Is there a mental exercise which has a tendency to order the mind such that the understanding and retention of mathematical concepts, symbology, and algorithmic procedure is increased?

Is there a mental exercise which has a tendency to order the mind such that the understanding and retention of mathematical concepts, symbology, and algorithmic procedure is increased?
Sometimes it seems to help if one is sober.Originally Posted by GiantEvil
You might want to look up recreational mathematics. It's not an exercise, but it might increase interest, which is probably more important.
While I intentionally enjoy the occasional buzz, I am by no means a drunkard. The only substance's I have ever meet the criteria for psychological and physiological addiction to is nicotine and caffeine, and I have wholly surrendered the use of nicotine, though I suspect that addiction itself will be with me to the end of my days. I am currently drinking a cup of coffee right now.Originally Posted by DrRocket
Now, back to my original query.
Say one want's to learn to juggle. It is good to start with one object, preferably uniform, and not to hard or bouncy. Then one assumes a rigid posture and practices throwing the object from one hand into the other with as minimal a movement as possible. This is the basic drill for developing the fundamental skill necessary to juggling.
I am asking for a drill that strengthens the fundamental skill's necessary to the study and understanding of pattern and order.
As an example, while travelling, picking two points on the horizon and using them, and myself as a third point, to visualize a triangle which would close down to a line segment as my ray of travel intersects the line segment formed by the two original points?
Or, whenever a discernable group of object's fall's within my vision, I could consider them a set, count them to discover the cardinality of the set, then group the set into subsets based on arbitrary equivalence classes?
Perhaps to never do a one to one count of objects in a geometric arrangement, but to always deduce their number from edge count and formula?
Thank you MagiMaster for the suggestion, though I can confidently assert that dearth of interest is not an issue.Originally Posted by MagiMaster
If you want exercises, get a copy of Topology: An Outline for a First Course by Lewis E. Ward and prove the theorems and work out the examples. It is relatively inexpensive on the used book market  Amazon.com or Alibris.comOriginally Posted by GiantEvil
This book is specifically designed for the reader to do his own proofs.
It may help to brew a new pot of coffee.
I have ordered the book. It was affordable. Thanks.Originally Posted by DrRocket
While were on books, I obtained a copy of Halmos's Naive Set Theory and found the exposition to be a little "fuzzy".
I have found this alternative; http://www.amazon.com/TheoryContinu...9743025&sr=12
I find it to be more rigorous, in depth, and clearer.
And after all the calculus texts I've failed to get anything from, I find that I am getting somewhere with Morris Kline's Calculus book. Although his Wiki page gives me the impression that most mathematicians might consider him a pariah.
I am a bit surprised at your feelings with regard to the book of Halmos. He is generally a model of clarity. I quite like that book. That said, it is naive set theory, not axiometric set theory. However, most mathematicians think in terms of naive set theory, and leave axiomeric set theory to logicians. There are exceptions, of course, and when real issues in set theory arise, there is no substitute for the axiometric approach. I do not know the book of Smulyian, yet. He is a logician. But I did order his book from Amazon, as well as Paul Cohen's book on the continuum hypothesis. Cohen is far more than a logician, and while he proved the independence of the continuum hypothesid, he also dif seminal work in harmonic analysis.Originally Posted by GiantEvil
Don't give up on Halmos. That is still a very good book and an excellent exposition of cardinal and ordinal numbers. In mathematics, I would describe it as "light reading".
The only "good" calculus book of which I am aware is the (seldom used) book by Mike Spivak.
BTW do not expevct to read Ward's book quickly. The proper waybto go through it is one theorem at a time, and proving each theorem before you go to the next one. You can expect to take a year, working hard and steadily, to get tjhrough the book.
I second these comments on the Halmos book, I enjoyed in immensely.Originally Posted by DrRocket
It is often said that most mathematicians, real or aspiring, learn set theory by osmosis, a silly thing to say in my opinion, as it underpins all modern mathematics (though category theory is grooming itself for that role  fat chance!). Read and enjoy, it is not hard going.
Forget axiomatic set theory, it is a very dry subject.
Yeah, why is it "seldom used"? What little calculus I ever had stomach for was taught me from James Stewart's book of the same name, which is thorough, but somewhat mechanical.The only "good" calculus book of which I am aware is the (seldom used) book by Mike Spivak.
I have Spivak, and, unlike Stewart, he explains the theory very well. I like it (but not enough to make me enjoy calculus).
I also recommend Birkhoff & Mac Lane A Survey of Modern Algebra, which starts from the very "bottom" and covers just about everything except topology and differential geometry. Every statement is proved, plus there are loads of exercises (always a Good Thing)
I got my copy from Alibris for about 10 euros.
BTW, it was Saunders Mac Lane who founded category theory, and one can almost sense the idea of this starting to come to him as he writes this book. As I like category theory, maybe I am biased, but I recommend the book thoroughly.
Silly or not, it is true.Originally Posted by Guitarist
One can read and understand the Halmos book ober w weekend. Other than (yawn) axiomatic set theory, I have never seen a set theory class offered, and I have no idea what would be left to talk about after the first week if one were offered.
The necessary topics are more than adequately covered in the introductions to real analysis, topology and algebra, including one's preferred version of the axiom of choice  the axiom itself, Zorn's lemmsa, Hausdorff maximum principle.
Category theory is useful language, but is not likely to ever be regarded as "underpinning everything".
Birkhoff & MacLane is a fine algebra book, if a bit dated, for an introduction to algebra and number systems. It is not a substitute for books on analysis or topology. It does tend to dwell on the obvious at times.Originally Posted by guitarist
I don't know if you're lurking around Doc, but I got that topology book the other day. I've given it a cursory examination but haven't dived in yet. I was expecting some pictures of donuts, but don't see any. Oh well.
I'm currently rather indulged in "A Book of Abstract Algebra" by Charles C. Pinter.
Anyhow, I was wondering if you had received your "Set Theory" by Smullyan yet. What do you think?
Set theory isn't hard, but it is a little tedious. The first order logic symbology in the Smullyan book threw me off a little, but there's a lot of context available, and The Wiki to reference, so I think I got a good handle on it now.
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