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How Euler Did It, by Ed Sandifer

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Re: How Euler Did It, by Ed Sandifer

#41

Post by jterrys »

ufocia wrote: post #14
norir wrote: post #7

It's interesting to me that I can't think of anyone remotely comparable to Euler in the public consciousness today. Even the work of someone like Erdos seems very esoteric by comparison and also was largely done in collaboration. Was Euler just born at the right time and picking all the low hanging fruit? Or maybe his immense creative production was a unique consequence of wealth plus limited distractions? I'm inclin…

Maybe today's emphasis on "well rounded" education is a distraction from specialized talents.

Absolutely HARD disagree. Pythagoras, Aristotle, Pascal, Descartes, Newton, Franklin, DaVinci, (just to name a few) were all polymaths. They all left lasting impacts outside of a singular domain. It's because they were such well rounded individuals that they were so successful and capable of revolutionary discoveries and inventions. Our world is made up of models that are related with other models that are related based off of that relatedness. It's maddening. Being "well rounded" is vital.

Today that concept is watered down. A "well rounded" education is just taking a few classes that people hate and will blow off because to graduate they need to check some boxes so they can focus on doing one thing moderately well and finding their place as a cog in a machine that will abuse their ignorance. It's all mass produced conveyor belt education that manufactures young adults with little conventional wisdom. The more you lean into behaving like a part, the more you will be treated that way.

Re: How Euler Did It, by Ed Sandifer

#42

Post by lp4vn »

feoren wrote: post #15
norir wrote: post #7

It's interesting to me that I can't think of anyone remotely comparable to Euler in the public consciousness today. Even the work of someone like Erdos seems very esoteric by comparison and also was largely done in collaboration. Was Euler just born at the right time and picking all the low hanging fruit? Or maybe his immense creative production was a unique consequence of wealth plus limited distractions? I'm inclin…

Part of it is low-hanging fruit, certainly. Euler lived at a time when it was still possible to "know all math". That breadth of knowledge is simply not possible for a single human anymore; the discipline of mathematics is orders of magnitude larger. Comparable mathematicians today like Erdős and Terence Tao collaborate because they really can't learn the intricate details of every corner of math, so they collaborate…

The low hanging fruit argument only takes you so far. How many other mathematicians in his epoch or before were able to pick as many low hanging fruits as him?

Re: How Euler Did It, by Ed Sandifer

#43

Post by vacuity »

testless wrote: post #19
norir wrote: post #7

It's interesting to me that I can't think of anyone remotely comparable to Euler in the public consciousness today. Even the work of someone like Erdos seems very esoteric by comparison and also was largely done in collaboration. Was Euler just born at the right time and picking all the low hanging fruit? Or maybe his immense creative production was a unique consequence of wealth plus limited distractions? I'm inclin…

Ramanujan died early. He is another one which comes to mind.

Riemann died young, too. I only really know him for the Riemann sum formulation of integrals (I have yet to learn complex analysis), but I wouldn't be surprised if he has some popular reach through the Riemann hypothesis.

Re: How Euler Did It, by Ed Sandifer

#44

Post by norir »

feoren wrote: post #15
norir wrote: post #7

It's interesting to me that I can't think of anyone remotely comparable to Euler in the public consciousness today. Even the work of someone like Erdos seems very esoteric by comparison and also was largely done in collaboration. Was Euler just born at the right time and picking all the low hanging fruit? Or maybe his immense creative production was a unique consequence of wealth plus limited distractions? I'm inclin…

Part of it is low-hanging fruit, certainly. Euler lived at a time when it was still possible to "know all math". That breadth of knowledge is simply not possible for a single human anymore; the discipline of mathematics is orders of magnitude larger. Comparable mathematicians today like Erdős and Terence Tao collaborate because they really can't learn the intricate details of every corner of math, so they collaborate…

To be clear, I am not saying that Erdos and Tao are less talented than Euler. That seems impossible to say. But the magic of Euler is that his fingerprints are all over so many of the fundamental things that can be understood by a bright high school student but were mostly unknown before him. The work of figures like Erdos and Tao seems far, far less accessible in its present form at least and thus more limited in its overall impact.

Re: How Euler Did It, by Ed Sandifer

#46

Post by vacuity »

liendolucas wrote: post #36
testless wrote: post #29

Earlier quoted context omitted.

Let's hope education indivualizes to the need and talents of the individual partially via software. ChatGPT is my hope here. 15% failure rate is optimal for learning https://www.nature.com/articles/s41467-019-12552-4

Seriously? Oh god, please no. We need less of all that BS. People are thinking that are being "tutored" by AI, when in fact is just the output of a number crunching program. Reading books will get you way way closer to solid knowledge than the output crap of this so called "AI".

I don't think such an attitude is warranted. While I'm skeptical about the potential of LLMs as AGI (whatever that means), being able to summarize subjects and even come up with test questions can be very valuable for learning. I am concerned about the confabulation aspect, though. I wonder if there could be a uncertainty metric for that.

Re: How Euler Did It, by Ed Sandifer

#47

Post by norir »

testless wrote: post #25
norir wrote: post #7

It's interesting to me that I can't think of anyone remotely comparable to Euler in the public consciousness today. Even the work of someone like Erdos seems very esoteric by comparison and also was largely done in collaboration. Was Euler just born at the right time and picking all the low hanging fruit? Or maybe his immense creative production was a unique consequence of wealth plus limited distractions? I'm inclin…

Is Euler really well known in public? Einstein is way more well known than Euler, I'd guess. Even if you will ask most famous mathematician, it is unlikely that he is named. Probably Pythagoras. I have no data to back it up.

Sure, I didn't really mean the general public but meant scientific/mathematically oriented public. When I studied math in college, I found Euler everywhere to the point that it was hard not to stop and wonder how the hell this guy had so many fundamental discoveries. By contrast, I have awareness of present day figures like Tao and Erdos, but I don't really understand their work and perceive it to be specialized enough that it is unlikely to be widely understood in the way that say e^(pi * i) + 1 = 0 is.

Re: How Euler Did It, by Ed Sandifer

#48

Post by Duanemclemore »

ufocia wrote: post #14
norir wrote: post #7

It's interesting to me that I can't think of anyone remotely comparable to Euler in the public consciousness today. Even the work of someone like Erdos seems very esoteric by comparison and also was largely done in collaboration. Was Euler just born at the right time and picking all the low hanging fruit? Or maybe his immense creative production was a unique consequence of wealth plus limited distractions? I'm inclin…

Maybe today's emphasis on "well rounded" education is a distraction from specialized talents.

Euler was nothing if not diverse. Having a "liberal arts" / "great books" education from high school I have (anecdotal) first hand evidence in the impressive successes of my classmates in a WIDE variety of fields.

Re: How Euler Did It, by Ed Sandifer

#49

Post by agnosticmantis »

What I really like and admire about Euler is how masterfully he handled infinities and infinitesimals to arrive at correct conclusions (the vast majority of times anyway) even though analysis hadn’t been made rigorous yet (by Cauchy and Weierstrass and friends), so some of what he did was pure magic and took a lot of good intuition.

An example of this is when, in solving the then notorious Basel problem, he factors trig functions into infinite products of (x +- k*pi) terms just by analogy with root factorization in finite polynomials.

Re: How Euler Did It, by Ed Sandifer

#50

Post by testless »

liendolucas wrote: post #36
testless wrote: post #29

Earlier quoted context omitted.

Let's hope education indivualizes to the need and talents of the individual partially via software. ChatGPT is my hope here. 15% failure rate is optimal for learning https://www.nature.com/articles/s41467-019-12552-4

Seriously? Oh god, please no. We need less of all that BS. People are thinking that are being "tutored" by AI, when in fact is just the output of a number crunching program. Reading books will get you way way closer to solid knowledge than the output crap of this so called "AI".

Happy to disagree here. I am hopeful that students get apps which provide them with - fast feedback loops - better adjustment to their learning speed - more patience - gamification - oppurtunity to ask endless questions

Of course, only as an addition to the current mix. For math problems, this will be easier than for other contexts.

I like text books for pop science or really hard things - like university level education. But I am surprised to see them as an option for basic education.

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