Can high school students know calculus better than Newton?

[Nereid]

Or, to write the question in slightly fuller form:

Assuming a sufficient level of interest and competence, and very good teachers, could a high school student today understand the calculus better than Newton or Leibniz ever did in their lifetimes? And if not a high school student, then maybe an undergrad university one?

The trigger for this question comes from Sean Carroll's blog, The Alternative-Science Respectability Checklist, and specifically his comment:

By almost any standard, I understand general relativity better than Einstein ever did. (Most parts of it, anyway.) Not because I’m anywhere nearly as smart as Einstein, but because we’ve learned a lot about GR since Einstein died. Once the theory was invented, he didn’t have a monopoly on it; it was out there for anyone to understand and move forward with.

In at least one respect, a high school student today can understand the calculus better: she can get a handle on limits and infinitesimals that is far better than Newton's was.

So, I'm not all that interested in whether the answer is "yes", or "no", but rather in what respect, which aspects, can we say that a dozen years' or so of dedicated study can give you a better understanding of the calculus than that which either of the geniuses who first discovered (invented?) it ever had?

 

[ank_gl]

i think its true. the people who discovered these coveted stuff got them after years of study and study and study. today a 16-17 year old kid start learning them. we have better understanding now. lot has been done after the greats died, not taking anything away from them, people these days are generally more smarter than them(but they couldn't have been so without them).

 

[enricfemi]

long ago,i have the chance to read Newton's genius work,mathematical principle.it's really difficult for a college student to understand it.not because its contents out of the course in university,but the calculates methods' really nuisance.

so i think calculus,in this hundreds of years,really transformed enormously.and as far as i know, even Leibniz made some mistakes on calculus.

so we surely can know much more than them.

 

[Kummer]

A person today might know more math "the gods". But, no way, will that person be more brilliant then them. A simple test. Take a person from college of todays time and sent him back in time 400 years before Newton. Will he be able to teach the world, affect the world, just as Newton did? No, he shall not.

 

[morphism]

So, I'm not all that interested in whether the answer is "yes", or "no", but rather in what respect, which aspects, can we say that a dozen years' or so of dedicated study can give you a better understanding of the calculus than that which either of the geniuses who first discovered (invented?) it ever had?

Well, calculus didn't end with Newton and Leibniz. Mathematicians have taken the ideas much, much further, bringing lots of insight to what "the bigger picture" is. So I would say that the answer is definitely "yes" in the sense that you can understand the inner-workings of calculus (a la analysis) much better than Newton and Leibniz did.

 

[ank_gl]

Take a person from college of todays time and sent him back in time 400 years before Newton. Will he be able to teach the world, affect the world, just as Newton did? No, he shall not.

i disagree, that perso has faaaar more understanding than Newton or Einstein did, he ll be more capable of making people understand.

but yes, Newton, Einstein and many more were brilliant minds. it takes a lot of brain to understand concepts which students do have, but it takes a lot lot more to discover them like Newton and some others had.
it seems too easy why apple falls downward or Earth revolves around sun now but it was not so earlier. it took the genious minds of Newton, gallileo etc etc to know that

 

[arildno]

Remember that the ancients were encumbered with many hindrances we are not:

a) Not knowing where and how calculus has been applied, or can be applied
b) Persistent beliefs in numerology
c) The lack of a ready-made notation that simplifies learning
d) The lack of peers that can answer your questions, dispel your flawed thinking and guide you along the best path
e) A lack of recognition amongst others, who would think it "strange and incomprehensible" what you were doing.

And probably a lot more..

 

[Kummer]

i disagree, that perso has faaaar more understanding than Newton or Einstein did, he ll be more capable of making people understand.

Oh no he will not. I can see how stupid most college students are. All they do is just memorize formulas, no understanding, no appreciation. These type of people cannot teach others. They will not be able to achieve the same things what the gods have.


People do not care about how to solve it, but rather why it solves it. And my parable about college students back in time explains it. A college student might be able to explain how to solve the heat equation but he will not able to explain to people why this works. And this is his weakness. This is what the gods did not have, they knew what they were doing.

 

[ank_gl]

Oh no he will not. I can see how stupid most college students are. All they do is just memorize formulas, no understanding, no appreciation. These type of people cannot teach others. They will not be able to achieve the same things what the gods have.


People do not care about how to solve it, but rather why it solves it. And my parable about college students back in time explains it. A college student might be able to explain how to solve the heat equation but he will not able to explain to people why this works. And this is his weakness. This is what the gods did not have, they knew what they were doing.

it can't be said about all the college students. moreover all students can't be brilliant, most of them are supposed to be stupid but there are good students too

 

[Klaus_Hoffmann]

What is easy is just very easy, i wish math problem were soooo hard as finding 'Calculus' , 'Fourier series' , 'Borel resummation' or similar things that anyone can understand

 

[enricfemi]

Oh no he will not. I can see how stupid most college students are. All they do is just memorize formulas, no understanding, no appreciation. These type of people cannot teach others. They will not be able to achieve the same things what the gods have.


People do not care about how to solve it, but rather why it solves it. And my parable about college students back in time explains it. A college student might be able to explain how to solve the heat equation but he will not able to explain to people why this works. And this is his weakness. This is what the gods did not have, they knew what they were doing.

i can't approve of your comments about college students,because you know my occupation in former.
but i just think we really have much insufficiency than this genius in some aspect.since they use intuition to reflect problems.and when they publish their work ,the intuition and the method can't explain in words.so much of it can't deliver to us.

 

[matt grime]

This question seems entirely silly. Kummer hit the nail on the head in post 4.

 

[turbo]

Or, to write the question in slightly fuller form:

Assuming a sufficient level of interest and competence, and very good teachers, could a high school student today understand the calculus better than Newton or Leibniz ever did in their lifetimes? And if not a high school student, then maybe an undergrad university one?

The trigger for this question comes from Sean Carroll's blog, The Alternative-Science Respectability Checklist, and specifically his comment:In at least one respect, a high school student today can understand the calculus better: she can get a handle on limits and infinitesimals that is far better than Newton's was.

So, I'm not all that interested in whether the answer is "yes", or "no", but rather in what respect, which aspects, can we say that a dozen years' or so of dedicated study can give you a better understanding of the calculus than that which either of the geniuses who first discovered (invented?) it ever had?

No. A high school student can learn the rules and tricks of calculus to do things that Newton did not anticipate, but they cannot aspire to gain the insight that inspired Newton's invention of the field. For the same reason, Sean Carroll cannot claim to "understand" relativity to a greater degree than Einstein. Carroll has "learned" (absorbed, processed and regurgitated) what learned men of physics believe about their understandings of relativity, but he does not understand the concepts that vexed Einstein and that plagued him to the end of his life. Truth lies in the questions about the basics. GR does not address the mechanics of gravitation, inertia, or the behavior of EM, and Einstein was painfully aware of this all the rest of his life.

 

[Kummer]

I purposely chose the word "most" in my post to not include all college students. There are the very smart collges students. But most are not smart enough.

My point is that, yes a person (college student) can know more math than Newton, but he cannot possibly outrank him. Newton's brilliance will be evident immediately. When faced with an extremely difficult problem, I doubt the college student will succede to solve it if he never encountered such a problem before. But Newton, though limited in his mathematical tools, shall find a brilliant solution to this problem. That is my point. It is not how much you know, rather how good you are at what you know.

For example, the mathematician Ramanajuan did not know much compared to Gofried Harlod Hardy, but he certainly outranked him in his brilliance.

Kummer hit the nail on the head in post 4.

What does that expression mean in English? Does it mean you agree with me?

-Wolfgang

 

[Nereid]

This question seems entirely silly.

I'm curious to know more of why you think so, matt grime.

To what extent is it because it seeks to ask about Newton and the calculus (rather than Einstein and GR)?

... about maths (rather than physics)?

... is on a topic that has centuries of development (rather than not quite 100)?

... something else?

Perhaps I worded the question poorly ... I'm curious to know the extent to which it is, indeed, possible to stand on the shoulders of giants.

Kummer hit the nail on the head in post 4.

Hmm ... Kummer focussed on a point that Carroll explicitly excluded ("Not because I’m anywhere nearly as smart as Einstein") - perhaps the OP should have been more explicit wrt excluding how smart Newton was compared to any high school student.

 

[Nereid]

No. A high school student can learn the rules and tricks of calculus to do things that Newton did not anticipate, but they cannot aspire to gain the insight that inspired Newton's invention of the field.

Interesting point of view, turbo-1!

I think you only addressed part of the question though, or gave only two aspects (practical application and insight/inspiration, as elements of understanding) - what about infinitesimals and limits (for example)? A sufficiently competent and interested high school student, with very good teachers, can surely understand these key aspects far better than Newton ever could ... or not. What do you think?

Oh, and I note that you think Newton (and, presumably, Leibniz) "invented" the calculus, rather than "discovered" it; to what extent does 'maths as invention' lead to an inability of students many centuries later being unable to understand the relevant math better, in any way, than its inventor?

Suppose instead of Newton and Leibniz and the calculus I had used Euclid and geometry; would your answer still be the same?

For the same reason, Sean Carroll cannot claim to "understand" relativity to a greater degree than Einstein. Carroll has "learned" (absorbed, processed and regurgitated) what learned men of physics believe about their understandings of relativity, but he does not understand the concepts that vexed Einstein and that plagued him to the end of his life. Truth lies in the questions about the basics. GR does not address the mechanics of gravitation, inertia, or the behavior of EM, and Einstein was painfully aware of this all the rest of his life.

My use of Carroll's blog was, merely, to record what triggered me to start this thread; I had not intended that we would discuss Carroll, Einstein, or GR in the Mathematics section ...

However, I think it would be an interesting discussion, so I'll start another thread, in the relevant part of PF. Am not sure where we'd cover the extent to which any Newton/calculus vs Einstein/GR differences could be attributed to maths vs physics, many centuries of development vs not yet one, etc.

 

[Nereid]

Oh no he will not. I can see how stupid most college students are. All they do is just memorize formulas, no understanding, no appreciation. These type of people cannot teach others. They will not be able to achieve the same things what the gods have.


People do not care about how to solve it, but rather why it solves it. And my parable about college students back in time explains it. A college student might be able to explain how to solve the heat equation but he will not able to explain to people why this works. And this is his weakness. This is what the gods did not have, they knew what they were doing.

This goes to one of the bounds I set ("a sufficient level of [...] competence") - perhaps we could quantify it somewhat? Approx 1% of the relevant high school student population? 0.01%?

And let's not overlook the two other bounds - "a sufficient level of interest" and "very good teachers". Sure, there will be very few with sufficient interest, and by definition very good teachers are at least somewhat scarce ... and maybe the context of the question is more realistic wrt university undergrads who are majoring in a relevant field?

As you've probably gathered by now, my interest is more general than just Newton, Leibniz and the calculus ... but they make for a good example, if only because no one doubts Newton's brilliance (wrt the calculus) and, equally, no one doubts that the work he did has been deepened and extended substantially in the intervening centuries (so, in some sense, we collectively understand the calculus far, far better than Newton or Leibniz ever did).

 

[matt grime]

The fact that you keep referring to 'the calculus' makes the question even sillier...

Look, calc, as it is taught now, bears little relation to what Newton wrote. You are comparing apples and oranges. The point is that Newton knew very little of what we now treat as analysis, but that has no relation to what he would have understood, or not understood, of the subject as it now stands.

 

[Klaus_Hoffmann]

I purposely chose the word "most" in my post to not include all college students. There are the very smart collges students. But most are not smart enough.

My point is that, yes a person (college student) can know more math than Newton, but he cannot possibly outrank him. Newton's brilliance will be evident immediately. When faced with an extremely difficult problem, I doubt the college student will succede to solve it if he never encountered such a problem before. But Newton, though limited in his mathematical tools, shall find a brilliant solution to this problem. That is my point. It is not how much you know, rather how good you are at what you know.

For example, the mathematician Ramanajuan did not know much compared to Gofried Harlod Hardy, but he certainly outranked him in his brilliance.


What does that expression mean in English? Does it mean you agree with me?

-Wolfgang

But the problems Newton faced were easier than anything you can find today, and a similar thing happens with Einstein's SR, Newton only used Trigonommetry and Euclidean Geommetry and Einstein (regarding SR) only used rotations and vector analysis, math tools that almost every high school student uses everyday and that anyone can understand, at least for me it has more merit a theory involving complicate math calculations although is not completely correct (approximation) and several pages and theorem to show it rather than a theory that is a direct result from previous one (for example Newton's law of Gravitation is a direct conclusion of the F=ma known by Galileo and Kepler's second law about orbits) and if Einstein were so smart he should have discovered Quantum Gravity or at least the Semi-classical Quantization of his own theory.

Of course i agree the inmense merit of Ramanujan,.. without almost any college (did he take some HIgh-School or similar ??) he invented many beatiful formulae

 

[turbo]

Oh, and I note that you think Newton (and, presumably, Leibniz) "invented" the calculus, rather than "discovered" it; to what extent does 'maths as invention' lead to an inability of students many centuries later being unable to understand the relevant math better, in any way, than its inventor?

Newton needed a mathematical tool that did not exist to solve the problems that faced him, and he invented it. The thought that he had "discovered" this tool brings with it the philosophical assumption that the tool exists independently of the creator of the tool and only needed to be stumbled on by somebody. I do not share this view of mathematics, nor its applicability to physical phenomena.

This point of view ascribes a sort of physical reality to a mathematical tool that is unwarranted. This has a parallel in the common attitude that the mathematics underlying GR have some sort of objective reality instead of simply being a tool for predicting the behavior of mass embedded in space and EM propagating through space. Einstein did not believe in the objective reality of the field equations and was quite frustrated by this attitude that grew amongst his contemporaries. (I almost typed "peers", but that would have been a very unfortunate choice of words.) Until the end of his life, he tried to determine the origin of gravitation, inertial forces, and the nature of EM propagation through the vacuum. Nobody teaches this these days - only the convoluted math that is demonstrably incompatible with quantum theory.

 

[mathwonk]

this discussion belongs in the lounge section, or philosophy, as there is no mathematics in it.

 

[Gib Z]

But the problems Newton faced were easier than anything you can find today, and a similar thing happens with Einstein's SR, Newton only used Trigonommetry and Euclidean Geommetry and Einstein (regarding SR) only used rotations and vector analysis, math tools that almost every high school student uses everyday and that anyone can understand, ...(for example Newton's law of Gravitation is a direct conclusion of the F=ma known by Galileo and Kepler's second law about orbits) and if Einstein were so smart he should have discovered Quantum Gravity or at least the Semi-classical Quantization of his own theory.

Klaus, there is so much wrong with that post. It disgusts me.

Why don't you go back to Newton's time with his mathematical tools and leave me a rock buried somewhere engraved with the news of your amazing mathematical insights?

Also, Special Relativity makes use of more complex tools, if you actually knew Jack about it then you would know.

I have no idea what high school you went/go to, but my classmates definitely do not know rotations and/or vector analysis. And We all know that you are very very smart Klaus, but topics of that level are not within every persons reach. You seem to be trying very hard to make things seem easier than they actually are.

Not to mention, Newton's law's of motion are Required to prove Kepler's previously empircal results! Do go around stating otherwise!

The last part guts me the most. What the hell are you talking about? Are you taking a go at Einstein? You are saying that any smart person should be able to derive Quantum Gravity? Even though no currently accepted theory exists (not due to any lack of people working on the problem)? Like General and Special Relativity wasn't enough, Browain motion and the photoelectric effect to kick it, you still think that he should have "at least" found QFT? Like all of that would have been easy for you? Einstein was on the cutting edge of physics for 50 years.

Don't you dare try to match up to Einstein, because you'll lose.

 

[Klaus_Hoffmann]

The question not only for 'Gib-Z' but all is just take a look at the papers published in 17-th 18-th or even 19-th century (of course there may be many exceptions) or the Papers involving "Photoelectric Effect" or similar and watch the papers on Arxiv to realize how complicate they are.. i think that many Graduate or Ph.D student can solve 'Blackbody radiation' (in the same fashion Planck did) or 'Bohr's MOdel' since math for these were very very easy however take a look to standard model QCD or QED or similar What is easier to understand and solve ?

For me is just like the difference Who's smarter ??(for you) the man who invented 'lollipops' (or other candy stuff) or the man who developed the rockets or movil phone?

 

[Gib Z]

The complexity of the problem is relative. Not to mention due to the more advanced nature of the problems, of course it will look more difficult. However it takes more Genius to find a geometric proof to Kepler's laws, as Newton did, than it does for someone to compute a somewhat difficult integral. The latter will seem like a more difficult task, but It actually isn't. Some integrals require insight, a lot don't. If there is 1 word to describe Newton, it is Genius.

I'm not sure you understand my point, and really I don't expect you to "Klaus". Maybe this will make it more clear: Smarter isn't really even the right word, but I would say the person who invented the wheel is 'smarter' than the Chinese people who invented the rocket, or the Person who invented the mobile phone. Can you appreciate that?

PS. Have you actually read the papers you talk about? If it is allowed to use the same argument Planck and Bohr used, then 15 year olds like me are able to do it. Not because I'm just as smart, but because I've absorbed the knowledge that they have had the genius to originally obtain. Note than I haven't been to Uni, or have a Ph.D.

And I've seen some of the math you've have problems with on threads here Klaus, don't lie to me when you say you even vaguely understand QED.

 

[arildno]

You are so dead wrong, Klaus!

Those topics that Newton and others struggled with has been HONED INTO PERFECTION and MAXIMAL PEDOGOGICAL EFFICIENCY.

That is WHY they seem easy to us, and WOULD HAVE BEEN SEEN AS CHILD'S PLAY TO NEWTON!

You really don't know anything about real development and inventive research in maths or the sciences, given that view of yours.

Inventors stumble, fumble, get the occasional spark that goes out again.
They are fumbling in the dark, most often never getting out of it.

The actual discovery process is completely different from understanding a well-thought out text designed for pedogogical purposes.

 

[mathwonk]

this is like asking whether mohammed ali would have beaten rocky marciano, it is endless, negative, and depressing. look where we are letting ourselves be led by this.

 

[rockytriton]

hmm, I don't know, personally I think that comparing the understanding of the theory of relativity and the understanding of calculus is like comparing apples and oranges.

 

[Klaus_Hoffmann]

If possible i would like to write some arguments supporting my ideas

1) Upto 1800 (but some exceptions) many math problems involved Calculus II or Algebra (without groups and similar) as you will have seen through the forum many users claimed having 'discovered' identities involving $$\zeta (s)$$ and prime-generating functions.. many of them proved by Euler or others, --> If a problems involves 'simple' math anyone can give the solution)

2) Many recent theorems or proofs of 'Poincare Conjecture' (Perelman) or 'Fermat theorem for every n>3' (Wiles) involved hard math , that is not avaliable for many of us, however the math at Newton's time was easier to understand and work with, you and me can understand reading a book the Zeta regularization (Ramanujan sum), or Borel resummation.. but i really believe that there will be only a few people understanding Cohomology, Diff. Geommetry or C*-Algebra and Functional Analysis.

3) Are Newton an Einstein really GENIUSES?.. calculus had been previously defined and invented by others such us Fermat ( a lawyer ¡¡) Leibniz,Gregori,Descartes... and SR was also 'invented' by Poincare, Lorentz, MInkowski, Hilbert himself even derived Field equation with a Variational principle

 

[rockytriton]

3) Are Newton an Einstein really GENIUSES?.. calculus had been previously defined and invented by others such us Fermat ( a lawyer ¡¡) Leibniz,Gregori,Descartes... and SR was also 'invented' by Poincare, Lorentz, MInkowski, Hilbert himself even derived Field equation with a Variational principle

I wonder what your definition of a genius is. If it's someone who can understand higher level math and that's it, then yea I guess they weren't geniuses. Though I would consider that a math genius, not the definition of a genius in general.

I would say yes they were geniuses. They were most certainly physics geniuses. Most likely not math geniuses though, I believe that Einstein himself had claimed to great ability at math.

 

[Johan de Vries]

Or, to write the question in slightly fuller form:

Assuming a sufficient level of interest and competence, and very good teachers, could a high school student today understand the calculus better than Newton or Leibniz ever did in their lifetimes? And if not a high school student, then maybe an undergrad university one?

The trigger for this question comes from Sean Carroll's blog, The Alternative-Science Respectability Checklist, and specifically his comment:In at least one respect, a high school student today can understand the calculus better: she can get a handle on limits and infinitesimals that is far better than Newton's was.

So, I'm not all that interested in whether the answer is "yes", or "no", but rather in what respect, which aspects, can we say that a dozen years' or so of dedicated study can give you a better understanding of the calculus than that which either of the geniuses who first discovered (invented?) it ever had?

I agree. If Math were taught properly in primary and high school, ten year olds would be able to do calculus. Unfortunately, society has decided that Math is not as important as Art and Literature, so most people either won't learn calculus at all. Those who will learn it will have to wait until they are in the late stages of high school.

 

[Kummer]

It seems the more experienced mathematicians here: matt_grime and mathwonk agree with me.

This is because, I believe, is that one who studies much math sees the beauty and the difficulties behind it. Hence they agree with me.

I find this thread absolutely insulting to Newton to the other gods. It should be closed immediately.

 

[matt grime]

i think that many Graduate or Ph.D student can solve 'Blackbody radiation' (in the same fashion Planck did) or 'Bohr's MOdel'

If I can see further it is because I am standing on the shoulders of giants...

 

[matt grime]

If possible i would like to write some arguments supporting my ideas

A simple test to verify your assertion is to see how many theories have been written by graduate students in the last 20 years that are comparable to the output of Bohr, Einstein, Newton, Leibniz, et al.

Or just produce something groundbreaking youself, Jose.

 

[Kummer]

In science one generation tears down what the other has done, but in mathematics one generations adds to the existed old structure.
-Herman Hankel

This is the list.

Mathematicians of today, stand on the works of Cantor/Kummer/Kronecker/Poincare/Hilbert... Those mathematicians stood on the works of Weierstrass/Galois/Abel/Riemann/Gauss... Those mathematicians stood on the works of Laplace/Lagrange/Legendre/Euler/Bernoulli/Fourier... Those mathematicians stood on the works of Newton/Leibniz. Those mathematicians stood on the work of Fermat.

I stop at Fermat because the era of Fermat and Descrate was the rise of the modern age of mathematics.

 

[mathwonk]

the answer is no.