Posts tagged with cohomology
M. C. Escher’s painting Ascending and Descending illustrates a non-conservative vector field, impossibly made…. In reality, the height above the ground is a scalar potential field [the scalar (single number attached to a point) being the height above the ground]. If one returns to the same horizontal place, one has gone up exactly as much as one goes down.
So that’s that picture related to
Conservative vector fields obey the product rule:
conservative scalar field is also the output of a derivative operation…just a different dimensionality)
- not only because that’s playing into the presentist viewpoint of American fundamentalists fighting to teach creationism in science class versus
/r/atheism, but because
- scientists don’t choose their research programmes at random. They “have a hunch” — or an aesthetic sense impels them. But staking your career on the belief that a particular line of investigation will be fruitful, both in a scientific sense and in a value-to-humanity sense, requires stronger language than merely “I think so” or “I have a hunch”. I think it’s fair to say that scientists have faith in their research programmes.
I’ll give an example of a research programme that I have faith in. Mostly unjustified faith, but I believe it nonetheless. (I could be wrong, of course — but still I can’t approach the world with no beliefs whatever — although some views of rationality would suggest that this unlivable mental life would be the most honourable way to live.)
- I believe there’s something wrong with economic theory. Call it a dark age on the way to enlightenment, call it an obsession with equilibrium-and-optimisation, call it the undue influence of Milton Friedman essays on the deeper, unspoken beliefs of economists vis-à-vis effect of careful studies or creative mathematics. ∃ many ways to describe the malaise | muddle | distraction | not even really sure what to call it.
This is not based on "Economists didn’t foresee the financial crisis!" or a critique of the Washington Consensus. It’s not about Objectivists or people who don’t understand what a model is, but rather at real, non-crazy economists. It’s more based on statements like "Economics is in a terrible state"—Ariel Rubinstein. Or questions like: since information and search costs and other such things dominate the f**k out of the normal incentives-based thinking we use to armchair-speculate—then what is even the use of the partial-equilibrium intuitions or DSGE or anything like that?
I also don’t think this idea would necessarily change the focus to more sociological or historical or cultural issues (like economists ignoring how utility functions come to be, or larger questions about history and culture and family norms … I actually think a lot of economists are already prepared to focus on those issues, they just need to make them mathematically tractable). Rather my gut instinct tells me that this research programme is “far upstream”—redirecting the river by diverting the water long before it becomes a rushing channel (sometimes called the MSNBC channel) that’s too powerful to redirect.
- I don’t know enough about sheaf theory or cohomology to say for certain whether they can be used for this or that. It’s just my spider sense tingling when I look at the ideas there. Most of the applications I’ve read about are to either physics problems or logic, or to higher mathematics itself (algebraic topology, algebraic geometry, topological analysis, … stuff that’s named as (adjective = way of thinking + noun = subject matter)).
That said I think there’s something to be found here in terms of new viewpoints on economic questions.
- Consider the Leontief input-output matrix (Cosma Shalizi recently wrote a lot about it in his book review of Red Plenty on Crooked Timber blog).
Mathematically savvy people know that every graph can be encoded as a matrix, and furthermore with the right base corpus and some knowledge of “characters” we can do one-directional graphs.
- What’s the [putative] application to economics? Well instead of thinking about all this stuff we can’t observe or interpret yet—utility curves, willingness-to-pay outside the lab, valuations, etc.
(we don’t even know experimentally if there is such a thing as a valuation—and it’s kind of dubious—yet we go on as if these things exist because they’re axiomatic keystones of the only tractable theory around). Instead of continuing to rely on the theoretical stuff handed down from Bentham, let’s think about all the things we can measure—like transactions—and ask how we can use mathematics to make theories about those things and possibly infer back to the stuff we really want to know, like is capitalism making the world a better place.
- Transactions are one place to start. Prices (like the billion price project) are another. And the web now generates huge amounts of text—maybe we can do something with that. But let’s start by going back to the Leontief matrix.
- In the formulation I learned in school, there’s a fixed time unit—like a year—and each dimension corresponds to an exactly comparable item class—so like a three button shirt and a four button shirt would be separate dimensions, but once we finally get down to a dimension, everything along that dimension is equivalence-classed.
- I can see three things missing from that picture.
First of all, I want to be able to “zoom in” to different timescales and have my matrix change in the sensible way. In other words I want a mathematical object that operates on multiple timescales at once, with a coherent, consistent translation between the Leontief matrix of October 17th between
21:13 GMT, and the Leontief matrix of
1877 A.D.I believe things floating around sheaf theory are the place to look for that.
- Second of all, I want neighbourhood relationships (and even distances) between the items—so that a three-button blue blouse is “closer to” a four-button blue blouse than it is to a ferret named Bosco the Great sold at the Petco in Moravia, Illinois. So something from algebraic topology is necessary here.
Maybe a tie-in to “lumpy human capital”—the most important kind of good because it’s what humans use to sustain themselves and help others. It’s acknowledge to be “lumpy” in that ten years of studying economic theory doesn’t prepare you to be a laundress or even necessarily to trade OTC derivatives. But we also know that in terms of neighbourhood relationships, economic theory studying is “closer to” finance than to farming. (Although most economists are not as close to finance as seems to be generally thought.)
- Both of those two points are more just æsthetic problems or issues with foundations. Like philosophical gripes could be solved, in the same way that a transition from cardinal utility to ordinal utility, even though I don’t think the outcomes of the ordinal utility theory were very different.
- Third, I want my matrix to be time-varying or dynamical. New trade partners come into existence, some businesses shutter their doors and file their dissolution papers, others are broken up and sold in parts, and even with an existing vendor I am not going to do the same business each year. Some of these numbers are available in
XBRLformat because public companies sometimes do business with each other.
- Fourth, and here is where I think it would be possible to get new ideas of things to measure. If I have some kind of dynamical, multi-level, “coloured” graph of all the trades in all the currencies and all the goods types in the world over the right number corpus, then I have a different mathematical conception of the world economy.
I can draw boundaries like you would see in a cell complex and denote “a community” or “a municipality” or “a neighbourhood” or “a province” and when I perturb those boundaries some rationality conditions need to hold.
Taking this viewpoint and applying only the maths that’s already been invented, people have already found a lot of invariants on graphs—cohomology invariants, generalisations of Gauss’ divergence theorem, different calcula on the interesting objects (like fox calculus)—and applying those theories to the conception of the super-duper Leontief matrix, we might find new things to measure, or new ways to make different sense of some measurements we already have.
If you remember this Perelman quote about calculating how fast Christ would have to run on the water to not sink in, or various nifty cancellations in the vacuum states of a gnarly physics theory — that is the kind of thing I’m thinking could be useful in theorising new invariants to measure from an überdy-googly Leontief trade matrix.
Or from www.math.upenn.edu/~ghrist/preprints/ATSN.pdf we learn "The Euler characteristic
χof any compact triangulable space is independent of the particular [finite] simplicial structure imposed, as well as independent of the topological type.”
Yum. Tell me more.
For example we know some Gaussian-divergence ∑ relations that happen within the grey box of a firm—all the internal transactions have to add up to what’s written on the accounting statements. But what about applying this logic to a group of three firms that circularly trade with each other and also each has a composite edge (with different weights) adding up all of their trades to “outside the cycle”?
Seems like some funky abstract nonsense could simplify problems like that and, crucially, tell us invariants that give us new ideas of what to measure.
- Fifth, this is not really related. I think the concept of symplecticity from physics nicely captures the essence of what tradeoffs are about.
But I’m still looking into this—I won’t definitely say that, it just seems like another fruit-lined avenue.
- There are tie-ins to categories, causal diagrams, and other stuff wherein I may be just lumping together a lot of seemingly-related ideas.
So I’m not sure if looking at a super-duper Leontief matrix like described above would have nice tie-ins to causal graphs / structural equations à la Judea Pearl, but hey it might. At least one tie-in I can already think of is that all the goods actually transacted doesn’t tell you enough because there are threats and possible counterfactuals and CV’s that are sent in but get ignored or rejected, or smiles and pats on the back which are a kind of transaction that influences the economic outcomes without being tied directly to money or a goods transaction.
- Why go for even more abstraction, even “more” maths, when so many of the critiques of economics say it’s become too mathematical? Simple answer. More abstract mathematics requires fewer assumptions. So conclusions drawn using those tools are more likely to actually hold true in the real world. For example, is it more plausible that someone’s utility increases linearly, or monotonically with good X? Monotonically of course is much more realistic, although we could infer much more if linear were the case. But what’s the point of making easier inferences if they’re wrong because the assumptions don’t hold? Hence the interest in more general, more abstract mathematics.
Now, realistically? The scale of investigating this “hunch” in terms of concrete steps that lead A → B → C → D are way beyond what I will probably accomplish. Even if I dropped all side interests and all work, it would take at least a couple years to get publishable material out of these hunches.
But that’s exactly my point about science. I was told by a Zaazen practitioner that this is kind of a Zen-like paradox. In order to investigate the premise that there are useful applications of sheaves & cohomology to economic theory, I first have to accept the premise that there are probably useful applications of sheaves & cohomology to economic theory.
Glancing at the text above you can probably tell that my thoughts on this issue are formless, probably mischaracterising the mathematics I’ve only heard about but don’t yet understand. My mental conception of these things, if it could be understood via a perfect future theory of mental representation and fMRI snapshots of my mind thinking about this stuff, would be some mixture of formless and inaccurate.
So the important decisions (decisions of major direction, not adjustments or effort) are made amongst the formless, but can only be harvested as a form. Like the beginner’s mind, with its vagueness and formlessness, giving way to the expert’s mind, its definition, choateness, and exactitude. (Form and formlessness being complementary in the QM | vNA sense.) I think that’s Zen as well.