About six weeks ago, Ne’Kiya Jackson and Calcea Johnson gave a talk at a meeting of the American Mathematical Society in which they presented a new trigonometric proof of the Pythagorean Theorem.
Even though I’m a pretty standard liberal, I like to dip my toes into heterodox waters to challenge my priors....which inevitably leads to Charles Murray. I keep trying to see what they think exonerates him, but I keep coming up underwhelmed. So for better or worse it’s good to hear that you come away with a similar impression.
I was wondering how I knew the name Charles Murray…
Charles Murray is the author of The Bell Curve, a book which has done much to popularize pseudoscientific race essentialism.
Stephen Jay Gould (an actual evolutionary biologist, unlike Murray) wrote The Mismeasure of Man specifically for the purpose of debunking The Bell Curve.
I’m not really sure why I should trust Murray on anything else, when his most prominent work can be charitably called a misunderstanding of statistics (and uncharitably, a nefarious manipulation of data to reinforce his own preferences for racial hierarchy).
I understand the Know Your Enemy argument you put forth, but why read Charles Murray when there’s plenty of thinkers out there producing correct information to learn about?
Why is everyone only mentioning one of the two proofs? They each independently figured out a proof. One with the infinite triangles, the other with a circle. Why doesn't anyone show the circle proof? My goodness.
Their accomplishment is unlikely. But, I know enough to see that their supposed method of proof manifests cleverness and an inventive mind (or two minds). At least one, if not both, of these kids is clever enough to warrant great opportunities at further education. She/they might even prove valuable leaders in academia.
The law of sines relies on the definition of an angle, which is the arc length of a unit circle using the metric space defined by x^2+y^2=r^2, which is in effect the pythagorean theorem. This makes the law of sines not independent of the Pythagorean Theorem
I'm looking for whether people have found anybody who already done the proof. It would be no reflection on the girls if they did; that often happens in academia, and it's an interesting question who should get the credit, the older or the better-written discovery.
In any case, can we get the law of sines in some other way? Defining a metric space sounds like heavier machinery than needed, tho maybe the Greeks or whoever got the law of sines in essentially the same way.
There's a trigonometric proof by Jason Zimba in 2009 which I think is the first, it's simpler but less elegant in my opinion than the waffle cone idea:
Thank you. Good webpost. I liked "A classical book by Elisha Loomis, The Pythagorean Proposition, contains some 370 different proofs of Pythagoras’ Theorem. The book has a section entitled “Why No Trigonometric, Analytic Geometry Nor Calculus Proof Possible”, but this was published in 1940 and was shown to be wrong when, in 2009, Jason Zimba published a proof using trigonometry."
There is an interesting issue of whether we should be readily spreading the idea of essentialism in areas such as IQ even IF it is true. There may be a good argument to suggest that while we need not hide information we should not necessarily look to remove the friction involved in understanding outcomes supporting the essentialist view by simply putting it in the form of a simple conclusion of the average capacity of group A being higher than that of Group B.
The friction in having to go through research and potentially come across such findings on the greater relevance of other factors when it comes to "lifetime success" and may help avoid to some extent the negative impact of the information being shared in its distilled form.
I'm mindful that this may smack of a sort of elitism regarding what information should be disseminated and how ,but I think it could be likened to not teaching students advanced chemistry without informing them of how some chemicals may be explosive.
This line of thinking makes axiom of anti-essentialism a better solution regardless of the truth of proposition that differences in group outcomes are not essential to the groups.
Even if all of the above is nonsense, the axiom of anti-essentialism of course makes great sense from a policy perspective because there is little benefit in looking for differences that we can't really address unless we have already achieved utopia where all groups have maximized their "essential" potential.
Thanks you for the interesting write up. Wishing the best for the young brilliant mathematicians.
Even though I’m a pretty standard liberal, I like to dip my toes into heterodox waters to challenge my priors....which inevitably leads to Charles Murray. I keep trying to see what they think exonerates him, but I keep coming up underwhelmed. So for better or worse it’s good to hear that you come away with a similar impression.
I do think he has useful things to say on some topics but also such horrific blind spots and a smug sense of superiority
I was wondering how I knew the name Charles Murray…
Charles Murray is the author of The Bell Curve, a book which has done much to popularize pseudoscientific race essentialism.
Stephen Jay Gould (an actual evolutionary biologist, unlike Murray) wrote The Mismeasure of Man specifically for the purpose of debunking The Bell Curve.
I’m not really sure why I should trust Murray on anything else, when his most prominent work can be charitably called a misunderstanding of statistics (and uncharitably, a nefarious manipulation of data to reinforce his own preferences for racial hierarchy).
I understand the Know Your Enemy argument you put forth, but why read Charles Murray when there’s plenty of thinkers out there producing correct information to learn about?
Why is everyone only mentioning one of the two proofs? They each independently figured out a proof. One with the infinite triangles, the other with a circle. Why doesn't anyone show the circle proof? My goodness.
Their accomplishment is unlikely. But, I know enough to see that their supposed method of proof manifests cleverness and an inventive mind (or two minds). At least one, if not both, of these kids is clever enough to warrant great opportunities at further education. She/they might even prove valuable leaders in academia.
You'd like this Substack of mine on Euler and God: https://ericrasmusen.substack.com/p/abnn-x-therefore-god-exists
The law of sines relies on the definition of an angle, which is the arc length of a unit circle using the metric space defined by x^2+y^2=r^2, which is in effect the pythagorean theorem. This makes the law of sines not independent of the Pythagorean Theorem
I'm looking for whether people have found anybody who already done the proof. It would be no reflection on the girls if they did; that often happens in academia, and it's an interesting question who should get the credit, the older or the better-written discovery.
In any case, can we get the law of sines in some other way? Defining a metric space sounds like heavier machinery than needed, tho maybe the Greeks or whoever got the law of sines in essentially the same way.
There's a trigonometric proof by Jason Zimba in 2009 which I think is the first, it's simpler but less elegant in my opinion than the waffle cone idea:
://thatsmaths.com/2023/06/01/the-waffle-cone-and-a-new-proof-of-pythagoras-theorem/
Thank you. Good webpost. I liked "A classical book by Elisha Loomis, The Pythagorean Proposition, contains some 370 different proofs of Pythagoras’ Theorem. The book has a section entitled “Why No Trigonometric, Analytic Geometry Nor Calculus Proof Possible”, but this was published in 1940 and was shown to be wrong when, in 2009, Jason Zimba published a proof using trigonometry."
Sorry link here:
https://thatsmaths.com/2023/06/01/the-waffle-cone-and-a-new-proof-of-pythagoras-theorem/
There is an interesting issue of whether we should be readily spreading the idea of essentialism in areas such as IQ even IF it is true. There may be a good argument to suggest that while we need not hide information we should not necessarily look to remove the friction involved in understanding outcomes supporting the essentialist view by simply putting it in the form of a simple conclusion of the average capacity of group A being higher than that of Group B.
The friction in having to go through research and potentially come across such findings on the greater relevance of other factors when it comes to "lifetime success" and may help avoid to some extent the negative impact of the information being shared in its distilled form.
I'm mindful that this may smack of a sort of elitism regarding what information should be disseminated and how ,but I think it could be likened to not teaching students advanced chemistry without informing them of how some chemicals may be explosive.
This line of thinking makes axiom of anti-essentialism a better solution regardless of the truth of proposition that differences in group outcomes are not essential to the groups.
Even if all of the above is nonsense, the axiom of anti-essentialism of course makes great sense from a policy perspective because there is little benefit in looking for differences that we can't really address unless we have already achieved utopia where all groups have maximized their "essential" potential.
Thanks you for the interesting write up. Wishing the best for the young brilliant mathematicians.