won't come to yours
Reminds me of Goldwynisms :P
In the TV animation series Archer, the spy Archer has a cyborg nemesis called Barry -- who often engages in dialogue with himself as 'other Barry'. I think Pappus' proof is like that -- an isoceles triangle talking to the other isoceles triangle with the same sides and angles as itself. :D


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. :D
Haha, I agree with all those, particularly the last line. :)

Well, Gauss went a bit up-market from my idea. He felt that the best pons asinorum in modern math was the Euler identity:

e ^ (i * pi) + 1 = 0

Aside from being beautifully simple, it connects logarithms, complex analysis, geometry and number theory. Gauss' view was that if you could see right away why it was true you'd be a great mathematician. If you couldn't, you kinda shouldn't bother. :D

A simple argument for why it's true comes form Euler's formula in complex analysis:

e ^ (i * x) = cos x + i sin x.

When x = pi, the result drops out easily. But you need to know complex analysis, or it'll still be mysterious. However, there are number-theoretic and geometric explanations too on the Wikipedia page [https://en.wikipedia.org...'s_identity]
Yeah, Euler's formula is the obvious route :P Though proving Euler's formula (to reduce this to more elementary axioms) would be harder, as it's not mere substitution.
Yet I still like the linear algebra example simply because just about anyone with high school math can attempt it, yet (I suspect) few can complete it despite its apparent simplicity.

3x3 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 = 3
3x + 1y + 2z = 2
2x + 3y + 1z = 1


Solve for x, y, z.

I would make the coefficients a tad more complicated than that, personally. I think a 12-year-old can solve this fairly easily using the substitution and elimination methods that they've already learnt.
It's possible that a twelve year-old might do a better job than most adults on this, DC, but if I handed this problem to my clients (most of them white-collar professionals in senior business roles) and demanded they solve it or I'll fire-bomb their Volvo, I reckon most would be hunting and pecking at guessed numbers. :p

But I agree that if given necessary instruction, simultaneous equations could be an instructive introduction to the world of linear algebra and 'higher' mathematics in general (in fact, it was one of the candidates I mentioned in the OP XD).
Yes -- I realised after I pushed 'send' that you had already listed a matrix inversion I hadn't acknowledged -- so let me do so now. :D

I hope that may be more constructive than my last post. :D
It was very insightful :)
Thank you for a fun question -- and congrats on getting your thread stickied (however briefly) as an exemplary Science topic. :D
Thanks :)
The thing is, I hate relativism. I hate relativism more than I hate everything else, excepting, maybe, fibreglass powerboats... What it overlooks, to put it briefly and crudely, is the fixed structure of human nature. - Jerry Fodor

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Diqiucun_Cunmin
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2/12/2016 5:05:53 AM
Posted: 9 months ago
At 2/11/2016 10:00:19 PM, keithprosser wrote:
But times have changed and so have mathematics. What should the new pons asinorum be?

Whether times and maths have changed I will not consider, but human minds haven't changed. The point of the original PA is that it is a test for an untutored mind, for 'mathematical virgins'. I think that anything requiring even minimal exposure to mathematical concepts can't be a PA, ruling out advanced algebra and calculus.
Well, I think Euclid's fifth does still require some knowledge of the previous axioms and theorems as well as simpler maths. Euclid's proof may seem simple for modern humans (taught at the age of 12-13), but for the Ancient Greeks, it may already be pretty advanced, and probably represented a transition to higher mathematics comparable to the transition from trigonometry and elementary algebra to calculus and linear algebra today. Just my take on this ;)
I think its still true that a mind capable of appreciating Euclid's 5th can learn much of mathematics (ancient and modern), and a mind that can't get it is probably best off hoping for a football scholarship. And of course footballers are paid more than mathematicians, so who is the a$$ anyway :( ?
Aww, don't say that. You don't have to be a mathematician by profession to appreciate the beauty of maths (I'm not a mathematics major myself). Lorenz, Feigenbaum, etc. are not mathematicians by profession either, but their contributions to maths are immense. I think even footballers can benefit from mathematics sometimes...
The thing is, I hate relativism. I hate relativism more than I hate everything else, excepting, maybe, fibreglass powerboats... What it overlooks, to put it briefly and crudely, is the fixed structure of human nature. - Jerry Fodor

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...
tejretics
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2/13/2016 4:00:24 AM
Posted: 9 months ago
At 2/10/2016 7:53:38 PM, Cody_Franklin wrote:
You know, it's the damndest thing--until today, I had never heard the phrase "pons asinorum" before.

^This
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