So much for posting regularly at this blog.I may as well just shut it down and start again.

But I won't. I WILL keep trying to post on a regular basis for the rest of the summer until the blog catches on. Or it doesn't. A blog is for the author,no one else.Anyone else reads it,that's a plus.

I AM hoping it does catch on,though. I have a lot of thoughts on many things ongoing-but now's not the time. If anything, small posts will begin appearing regularly.

This summer-my last one before applying to PHD programs has not gone well. Sleep has eluded me for the better part of a month-stolen by gut pain combined with frequent urination. And the wonderful health care system of America has assured my internest can't see me.

I don't have a right to live according to the AMA, you see-not enough money to buy good health.

That's why they let my father die of agonizing prostate cancer at the end-they crunched the numbers and thier profits simply outwieghed my dad's treatment. So they gave us the bullshit story that "There's nothing more we can do." The cancer metastisized througout his bones over his last few weeks, giving him a death you wouldn't wish on Bernie Madoff.

Meanwhile, if he was a drug kingpin who dropped off 5 million in CASH,I wonder if a miraculous treatment they suddenly remembered about would have appeared and extended his life by 5-10 years. Since corporations now control the publication of most medical research as well as the mass media, we'd never know if one existed no matter how much you researched.

I can get fully into this here, but I WILL say this: The fact that Yale Medical School considered seriously adding ACTING CLASSES to it's required cirriculia for the M.D. for all students entering after 2011 to "improve maximally productive patient-practitioner interaction"(translation:to make the doctors the best con-artists possible) speaks volumes of the age of medicine we live in-and why I turned my back on that world years ago. I consider myself VERY lucky to have good and trustworthy doctors-but I can't tell you how hard my family searched to find them.

100 monsters for every one like them.

*"We're coming for your money and we'll GET it all. We're the only real winners.The players don't stand a chance." -from the screenplay of Martin Scorsese's CASINO **Changing the subject to something mathematical, something on the web caught my eye yesterday and I just need to share it with the house. Ever wonder what well-defined means? It's amazing how many graduate students-particularly those working in category theory and the higher altitudes of algebra,where the phrase probably comes up most-never ask what that means. It's kind of accepted everyone "sorta" knows what it means. And for most people,that's good enough.*

I remember the first time I ever wondered about it-it was in Kenneth Kramer's honors abstract algebra course a few years ago at Queens College. He was sketching the proof of Cayley's theorum on the fact that every group is the same as some group of permutations on a set (i.e. they're isomorphic). ( Actually, he wasn't proving it,he just wanted to sketch the proof because he'd rather spend the classes' time developing the theory of group actions on a set, of which Cayley's theorum is a special case-i.e. a group acting on itself. But I digress.............)

He was constructing the composition map which is the isomorphism of a group G onto it's corresponding permutation group acting on it's underlying set S -I forget what he denoted it as,call it P(S). He commented the map was clearly well-defined. I raised my hand in frustration since I'd asked the question before and never gotten a straight answer from any professor (some of them actually got annoyed with it and made unkind remarks about my age as a student)

What followed was one of the most impressionable moments of my student career as Dr.Kramer and I exchanged comments on what exactly it meant to be well-defined. "It means it's not ambiguous what the value assigned is, Andrew-that we don't get 2 values for the same arguement." "Oh, you mean the relation actually specifies a function?" "Well, not exactly-if the formula IS a function, you're absolutely right. But this may not be a function and still be a well defined mathematical object." I didn't get it. After a few minutes of him giving a few examples, no progress was made. He ultimately asked me to table the question so we don't waste any more of the classes' time.

I did so,but ultimately,it disturbed me. Dr.Kramer is a gifted teacher on all matters mathematical-an early student of John Tate's at Harvard-and usually the most pleasant and patient of people with even the stupidest of students' questions. In fact,I'll be taking a course on elliptic functions with him at the City University Of New York Graduate Center this fall. The sheer wieght of the subsequent coursework-the first 6 chapters of Herstein's classic *Topics In Algebra *in a VERY intensive 2 semesters,plus his own notes-prevented us from broaching the subject further. All that really got settled was that it was pretty clear what "well-defined" meant if the object under consideration was a function-in fact, it's almost redundant. But how would you describle a general mathematical object as being "well defined"?

Leave it to Tom Gowers to make everybody happy.

There are several blogs online I try so hard not to miss. Peter Woit's *Not Even Wrong*, Terrance Tao's, John Baez's *The N-Category Cafe'* , *The Secret Blogging Seminar *and a few others. But nothing matches Gower's blog for sheer beauty of writing and ** thinking **about mathematics. A lot of people can do mathematics, a lot more people can

*teach*mathematics, and even more people can

*talk*about mathematics .(Sadly, this is whether or not they know what the fuck they're talking about or not...........)

There are so few who can do all of the above.

Elias Stien can do it (sometimes).

Melvyn Nathanson can do it.

James Stasheff can do it. Better then anyone I've ever heard.

William Thurston can do it.

But for my money, no one does it better currently and consistently then Tom Gowers. His blog should be required reading for all mathematicans and serious math students. (By the way-his old teacher at Cambridge, Bela Bollabos-is also great at all of the above. I doubt that's an accident. )

Anywho, I was reading Gowers' blog and low and behold, Gowers also wanted to know, after grading the exams for the year at Cambridge and discovering NONE of his students understood it,either-what's it mean for something to be well defined?

People who know me know I'm Socratic to a fault, to the point of making people violent. I almost NEVER agree with EVERYTHING someone says.

But this is rare occasion when I'm speechless with complete conviction and agreement with someone else's analysis. As I said, leave it to Gowers to give the perfect answer to a great question.

I'll simply let the beauty,depth and simplicity of Gowers' blogpost speak for itself-I simply have nothing to add to it. Nothing at all. Anyone asks me this question in the future, I'll simply give them a copy of Gowers' post. For all basic mathematical discussions that may come up in the future, I seriously doubt anyone can debunk this discussion.

It's THAT good.

Oh,screw the self-engratiating pontification,here's Gowers. And if you don't bookmark his blog, shame on you.

Good night to all,fellow travelers. Until next time.

http://gowers.wordpress.com/2009/06/08/why-arent-all-functions-well-defined/#more-605