A Mathematician’s Lament (2002) [pdf]
maa.org
maa.org
As a Mathematician I've always identified as a creative person, yet struggle to convince "artsy" persons that what I do is creative. As soon as I utter the word math they convulse and shutter as if they were afflicted with a PTSD flashback. I try explaining what the idea involves and draw a diagram or three, but all they do is just nod and say "yeah, yup, okay.." Are the ideas of math that inaccessible to the general population when compared to a work of Art?
I'd guess at least some fine arts (composing classical or modernist orchestral music, modern abstract painting and sculpting) would also be pretty inaccessible to general population.
Or at least I think that'd be true.
(Well I used rock/jazz terms, since I am not familiar with classical music.)
If we had a specific, unique, and widely used word for the "artsy" free-expression type of creativity, I think the confusion many people express when you try to convince them that mathematics is a creative endeavour would be greatly diminished.
[1] http://oxforddictionaries.com/us/definition/american_english...
I do not find this to be the case. Art is full of structure, rules and logic. When artists "break" the rules, it usually means they been able to operate using underlying rules, and understand well the rules they break.
Yes, if only because many (if not most) don't want access. Been there, done that, got the medications to prove it.
Please, elaborate?
As another mathematician I've almost found the exact opposite. As soon as I mention math to an arts person they instantly start babbling about fractals and chaos and Fibonacci and all kinds of other vague pop-culture terms they've heard of but don't really understand. Artsy types almost seem to find math much more artistic than I do.
So there is this idea of different arts being more or less accessible to the general population.
So painfully spot-on. My mathematical education was horrible. Meanwhile I had been writing code since I was a little kid. It wasn't until I was an adult that I realized how much math I had been learning while programming. And worse, that I had been completely miseducated about what math actually is.
I was always into computers and tech but never really programming.
I was a math major in college, however, and after graduating I started programming. The transition was almost seamless, I picked up programming really quickly, it was surprising to me how much the "ways of thinking" are alike.
A teacher's job is not to teach. It's to provide a space in which a student can learn. A focus is useful for this, but the focus doesn't need to be an abstract subject. It's a MacGuffin; it can be anything.
As I understand, classifying classes into "disciplines" is for the convenience of administrative systems, like university deans. This classification is not a law of the universe, nor of the human mind. One can imagine the negatives of fitting learning into hierarchical administrative models.
Donald E. Knuth lived two separate lives in the late 1950s. During daylight he ran down the visible and respectable lane of mathematics. During nighttime, he trod the unpaved road of computer programming and compiler writing. Both roads intersected! -- as Knuth discovered while reading Noam Chomsky's book Syntactic Structures on his honeymoon in 1961. "Chomsky theories fascinated me, because they were mathematical yet they could also be understood with my programmer's intuition. It was very curious because otherwise, as a mathematician, I was doing integrals or maybe was learning about Fermat's number theory, but I wasn't manipulating symbols the way I did when I was writing a compiler. With Chomsky, wow, I was actually doing mathematics and computer science simultaneously."
Which makes you wonder, what kind of person reads this stuff on the honeymoon.
I am working on an iPad app that pairs up people to mentor each other through books like this. It has video chat and a shared whiteboard, so it is ideally suited for discussing math. If anyone is interested in reading the book with someone, email me at bridger@understudyapp.com. I could really use some beta testers for the app!
If you have already read it, you could still mentor someone in it to review the material again.
"This is beyond doubt the most interesting book on the evolution of mathematics which has ever fallen into my hands. If people know how to treasure the truly good, this book will attain a lasting place in the literature of the world. The evolution of mathematical thought from the earliest times to the latest constructions is presented here with admirable consistency and originality and in a wonderfully lively style."
—Albert Einstein
http://www.amazon.com/Number-Language-Science-Tobias-Dantzig...
http://www.engineering108.com/Data/Engineering/Maths/Number_...
I'd first recommend "Men of Mathematics" by E.T. Bell. It's a collection of short biographies on 20 or so Mathematicians, also discussing a few of the most salient points of each's work. It's an enjoyable introduction, useful for getting a broad view of what math is made of and how mathematicians think. Bell was a serious mathematician himself (not of the rank of anyone he's writing about, of course), as well as a sci-fi author apparently :)
edit: could also try "Mathematics and the Imagination" as an alternate introduction.
After that would be "What is Mathematics?," by Richard Courant and Herbert Robbins. This one's a bit tougher, and I have to admit I had the experience of being perplexed at the selection of topics, and that it didn't tell me immediately what mathematics is -- but! Without too much time passing, I now appreciate the selection and think it could be read profitably by trusting that the selection is good and trying to answer the question why that's the case while reading.
At the moment I'm trying my second book from E.T. Bell, The Development of Mathematics, and like it quite a lot so far, though it assumes a little more math knowledge. This one's probably great if you did a mathematics undergrad, or similar, but would like to see the various topics related and given context.
Another I believe worth checking out, if none of the others fits exactly, is William Kingdon Clifford's "Common Sense of the Exact Sciences." I've only skimmed sections in this one, but it looks extremely promising; and from what I've read about it and about Clifford, I think it could be an important piece of pedagogy along the lines of what Lockhart's into. Not too long and pretty accessible I think.
Lots of interesting stories, many of applied mathematics. And, to my surprise, available for free from archive.org.
My brother and his wife have since started an organization called Math For Love (http://www.mathforlove.com) focused on changing the way math is taught. They run workshops for teachers and provide great material for students.
If you're in Seattle and interesting in pedagogy and math, you should check them out.
I'm really asking -- my friend is about to start as a high-school math teacher.
I guess the first recommendation would be: motivate every new technique by starting with one or more problems that the technique helps to solve. (Here "problems" is meant in the Lockhart sense -- real puzzles, not exercises.)
But how often are "techniques" actually taught in high school math, especially algebra and precalculus? A lot of high school math consists of digesting new definitions, or the generalization of old definitions. A fair amount of it consists of learning theorems that go unproven, or that are proven (by the teacher) too quickly for students to understand where they come from -- and in general it isn't satisfying to solve a puzzle with a theorem that one doesn't actually understand.
On top of that... students have to spend time with problems before they become genuinely interested in their solutions, so progress would be slower with this method. It's not clear that you could teach a whole year's curriculum in one year like this. (And if you fail to do that you'll eventually get fired.)
Any insight? I believe that it's possible to teach math, even standard high school curriculum, in such a way that students are at all times intrinsically interested in what's presented. But it would be awfully hard to do at scale, at the standard pace, as a high school teacher would have to. How might a teacher start in that direction?
Here are my scattered thoughts, though. I'm going to try to not suggest a pie-in-the-sky solution like "new curriculum!"
First, I majored in mathematics at the University of Chicago, but I hate, hate, hated mathematics in high school. Take something you'd see in Algebra II like matrix multiplication, matrix inverses, and solving systems of linear equations. You're presented with these things called matrices and taught a bunch of rules. Where did these rules come from? Why are we calling this "multiplication" when it doesn't look or act anything like multiplication?
And sure, I see that when I go through the steps you tell me to go through like a monkey I get an answer that works, but how do we know there aren't more correct answers? How did anyone even come up with these steps in the first place? It's not like someone sat down and tried a trillion random combinations of symbols and steps until one of them happened to work.
Augh. In that world the only recourse for students is to memorize, usually just enough to do the homework or pass the test, and then promptly forget. The only experience they associate with math is the utterly humiliating feeling of being terrible at it.
So, I think that's one of the root problems. People remember what they feel and most people remember feeling stupid, humiliated, and possibly ashamed when it comes to mathematics. It's only a matter of time before that becomes part of their identity. "Oh, I'm terrible at math. Oh, I'm not smart enough to do math." and so on.
If I were a HS math teacher my top priority would be to watch out for when those counterproductive, self-defeating beliefs were forming and do whatever I could to preempt them.
Second, I think the way math is taught is overly symbolic. What most non-mathematicians don't realize is that when most mathematicians look at a set of abstract symbols they don't "see" the symbols per se, they see what those symbols are meant to represent. They freely move between a geometric and algebraic picture of the world, but the algebraic picture is usually incredibly compressed.
I think the key thing is not to pick a side -- algebra vs. geometry -- but to show the relationship between the two. Geometric objects admit a symbolic representation and vice versa.
Third, students have this idea that math is all about being "right" or "wrong", that it's "black" or "white", that there's some universe of Proper Math that is insisting on certain rules for no rhyme or reason
Here's a silly but illustrative example that I think students would cover in 6th or 7th grade: order of operations.
Hey class! Look at this expression: 45+6. What does it equal?
A bad teacher says "It's 26 and any other answer is wrong." An ok teacher says, "Remember the order of operations. If we apply those rules we get 26, so that's the right answer."
A great teacher shows their students that some things are necessarily true and other things are definitionally (or conventionally) true. This teacher would do something more like...
Who got 26? Who got 44? Students who said the answer was 26, how did the students who got 44 arrive at their answer? Students who said the answer was 44, how did the students who said 26 arrive at their answer? Neither of you are wrong per se. We could have chosen to live in either world, but we have to choose one consistent set of rules.
These rules lead us to 26. If we chose the other set of rules, we'd get at 44. We only do this because we don't want to have to write down parentheses all the time, but without them it's unclear what order we're supposed to apply + and . So we need to agree on a set of rules so that two people looking at the same expression both understand how to make sense of it.
It's like traffic laws. There's nothing stopping people from driving on the left side of the road. In fact, there are countries where everyone does drive on the left side of the road. The important thing is that everyone agrees on a convention -- left-side or right-side. It works as long as everyone agrees and breaks if people don't.
I could go on, but I'll stop here. Like I said, these are my scattered thoughts. :)
Might we benefit from a different set of symbols that actually convey the geometrical meaning behind them? If instead of π, we used a glyph that shows a circle over a diameter, instead of x for a variable, we show an empty rectangle that shows that it's a placeholder for a value?
> Students who said the answer was 44, how did the students who said 26 arrive at their answer?
I came across a great example of this approach in Chess: The Complete Self-Tutor by Edward Lasker. Instead of just showing the right answer for a chess puzzle, he tells you what you did right in your answer and what you missed in getting an even better answer. This was a printed book that was completely interactive.
(Here, I use spaces to "escape" the * . '\' doesn't work as an escape character.)
One thing I noticed during in math classes is that I don't really need to know anything. For me, and most of my classmates, most of formulas could be easily derived from simple and intuitive principles. For example, almost no one in my class actually 'knew' the quadratic equation, or the common trig values (ie. sin(30)). What we did know was how to quickly find those if we needed them.
As to your question of a standard pace, groups tend synchronize themselves. If every comes in at a similar place, and you have alot of group work and full class collaboration, then the slower students will gennerally still be able to follow the groups discovery trail, even if they do not contribute as much. The important thing here is you make sure that students are comfortable to ask questions, and that you do not have a few students dominate the discussion such that they loose the rest of the class.
Again, that comes from the perspective of a student at a school where the teachers had a lot of leeway in how and what to teach.
Award participation credit, etc as relevant to help keep people engaged who need it. The fact that it's "future" material can also help students who need the extra goals pay attention.
Most likely you can't actually cover the required curriculum like this— as you note, a lot of things do not lend themselves to compact discovery. (It's all ashame, it's not like the students actually retain into adulthood all those procedures that they don't really understand in any case :( ) But maybe you can still inspire people with a few things which do lend themselves to compact discovery, and that inspiration may also make the rest of the subject more accessible to them. "This things have a reason and a pattern to them, even if I don't know what it is right now."
I had some challenges in math in school because I studied calculus, analysis, linear algebra, discrete math, etc. on my own and would derive solutions— sometimes the same as they wanted me to memorize, sometimes not— on my own instead of memorizing the fixed routines, and this was unwelcome. It would be nice if more teachers made an effort to at least not penalize students that were independently interested.
One of the major themes is the relationship children have with mathematics and ways teachers can change it.
That's almost Douglas Adams levels of dry wit.
This is a fascinating article, as someone who's never really contemplated the playfulness of maths....I mean, for sure the wonder of maths or the power of maths - I've just never really put the pieces together to link that back to my grounding in maths, beyond the practical, functional stuff I incorporate into my daily life, without considering it's mathiness. Very interesting.
It's worth noting the times when HN really delivers. I doubt I'd have come across this anywhere else.
At risk of getting to meta, this piece seems to pop up everywhere. I don't mean that in a bad way, but this is at least the fifth time over the course of many years that I have seen this piece pop up on completely unrelated sites.
"This course was like climbing Mt. Everest. It's difficult, and sometimes slow going, but at the end, it's breathtaking how much you've learned and accomplished."
On the other hand, I think everybody really did see the beauty in geometry. Yes, the initial manipulation to prove something about symmetric angles is a bit silly, but you aren't being taught symmetric angles here, you are being taught how to do a geometric proof. Which is not easy to learn, so you start with something super simple and obvious. I didn't observe anyone in my classes (well this was back in 1982, so memory is a challenge here) confused about that point. And, as soon as we learned it, the requirement for formalism was dropped. It was the same in algebra. In the first weeks you weren't allowed to go directly from x + 3 = 1 to x = -2. You had to do something like x + 3 - 3 = 1 - 3; x + 0 = 1 - 3; x = 1 - 3; x = -2; with all the rules that you are using written out. Annoying yes, once you grasp it, but once you proved you grasped it that was the end of that, and we never had to do it again.
But, I had good teachers that always tried to explain the 'why' of what we were doing, and did not make us engage in pointless formalisms. But you need to understand terminology like ABC in geometry; when you get to tough problems you'll be using it. So, learn it with the easy problems.
The question I struggle with is... How do you really get by without math as a mandatory topic? How can you teach science? How can you teach someone to balance a checkbook? The current system is awful, but do we push the subject to electives as a result?
That's the wrong question, because "mandatory learning" is a contradiction in terms. It very clearly doesn't work. Beyond the most primitive pavlovian conditioning, you can't force people to learn if they don't want to. Intrinsic motivation is the dominating factor in how well a person can master an intellectually demanding task.
The right question is, how do you inspire people to want to learn?
The proper next step, it seems to me, is to figure out what bits of science we really want to teach and then figure out how to explain math through that.
> How can you teach someone to balance a checkbook?
Outside of America, no one uses checkbooks anymore. There's no reason to use exact change for tipping, either: just estimate a percentage and then round up. Or splitting the bill: the servers virtually always have a calculator at their disposal.
He is 100% right. We should not be teaching the formulaic rules of basic arithmetic until high school, and only then as a part of life skills class. It should be taught as the art form that it is.
While SALVIATI is completely correct in his analysis of the situation, he offers no solution, and I identify with SIMPLICIO more.
Perhaps the problem is that in our schools we conflate arithmetic with mathematics. Surely, they are related, but perhaps they need to be delineated and the difference understood.
That's not actually true.
(1) The subprime mortgage crisis could not have been averted by a larger percentage of the population understanding sums and compounding processes.
(2) The advent of civilization was a contributing factor to everything that's happened in the last several thousand years, including the paper cut I just got.
But let's not only blame the borrowers of subprime loans, let's also ask whether the banks didn't entirely understand the risk models of the derivatives they were compiling out of subprime mortgages because many of their senior managers also didn't understand compounding processes, or, possibly, sums...
Oh, it's relevant. It's sort of like how being able to load and fire a gun is relevant to the decision whether or not to commit suicide. Sure, it can modify one of the many, many details, but some people just make a noose and hang themselves.
Keep in mind, this is your argument: if more people in the world understood sums and compounding processes, this would have prevented the subprime mortgage crisis and thus the many deaths that were inspired by the resultant fallout. There is no possibility that anything else caused the crisis, and no possibility that anyone is at fault for these deaths other than parents and teachers.
This claim, if true, actually absolves the lenders that you talk about below, because they can be held responsible only for their own understanding of sums and compounding processes.
Amusingly, if you accept your argument, you can also make an interesting inverse version. The fact that lenders understood sums and compounding processes led to their employment at unscrupulous institutions which then mandated their sign-off on high-risk mortgages, which then caused the deaths of all those people.
Math, apparently, kills.
A Euclidean domain is an integral domain where the Euclidean algorithm works. For the Euclidean algorithm to work, you need to be able to divide two elements and produce a remainder that's smaller than the divisor.
So an ED requires a notion of "smallness" (the Euclidean norm) which interacts with division in a way that makes the Euclidean algorithm work (remainder is always smaller than divisor).
A principal ideal domain is an integral domain where every ideal is principal (can be generated by one element). It can be useful for you to know that certain situations cannot happen, e.g. in a PID you can say, "Let I be an ideal of D, then I = <g>..." and do something with the generating element g. It lets you pass from an ideal to a single generating element in a proof, which may be a useful capability. The PID concept is also part of a taxonomy, since Euclidean domains ⊆ principal ideal domains.
I really want to experience the kind of math the author writes about; can anyone recommend a place to start as someone who has only ever done "fake" high school math? I'm in college now and I'm halfway through a computer science degree; I've tried a few times to break into theoretical math classes but I've found the bar for entry pretty high (especially when I only have room for one or two courses), with most classes and even peers asking for years of experience and "mathematical maturity." Have any of you ever succeeded in learning some math outside formal curricula?
It's hard to keep at it, learning on your own. I sometimes find problems where the usual solutions or explanations feel kind of ugly, and try to make them cleaner, like rewriting someone's code. (A couple days ago it was Snell's law: this optics tutorial http://www.bigshotcamera.com/learn/imaging-lens/refraction linked from HN just dropped this formula down, and to most kids it's going to be magic. Can you formulate the law of refraction in a more elementary way and derive it from some simple assumptions? I did come up with a version that never mentions sines, but I'm not really satisfied and it's gone back on the to-do list to try to take it further. See: hard to keep at it.)
More generally, this kind of work can come up all the time when programming if you say, "No, I'm not going to look up the algorithm, I'll work one out for myself and then see what's been done." Occasionally you find something kind of new that way, besides often deepening your appreciation of the usual solutions. For example, last week I found a new way to avoid the epsilon-loops in Thompson's regular-expression search algorithm -- new to me, at least. This has minor significance and came out of a ridiculous amount of work rediscovering things taught in automata-theory classes, but Lockhart wasn't kidding: it's a rush when you figure it out.
I mention "Gödel's Proof" because it's the first thing I came across that informed me I am in fact interested in mathematical systems. I'm more interested in architectures than problem solving though. If you suspect that might be the case for yourself, might check it out -- it's about 100 pages.
“Physics is like sex: sure, it may give some practical results, but that's not why we do it.”
Induction doesn't just involve numbers and equality signs, it involves _statements_ with variables inside of them, and non-programmers need to be made well-aware of this (and taught Boolean logic early, PLEASE)
The reductionist approach of our education is it's major failure.
I could say that this or that argument is flawed. But really, the only valid arguments are data. His data is severely lacking, and many modern methods of teaching are much more supported by experimentation. Logic is a tool for finding logically consistent imaginary realities. Science is a tool for finding out about this reality.
I wince every time I see these examples in my niece's textbook. So ridiculous.
+1
Also s/math/science/ +1
When coding is presented as: - here is a problem - how would you solve this problem? - here are some hints to get you started
it is incredibly fun for me. when it is presented as:
- follow along - look what you did!
it can be a bit dry.
(It's not quite finished and I'd love to get feedback.)
CALCULUS: This course will explore the mathematics of motion, and the best ways to bury it under a mountain of unnecessary formalism. Despite being an introduction to both the differential and integral calculus, the simple and profound ideas of Newton and Leibniz will be discarded in favor of the more sophisticated function-based approach developed as a response to various analytic crises which do not really apply in this setting, and which will of course not be mentioned.
"Mathematics of motion", which makes it sound so simple, has in fact perplexed philosophers and mathematicians for centuries and continues to perplex a great many people even today, consider for example the Zeno paradox:
http://en.wikipedia.org/wiki/Zeno%27s_paradoxes
The ideas of Newton and Leibniz were hardly simple, they had some valid intuitions and managed to do formal manipulations that led to correct results, but in their day it was impossible to at all logically understand why what they are doing works, and not for some god knows how complicated things, but even for most elementary ones. You don't even have to go back to writings of Newton or Leibniz, just have a look at a 19th century textbook of calculus to see how noticeably strange and illogical the exposition of the subject was even then, with "infinitely small quantities" and an air of mysticism about it:
http://archive.org/stream/elementsofdiffer00woolrich#page/n1...
This kind of approach simply doesn't make sense, even though it happens to apparently produce correct results sometimes. Now, "function-based approach" is a weird phrase, but I guess he means the common modern exposition of elementary calculus using limits. This however wasn't developed in response to "various analytic crises". The only explanation of this statement I see is that he knows history of mathematics poorly and confuses the latter developments by Lebesgue, Jordan etc. that led to what we now call real analysis (inspired by considerations of nowhere continuous functions, continuous but nowhere differentiable functions etc.) with the earlier and more general lack of any decent understanding of how calculus works at all that was solved by Cauchy, Weierstrass and others. It is their introduction of what the author considers "unnecessary formalism" that made us finally really understand "mathematics of motion" and satisfactorily resolved things like the before-mentioned Zeno's paradox.
If it is only motivated appropriately, the concept of a limit is actually very interesting and powerful. There is a ladder of granularity with which you can treat computational problems, with the most elementary approach being always trying to get the exact answer. However, the class of problems that can be solved this way is very narrow. You can jump over this severe restriction by getting a bound, with inequalities for example, or you could try to get an equality in the limit (when n approaches y, the sought thing x approaches w*z). Unfortunately in school people almost exclusively learn to look for the exact answer, while in mathematics proper and in real world it is much more common to look for approximations and limiting behaviour. Furthermore, since the limit concept so powerfully extends the range of problems for which we are able to state anything interesting, there are lots of mathematical disciplines that rely on it to a great extent, for example probability theory (laws of large numbers, central limit theorem, ...). You won't understand almost any higher mathematics without learning limits first!
One can get an excellent and well motivated introduction to reasonably rigorous calculus using limits in Courant's "What is mathematics?" in less than a 100 pages, up to the point of understanding basic differentiation and integration, the exponential function, power series etc. The problem is not the formalism, but the teachers who can't motivate the material well enough both mathematically and physically and students who are not always mature enough to put in the amount of work necessary to understand calculus, which for most of them will be by far the most difficult thing they ever attempted to learn.
When I help students with calculus most of them have no trouble with the ideas but the implementation that they are required to perform. I spend a lot of time getting them to understand the simple particulars. There are ways of teaching calculus that would dispense with some of the more rigorous aspects and make it a much more bearable experience for most.
you're also wrong about your examples. it wasn't nowhere differentiable functions, but fourier series that motivated lebesgue. that's what the author is referring to regarding analytical traps ie, monotone convergence.
it is also quite a leap to assert the arithmetical definition of limits solves the zeno paradox!!! i few of my colleague's might disagree with you.
don't led your initial fascination with rigour (it can be addicting) get in the way of your intuition. rigour is necessary, but it comes after - sometimes to the chagrin of some. look at the teaching of modern algebra. pure abstraction and rigour, with complete detachment of all the wonderful ideas, and experiences that gave it rise.
up with triangles i say :)
Now any hard-working student of university calculus can without problems understand and reproduce the results of those treatises. I think this would not be possible without the modern systematic methods, including the epsilon-delta approach.
you're also wrong about your examples. it wasn't nowhere differentiable functions, but fourier series that motivated lebesgue. that's what the author is referring to regarding analytical traps ie, monotone convergence.
I mentioned fourier in another comments. Those weird functions were postulated in the discussion that arised somewhere in the same time period. I don't know what the author is referring to, because he is irritatingly vague, especially for a mathematician.
it is also quite a leap to assert the arithmetical definition of limits solves the zeno paradox!!! i few of my colleague's might disagree with you.
If you are familiar with the concept of a limit, you can notice that Zeno considers a limiting process of two related quantities, time and the difference in the position of Achilles and the Tortoise. Since in this limiting process time get arbitrarly close to some definite value (the meeting time of Achilles and Tortoise), but never gets equal to it, it stops being so surprising that the distance between them never reaches zero, though it gets arbitrarly close. The formalism clarifies what seems parodoxical when described in natural language. I find this quite convincing, and I never found a better explanation, altough I know philosophers still dispute this.
I never said rigour takes precedence over intuition. I just think it's the inherent difficulty of calculus that stops students from understanding it, and not the epsilon-delta stuff.
99.9% of calculus students are there to expand their mind. They will never design a quantum mechanical reactor and think "Wow, I'm getting all these weird results, my calculus education must have held me back."
It's generally preferable to give students a first order approximation at first, the successively refine with additional terms down the road. The fact that most calculus students will have no idea I'm referring to a Taylor series explanatory strategy highlights the problems with an overly rigorous introduction. Math at this high level should primarily expand your thinking, and secondarily your storehouse of previously proven rigorous statements.
How many students make it through calculus without learning that it is the mathematics of motion at all? I'd wager that it's more than half.
It was poorly explained there, but essentially that style of reasoning does work: http://www.amazon.com/Primer-Infinitesimal-Analysis-John-Bel...
And I at least find that approach easier and more useful. (I learned it from the Feynman lectures on physics, where he didn't axiomatize it; the above link does.)
Most of them could get a better intuitive understanding of Calculus. (We should of course conduct a rigorous study comparing the effects of using different approaches to teach Introductory Calculus.) Then those who need to use the limits approach for other courses could use the intuition to learn it faster. In addition, they would understand Calculus from two different angles and could select the more suitable approach for each problem.
I don't think so.
99.99% of the mathematics of motion consists of continuous functions with continuous derivatives of all orders. Infinitesimals are just fine for that. I find the epsilon-delta construction (EDC) to be not only inordinately clunky but to provide no insight whatsoever insofar as motion is concerned. From a pedagogical standpoint EDC is roughly the equivalent of tossing a monkey wrench into the smoothly oiled and finely-working innards of the mathematics of motion. The student's mind grinds to a screeching halt as (s)he attempts to come to terms with a new construct that not even a mathematician's mother could love. I consider any praise of the epsilon-delta construction of calculus to be little more than turd-polishing and while it is unfortunately necessary, there is no need to call attention to it more than once. And once pointed out, best forgotten.
One of the most underrated advancements in mathematics is Descarte's introduction of Cartesian coordinates. The formalization of the calculus, in contrast, was a step backwards in creative terms. Although necessary, it is of interest mostly to pure mathematicians.
There were logical contradictions even in Newtons and Leibniz works, far before anyone considered continuous functions without derivatives etc., they were basically making decisions about when a given operation or transformation can be applied based on intuition alone and not any logical deductions, and it wasn't rare they arrived at incorrect conclusions. Also, Newton himself did epsilon-delta reasonings, he just did not notice their generality:
http://www.sciencedirect.com/science/article/pii/S0315086000...
Again, you are confusing the work of Cauchy and Weierstrass and the epsilon-delta stuff with all the latter even more formal approaches to treat more complicated functions, which by the way were developed in response to Fourier examining heat transfer and trying to describe it mathematically (so it's actually rooted in physics).
People were saying the same things you are just saying about limits about the geometry of Euclid, in fact Newton at one time was of the opinion all the formal development of geometry is useless. He reconsidered after obtaining nonsense geometrical results a few times...
The formalization of the calculus, in contrast, was a step backwards in creative terms. Although necessary, it is of interest mostly to pure mathematicians.
I wish you luck doing quantum mechanics with Newton-style calculus.
Perhaps physicists (e.g., your example of Newton above) somehow intuitively step over the "holes" in the underlying analytical frameworks that mathematicians fall into with regularly. Mathematicians are wont to build "manholes" to cover those holes and physicists have no desire to stop them, seeing that it is honest labour and keeps the mathematicians busy.
All of which is besides the point (he makes), that they are simpler than the current formalism.