Foundations of mathematics

[sarcina]

Hi,

 

Over the course of the last few weeks I've been attempting to learn about the foundations of mathematics and yet I've been unable to find out what the single most fundamental branch of mathematics is. I've read a few books that covered mathematical logic, mathematical philosophy, and elementry set theory, and yet the answer still eludes me.

 

Could someone please point me in the right direction and tell me what the single most fundamental branch of mathematics is, and what branch of logic does the whole of mathematics stem from? Any help would be very much appreciated.

 

 

Edit: Clarified question

Edited February 18, 2014 by sarcina

[John]

Mathematical logic is probably what you're looking for if you're trying to dig deep into the foundations of mathematics. It encompasses things like set theory, model theory, and proof theory, on which much of mathematics is built. Depending on how far down the rabbit hole you wish to go, there's also metamathematics to consider, and of course you can then get into the philosophy of mathematics itself.

 

"Foundations of mathematics" classes vary by school. Mine essentially consisted of propositional logic, methods of proof, set theory, and a bit of group theory. The latter doesn't strike me as something that really belongs in a foundations class, but there it was, anyway.

Edited February 18, 2014 by John

[sarcina]

I guess what I'm looking for is the single branch or sub-branch of study that cradles the very edge between philosophy and mathematics -- if it even exists.

 

Part of my confusion stems from the wide range of sub-disciplines (ie. Predicate Calculus encompasses first-order logic, second order logic, etc).

[John]

Well, again, mathematical logic and metamathematics are probably the areas to check.

Rautenberg's A Concise Introduction to Mathematical Logic is decent reading so far, though I admit I haven't gotten terribly far into it. It seems much nicer than the text my logic class used, which is now out of print anyway.

Something like Kleene's Introduction to Metamathematics may be closer to what you're seeking, but I haven't read it at all.

It's difficult to really recommend very specific things to study. The various subfields of mathematics connect in sometimes surprising ways, and while foundational areas may be somewhat more isolated than others, the fact remains that the fruitful study of mathematics involves exposing oneself to a variety of topics. It sounds like you've already taken some initial steps. Now just strike out on some interesting path and see where it leads.

Edited February 18, 2014 by John

[ajb]

The fundamental issues in mathematics border on philosophy and logic, as already stated. But mathematical logic can be very removed from the standard practices of most mathematicians. For example, I do not worry about very fundamental issues.

 

Set theory (+ axiom of choice) is one of the single most important and basic theories of mathematics.

 

Beyond that we have category theory, which is "the mathematics of mathematics". To me, category theory seems the most fundamental and encompassing framework we have.

[LaurieAG]

Could someone please point me in the right direction and tell me what the single most fundamental branch of mathematics is, and what branch of logic does the whole of mathematics stem from? Any help would be very much appreciated.

 

Geometry is the oldest. http://en.wikipedia.org/wiki/Geometry

[studiot]

What makes you think there is a single most fundamental branch of mathematics?

Why should this be so, after all the Hindus thought that the world was carried on four elephants and even neolithic man placed his lintels on two supports.

 

There are many different ways you could divide maths into branches.

One such is maths is the study of number and shape.

This was certainly true early maths, although the study of Geometry was indeed codified first, this could not have happened without some numebr theory.

 

 

Here is an interesting viewpoint from one of the world's most famous mystical poets.

 

 

TO THE REV. GEORGE COLERIDGE

Dear Brother,

I have often been surprized, that Mathematics, the quintessence of Truth, should have found admirers so few and so languid.--Frequent consideration and minute scrutiny have at length unravelled the cause--viz.--that though Reason is feasted, Imagination is starved; whilst Reason is luxuriating in it's proper Paradise, Imagination is wearily travelling on a dreary desart. To assist Reason by the stimulus of Imagination is the design of the following production. In the execution of it much may be objectionable. The verse (particularly in the introduction of the Ode) may be accused of unwarrantable liberties; but they are liberties equally homogeneal with the exactness of Mathematical disquisition, and the boldness of Pindaric daring. I have three strong champions to defend me against the attacks of Criticism: the Novelty, the Difficulty, and the Utility of the Work. I may justly plume myself, that I first have drawn the Nymph Mathesis from the visionary caves of Abstracted Idea, and caused her to unite with Harmony. The first-born of this Union I now present to you: with interested motives indeed--as I expect to receive in return the more valuable offspring of your Muse--

Thine ever,

S. T. C.
Christ's Hospital,] March 31, 1791.

 

Coleridege was also responsible for this poem about the first proposition on Euclid's first book of fomal geometry.

 

http://blogs.ams.org/mathgradblog/2013/06/05/euclid-coleridge-poem-2/

 

More recently we have categorised the subject into two branches Analysis and Synthesis.

 

Synthesis is interesting because it leads to the idea of cmputability, a modern notion.

 

In modern times a good source for your research would be the Cambridge University text

 

"Computability and Logic"

 

by Boolos and Jeffrey

 

go well

[Acme]

http://fair-use.org/bertrand-russell/the-principles-of-mathematics/

 

Russell's The Principles of Mathematics seems a reasonable foundational menu even today. Bon appétit!