Pure mathematics's marriage with sets

Is all of pure mathematics tightly coupled with sets ? I love mathematics but for over 2 weeks now all i have read has been somehow tied with sets. i am having such a hard time dealing with constant involvement of sets and proofs in all the current books. Is there a way to study these subjects without such heavy reliance on sets. I own a few copies of Analysis text books, all use sets left right and center except "A course of Modern Analysis" by E.T. Whittaker and G.N. Watson. Would it be enough for me to just study this book since it goes light on involving sets everywhere instead of Rudin, Royden and binmore's books. ? I am studying towards learning rigorous probability theory, so is there any hope for me to be able to learn measure theory without being driven insane by sets ? I apologize if this question is too vague but i think i am little frustrated with you can guess involvement of sets everywhere.

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Whether Kolmogorov's measure-theoretic foundation of probability is the last word is a question that will some day get a lot of attention. –  Michael Hardy Feb 11 '12 at 0:42
@MichaelHardy Buddy i am not sure if i follow what u said. I thought his work has gotten a lot of attention all ready has n't it ? I guess u were implying that there may be more to probability theory than what became of it from his axioms. –  Hardy Feb 11 '12 at 0:45
Learning measure theory without 'heavy reliance on sets' is impossible. A measure is a function that takes a set and ascribes a non-negative real number to it. –  user23211 Feb 11 '12 at 0:59
Some branches of mathematics are inherently set-theoretic in nature. Many (Number Theory, Numerical Analysis, plenty of others) are not. That said, elementary set-theoretic language is omnipresent in mathematics. After not very long, you will become fluent in that language. –  André Nicolas Feb 11 '12 at 2:11
Kolmogorov's work has received immense amounts of attention. What I said was that the question of whether his proposed foundation of probability theory is the last word will some day get lots of attention. –  Michael Hardy Feb 11 '12 at 13:15

1 Answer

What does the "measure" in "Measure theory" measures? It measures sets. You'll be hard pressed to do any measure theory at all without sets.

Actually, as you ask at the beginning of your question, "all of pure mathematics is tightly coupled with sets". Sets became the natural objects to use to define most (if not all) mathematical objects, and that use has only increased along the last 100 years.

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How come people before then did not need to define all mathematical objects using sets ? I guess why was this done probably be a better question ? –  Hardy Feb 11 '12 at 1:53
@Hardy: that's a very good question. There are many notions in mathematics that were used for centuries without mathematicians really understanding what they were; that still happens with physicists, who usually use lots of math following their physical intuition and without really caring about the formality of their objects and operations. That was the case with math until the 19th century, where concrete efforts to formalize the subject were made. –  Martin Argerami Feb 11 '12 at 2:00
(continuing from the previous comment) The fact that one has an intuition about an object, or a desire for the object to exist, is not enough. For instance, you can say "let $n$ be a number that is both even and odd" all you want, but such a thing doesn't exist (but first you need to say what a number is!). Another example is $i=\sqrt{-1}$; mathematicians started using it because it made sense in the right context, but there was the need to guarantee it did exist! The absolute majority of these constructions are done with sets. –  Martin Argerami Feb 11 '12 at 2:05
@Hardy: it's exactly the opposite, the use of sets allowed for formal constructions and proofs that were not available before. More than that, new ideas and constructions were possible, and math has greatly evolved since then. Measure theory, for example, is 100 years old; since then, lots of new things have been developed and proven. Old results have been settled and new areas have been created that would be impossible to deal with without the language of sets. –  Martin Argerami Feb 11 '12 at 2:07
@Hardy: no, most applied mathematicians (and many physicists, as I mentioned) usually work with their concrete objects that are mostly numerical one way or another, and often they don't need to deal with the formalism. But that varies greatly depending on different areas. –  Martin Argerami Feb 11 '12 at 2:09