So far?

Another overcast morning.  I feel as though I must put on the lights to read at 7:16 in the AM.  But that is not what I want to talk about.  What I really want to talk about is mathematics and my obsession with it this year.  (Yes, I am always obsessed with something as you well know by now.)

This is what began all the trouble around 300 BCE.

Euclid’s Fifth Postulate:

That, if a straight line falling on two straight lines make the interior angles on the same side less than two right angles, the two straight lines, if produced indefinitely, meet on that side on which are the angles less than the two right angles.

Thomas L. Heath translator

Twenty-two hundred years later, Gauss, Bolyai, and Lobachevsky had the audacity to negate the postulate and arrive at the conclusion that its consequences produced a geometry as consistent as Euclid’s although containing propositions strange and contrary to Euclid’s.  Several decades later, Betrami and Klein established that if hyperbolic geometry is inconsistent, then so is Euclidean geometry.  Today, hyperbolic geometry produces the richer and more useful geometry for mathematics.

The history of this, its impact on philosophy of mathematics, philosophy in general, the connections between mathematics, science, and art, and how mathematics is an art and creative activity is what I have been working on this year.

It is the best year of my life so far.

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Published in: on July 29, 2011 at 8:36 am  Leave a Comment  

Desire or lack thereof

So fucking awesomely bored–maybe, that’s why I spend 10 to 12 hours each day writing a stupid geometry book. (And you’ve to get up early in the morning to do that.)

But I like it. At one point in my dismal life, it was a dream.

Euclid, Hilbert, and Alone

I have Euclid’s Elements on one side of me and Hilbert’s Foundations of Geometry on the other side.  I’m trying to build a bridge in my mind between the two.  I find it desperately lonely and hard work.  I labor on it for at least 12 hours each day.

The thing that makes it even harder than it should be is that I have no one to talk to about it.  The major parts of my thoughts are locked in solitary confinement.

That’s the nature of desire though.  It’s the thing most personally felt even though the least most noticed by others.

Let’s call the situation a secret romance.

Published in: on April 26, 2011 at 2:04 pm  Leave a Comment  

Damn you, Tarski

I’ve been reading this totally smokin’ biography of Alfred Tarski which has made me interested in math logic again–something I said I’d never do again.

Oh well, it’s just a benign activity except for the valuable use of time, time I must marshall and conserve at my advanced age.

Published in: on January 11, 2011 at 9:49 pm  Leave a Comment  

Tarski and Godel

You are reading a biography of Alfred Tarski. In it there
is a photo of Tarski and Godel laughing together in Vienna in the
early 1930’s. Apparently, even the gods of logic enjoy a good
joke.

Published in: on January 2, 2011 at 9:32 pm  Leave a Comment  

Rotating

Before daybreak Sunday Morning: I’m rotating between Pamuk’s The Museum of Innocence, Richards’ Life, and Dijksterhuis’s translation and commentary of the works of Archimedes.  Pamuk takes me where he wants me to go.  Richards takes me where he wants me to go.  Archimedes takes me to another world.  Coping with the ancient Greek way of doing mathematics is hard work, especially since it’s Archimedes and Dijksterhuis both with whom I’m coping.

The sun is up although you’d hardly know it for the overcast low in the sky.  I must focus my reading for the rest of the day.  I’ll work on Archimedes until my brain rebels.  Then I’ll read Pamuk and Richards as if I were eating desert after dinner.

Published in: on November 14, 2010 at 10:27 am  Comments (2)  

Math library

I now have an extensive mathematics library downloaded to my iPad–probably enough to keep me entertained forever.

Yeeeeh-aaaah!

Published in: on July 18, 2010 at 10:49 am  Leave a Comment  

Philosophy of…

I figured out philosophy of mathematics.  But how hard can that be?

Now, what remains is the philosophy of desire to be reckoned–the last thing on my list as philosophy of stuff goes.

Maybe, there is just desire and philosophy has nothing to do with it.

Published in: on January 6, 2010 at 11:22 pm  Comments (1)  

The math thing in your head

You and I are sitting together talking.  You leave and I can no longer see you.  That’s an instance of one minus one equals zero.  It has been shown that babies have some facility with and some understanding of small arithmetic even though they have not been taught arithmetic.

Beyond our ability to do very small arithmetic, I believe mathematics is a product of our imaginations.  That means we create mathematics using our abilities to create and manipulate metaphors.

One million is a concept in our heads.  We won’t find it outside our heads.  Even if you show me precisely one million grains of sand, the million part is in my head.

Good poets and good mathematicians have a lot in common.

Published in: on January 4, 2010 at 2:20 pm  Leave a Comment  

One more demon joins the party

I shouldn’t have done it.  Started reading philosophy of mathematics stuff again, that is.  Now, an old obsession has returned to join ranks with all the other unproductive obsessions demonizing me.

Damn it, Larry.  It’s your fault.

Published in: on December 29, 2009 at 11:46 pm  Leave a Comment  

Down the road

On this solitary afternoon, in a fit of madness or stupidity, I thought about taking one more run at mathematical logic and mathematical philosophy just to meditate on it in general one more time before I die–just to get a little bit farther down the road.

Published in: on December 13, 2009 at 5:36 pm  Leave a Comment  

Still ticking

While browsing the mathematics section at Border’s the other day, I noticed they had a copy of Russell’s Introduction to Mathematical Philosophy on the shelf.  The book apparently still has some legs.

Published in: on December 9, 2009 at 11:03 am  Leave a Comment  

Drawings and the future

I spent the afternoon sitting in a corner at Pippin’s drawing diagrams on the computer for my geometry book.  At first, it was drudgery, but then my mind wandered off the task at hand.  The diagrams started to look pretty.  Other drawings came to mind and I explored them.–things I had not thought about.

Questions recurred.  What if this silly geometry book is the only thing I really work on or care about?  Would my life be any different?

Probably not.

Published in: on October 22, 2009 at 5:39 pm  Comments (2)  

What to write about?

On a given finite straight line to construct an equilateral triangle.

Proposition I.1, Elements, Euclid

You might naively ask: what are all the interesting propositions you can prove about equilateral triangles?  I suspect enough propositions to create a quirky yet interesting book for the mathematically inclined.  Of course, one can generalize to regular polygons, polyhedrons, and polytopes, but by then you would have Coxeter’s splendid book Regular Polytopes.  Let the first project consist of equilateral triangles.

It’s warm and sunny.  I dream about and drift through a land whose significance lies in its interesting propositions.  Or to paraphrase Wallace Stevens: not dreams of life itself, but dreams of propositions about life.

Published in: on October 21, 2009 at 12:23 pm  Leave a Comment  

The days with Gauss

I picked up a notebook at Walgreen’s yesterday.  It’s just for my notes on Disquisitiones Arithmeticae.  I liken the Gauss experience to reading Hume.  It doesn’t get any better.

Published in: on October 3, 2009 at 10:06 am  Leave a Comment  

Disquisitiones Arithmeticae and me

Gauss wrote his classic book on number theory, Disquisitiones Arthmeticae, in 1798 when he was 21 years of age.  The book collected and organized previous results in number theory.  To that he added many stunning new results of his own.  The book set a new standard in logical demonstration and rigor.  The book led the way to further fruitful investigations in number theory.  Thus, we may consider it one of the true classics of mathematics.

In the summer of 1974 I purchased a copy of Disquisitiones Arthmeticae at a used book sale.  A little later, I purchased a notebook and began taking notes as I studied the book.  While rummaging through the book closet yesterday, I found the tattered book.  The notebook seems to be lost though.  Oh well, empty notebooks are easy to come by if not so easily refilled with results.

Mathematics and number theory have advanced far beyond Gauss during the past 200 years.  As i thumb through Disquisitiones Arithmeticae, I follow the mind of one of mathematics’ preeminent geniuses.  I find it exciting that with much effort I can follow what he is doing.

Just maybe, another notebook devoted to the book is in order.

Published in: on October 2, 2009 at 10:57 am  Comments (1)  

Numbers and arithmetic …

Something you can play with while pretending you are doing something meaningful.  Let’s call them toys.

Published in: on September 30, 2009 at 11:41 am  Leave a Comment  

Math extra credit 3

How many twin primes are there?

Happy hunting.P

Published in: on September 11, 2009 at 4:19 pm  Leave a Comment  

Top of mind

Some questions nag and persist.  Why are there many  seemingly easy questions about the numbers 1, 2, 3, 4, … that have not been resolved?  I suppose much of mathematics as we know it would not exist if the easy problems to state were also easy to resolve.

I can’t put the question out of mind this week.  That leads to another unanswered question: why?

Published in: on September 10, 2009 at 11:06 am  Leave a Comment  

Math extra credit problem 2

Can every even number be expressed as the sum of two prime numbers?  If so, prove it.  If not, provide a factored counterexample.

If you can’t resolve the question, Google Christian Goldbach and Leonard Euler and turn in a short report about them for partial credit.

Or cop a plea.  Explain why you are not necessarily innumerate just because you could not resolve this seemingly trivial problem.

Published in: on September 9, 2009 at 11:49 pm  Leave a Comment  

Frustrated by opacity

Sunday morning and no noise out of doors, but a lot of non-Euclidean geometry clanks inside my head.  I finally see how to map an infinite non-Euclidean plane into a disk without boundary to get the Beltrami-Klein model.  What shall I sacrifice to the gods on this holy day for granting me this vision?

I wish it did not take me so long to see things that are easily visible to others.

Be that as it may, now that I see the mapping, I can think more clearly about the Riemann Hypothesis in the Beltrami-Klein model.  In fact, I don’t know doodly about number theory in that kind of model, so meditating on Riemann is a little premature.

No matter.  This shit is fun.

Published in: on August 30, 2009 at 11:41 am  Leave a Comment  

More dawdling

My current walking around the street book is William Dunham’s Journey Through Genius: The Great Theorems of Mathematics.  I read it shortly after it came out in 1990 and was impressed.  Working my way through it again, I am even more impressed.

Dunham takes a tour of some splendid mathematical theorems proved by the giants of mathematics from ancient times onwards.  Assuming the reader has a high school understanding of algebra and geometry, he gives detailed proofs of the theorems and presents the interesting mathematics, history, and biographies required to appreciate them.  The book is charming, a perfect book over which to dawdle.  (And don’t we all like dawdling over something pretty on occasion?)

Just as one goes to the museum to view one’s favorites works of art and returns with renewed appreciation, one can do the same thing with math theorems, for they are works of art too.

Published in: on August 29, 2009 at 2:55 pm  Leave a Comment  

The art of negation

Euclid in his Elements first uses his fifth postulate, the parallel postulate, to prove proposition I.29.

Exercise: negate the parallel postulate and determine the consequences for all the propositions in Elements from I.29 onwards. Hint: be sure to have have plenty of paper, ink, and patience before you start.

Another hint: take as much time as you need, think hard, and enjoy the ride, for this is a no-grade and no-credit course.

Published in: on August 29, 2009 at 2:30 pm  Leave a Comment  

Do what you want, but before you start, meditate on whether you can do it

I went to Starbuck’s this morning with a math notebook in hand.  My intent was to work on some elementary notions and results regarding elliptic curves.  As I drank my coffee at the bar and stared out the window running along State Street (and isn’t math  like the rest of life–a lot of staring into space?), I fixed upon the conceptual outline and style of the geometry book I have been writing.  It fits with what I can actually do when writing a geometry book.  Wow, how did I ever come up with that idea!

And with that, I decided what I want to do with a significant portion of the rest of my life.  I’d tell you, but it’s a secret until I actually do some of it.

Published in: on August 27, 2009 at 1:04 pm  Leave a Comment  

I think you’re pretty

No sense in saying more than what can be said.  I’m studying some math theorems again just because I think they’re pretty.  They are like a beloved who steals your heart.  When someone asks you why you are with the beloved all you can think to say is your heart is filled as you contemplate her and feel her splendor.

Published in: on August 25, 2009 at 11:04 pm  Leave a Comment