Posts Tagged 'math'

In remembrance of topology past

I’m so glad this exists:

À la Recherche de la Topologie Perdue

It’s in two parts: I. Du côté de chez Rohlin. II. Le côté de Casson.

I’m inspired to re-title my thesis in homage to some masterpiece of French lit. A special prize will go to the best suggestion I receive, so fire away! If you can incorporate hyperbolic rank-rigidity or homogeneous spaces somehow, bonus points for you.

Zero is the number of yearning

The promised report on the Poetry of Love and Mathematics reading here at the Joint Meetings:

I enjoyed the reading, more for the break from the conference, the eagerness of the readers and the generous appreciation of the crowd than for the quality of the poetry. It was a nourishing experience.

  • Judith Baumel was maybe my favorite poet reading tonight.
  • The best poems read were not from the readers present, but were others included in the anthology Strange Attractors, whose editors organized the reading. The selection from Eryk Salvaggio’s “Five Poems about Zero” might have been my favorite treatment of mathematical theme.
  • Bob Grumman’s visual poetry in the form of long division problems was unpretentiously clever formally, I thought.
  • A poignant moment was the reading of the late Ron Mosier’s “Finishing the Math” in which he contemplates sleep and death as great places to enjoy doing math free from the worries about correctness.
  • And the most entertaining moment was Israel Lewis’s reading of his “Cantor: Not Eddie”, a poem for two voices. The poem is a mid-intercourse conversation between two lovers about cardinalities of infinite numbers. Israel, who is an elderly white man, was missing his usual co-reader; her spot was ably and amusingly filled by Deanna Nikaido, a young and attractive asian woman.

But the best and most interesting poem of all was read by John Vieira, who very astutely chose a word problem from Bhaskaracharya‘s Livati, a 12th century Indian treatise on arithmetic. It was the most beautifully written piece read all night! Here it is. Enjoy and solve with memory and longing!

Whilst making love a necklace broke.
A row of pearls mislaid.
One sixth fell to the floor.
One fifth upon the bed.
The young woman saved one third of them.
One tenth were caught by her lover.
If six pearls remained upon the string
How many pearls were there altogether?

Sit still and think

A new ranking of the best jobs puts mathematician at number 1, followed at 2 by actuary and at 3 by statistician. Here’s why:

According to the study, mathematicians fared best in part because they typically work in favorable conditions — indoors and in places free of toxic fumes or noise — unlike those toward the bottom of the list like sewage-plant operator, painter and bricklayer. They also aren’t expected to do any heavy lifting, crawling or crouching — attributes associated with occupations such as firefighter, auto mechanic and plumber.

Good for us, I guess.

Also, breaking news from the Joint Math Meetings in DC: The figure-eight knot beat out the Euclidean algorithm for president of the United States of Mathematics after their lively debate here last night. Figure-eight promises an increased supply of sub-prime numbers to alleviate the ongoing crisis. That the Euclidean algorithm’s running mate can see the Chinese Remainder Theorem from her porch did not carry the day.

Maybe ability to pun was also a consideration in ranking careers?

Stay tuned for a report from tomorrow night’s Mathematics and Love: a poetry reading.

Transphysical!

The nature of the future resurrection body is further clarified: it will be incapable of dying or decaying, thus requiring a transformation not only for those already dead but for those alive. This new mode of embodiment is hard to describe, but we can at least propose a label for it. The word ‘transphysical’ seems not to exist, surprisingly enough (one might have thought some enterprising ontologist would have employed it long since), and I proffer it for inclusion between transphosphorylation and transpicuous in the Oxford English Dictionary. The ‘trans’ is intended as a shortening of ‘transformed’.

— N.T. Wright, The Resurrection of the Son of God

Perhaps not yet in the tomes of the OED, but we are amused/dismayed to report that ‘transphysical’ in already in use in the world of amateur mathematics. This remarkable study addresses the ‘transphysical problem’ in its project of an “extension to symbolic logic – the temporal propositions, as well as the intuition on which they are based … allowing statements that emulate the entire breadth of human thought.” This extension apparently exceeds the limitations of that most metaphysical of theorems, Gödel‘s Second Incompleteness Theorem: Any (formally recursively enumerable) system of logic strong enough to encode basic arithmetic (and some basic things about formal provability) can prove its own consistency if and only if it is, in fact, inconsistent. (Consistency means that a statement ‘A’ and its negation ‘not A’ are not both provable.)

Judging from Google results, Wright’s ‘transphysical’ will have to contend against much usage in the vein of the physical/spiritual dualism he argues against throughout RSG, let alone the mathematical wackos. We should get to work popularizing his usage! (Or suggest something better?) Currently the first hit it gets isn’t even the first eschatological use of the word — it gets beat out by some architectural/cybernetic notion of the Transphysical City.

Frenemy isomorphisms

I was recently introduced by some friends to the concept of the frenemy, a term which has recently even entered public political discourse. I was excited to learn about this new class of social relationship and feel this may revolutionize how I classify my relationships.

However, I wondered whether this concept has yet been fully described. Are its relationships to other categories well-defined, and is it robust enough to undergo the same usages and manipulations as its constituents, ‘friend’ and ‘enemy’? For example, it immediately occurred to me to ask, “Is the frenemy of my frenemy my frenemy?” The answer is not immediately clear.  What about the friend of my frenemy, or the frenemy of my enemy?

It struck me that more work on this term was needed. And it struck me that the proper arena for rigorously standardizing such manipulations of interpersonal relationship terminology was mathematical. Specifically, the field of group theory suits such a task perfectly. 

In pursuance of this line of thought, I have written a research article on the subject, to which I link here:

My paper, "Frenemy Isomorphisms and Related Results"

My paper, "Frenemy Isomorphisms and Related Results"

 

In it you can find the details of this work, but I would like to summarize some of the main results here for the lay reader, and lay open the exciting conclusions my preliminary work suggests to a broader, non-technical audience.

Let us begin with an example we know and understand well; let us restrict ourselves to the world of friends and enemies.  Consider a map, or assignment, that associates to ‘friend’ the number 1 and to ‘enemy’ the number -1.  This assignment relates the friend-enemy dynamic to the algebraic structure of multiplication.  For example, who is the friend of my enemy?  ‘Friend’ is assigned to 1 and ‘enemy’ to -1, 1 x -1=-1, hence the friend of my enemy is my enemy.  This reflects our previous understanding of friend-enemy dynamics.  The equation -1 x -1 = 1 reflects the famous maxim “the enemy of my enemy is my friend.”  The algebraic structure perfectly describes the interpersonal dynamics.

In my paper I consider extensions of this framework to the friend-enemy-frenemy situation.  I conclude that there are two ‘frenemy structures’ that could work.  One (the Z4-structure) corresponds to assigning ‘friend’ to 1, ‘enemy’ to -1 and ‘frenemy’ to the imaginary number i.  In this case the frenemy of my frenemy can be calculated: i x i = -1.  The frenemy of my frenemy is my enemy.

In Section 4 of my paper the two possible frenemy structures are described via the following tables.  These tables are read as follows: to find what the frenemy of my friend is, find the entry in the row beginning with ‘frenemy’ and the collumn beginning with ‘friend’.  For either structure we see that the frenemy of my friend is my frenemy.

Model 1, the Z4 model
  friend frenemy enemy ?
friend friend frenemy enemy ?
frenemy frenemy enemy ? friend
enemy enemy ? friend frenemy
? ? friend frenemy enemy
                

Model 2, the Z2xZ2 model

  friend frenemy enemy ?
friend friend frenemy enemy ?
frenemy frenemy friend ? enemy
enemy enemy ? friend frenemy
? ? enemy frenemy friend

There are several striking implications of this work, and it indicates some problems that should be addressed.  First, the tables show that the frenemy of my frenemy is never my frenemy (Theorem 4.1 in my paper).  He or she is either my friend (in Model 1) or my enemy (in Model 2).  Thus, to determine which model actually reflects the friend-enemy-frenemy relationships, we need to determine who the frenemy of my frenemy is.  Model 2 seems more likely to me, but such relational analysis is a bit beyond my expertise as a mathematician, so I solicit your help.  Is the frenemy of my frenemy my friend, or my enemy?

Second, my work shows that there is no algebraic structure for the friend-enemy-frenemy dynamics that has only three objects (Lemma 3.2).  In either structure there is a fourth element, suggesting that the friend-enemy-frenemy categorization is incomplete!  There must be fourth, as yet undiscovered, relationship category.  I have denoted this category by ? in the tables above.  (Lest you be incredulous of this claim, I note that my procedure here is analogous to that which allowed Murray Gell-Mann, after noting the algebraic structure underlying known elementary particles, to predict the existence of new types of quarks.)  The behavior of this new category will be described by the proper table above.   For example, if the first table represents the correct model for the friend-enemy-frenemy relationships, we must look for a relationship category so that the enemy of my ? is my frenemy and so that the ? of my frenemy is my friend.  Again, I solicit your help. Perhaps this relationship is the enemend?  How can we understand this new category?

The ethical lives of mathematicians

For your comparison:

John von Neumann   alexander-grothendieck
John von Neumann             Alexander Grothendieck

The biographical info linked via the captions above is instructive, I think. (Apologies for the very slow-loading pdf on Grothendieck from the AMS Notices). I find von Neumann’s ethics rather terrifying. Grothendieck is a stark counterpoint, though not much of a model on the whole. Read about Grothendieck at least for the story of his father — the little that seems to be known about him is utterly fascinating!

On this theme, this is very encouraging.