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February 21st, 2014, 09:35 PM   #1
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Need recommendations for a good foundations of mathematics b

Hello everyone. I'm new here. I was hoping someone on this forum could recommend a few good books on the foundations of mathematics. I recently read a very short biography on Georg Cantor and his works online and it just happened to ever so slightly quench a "thirst that knows not where the water is" I've had for sometime now. I've always wondered about what lies at the foundation of all the mathematics ever developed. From what I gather all of math is built on axioms, which are small and very simple statements that everyone accepts to be true because they seem so intuitively valid and always seem to stand up to intellectual scrutiny. Once enough axioms are laid down logicians, who have already agreed on what constitutes well-founded, precisely defined, and defensibly correct (by some standard of reason) rules of logic, can then use those axioms and rules of logic to prove higher mathematical statements known as theorems. It's my understanding that although it may be inspired by truths that seem to be evident in nature, and certainly has found much application to the study of the universe, mathematics is born out of human reason and philosophy, and is therefore, at least at its roots, not a part of empiricism. I was wondering if there are any good books out there that explain the foundations of mathematics. That is to say, explains these fundamental axioms, what inspired them, the scrutiny they have been put under, how they have held up to that scrutiny, the rules of logic that can be used to put them together to prove higher statements in mathematics, the definitions and origins of those rules of logic, how they have withstood criticism, list each and every fundamental axiom which exists, and maybe even show how some of them can be used to prove higher theorems. I know that Euclid's Elements is one such book that already has a few of those axioms and other proofs when it comes to the foundations of planar geometry. I do have "The Elements", but much of it is kind of hard to follow and really it was written for people of a much different time. I would like something written by an English speaking author (no translations please) and which is no more than 30 years old. If I could find literally all of the fundamental mathematical axioms, easy to follow explanations of each, simply defined and explained rules of logic, and a few simple proofs of fairly easily grasped theorems using the axioms and logic already explained that would be great. I'm assuming, and I realize that this assumption may be wrong, that these axioms are few enough in number and the rules of logic condense enough to all fit into one good book detailing and explaining to the layman the foundations of math. If anybody could make some good recommendations I would appreciate it. Thank you.
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February 22nd, 2014, 08:07 AM   #2
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Re: Need recommendations for a good foundations of mathemati

I think Enderton has a good book along those lines.
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