Remember the Parasite: Presentation-Invariance and Degrees of Weakened Equivalence

March 7, 2012

Parasites:  Presentation-Invariance and Degrees of Weakened Equivalence

Last night in the seminar on Legal Aspects of International Finance, we had each student read their draft assignments on various aspects of international financial regulations, e.g., the Orderly Liquidation Authority, Whistleblower Incentives and Protection, OTC derivatives under the Dodd -Frank Act, the EC proposal to regulate the OTC derivatives markets, the Volker Rule under Dodd-Frank and the Vickers Report in the UK.

I said to this small but very able group, "I'll be lecturing on Category Theory for the next ten years and the importance of having this view in doing comparative law studies and global financial meditations, and not one student will ever even try to apply these ideas to their work.  C'est dommage, what a pity1"  

I then said that legal and financial studies are in a quandary because the ambition of both are quite clear, and the honest academic knows quite well that the so-called "theories" in each are not doing their job which is to give a sense of "presentation-invariance structure of theories." [See, Marquis, Jean Pierre (2009)  From a Geometric Point of View, at 234.]  What does this mean?  It means that somehow the idea, and here I must insist on the Platonic concept of idea, simply does not change!  It is invariant against all presentations!  But this has no meaning at all unless we relax to the nth degree what we mean by equivalence.  The clarity coming out of Category Theory is the discovery of a way of showing how we can move from a simple and absolutely intuitive sense of "equals to" as in A = A, to more and more weakened senses of "equivalence," i.e., in order of degrees of weakness, (1) isomorphism, (2) natural transformation, (3) natural isomorphism, (4) adjunction.  As we move to weaker and weaker forms of equivalence, we start to realise what we really mean by "identity."

Now, I had prepared mentally for this point in the seminar.  I knew that my students would look at me as if I were an alien who may or may not have an evil intent on their immediate future.  So, I said, "Look, I've been thinking about this for a while, and I think I have an example which you all can understand about how 'weak equivalence' works in the real world.  PARASITES."

At this point, I get a look of embarrassed silence.  Some look at me in politeness, others are giving me that "yuk" look.  I've got their attention, so I continue:

"You all know how parasites work.  You walk barefoot in a beautiful paradise, say Hawaii, and you don't mind walking in puddles.  But as your foot enters the puddle, a little screw-shaped parasite enters your heel.  It burrows deep, gets into your circulatory system, going round and round your blood system, until it lodges idyllically in your liver or your lungs, and there it grows to an enormous size and proliferates seeds of itself, which then are re-deposited to the environment through your faeces."

They grimace at faeces.  They don't like me taking them down THAT path, so I take them up the theoretical track.

"So, look at what's going on here.  How is it that such a simple small organism is able to travel and adjust to such exotic and hostile environments?  Is it intelligent?  Does it have some kind of global navigator?  Does it depend on the magnetic or coriolli forces of the sun?  No.  The parasite has a simple linear instruction set which it can simply follow because the environments which it enters are extraordinarily stable over millions of years.  So, imagine you are a parasite and this room is the "heel of a human's foot".  Your instruction set says something like, "wiggle like hell" when you come into this environment, and this environment is absolutely stable for millions of years because for millions of years the environments of heels are always the SAME.  By the way, this is the kind of IDENTITY that we say we must have in Category Theory.  Now what?  As a parasite, you burrow to the edge of this environment and you break through to the blood circulatory system.  Now this environment has also been the same for millions of years, and your instruction set says, "turn off all flagella and float" and wow, you float and float for a couple circulations -- this is like going round the world in a lifeboat for humans -- until you get to a particular chemical environment that "smells like" a liver or a lung, and then and there, "throw anchor."  All of these seemingly complex behaviours are reduced to a instruction set where each element corresponds to a particular stable environment.  We have an identity of the parasite moving through identical environments for millions of years.  This is an example of how IDENTITY works in a WEAKENED EQUIVALENCE.  In other words,  the parasite through its life cycle shows us an INVARIANCE of PRESENTATION, it will follow a particular path for itself but it is a path that is merely a presentation of a much more general set of equivalent relations.  Remember the parasite behaviours from one level and they look "random"  but at another level, they are all connected through equivalences, and taken all together those equivalences are invariant."

I got a bunch of "bug eyes" so, I ended this part of the lecture with, "OK, remember the Parasite."