Jerry, there is so much to say about the evolution of rationales of exploitation that just talking about what Marx has to say on the subject would make this series of posts endless [which it well may be, anyway!] Tomorrow I will say a bit about some of the really dopey explanations given by Marx's contemporaries for the existence of profit. Speaking generally, Marx thinks [and I agree] that the principal task of philosophers, theologians, economists, and, I would add, political scientists in any era is to conjure up rationalisations for the manifest inequality in the distribution of the social product in that era. To their great credit, Smith and Ricardo do not offer facile justifications, and that is one of the reasons why Marx thought so highly of them. The rest he referred to as "vulgar economists."

Seth, I freely confess that I had never so much as encountered the word "holonomy" before reading your comment. Can you say just a word about the sense in which profit is, as you call it, a "holonomy effect"? Is it because it involves mapping a multi-dimensional magnitude onto a one-dimensional array? As for the time I spend explaining things that are immediately obvious to some people, this is the price I pay for trying to weave together into a single narrative arguments taken from mathematical economics, history, sociology, philosophy, and literary criticism. My deepest passion, during my entire life, has been to make difficult ideas clear and simple to anyone willing to follow the line of my exposition, with no assumptions made about their prior expertise.

## Saturday, February 9, 2013

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## 7 comments:

I'll need to spend a little time putting together a concise explanation that can put the idea across. You should take a few minutes to examine the pictorial example on the Wikipedia page I cited as a start. That will help to fix the word "holonomy" with a specific meaning.

There isn't a direct connection between holonomy and the indexing topic. I jumped from holonomy to indexing too hastily. The only connection is the common pattern of "talking about mathematical analogies" -- both holonomy and indexing are mathematical ideas with application to political economy. Analogies that require explanation.

I just found this paper in which a distinguished physicist makes the connection btw econ and gauge theory more carefully (haven't read it yet):

http://arxiv.org/PS_cache/arxiv/pdf/0902/0902.4274v1.pdf

A further down payment on holonomy:

In flat space we take it to be obvious how to compare vectors because we intuitively define to vectors as the same if they "point in the same direction". Two vectors which point North are going the same way. But on a sphere, this approach breaks down. At the North Pole ALL tangent vectors point due south! To enable us to compare vectors at different points (so we can do calculus, differentiate, solve diff'l equations, etc.) we have to be more precise about how to subtract vectors at nearby points. We define "parallel transport" as a process of carrying a vector "in a fixed direction" as we move along a geodesic (great circle path) from one point to the other. The picture in the Wikipedia page about holonomy illustrates this.

In flat space this process is always boring, You go around any sort of path carrying vectors "in parallel" and you wind up back home with the same vector. But in curved space (as in gauge theories associated with particle physics) something fun can happen. You carefully walk around a closed loop, diligently carrying your vector in "the same direction" and when you get home, the vector is pointing in a DIFFERENT direction. The difference can "measure" something about the path you took -- like the area inside it, or the field flux across it, etc.

Another deep example would be the Aharonov-Bohm effect, which you can google.

For financial markets, a handful of creative physicists are exploring a rigorous application of gauge theory to price movements (Kirill Illinksi is one who has written a challenging book about it). For economics more generally, there is a further trouble that the geometry involved probably needs to be done on a vast graph (like the Facebook social graph) -- mathematical techniques for the right way to do gauge theory in that context are still lacking, though I've seen smart people working in that direction.

I just found this interesting paper by another physicist making the analogy more carefully (haven't read it yet):

http://arxiv.org/PS_cache/arxiv/pdf/0902/0902.4274v1.pdf

To close back with the original puzzle: where does the profit come from? Entrepreneurial activity takes a vector of items with specific 'money values' and takes them through a 'path' of exchange activities, ultimately flowing from capital all the way back to (more) capital. I don't recall the terms Marx used for this (C-M-C cycle?) but the point is that it is hard to follow just how the capitalist can wind up with MORE at the end than at the beginning. Holonomy is a mathematical analogy for this mysterious 'accounting anomaly'.

If you have questions, they may help me to understand what part of this account needs the most work ;)

I should stop trying to compose comments directly on your blog (or not use an ipad). Some kind of accidental copy/paste mangled my already over-hasty composition.

There is a nice animation showing holonomy on a sphere at Wolfram's MathWorld site:

http://mathworld.wolfram.com/HolonomyGroup.html

You have to watch it go through several cycles (it runs a little too fast) to start spotting how the pair of vectors gets rotated a bit on each cycle through the triangular track.

To relate this to indexing a bit: the rotation of the pair of vectors could be taken to represent a change of the 'price vector' or 'index weights' used to compute the value of the set of commodities the trader is 'moving' or the weights used to compute the index.

Reading the paper I cited above slightly more, I noticed that P N Malaney's 1996 PhD Thesis makes the connection between the Index Number Problem and the holonomy concept very directly:

http://lists.perimeterinstitute.ca/pipermail/piuwcomplex/attachments/20090205/7f2cccd3/attachment-0002.pdf

Sadly, Malaney never actually uses the word holonomy -- but the machinery is all there.

Ah, and here is where I first discovered Malaney and Eric Weinstein: A Perimeter Institute conference. Videos here:

http://pirsa.org/C09006/2

There are two pages of talks, but Malaney's might be most immediately relevant to your current line of thought and her's appears on the second page.

Your ambition "to make difficult ideas clear and simple" is one I admire. But it takes more effort -- and EDITING -- than I can spare at the moment :(

All I have time for is to pass along a rich source of thought provoking work at the intersection of economics and physics.

I'm pleased to know that both you and Marx think that the principal task of philosophers, et al,has been to conjure up rationalizations for the manifest inequality in the distribution of the social product in any given era - because this is consistent with my belief that career opportunities made available to painters, from the Renaissance to the present (and possibly forever, with few exceptions) have turned precisely on the promulgation of such rationales.

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