Useful math as a foundation which prevents the corruption of math

(Below, I refine what the commenter had to say with this comment.)
You cannot corrupt math, although it’s not because there aren’t mathematicians who wouldn’t corrupt it if it could be corrupted. Once mathematicians stray from math, their ability to be logical and unbiased is nothing but average. What can be corrupted is the application of [...]

Wolframalpha.com: clueless how to use it until I figured out it’s free Mathematica without having to fire up Mathematica

Because of all the media hubbub, I went to check out the search engine capabilities of wolframalpha.com.
Not having any profound question on my mind at the time, I typed in “multisets,” and it came back with Wolfram|Alpha isn’t sure what to do with your input.
So I said to myself, “I thought you were supposed to [...]

A free ordinary differential equations book

There’s this free ODE book by Gerald Teschl, Ordinary differential equations and Dynamical Systems.
An instructor who uses the book for a class says this:
I can point out some other sources for the same material upon request.    For example,  Wolfgang Walter’s Springer GTM  Book Ordinary Differential Equations covers much of the same material with an insane [...]

Gower says examples first

Post: My favourite pedagogical principle: examples first!
For anything complex, it’s six of one or a half dozen of the other. Examples might be best first. It depends on if it’s your very first exposure.

Chu-Carroll reviews “The Faith Equation” and screed against set theory

Book Review: The Faith Equation (part 1), The Faith Equation: Part Two of the Review,
A few weeks ago, I received an email about a new book, “The Faith Equation”, by Marvin Bittinger. Bittinger is an author of math textbooks – including, I think, my first calculus text. The book is supposed to be Bittenger’s explanation [...]

Yet another axiomatic set theory book

I still haven’t made any significant headway through a book on axiomatic set theory. I thought I was going to work through Jech’s Introduction to Set Theory, but then I stumbled onto Classic Set Theory: For Guided Independent Study by Goldrei.
This will be the fourth introductory level axiomatic set theory book I’ve found this year. The [...]

God made 1, and all else is the work of man?

God made the integers and elementary sets
Leopold Knonecker said, “God made the integers, and all else is the work of man.”
I reduce that down to “God made 1, and all else is the work of man.”
I partly reduce it down because I believe that 1 is what ties math to the physical world. I used [...]

Messy heterogeneous vs. clean homogeneous addition, taking the mystery out of primes

Problem 25 of Chapter 1, Discovering Number Theory, by Holt: Let . What are the possible values of .
This problem brings to my mind how messy integer addition is compared to integer multiplication.
To prove the above, it helps to prove this lemma: if , then , which is a little easier to try and visualize, [...]

Murray Spiegel, a mathematician’s axiomatic approach to physics

In Schaum’s Theoretical Mechanics by Murray Spiegel, Spiegel demonstrates a mathematician’s approach to physics. He writes:
Axiomatic Foundation of Mechanics
An axiomatic development of mechanics, as for any science, should contain the following basic ingredients:

Undefined terms or concepts. This is clearly necessary since ultimately any definition must be based on something which remains undefined.
Unproved assertions. These are [...]

Logic over experiment; experiment over logic

Logic, though it provides the basis for reason, does not always take precedence. Experience, though it provides hypotheses for logic, does not always take precedence.

With math, logic takes precedence over experiment because many conclusions based on experiment alone have been shown to be faulty.
With experimental science, experiment takes precedence over logic because many conclusions based [...]