Struggling with Spivak's Exercises: Building Mathematical Maturity

[Bearded Man]

For around 90% of Spivak's exercises, I need to look in the solutions manual. I'm on chapter 5, which covers limits, and the proofs just aren't coming to me. It's as if I just can't think like him. The question is: Should I come back later, or should I just trudge through until I have "mathematical maturity"?

 

[Jorriss]

What was your math background prior to forgoing Spivak?

 

[Bearded Man]

What was your math background prior to forgoing Spivak?

Fairly rigorous PreCalculus.

I learned the properties of real numbers, proof by induction, by contradiction, and direct proof.

Subjects were limits, sequences, series, vectors, logs and exps, functions and their inverses, and trigonometry.

Perhaps this just has a steep learning curve and I'm adjusting.

 

[Jorriss]

Fairly rigorous PreCalculus.

I learned the properties of real numbers, proof by induction, by contradiction, and direct proof.

Subjects were limits, sequences, series, vectors, logs and exps, functions and their inverses, and trigonometry.

Perhaps this just has a steep learning curve and I'm adjusting.

So Spivak is your first exposure to calculus then?

 

[Bearded Man]

So Spivak is your first exposure to calculus then?

Yes.

It should be noted I understand the proofs after looking at them, but I can't think of them myself.

 

[Jorriss]

Yes.

Then you likely just need more and more practice. Spivaks problems are hard. Some of them REALLY hard. You may also want to get a book on proof writing, say, An Introduction to Mathematical Reasoning by Peter Eccles.

 

[lugita15]

It's probably a good idea to first go through the calculus material in a less rigorous setting, a conventional textbook like Stewart or Anton, and then study Spivak as a stepping stone to introductory analysis texts like Rudin (or my favorite, Carothers).

 

[slider142]

That's true. a lot of Spivak's problems become almost trivial once you already know the fact that you are trying to prove is true, and the general gist of why it should be true.

 

[deluks917]

I could do about 65% of them eventually. I remember thinking it took about 2.5 hours an exercise on average (some took about 6-8 hours before I got them or gave up, some only about 30 min). How long are you giving yourself per question on average?

 

[Bearded Man]

I could do about 65% of them eventually. I remember thinking it took about 2.5 hours an exercise on average (some took about 6-8 hours before I got them or gave up, some only about 30 min). How long are you giving yourself per question on average?

I'm most certainly not giving myself enough time then. I suppose my background in mathematics has led me to think that you should be able to get most problems within 5 minutes, but this is clearly not the case with higher level mathematics. I will give myself more time and put more effort in.

 

[truth is life]

I'm most certainly not giving myself enough time then. I suppose my background in mathematics has led me to think that you should be able to get most problems within 5 minutes, but this is clearly not the case with higher level mathematics. I will give myself more time and put more effort in.

Oh no, not at all. A significant amount of time per problem is expected at higher levels, in mathematics and physics both. It often takes quite a bit of wrestling before you can see the way through.

 

[Mépris]

Then you likely just need more and more practice. Spivaks problems are hard. Some of them REALLY hard. You may also want to get a book on proof writing, say, An Introduction to Mathematical Reasoning by Peter Eccles.

Would *any* intro to set theory and logic work just as fine? I believe Apostol has a section on this before starting integral calculus on his book.

Also, is the appendix on Spivak's book not just a rigorous pre-calculus course?

 

[Jorriss]

Would *any* intro to set theory and logic work just as fine? I believe Apostol has a section on this before starting integral calculus on his book.

Also, is the appendix on Spivak's book not just a rigorous pre-calculus course?

The book I suggested isn't meant as an intro to set theory and logic, it's meant to teach people how to think mathematically and to do this it introduces a lot of set theory and number. If you want a book that just teaches logic and/or set theory I don't have suggestions I'm afraid =/.

 

[Mépris]

The book I suggested isn't meant as an intro to set theory and logic, it's meant to teach people how to think mathematically and to do this it introduces a lot of set theory and number. If you want a book that just teaches logic and/or set theory I don't have suggestions I'm afraid =/.

Fair enough. I suppose that this is what I wanted. I've been suggested that book in another thread, so will try it out. The first three parts are available for free (legally, it would seem) on the website of the University of Manchester's math department.

 

[Mike706]

I had to rely heavily on the solutions manual at first, as well. I would try to just look at it one line at a time to give me a bump in the right direction, but at first I almost completely relied on it. Specifically I had trouble seeing why things even needed to be proved in the first place, when intuition makes the truth of the statement appear obvious. Gradually, I needed it less and less. Possibly I didn't get as much out of the beginning as I would have if I was properly prepared, but that's life.

As long as you're working through every problem and making an honest effort without using the solutions manual, I think you will find that eventually you won't need it anymore.