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Calculus Why read Spivak's Calculus?

  1. Mar 30, 2015 #1
    I'm currently an A-Level Maths and Physics student looking to get ahead before university maths/ physics. Looking on the internet I see people making a big deal of Spivak's Calculus but on looking thought it I can't see how it could possibly be useful for future maths and physics. To me it seems in a way like learning geometry using euclids elements. Sure it's quite interesting and satisfies extreme skepticism but if you just want to learn about it you can get lost in the details. I feel like I might come out from reading it thinking 'why am I learning all this stuff?!'. And that's what I'd really like to know. What will I get out of reading Spivak's Calculus book? If I work my way through it what advantage would I have over the me in a parallel universe who didn't work through it? Thanks :)
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  3. Mar 30, 2015 #2


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    You read it because it interests you. If it doesn't interest you to read a book like that, then don't read it. You will certainly not miss much.

    Personally, I enjoy getting "lost in the details" and to "satisfy extreme skepticisim". If you don't, then you probably shouldn't go into math. There is no real other reason to go into math besides that.
  4. Mar 30, 2015 #3
    The main reason I like to learn something is because it is a bridge to solving an interesting problem of some sort. I mean I can see the purpose of learning integration because I like that it can tell you the area under a graph between two intervals and I like combinations because I can tell tell what number of arrangements of objects I can have. And then I can see you can use this knowledge to go to something else and something else etc. But when I see a proof of 'for a<c<b integral of f from a to b equals the integral of f from a to c plus the integral of f from c to b' with extravagant notation and such when it's obvious to start with! And after you've finished with all that stuff what was the point? Does it help you to solve more difficult problems and what kind of problems does it help you solve. Or does the style in some way prepare you for something later on?
  5. Mar 30, 2015 #4


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    Yes. But it depends on what kind of problems you want to solve. If you want to solve problems in physics, then you don't need all that math. But if you want to do math, then Spivak is very necessary. But if you already don't see the point of Spivak, then you'll likely won't see the point of abstract math. However, if you want to go deeper into math and know the deep reasons of why something is true, then Spivak is necessary.

    Yes, more abstract math. Whether you want that or not is up to you.

    But it is true, many things in Spivak are trivial to see geometrically, but with difficult proofs. So I know why you find it useless. However, the further you go in physics, you'll see math that isn't trivial to see and is in fact quite subtle. For example, you'll find functions ##\delta## such that ##\int_{-\infty}^{+\infty} \delta(x)dx = 1## but such that ##\delta(x)=0## for nonzero ##x##. Such a function obviously does not exist. But in physics we often pretend that it does. So either you can go on and pretend it exists, which is ok. But you can also try to find the deeper reasons why it works. In that case, you need to study the difficult proofs of math. And then Spivak is a necessary first step.
  6. Apr 4, 2015 #5


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    Here is my take; I'll use the example of riding a bicycle. Riding a bicycle certainly seems difficult in the beginning, partly because one doesn't innately know that slower is more hazardous. One might think that one should start off slow and master it before speeding up, but actually it works the other way round. The intuition is wrong and makes it seems much more difficult than it is.

    If we extract this wisdom, any new thing will seem difficult at first because our intuition is wrong, but by the simple fact that many other people succeed at it, there must be a way to succeed and we just need to correct our mistaken intuitions.

    I'll say the same about Spivak's book, it is difficult because our intuition is wrong. Our intuition is that a person should start at page 1, read everything carefully, answer every question, never turn a page without understanding what it contains, etc, a linear approach. And it is appealing that we can learn every definition and prove every theorem and then know the subject completely, so it seems that this is how one should proceed.

    But there is a philosophical problem here. What one is trying to do is learn the subject, and sticking to one particular method of learning can only be a hindrance, it can never be a help. So that's the first point: don't be limited to any one method. Hopefully this quells your worry about it being like learning from Euclid's Elements. It would still be learning, you wouldn't need to learn in any one particular way.

    Why this book? Because it contains rigorous calculus and is known for having challenging problems.

    Why rigorous calculus? I want to defer to Micromass's answer on this point.

    What I will say is, I like to think of math as a language. To be philosophical, words are just words, what is important is the things that they refer to. And mathematical symbols are just symbols, what is important is what they refer to, what they are about.

    Calculus, rigorous or not, is about a language of symbols to talk about rates of change, slopes or areas and volumes. Rigorous caclulus is careful to define those symbols rigorously, in excruciating detail so as to make the meaning very clear, and that is interesting of course, but really I find the use of the symbols to be more interesting.

    One can ask, what is a continuous function? And if one is being precise, one must use the rigorous definition. But why was it defined that way? Because we want it to define what was non-rigorous before. Because if continuity was anything else, it wouldn't be called continuity. If you understand that, that the definition is meant to define something that is intuitively obvious, that it is meant to be somehow simple, and the proofs themselves, although complicated, have simple start and end points because they connect ideas that should be simple, you should start to see that there is a simplicity criterion to the theorems and the complication is buried in the proofs.

    But realize that the theorems are chosen so that the proofs have a simplicity criterion as well. Proofs that are too complicated will make use of lemmas to make them easier to understand. So proving any particular lemma or theorem will be to some degree simple or elementary as well. That said, the simplicity criterion for proofs applies only to how the proof looks, not to how easy it was to find the proofs. So any particular proof could be difficult to find.

    So hopefully this is a bit of a road map. The theorems themselves, if you don't prove them, are still interesting to know, I believe. And because the proofs have a simplicity criterion, it can be worthwhile to just look them up because they will usually be short and easy to memorize. Why spend hours when you can spend minutes?

    As a strange peculiar example, one of the things I wanted to learn from Spivak was how to do limit proofs. I devised a strategy that always worked, a universal procedure to tackle any limit proof (of the type that was in the book). It seems magical but facing a bunch of those questions, I was like, how can I make this more efficient?

    Consider this, limit proofs should be similar, very similar; if the function is suitably continuous, why wouldn't the proof be almost identical? So here again, intuition has led us wrong. What seemed magical, that there could be a universal procedure, now perhaps seems even obvious, of course, each problem is the same in a sufficiently small interval (unless the function is strange but no problem was of that sort).

    My point is, if you want to read it, it doesn't need to be some kind of pilgrimage to the Mecca of Math, you could read it just to learn about the theorems, to have a more structured knowledge, to reinforce what you know. That said, it is a definition-theorem-proof book so you would need to be able to handle that.
  7. Apr 4, 2015 #6


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    Spivak is a modern book and Euclid is not. If you want to learn calculus Spivak is a good book, but it is not necessary or sufficient to read it. Most people would read an easier book before Spivak and some other books because Spivak does not cover all topics, covers some topics in a strange way, and it not general enough. For example Spivak often hides the topology in his arguments, leaves out many methods, does not cover differentials and so forth. So read Spivak or don't who cares. Your question really seems to be if it necessary to learn calculus well. Sure you can learn to calculate some things without really understanding what you are doing. Sometimes you will get wrong answers and not know why. Spivak is an introductory calculus books (though among the harder ones). If you want calculus as a tool you will in practice cut your study of it off at some point. I do not see why you would choose such a low bar as Spivak though. That would be very limiting.
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