Archive for the 'math' Category

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.

Advertisements

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.

Questions for community

A recent exchange with roses brought the idea of this post to my mind.  In our exchange, roses raised the idea of the faith community. In part of my response I asked some questions about this community. I’m interested in fleshing these out as an item of independent interest to me, only peripherally related to the other discussion. (Roses originally meant something more like the idea of faith, and here I’m riffing on something that came up in our dialogue, not ripping her position.)

I want to propose a set of questions that any community should be able to answer. I poked around for resources on a theory of community, and found only this, so if anyone knows of a better resource let me know. (Alasdair MacIntyre‘s After Virtue is in the queue, and will hopefully soon make an appearance in some posts. In the meantime, let me just say that I know I’m woefully uninformed.)

I take it as axiomatic that any notion of community involves boundary, a sense of who’s in and who’s not, and that this need not be hostilely exclusive, but is necessary for a community’s self-definition to be coherent and not vacuous. (So “I am a citizen, not of Athens, nor of Greece, but of the whole world” is incorrect if Socrates wants citizenship to reference involvement in a community.)

Finally, since terminology like ‘the faith community’ or ‘the math community’ is pretty well established and yet not what I want to describe, let’s do what we do in math when the terminology well runs dry and coin ‘strong community’ or, better, ‘community in the strong sense.’

Any community in the strong sense should be able to positively and substantively answer the following questions:

  1. How is community membership defined? What defines the community’s shared identity?
  2. How and where are social networks between members created and used? (Here we need real interaction, preferably but not exclusively physical, face-to-face interaction.)
  3. What commitments are expected of community members? More particularly, what responsibilities to they have to one another? (Real, meaningful mutual obligation is necessary for community in the strong sense.)
  4. What behavioral norms are expected of the members?
  5. What symbols, rituals, habits or language characterize the community?

Lest it seem like I’m cooking these up to end up with church as the only community in the strong sense, take as examples some subsets of the math community: the community as a whole, the dynamical systems (here, DS) community, the community of my department. On question 1: membership is defined around shared expertise, and the dynamical systems community is more like a community in the strong sense than the math community. For question 2: conferences (the yearly Penn State-Maryland cycle for the DS community), and local seminars, colloquia and afternoon tea for the department. For question 3: writing letters, giving talks, writing reviews for MathSciNet, refereeing journals, attending seminars, answering questions — all more pronounced at the more specific DS level than at the general math community level — and normal department requirements. On 4: cooperation, friendliness, professionalism, participation lightly characterize the DS community and much more heavily the department. And yes, for 5, there are rituals and habits. We all applaud twice after a talk, once when the speaker concludes, once after the question time, prompted by the obligatory “if there are no further questions, let’s thank our speaker again.” A (minor) symbol: no one puts their own name on a theorem; you only self-reference with an initial, M. or perhaps M—. (It’s truly remarkable to me how pervasive this bit of symbolic humility is.) Each department surely has its own little habits, and it’s clear that a common language not shared by the outsider characterizes math as a whole, it’s subfields even more strongly. (Technical language, yes, but other usages as well — ‘community in the strong sense’ is itself somewhat a tongue-in-cheek reference to such language.) These aren’t strongly formative things, but they are definitely indicative of close and initiated involvement in the community.

Conclusion: There is probably no such thing as ‘the math community’ in the strong sense as it totally fails on 2 and only answers the rest weakly.  The DS community has a much stronger claim to such a distinction, and a healthy department could very reasonably be community in the strong sense. What abut the faith community? I think there’s even less reason to believe there is a ‘faith community’ in the strong sense than that there is a ‘math community.’ Individual faith traditions, denominations or sub-confessions therein and local congregations could all exhibit community in the strong sense increasingly well. Those of these that I belong to, however, do not always (often?) do so terribly well.

A proposal: the word ‘community’ is very popular in church usage. But in Christianity, where community is not just descriptive of something we like when we have it, but where strong community is constitutive of the faith itself, maybe we should reserve the word only for something more like community in the strong sense, or our attempts to attain it.

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.