Learning mathematics from the basics. Questions about order and books.

[Kandaron]

Before I begin, I apologize for any bad English, it isn't my first language.

I'm a medical student who didn't do any mathematics for 3 years. However, after finding out the "plug 'n chug" approach isn't really what mathematics is about, I became interested in learning it from the basics all the way to advanced stuff (I'm aware that will take years).

The reasons why I want to learn Mathematics are:

1- I'm planning to specialize in predictive medicine, which employs a lot of statistics and probability.

2- I love physics and want to (eventually) be able to comprehend even academic books.

3- I'm interested in learning computer science.

(I'm greedy I know, but I don't mind spending my life learning all that if I have to)

I have two question: 1- in what order should I study areas of mathematics? 2- which books are the best? I have done some research, and so far I'm planning on this:

Basic Algebra (using Lang's book) > Discrete math (Need suggestions') > Proofs (Velleman's book) > Calculus (Apostol's book).

What do you think of the order and the books? Thanks in advance!

 

[jbunniii]

Learning calculus early will probably be fruitful, given your interests. If you have never learned calculus before, I would not suggest the path you indicate. Apostol's book is great for a second exposure to calculus with greater rigor, but it's overkill (and will take a very long time to read) for a first exposure. Instead, I recommend the following:

Assuming you need an algebra refresher, by all means start with Lang, Basic Mathematics.

Then, assuming you like Lang's style, follow this up with his A First Course in Calculus. This will open many options for you, including probability/statistics and introductory physics.

I don't think you need to read a dedicated proofs book such as Velleman's.

Discrete math is not needed as a prerequisite for calculus, so you don't need to read it before calculus. It may be useful for computer science, however. I don't know any good books on that topic, hopefully others can recommend one.

 

[Kandaron]

Thanks, I'll start with Basic Mathematics. However, instead of going with Lang's book for Calculus, I'm thinking of using Hamming's Methods of Mathematics Applied to Calculus, Probability, and Statistics, it seems it'll be especially useful for me, do you have any experience with this book?I'd like to note I intend to go the Bayesian way with statistics and probability, although I'm unsure if that should have certain implications of which books I choose.

Thanks again!

 

[jbunniii]

Thanks, I'll start with Basic Mathematics. However, instead of going with Lang's book for Calculus, I'm thinking of using Hamming's Methods of Mathematics Applied to Calculus, Probability, and Statistics, it seems it'll be especially useful for me, do you have any experience with this book?

I don't know that book, but Richard Hamming was a distinguished applied mathematician who made many contributions in computer science and communication theory. He should certainly know what he is talking about when it comes to applying calculus.

As an amusing side note, he is also credited with the following quote:

Does anyone believe that the difference between the Lebesgue and Riemann integrals can have physical significance, and that whether say, an airplane would or would not fly could depend on this difference? If such were claimed, I should not care to fly in that plane.

I also recall reading an interview with him years ago, which included the following. Unfortunately I can't find a citation now.

Q: Favorite food?
A: Steak.

Q: Hobbies?
A: None.

 

[AlephZero]

Does anyone believe that the difference between the Lebesgue and Riemann integrals can have physical significance, and that whether say, an airplane would or would not fly could depend on this difference? If such were claimed, I should not care to fly in that plane.

The difference might be, that Riemann was arrogant enough to say "the plane will fly", but Lebesgue was humble enough to say "it will fly with probability 1"

More words of wisdom from Hamming here: http://todayinsci.com/H/Hamming_Richard/HammingRichard-Quotations.htm