[HN Gopher] Elementary Calculus: An Infinitesimal Approach (2000)
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Elementary Calculus: An Infinitesimal Approach (2000)
Author : mkl
Score : 139 points
Date : 2021-06-02 10:43 UTC (1 days ago)
(HTM) web link (people.math.wisc.edu)
(TXT) w3m dump (people.math.wisc.edu)
| marosgrego wrote:
| Another approach to infinitesimal calculus which allows a
| presentation in a relatively clear elementary way is smooth
| infinitesimal analysis and related synthetic differential
| geometry: http://home.sandiego.edu/~shulman/papers/sdg-pizza-
| seminar.p...
| eigenhombre wrote:
| This was the book my 1st sem. calculus class used in 1986, at UW-
| Madison (where the link is hosted and where the author presumably
| teaches). I remember it as providing a fairly gentle introduction
| to differentiation, providing a stepping-stone to more
| traditional notions of limits later on.
|
| I was just thinking the other day about how enjoyable I found
| learning calculus, compared to what others tend to report.
| Perhaps this book was part of the story -- very cool that it is
| being provided free of charge now!
| mcguire wrote:
| In roughly 1987-1988, I took the required 2 semesters of
| calculus at UT Austin. It actually took me 4 semesters, for a
| number of reasons including that I was seriously unprepared for
| calculus.
|
| But the worst part of the experience was the method of starting
| with epsilon/delta and limits, to "explain" what was going on,
| and then throwing that away to get on with solving problems
| using differentials and integrals. The lectures almost
| universally took the form of the professor going through a
| proof of the technique in question and then assigning a set of
| problems from the text.
| bachmeier wrote:
| I had to retake the first calculus class. Didn't have much to
| do with the difficulty of the material, but rather a
| professor doing things that would get him pulled from the
| classroom in 2021. Just one of the many things he'd do is
| call on someone to answer a question during the class. If he
| didn't like the answer he'd tell them they were lazy or that
| they should quit college.
| dkarl wrote:
| > But the worst part of the experience was the method of
| starting with epsilon/delta and limits, to "explain" what was
| going on, and then throwing that away to get on with solving
| problems using differentials and integrals
|
| I guess that was pretty normal once upon a time? I remember
| that was how my calculus textbooks did it. First pictures to
| develop some visual intuition, then a gentle foray into how
| to make the intuitions more rigorous using limits, sequences,
| and series, and then techniques for solving certain
| differentiation and integration problems. It makes sense to
| me; it shows that the techniques are based in math and not in
| magic, and it lays the groundwork to introduce Taylor series
| later. It also foreshadows the kinds of proofs that students
| might do in real analysis if they decide to pursue an
| engineering or physical science major.
|
| I guess every math class could be very different if you knew
| it was the last math class that students would take, but you
| don't know that, and it's problematic to separate students
| according to which ones will study the material further
| before they're even exposed to it. If a student takes a class
| and unexpectedly finds they want (or need) to take further
| classes in the area, they don't want to be find out later
| that, "Oh, sorry, we didn't think you would pursue this
| subject, so we gave you the version of the class that didn't
| prepare you for the next one."
| [deleted]
| ganzuul wrote:
| inf + 1 = inf, which is idempotent, so if you extend the complex
| numbers with infinity they become a semiring. Is the ire that
| infinitesmals draw related to this?
| iib wrote:
| From what I understand, you do not work directly with infinite,
| but infinitude, basically a hyperinteger that is bigger than
| any integer. For which H + 1 = H does not hold.
| hansvm wrote:
| There are lots of different "infinities" floating around in
| mathematics, and that probably doesn't help the matter.
|
| The extended reals or extended complex plane do behave as you
| describe (and infinite cardinalities have the same property),
| but infinitesimal treatments of calculus do not. You're instead
| working in an ordered field that contains the reals and some
| element greater than all reals.
|
| Supposing such a thing can exist (they can exist in ZFC), the
| fact that you're working in an ordered field gives you a lot of
| infinitesimals and infinities (and also a lot of numbers
| "between" the ordinary reals).
| pfdietz wrote:
| Terry Tao's blog has entries on nonstandard analysis, which he
| uses in various ways as a working mathematician.
|
| https://terrytao.wordpress.com/tag/nonstandard-analysis/
|
| https://terrytao.wordpress.com/2010/11/27/nonstandard-analys...
| kkylin wrote:
| People who like this may also be interested in Nelson's little
| book on probability using infinitesimals:
| https://web.math.princeton.edu/~nelson/books/rept.pdf
| billytetrud wrote:
| This is how all calculus should be taught.
| roydanroy2 wrote:
| Recent Annals of Statistics paper that uses nonstandard analysis:
| https://www.e-publications.org/ims/submission/AOS/user/submi...
| eqbridges wrote:
| Is there an online course covering this?
| mkl wrote:
| The key terms to search for are "nonstandard calculus" or
| "nonstandard analysis". I just found the following set of
| lectures (but I haven't watched any of them); maybe they are
| what you want:
| https://www.youtube.com/watch?v=ILDkYszP2lA&list=PLDXeoTykA-...
| andi999 wrote:
| How to show such numbers exist? Chapter 1 states they can be
| constructed from the real numbers and this construction is
| discussed in the epilog, but (I dont have enough time probably)
| there seem to be only some axioms in the epilogue but not a
| construction (or proof of existence).
| bikenaga wrote:
| Keisler wrote a monograph to accompany the textbook:
|
| https://people.math.wisc.edu/~keisler/foundations.html
|
| It's a bit more than an instructor's manual. It covers the
| construction of the hyperreals in two ways: One (in section 1E)
| is relatively short, and is like the compactness theorem
| approach Abraham Robinson originally used in his book ("Non-
| standard Analysis", Princeton U. Press).
|
| The second construction uses ultraproducts (in Chapter 1*).
| Keisler gives a quick introduction to the logic needed to
| understand the ultraproduct construction, beginning with formal
| languages and sentential logic. So the prerequisites are there,
| and explained concisely but clearly. The ultraproduct
| construction comes up elsewhere; for applications to other
| areas of math, see Martin Davis's book "Applied Nonstandard
| Analysis" (Dover Publications). (There's a review of the books
| by Keisler, Davis, and Stroyan and Luxemburg: https://www.ams.o
| rg/journals/bull/1978-84-01/S0002-9904-1978...).
|
| The rest of Keisler's monograph contains mostly nonstandard
| proofs of many of the results in the textbook. I like it a lot,
| and anyone who's interested in this stuff should grab the PDF.
|
| Another book that is short and moderately elementary is
| "Infinitesimal Calculus" by James Henle and Eugene Kleinberg,
| also published by Dover [which also publishes a paper version
| of Keisler's text, and is generally a great publishing
| company].
|
| When Keisler's book came out, someone decided that it would be
| reviewed for the Bulletin of the AMS by Errett Bishop, who (as
| I recall) was a noted constructivist. You can read the review
| here:
| https://www.ams.org/journals/bull/1977-83-02/S0002-9904-1977...
| Having a constructivist review a calc textbook that uses
| nonstandard analysis was probably not a great idea. The review
| ends with: "Now we have a calculus text that can be used to
| confirm their experience of mathematics as an esoteric and
| meaningless exercise in technique." I believe Keisler wrote a
| reply in a subsequent issue of BAMS, but I can't find it.
| andi999 wrote:
| I do not see how 1E shows existence. The definition of the
| natural extension needs the hyperreals in the first place.
| (reading on mobile, maybe I am missing something here)
| bikenaga wrote:
| Check the existence theorem on page 23 (I'm looking at the
| paper edition): "Let R be the ordered field of real
| numbers. There is an ordered field extension R* of R and a
| mapping * from real functions to hyperreal functions (i.e.,
| functions on R*) such that Axioms A - D hold." Note that he
| just starts with the reals. The actual definition of R* is
| in the middle of the proof: R* =
| {tau[epsilon] : tau(x) is an element of T(M)}
|
| Note that the proof uses Zorn's lemma, so it's definitely
| not "constructive" in the strict sense (in case that's what
| you meant).
| bikenaga wrote:
| Addendum to my post above - I didn't notice that mkl had
| linked to Keisler's "Foundations" earlier - my bad!
| mkl wrote:
| No worries, you provided useful details.
| pfortuny wrote:
| It is not complicated but it requires some work and more.
|
| Define "f(x)" is an infinitesimal if lim f(x)=0 when x-> 0.
|
| which is the same as saying "f(x)" is smaller in absolute value
| than any real number.
|
| Nothing more.
|
| The problem: you cannot divide by infinitesimals. Now you need
| to "create a field" from these numbers and the reals.: this
| requires the axiom of choice. There is also the sign problem.
|
| But in the end, "that is all you neeed". You think of
| infinitesimals as "numbers smaller than any true real number".
| andi999 wrote:
| Thanks! I know about axiom of choice (and I am fine with it),
| so how do you do the field construction?
|
| I am also a bit worried, if you have a field it is also a
| division algebra. And I thought we know that the only
| divisions algebras are R, C, H? Like
| https://math.stackexchange.com/questions/2020399/division-
| al... (The link does not 100% fit). So how do hyperreals go
| around that 'no-go' theorem?
| scapp wrote:
| The Frobenius theorem characterizes _finite-dimensional_
| associative real division algebras as being isomorphic to
| the reals, the complex numbers or the quaternions. The
| hyperreals aren 't finite-dimensional as a vector space
| over the reals (for example, if E is an infinitesimal
| hyperreal, then E, E2, E3, E4 ... are linearly independent
| over the reals).
|
| A simpler example of another real division algebra (and in
| fact, another field) is the field of rational functions
| with real coefficients. This field is also infinite
| dimensional over the reals (for example, 1, x, x2, x3...
| are linearly independent).
| andi999 wrote:
| Ah yes. But then if I think of infinitesimal als
| functions going to 0 at 0, I can have two infinitesimals
| (first one is constant zero for negative values other one
| constant zero for positive values), which multiplied give
| the constant zero function. How does a construction deal
| with that?
| matt-noonan wrote:
| Although the original statement about "infinitesimals
| being functions that vanish at 0" was stated with
| confidence, it is wrong.
|
| The usual construction of the hyperreals replaces real
| numbers with sequences of real numbers, and also
| introduces a nontrivial equivalence relation on the
| sequences, making two sequences equivalent if they agree
| on a "large" set of terms. The real numbers get
| represented by the constant sequences, infinitesimals get
| represented by sequences that approach 0, and infinite
| numbers are represented by sequences that grow without
| bound.
|
| The magic is in how "large set of terms" is defined. You
| need a "large set" relation with the property that finite
| sets are not large, and for any set either the set or its
| complement is large. Then we can resolve your question:
| say you had two not-always-zero sequences that multiply
| to give the all-zero sequence. Then the set of zero
| positions is large for one of those two sequences. And
| that means one of your sequences is equivalent to the
| zero sequence. The field axioms are saved!
| andi999 wrote:
| Thanks! Do you know of any source (textbook/paper) about
| this construction.
| matt-noonan wrote:
| I learned it originally from Jim Henle, and iirc he had a
| textbook on the hyperreals ("Infinitessimal Analysis",
| possibly?)
|
| This honors project has what looks like an accurate write
| up of the construction along with proofs of some of the
| main theorems: https://ideaexchange.uakron.edu/cgi/viewco
| ntent.cgi?article=...
| mkl wrote:
| Well, no numbers exist in any real sense. Integers, real
| numbers, complex numbers, hyperreals, etc., they're all
| imagined, and built from axioms. Keisler does have a companion
| book that goes into more of the theory of hyperreals, and its
| bibliography (p209) points to deeper theoretical work too:
| https://people.math.wisc.edu/~keisler/foundations.html
| Koshkin wrote:
| Well, the real existence of abstractions is a pretty involved
| topic in philosophy of knowledge, and so you can't simply say
| that, for example, integers do not 'really' exist. Because
| they indeed do, in a very important (and, in fact, obvious)
| sense: you need _two_ cats to have a fight.
| fpoling wrote:
| The notion of existence of ideas has been discussed in
| philosophy since Platon and has not settled. Consider that
| to express an idea even a simple one like the numbers one-
| two-three one needs to use a language. And language has
| infinite number of interpretations that may be entirely
| different. So how do one know that his ideas are not
| misinterpreted by others? Moreover, memory is based on the
| language, so how does one know that things that he
| remembers from a day ago mean the same things as it was
| then?
| andi999 wrote:
| Only finite amount of integers like 10^91 (or so, number of
| protons in the universe) exist in this sense.
| Koshkin wrote:
| On the other hand, it is interesting to think about the
| fact that, say, the Avogadro number is usually
| represented as a _real_ number; the problem with the
| number of protons or other particles is that their number
| is unbounded: given enough energy, another particle can
| always jump into existence...
| xyzzy123 wrote:
| I flip between agreeing and disagreeing with you! I'm not
| well qualified to comment so I expect an informed rebuttal
| to this will be instructive for me. So here goes:
|
| I still find myself in the "number systems are a cultural
| artifact" camp. We choose which details to keep and which
| to throw away in our abstractions. There are anumeric
| cultures and it's not clear to me at all that the existence
| of integers is obvious (beyond what humans can subitize)
| unless you're immersed in a culture that has impressed them
| upon you since early childhood.
|
| Do you have "2 cats" or actually just "this cat and that
| cat"? This cat is a bit bigger but that one looks meaner.
|
| Deep down I think that an insistence on the "reality" of
| integers versus reals (say) is purely aesthetic. (Ignoring
| for the moment that our conventional constructions build
| one from the other in a certain way).
| andi999 wrote:
| Anumeric cultures might not really have existed (not
| conclusive evidence, but some evidence:
| https://www.jstor.org/stable/10.1086/667452?seq=1 )
| Koshkin wrote:
| The integers represent perhaps the simplest case. Another
| example of their actual existence is the fact that
| properties of atoms and their nuclei critically depend on
| the _number_ of the nucleons of particular types in them;
| note that this fact does not in any way depend on whether
| a culture is anumeric or not. With some effort you can
| extend this idea to other abstractions (this process is
| usually greatly hindered by one 's leaning towards
| solipsism).
| xyzzy123 wrote:
| The existence proof you describe is in terms of a yet
| more complex abstraction; we can imagine aliens might
| describe a more useful atomic theory than our own which
| does not involve nucleons or their discrete counts at all
| (ok, unlikely as I acknowledge that may be).
|
| I don't want to descend into solopsism and apologise if I
| veer too far in this direction. Also thanks for taking
| the time to rebut what is probably a sophomoric argument.
|
| I suppose my fundamental objection is that if integers
| are "real" then it seems to me that quaternions must be
| similarly "real" (since I can describe useful things with
| them) and so on, I can't see a boundary which would let
| me say "ok integers are the real deal but infinitesimals
| are just a thing we made up".
|
| I think we have a zone of possible agreement if we decide
| either that all these abstractions are "real" or none of
| them are.
| Koshkin wrote:
| This may seem a bit extreme, but, in my opinion, while in
| mathematics (and especially in physics) there is a fair
| amount of scaffolding, what they do in mathematics is,
| actually, they _discover_ things that do exist in reality
| in one form or another. (It is important to understand
| that few, if any, things in mathematics are pure fantasy;
| the objects and relations studied there are pretty much
| _forced_ upon us - which, incidentally, is yet another
| evidence in support of their having real significance,
| i.e. as something lying outside our consciousness and
| _acting_ there independently from it.)
| naasking wrote:
| > Deep down I think that an insistence on the "reality"
| of integers versus reals (say) is purely aesthetic.
|
| I disagree! In fact, I think the difference between
| countability and uncountability is pretty huge!
| Y_Y wrote:
| I agree with you (for some values of "exist") and furthermore
| think this is a good reason why "imaginary" is a terrible
| name for the number set they refer to.
|
| (I prefer "pure complex", but that's got its issues too, of
| course.)
| andi999 wrote:
| They are not all build from axioms but from definition (apart
| from some models of natural numbers, or you go back to ZFC).
| Like C is build from R^2, R from the powerset of Q, Q from
| integer^2, integer from natural numbers.
|
| Otherwise you do not easily know if your axioms are actually
| consistent.
| mkl wrote:
| Yes, but what I meant is that there are axioms at the base
| of this structure, so everything is ultimately built on
| them. Some people put axiom foundations higher up too, like
| axiomatic real numbers.
| andi999 wrote:
| Yes, there are of course axioms, but if you have a rich
| enough set of axioms like zfc, then one should stop
| piling more axioms (if not absolutely necessary). So of
| course the question is existence in zfc
| mcguire wrote:
| A proof of existence for _any_ number?
| dwheeler wrote:
| There are various ways to construct real numbers and complex
| numbers. One approach is given in Landau's Foundations of
| Analysis (a.k.a. Grundlagen der Analysis).
|
| You might want to look at the Metamath Proof Explorer materials
| on constructing numbers, a good starting point is here:
| http://us.metamath.org/mpeuni/mmcomplex.html Metamath is a
| general tool that lets you specify axioms and proofs, and
| verifies that the proofs only depend on axioms and previously
| proved proofs. The Metamath Proof Explorer (MPE) is a
| particular application of it that uses classical logic and the
| ZFC set theory axioms. MPE shows how to construct numbers using
| these axioms, then proves a set of number axioms, and from then
| on uses only those axioms so that the details of any particular
| construction are not important. What's usually more important
| is showing that you can construct them.
| andi999 wrote:
| Cool link. I know some construction of the real and complex
| numbers, I was asking about the construction of the
| hyperreals used in the textbook. I couldnt find info about
| that on the page.
| Smaug123 wrote:
| The construction is fairly easy but requires some fairly
| hefty background knowledge to make formal. In brief,
| though: an ordinary real number can be implemented as a
| sequence of rational numbers, with two sequences considered
| to be "the same" if they eventually get, and stay,
| arbitrarily near to each other. A hyperreal can be
| implemented as a sequence of real numbers, with two
| sequences considered to be "the same" if they are equal at
| "very many places", for a certain strict and formal
| definition of "very many places". (Formally, fix a
| nonprincipal ultrafilter on the naturals; then the
| requirement is that the sequences be equal on a set of
| indices which is a member of that ultrafilter.)
|
| A standard real z can be viewed as a hyperreal, by taking
| that sequence to be z, z, z, .... Another equivalent
| representation of that same hyperreal would be the sequence
| 0, z, z, z, ... because that's equal to the first sequence
| in "very many places". There's an infinite hyperreal,
| implemented by the sequence 1, 2, 3, .... (In fact there
| are tons and tons of infinite hyperreals.)
|
| You can prove that the space of hyperreals is a field, and
| moreover with some model theory you can show that actually
| a _lot_ of structure (in fact, all first-order structure)
| transfers directly over to the hyperreals from the reals.
| andi999 wrote:
| Very cool, thank you. Do you know what happens to measure
| theory in the hyperreals? Or integration in general?
| Smaug123 wrote:
| I don't know much about it, I'm afraid. But there is the
| very cool fact that the usual Riemann integral can be
| written as basically just a _sum_ in nonstandard
| analysis; it 's covered very neatly in Andre Petry,
| "Analyse Infinitesimale: une presentation non standard"
| which is very readable. There's a lot more advanced
| stuff, including Brownian motion and (I think) some
| integration, in Hurd and Loeb, "An Introduction to
| Nonstandard Real Analysis".
|
| For the foundations of nonstandard analysis, and a wide-
| ranging overview, I strongly recommend Robert Goldblatt,
| "Lectures on the Hyperreals: an Introduction to
| Nonstandard Analysis", one of the Graduate Texts in
| Mathematics.
| ska wrote:
| I'm rusty, but did think about this a bit at one point. I
| recall you can't really do the usual constructions on
| hyperreals (problems with completeness?) but you can do
| something like approximate any real-valued measure
| closely.
| steve76 wrote:
| There's operator theory. Treat a spectrum of prime numbers like
| we treat particles:
|
| https://en.wikipedia.org/wiki/Riemann_hypothesis#Operator_th...
| rsj_hn wrote:
| If you are worried about epsilons and deltas being too hard, then
| you can write your calculus text to not be rigorous rather than
| trying to make it rigorous with non-standard analysis. I think
| giving up on some of the formalism while still sketching the
| ideas of proofs is the way to go from a pedagogical point of
| view.
| bgorman wrote:
| Have you compared the approaches? I learned with epsilon delta,
| but I would be interested in seeing data about learning
| outcomes with alternative approaches.
|
| In my experience, epsilon delta understanding was pretty much
| unrelated to passing calculus, where tests consisted of
| formula/substitution crunching.
| rsj_hn wrote:
| > In my experience, epsilon delta understanding was pretty
| much unrelated to passing calculus, where tests consisted of
| formula/substitution crunching
|
| That's a shame. There are so many exciting things to learn in
| calculus that you can skip the epsilon delta stuff and still
| do so much more than formula/substitution crunching. Calculus
| is the gateway to differential geometry, topology,
| mathematical physics, differential equations, taylor series
| which are useful for numerical approximations and so many
| other things, analytic number theory, complex analysis, many
| forms of statistics. So many wonderful things you can do with
| it.
|
| Imagine using your new found calculus knowledge to show that
| the orbit of the planets about the sun is an ellipse -- which
| is what Newton used calculus to do -- OR learning all about
| epsilons and deltas. Which would be more fun as a student
| learning calculus. Make the math exciting enough and people
| will put up with the drudgery of calculations.
| ska wrote:
| > you can write your calculus text to not be rigorous rather
| than trying to make it rigorous with non-standard analysis.
|
| e.g., many engineering calculus courses, and explicitly some of
| the texts.
| mcguire wrote:
| " _...you can write your calculus text to not be rigorous..._ "
|
| Oooh, the comments you'll get from other mathematicians...
| rsj_hn wrote:
| I can take the heat. School texts don't need to be rigorous
| and there are different levels of rigour. I think Bourbaki
| did more harm than good.
| Koshkin wrote:
| While infinitesimals lie at the very heart of the classical
| approach to calculus, in this day and age it is important to
| complement a course based on what is now considered _non-standard
| analysis_ with one of the more standard (limit-based) courses. (I
| think this is also true in other areas, e.g. if the students are
| taught a physics course based on, say, geometric algebra, they
| also should be trained in the more 'standard' ways of
| understanding things.)
| Smaug123 wrote:
| I claim that the first undergrad course in Analysis is not
| actually there primarily to teach you real analysis. Rather,
| it's there to teach you the difference between "forall exists"
| and "exists such that forall", and to hammer into you that
| intuition is not by itself a proof. Nonstandard analysis, I
| claim, is not as good a vehicle for those purposes as epsilon-
| delta analysis is.
| mkl wrote:
| I think most universities won't even have a non-standard
| analysis course available to complement the courses using the
| standard limit approaches. Your fear seems quite unwarranted to
| me, since standard analysis courses are usually the only
| option, and always outnumber nonstandard analysis courses.
| rnhmjoj wrote:
| Why isn't this approach to calculus more common? Does it lead
| to more complex proofs or have any serious drawback?
|
| I understand the e,d way of defining the limit is important
| because it extends to other context such as continuity in
| topological spaces etc; however, as a physicist,
| infinitesimal quantities reflect the way we really think
| about calculus intuitively, so it makes sense make them
| "first-class" numbers.
| JadeNB wrote:
| > Why isn't this approach to calculus more common? Does it
| lead to more complex proofs or have any serious drawback?
|
| Simply put, because there aren't more textbooks for it and
| because professors aren't as familiar with it, leading to a
| vicious circle where it's not taught because it's not as
| easy as teaching the standard way, and then it's not
| routinely learned because it's not taught ....
|
| (The advanced logic that goes into the underlying
| constructions, but that isn't necessary to _use_ non-
| standard analysis, also causes many non-logician
| mathmaticians to give it an unjustified bit of side-eye.)
| btilly wrote:
| Unjustified bit of side-eye???
|
| You really think that it makes sense to require the axiom
| of choice to prove that the derivative of x^2 is 2x as
| Robinson's ultrafilter construction does?
|
| I personally like understanding infinitesmals, and
| knowing what the d in dy/dx means. And knowing that the
| notation for a second derivative, d^2y / dx^2, is not
| just arbitrary. This occasionally has uses. For example
| if you have implemented a numerical d function, for
| example as lambda x: f(x+h/2) - f(x-h/2), then
| d(d(y))/d(x)^2 is an excellent approximation to the
| second derivative.
|
| But conceptually I think it is far better to understand
| approximation a lot more directly than using a complex
| construction for the infinitesmals.
| JadeNB wrote:
| > You really think that it makes sense to require the
| axiom of choice to prove that the derivative of x^2 is 2x
| as Robinson's ultrafilter construction does?
|
| This was just what I meant to say--the details of the
| _construction_ shouldn 't matter, only the axiomatics of
| the structure that's been constructed. You can _use_
| infinitesimals perfectly well without having to get into
| the weeds of how they 're constructed. To be sure, your
| results only apply within that structure, but that's the
| way of mathematics, that things are proven only within
| some structure.
|
| I think one wouldn't expect, for example, a
| constructivist mathematician to disparage classical
| mathematics because it relies on such inelegant logical
| machinery as the law of the excluded middle. Well, maybe
| some constructivists do that, but more often they offer
| the better response of showing how many of the same
| results you can recover _without_ requiring that logical
| machinery.
|
| The same response seems appropriate here: there's no
| reason to cast aspersions on people whose work rests on
| the axiom of choice; but, if it bothers you, then see how
| much of their work you can do _without_ the axiom of
| choice. If you can prove the equivalent, then great; no
| need to complain! If part of their work genuinely
| requires choice, then that 's an interesting fact, too.
| canjobear wrote:
| Learning math isn't just about solving problems. It's about
| solving problems in ways that other people can follow,
| evaluate, and understand. Since nearly no one knows
| nonstandard analysis, but most people with college
| mathematics know standard calculus, there is much less
| utility in learning nonstandard analysis. It might make it
| easier for you to solve certain problems, but no one will
| understand what you did.
| ibeckermayer wrote:
| The true telos of a mathematics education is gaining a
| genuine understanding of the principles of mathematics.
| Pragmatic concerns of the nature you describe should be
| treated as secondary concerns in an educational
| environment.
| billytetrud wrote:
| If we just do what our ancestors did because its what
| everyone already understand, we'd never improve.
| Koshkin wrote:
| Sure - this is a great way to think about calculus
| intuitively, but imprecisely; rigorous, mathematical proofs
| based on the hyperreals, on the other hand, may become
| unwieldy, and, as you rightly noted, they do not extend to
| the modern treatments of, say, differential geometry which,
| too, plays a huge role, as a framework, in the modern
| theoretical physics.
| iib wrote:
| From what I read on the subject, this _was_ the main way of
| dealing with calculus, but it was not rigorous. The first
| rigorous definition of a limit was given using the epsilon-
| delta approach, by Weierstrass, with of course many many
| contributions by his predecessors.
|
| Everybody switched during the 19th century, because
| epsilon-delta was in fact the only rigorous method.
|
| Abraham Robinson gave a rigorous foundation for
| infinitesimals only in 1960.
| tzs wrote:
| One factor I've heard cited is that if you can prove
| something in calculus using nonstandard analysis, you can
| always prove the same thing with standard analysis.
|
| If we switched to teaching calculus based on nonstandard
| analysis we'd have to also teach the standard approach
| because of all the existing material that uses it and all
| the people who already know the standard approach but not
| the nonstandard approach.
|
| Not many people are willing to commit to a few generations
| of teaching dual approaches and using both in their work
| until the people who only know standard are all dead or
| retired and all the old material has either been translated
| or is obsolete, when the advantage of the nonstandard
| approach is just that it might be conceptually easier or
| more intuitive.
| javbit wrote:
| We learnt some non-standard in a logic class (as a capstone
| application of FOL and model theory).
|
| Perhaps there's a perceived notion that to understand
| infinitesimals and such you need a logic background, but
| logic classes are generally an upper-half class that
| doesn't have a full track for undergrads. This would be in
| addition to the fact that most professors are familiar with
| e,d-proofs and AP calculus classes cover the limit approach
| (though not rigorously).
|
| The entrenched pipeline of mathematics students and
| professors would have to face a period of getting flushed
| out and refitted, and this is probably not attractive to
| administrators.
| TeeMassive wrote:
| I quickly skimmed through the book, this looks great for a much
| needed refresh after 10 years of a non-math intensive programming
| career. Question: are there solutions to the problems?
| mkl wrote:
| This is a first-year university calculus text book based on
| hyperreal numbers.
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