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Elementary Calculus: An Infinitesimal Approach (2000)

153 points by mkl 5y ago

mkl 5y ago
This is a first-year university calculus text book based on hyperreal numbers.
marosgrego 5y ago
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.pdf http://home.sandiego.edu/~shulman/papers/sdg-pizza-seminar.p...
eqbridges 5y ago
Is there an online course covering this?
- mkl 5y ago
  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-bnyOLtd6W5lazRl57SnvWuS https://www.youtube.com/watch?v=ILDkYszP2lA&list=PLDXeoTykA-...
andi999 5y ago
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).
- pfortuny 5y ago
  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 5y ago
    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-algebra-field 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 5y ago
      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 Ɛ is an infinitesimal hyperreal, then Ɛ, Ɛ², Ɛ³, Ɛ⁴ ... 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 5y ago
        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 5y ago
        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!
- mkl 5y ago
  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 https://people.math.wisc.edu/~keisler/foundations.html
  - andi999 5y ago
    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 5y ago
      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 5y ago
        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
  - Koshkin 5y ago
    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.
    - xyzzy123 5y ago
      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).
- dwheeler 5y ago
  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 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 5y ago
    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 5y ago
      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.
- steve76 5y ago
  There's operator theory. Treat a spectrum of prime numbers like we treat particles: https://en.wikipedia.org/wiki/Riemann_hypothesis#Operator_theory https://en.wikipedia.org/wiki/Riemann_hypothesis#Operator_th...
- bikenaga 5y ago
  Keisler wrote a monograph to accompany the textbook: https://people.math.wisc.edu/~keisler/foundations.html 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.org/journals/bull/1978-84-01/S0002-9904-1978-14401-2/S0002-9904-1978-14401-2.pdf https://www.ams.org/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/S0002-9904-1977-14264-X/S0002-9904-1977-14264-X.pdf 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 5y ago
    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)
    - 5y ago
- mcguire 5y ago
  A proof of existence for any number?
  - andi999 5y ago
    if you assume ZF, then 1={0} and existence follows from existence of empty set.
Koshkin 5y ago
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.)
- mkl 5y ago
  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 5y ago
    Why isn't this approach to calculus more common? Does it lead to more complex proofs or have any serious drawback? I understand the ε,δ 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 5y ago
      > 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.)
      - 5y ago
pfdietz 5y ago
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/tag/nonstandard-analysis/ https://terrytao.wordpress.com/2010/11/27/nonstandard-analysis-as-a-completion-of-standard-analysis/ https://terrytao.wordpress.com/2010/11/27/nonstandard-analys...
roydanroy2 5y ago
Recent Annals of Statistics paper that uses nonstandard analysis: https://www.e-publications.org/ims/submission/AOS/user/submissionFile/31623?confirm=381d7eb7 https://www.e-publications.org/ims/submission/AOS/user/submi...
TeeMassive 5y ago
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?
kkylin 5y ago
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 https://web.math.princeton.edu/~nelson/books/rept.pdf
billytetrud 5y ago
This is how all calculus should be taught.
ganzuul 5y ago
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 5y ago
  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 5y ago
  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).
rsj_hn 5y ago
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 5y ago
  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 5y ago
    > 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.
- mcguire 5y ago
  "...you can write your calculus text to not be rigorous..." Oooh, the comments you'll get from other mathematicians...
  - rsj_hn 5y ago
    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.
eigenhombre 5y ago
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 5y ago
  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 5y ago
    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 5y ago
    > 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 5y ago
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hilbert42 5y ago
This is a nice gesture by H. Jerome Keisler, every bit helps to promote a math subject that frightens off so many. That said, it's a shame the PDFs are in image format instead of text as it makes it more difficult to copy notes etc. from the text into one's workbooks. This would save the student time in taking notes. Another minor criticism I have is that like many texts on mathematics it lacks both detail on the background to mathematical concepts and their applications. For example, the discussion of Eula's Formula on page 879 is somewhat incomplete. In some ways the Wiki page https://en.wikipedia.org/wiki/Euler%27s_formula https://en.wikipedia.org/wiki/Euler%27s_formula does a better job in that it has a diagram of the 'Three-dimensional visualization of Euler's formula' (its application to circular polarization, etc.). Including this diagram and having some discussion about the significance of the relevance of the i, e and ᴨ in this famous relationship would have put the subject into better context. (I recall the significance was lost on me when I first studied the subject for the same reasons. I think that many mathematicians don't realize that many of us ordinary mortals just don't think the same way they do, so including more visual descriptions and diagrams, typical applications and historical background do actually help one's understanding. However, I acknowledge that including this extra material makes textbooks larger and thus more expensive. Perhaps that's the penalty we have to pay.)
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chobytes 5y ago
This helped me a lot when I was taking calculus. We were assigned Stewart, but I basically learned the from this. Analysis from Rudin, and later some manifold theory from Lee, have converted me to more mainstream views on the subject... However I think this is probably how calculus should be introduced at first, particularly to say highschoolers.

153 points by mkl 5y ago

mkl 5y ago

This is a first-year university calculus text book based on hyperreal numbers.

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.pdf http://home.sandiego.edu/~shulman/papers/sdg-pizza-seminar.p...

eqbridges 5y ago

Is there an online course covering this?

mkl 5y ago
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-bnyOLtd6W5lazRl57SnvWuS https://www.youtube.com/watch?v=ILDkYszP2lA&list=PLDXeoTykA-...

andi999 5y ago

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).

pfortuny 5y ago
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 5y ago
  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-algebra-field 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 5y ago
    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 Ɛ is an infinitesimal hyperreal, then Ɛ, Ɛ², Ɛ³, Ɛ⁴ ... 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 5y ago
      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 5y ago
        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!
mkl 5y ago
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 https://people.math.wisc.edu/~keisler/foundations.html
- andi999 5y ago
  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 5y ago
    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 5y ago
      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
- Koshkin 5y ago
  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.
  - xyzzy123 5y ago
    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).
dwheeler 5y ago
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 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 5y ago
  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 5y ago
    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.
steve76 5y ago
There's operator theory. Treat a spectrum of prime numbers like we treat particles: https://en.wikipedia.org/wiki/Riemann_hypothesis#Operator_theory https://en.wikipedia.org/wiki/Riemann_hypothesis#Operator_th...
bikenaga 5y ago
Keisler wrote a monograph to accompany the textbook: https://people.math.wisc.edu/~keisler/foundations.html 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.org/journals/bull/1978-84-01/S0002-9904-1978-14401-2/S0002-9904-1978-14401-2.pdf https://www.ams.org/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/S0002-9904-1977-14264-X/S0002-9904-1977-14264-X.pdf 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 5y ago
  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)
  - 5y ago
mcguire 5y ago
A proof of existence for any number?
- andi999 5y ago
  if you assume ZF, then 1={0} and existence follows from existence of empty set.

Koshkin 5y ago

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.)

mkl 5y ago
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 5y ago
  Why isn't this approach to calculus more common? Does it lead to more complex proofs or have any serious drawback? I understand the ε,δ 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 5y ago
    > 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.)
    - 5y ago

pfdietz 5y ago

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/tag/nonstandard-analysis/ https://terrytao.wordpress.com/2010/11/27/nonstandard-analysis-as-a-completion-of-standard-analysis/ https://terrytao.wordpress.com/2010/11/27/nonstandard-analys...

roydanroy2 5y ago

Recent Annals of Statistics paper that uses nonstandard analysis: https://www.e-publications.org/ims/submission/AOS/user/submissionFile/31623?confirm=381d7eb7 https://www.e-publications.org/ims/submission/AOS/user/submi...

TeeMassive 5y ago

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?

kkylin 5y ago

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 https://web.math.princeton.edu/~nelson/books/rept.pdf

billytetrud 5y ago

This is how all calculus should be taught.

ganzuul 5y ago

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 5y ago
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 5y ago
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).

rsj_hn 5y ago

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 5y ago
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 5y ago
  > 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.
mcguire 5y ago
"...you can write your calculus text to not be rigorous..." Oooh, the comments you'll get from other mathematicians...
- rsj_hn 5y ago
  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.

eigenhombre 5y ago

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 5y ago
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 5y ago
  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 5y ago
  > 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."

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hilbert42 5y ago

This is a nice gesture by H. Jerome Keisler, every bit helps to promote a math subject that frightens off so many. That said, it's a shame the PDFs are in image format instead of text as it makes it more difficult to copy notes etc. from the text into one's workbooks. This would save the student time in taking notes. Another minor criticism I have is that like many texts on mathematics it lacks both detail on the background to mathematical concepts and their applications. For example, the discussion of Eula's Formula on page 879 is somewhat incomplete. In some ways the Wiki page https://en.wikipedia.org/wiki/Euler%27s_formula https://en.wikipedia.org/wiki/Euler%27s_formula does a better job in that it has a diagram of the 'Three-dimensional visualization of Euler's formula' (its application to circular polarization, etc.). Including this diagram and having some discussion about the significance of the relevance of the i, e and ᴨ in this famous relationship would have put the subject into better context. (I recall the significance was lost on me when I first studied the subject for the same reasons. I think that many mathematicians don't realize that many of us ordinary mortals just don't think the same way they do, so including more visual descriptions and diagrams, typical applications and historical background do actually help one's understanding. However, I acknowledge that including this extra material makes textbooks larger and thus more expensive. Perhaps that's the penalty we have to pay.)

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chobytes 5y ago

This helped me a lot when I was taking calculus. We were assigned Stewart, but I basically learned the from this. Analysis from Rudin, and later some manifold theory from Lee, have converted me to more mainstream views on the subject... However I think this is probably how calculus should be introduced at first, particularly to say highschoolers.