Posts tagged with theory

From commensurability to commensuration is a long trek, and we should feel self-congratulatory at this juncture.

Historic events have turned [our] guild from theory toward—if not practice, then at least talk of practice.




Nature is the best teacher. Working on real problems makes you smart. … it is not by staring at a hammer that we learn about hammers.

Similarly, scientists who do nothing but abstract work in the context of funding applications are missing out. The best scientists work in the laboratory, in the field; they tinker.

By removing ourselves from the world, we risk becoming alienated. We become strangers to the world around us. Instead, we construct this incoherent virtual reality

Daniel Lemire (@lemire)

(Source: lemire.me)




But there were also more profound features, which took me a long time even to notice, because they are so at odds with modern experience that neither New Guineans nor I could even articulate them. Each of us took some aspects of our lifestyle for granted and couldn’t conceive of an alternative.

Those other New Guinea features included the non-existence of “friendship” (associating with someone just because you like them), a much greater awareness of rare hazards, war as an omnipresent reality, morality in a world without judicial recourse, and a vital role of very old people. …

Many of my experiences in New Guinea have been intense—a sudden encounter at night with a wild man, the prolonged agony of a nearly-fatal boat accident, one broken little stick in the forest warning us that nomads might be about to catch us as trespassers …

Jared Diamond, The World Before Yesterday

via University of David




Some cute applications of computability theory:

  1. We now know that, if an alien with enormous computational powers came to Earth, it could prove to us whether White or Black has the winning strategy in chess. To be convinced of the proof, we would not have to trust the alien or its exotic technology, and we would not have to spend billions of years analyzing one move sequence after another. We’d simply have to engage in a short conversation with the alien about the sums of certain polynomials over finite fields.
  2. There’s a finite (and not unimaginably-large) set of boxes, such that if we knew how to pack those boxes into the trunk of your car, then we’d also know a proof of the Riemann Hypothesis. Indeed, every formal proof of the Riemann Hypothesis with at most (say) a million symbols corresponds to some way of packing the boxes into your trunk, and vice versa. Furthermore, a list of the boxes and their dimensions can be feasibly written down.
  3. Supposing you do prove the Riemann Hypothesis, it’s possible to convince someone of that fact, without revealing anything other than the fact that you proved it. It’s also possible to write the proof down in such a way that someone else could verify it, with very high confidence, having only seen 10 or 20 bits of the proof.

by Scott Aaronson, via studeo




The credible hulk
via zogotunga




It’s wrong to say that faith and science are opposites,

  • 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.)

  1. 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.
    nerve of cover
  2. 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)).
    presheaf axioms
    That said I think there’s something to be found here in terms of new viewpoints on economic questions.
  3. 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).
    hypothetical transactions from a leontief input/output matrix
    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.
    social graph
  4. 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.
    indifference curves
    (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.
  5. 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.
    input output matrix
  6. 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.
  7. 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 19:29 and 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.
  8. 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.
    algebraic topology
    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.)
  9. 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.
  10. 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 XBRL format because public companies sometimes do business with each other.
  11. 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.
    a drawing of some foreign exchange flows I did whilst reading about cohomology ... this is the beginning of some ideas which are both graphical and hard to draw (because they're so convoluted).).
    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.
    cells with directed boundary
    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.
    fox calculus
    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.
    string diagrams for the Frobenius algebra axioms

    Or from www.math.upenn.edu/~ghrist/preprints/ATSN.pdf we learn "The Euler characteristic χ of any compact triangulable space is independent of the particular [finite] simplicial structure imposed, as well as independent of the topological type.”
    ok, not a triangulated space. but this gives you the idea.
    Yum. Tell me more.
    another drawing from the first time I was reading about cohomology and thinking about its possible applications to a dynamical graph of actual transactions. Looking back on it this isn't super clear either. Maybe you get what I was "suggesting" though.
    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.
  12. Fifth, this is not really related. I think the concept of symplecticity from physics nicely captures the essence of what tradeoffs are about.
    a symplectic manifold. drawn by r ghrist in the /preprints/ATSN.pdf
    But I’m still looking into this—I won’t definitely say that, it just seems like another fruit-lined avenue. 
  13. There are tie-ins to categories, causal diagrams, and other stuff wherein I may be just lumping together a lot of seemingly-related ideas.
    A causal diagram. (not to be confused with an acausal diagram!)
    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.
  14. 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.a reflexive node (CYCLIC graph, not a dag) 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.


cylical graph / circular logic

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.




1. Use mathematics as a shorthand language rather than as an engine of inquiry
2. Keep to them [your models/problems] till you have them done
3. Translate to english
4. Illustrate with examples important to real life
5. Burn the mathematics
6. If you can’t succeed in 4, burn 3




It’s not universally agreed that mathematics is the worst subject everyone has to study in school, but I would say the agreement is close to universal. Why is it so boring?

Aesthetically, I prefer non-miraculous explanations that don’t invoke unique, incomparable properties of the thing considered. So for example I wouldn’t like the explanation that $AAPL is a $10^8 company because of “the magic of Apple”. I would prefer an explanation that involves definite choices they made that others didn’t—like that the Walkman was quite old when the iPod came out and they correctly assessed what the average consumer wanted and spent the right amount on an ad budget, and so on.

Even if it’s about company culture, there are probably some mundane, tangible, doable actions or corporate structures that cause culture—Greg Wilson pointed out, for example, that code is less tangled when multiple programmers are less separated in the organisational chart.

 

So here’s a theory of that aesthetic kind, about why mathematics is different from other subjects and ends up being taught worse.

  • The model of one teacher with a chalk and a blackboard, is more insufficient to explain mathematics than it’s insufficient in other subjects.
Here are some differences between mathematics and other subjects—not incomparable differences like “Well mathematics has group theory" or "Mathematics is for logical minds"—but comparable differences.
  1. One doesn’t become conversant in mathematics—like knowing the basic grammar and syntax—until after 3 years of upper-level courses. Typically linear algebra, analysis, modern algebra, measure theory, and a couple applied topics are required before one could be said to “speak the language”—not to be like Salman Rushdie with English, but to be like an 8th grader with English. So whereas an English teacher who went to university was working on honing skills and developing to a level of excellence, a maths teacher who went to university was becoming functionally literate.
  2. Visuals are necessary to teach mathematics. An ideal lecture in geometry would have heaps of images, videos, and interactive virtual worlds. Virtual worlds and videos take a long, long, long time to create compared to for example a lecture in history. History can be told in a story, whereas talking about for example hyperbolic geometry is not really showing hyperbolic geometry. Sure, a history lecture is nicer with some photos of faces or paintings of historical scenes—but I can get the point just by listening to the story. See my notes on a lecture by Bill Thurston to see how ineffective words are at describing the geometries he’s talking about.

So it takes longer to program a virtual world or a video than it does to write a story, and it takes longer to become functionally literate in mathematics than it does to become functionally literate in history.

Suddenly we’re not telling a story about mathematics being a special subject area with unique problems that can never be overcome. We’re not talking about heroes or villains or “Mathematics just is boring”. (Which is a ridiculous thing to say. That’s saying the way things currently are, is the only way things could ever be. “Mathematics by some intrinsic, unique, incomparable property is more boring than, say, history.” In reality, either can be taught in a boring way and yet both topics have interested people for thousands of years.) Now we’re telling a story about a subject in which it takes a lot of resources to produce a talk, compared to a subject in which it takes fewer resources to produce a talk.




Interesting how Austrian economists see themselves:

  • Ludwig von Mises was a genius (obviously)
  • a Man of the Mind — and that’s a good thing
  • "dignified ruthlessness"
  • seriousness … also a good thing … Serious about Reality
  • quoting Ayn Rand is OK (erm, quoting her straightforwardly, not mockingly)
  • "We’re not dogmatic … we’re consistent
  • People who call us dogmatic believe in relative truth, not absolute truth

^ The last one is the only really confusing one to me.




[T]he Efficient Markets Hypothesis [is] in contention for one of the strongest hypotheses in the whole of the social sciences.

Strictly speaking the EMH is false, but in spirit it is profoundly true. Besides, science concerns seeking the best hypothesis, and until a flawed hypothesis is replaced by a better hypothesis, criticism is of limited value.

Martin Sewell

Um … the criticism might not be useful to science, but knowing that the EMH is false is certainly useful to market participants.

The Kelly Rule for optimal bet sizing says to leverage your bet in geometric proportion to your confidence. So a small reduction in confidence means a big reduction in (optimal) leverage.

I could also wheedle about how obvious modifications of the EMH are more correct than strong EMH, but I won’t. My main thought on this quote is that Sewell has effectively redefined “truth” so that his viewpoint is unfalsifiable. (What example could I cite to demonstrate that EMH is not “profoundly true in spirit”?) On the other hand, it would be naive of me to think that esthetics and organa are not driving the engine of any kind of theory—be it economics, physics, or Foucault.

Anyway … despite the fervour of the abstract (from which I’m quoting), Martin Sewell’s History of the Efficient Markets Hypothesis is an informative read. 

(Source: e-m-h.org)