Given our inevitably incomplete knowledge about key structural aspects of an ever-changing economy and the sometimes asymmetric costs or benefits of particular outcomes, a central bank needs to consider not only the most likely future path for the economy but also the distribution of possible outcomes about that path. The decisionmakers then need to reach a judgment about the probabilities, costs, and benefits of the various possible outcomes under alternative choices for policy.”

The product of a low-probability event and a potentially severe outcome was judged a more serious threat to economic performance than the higher inflation that might ensue in the more probable scenario.”

Mark Colyvan, an Australian philosopher of mathematics, published a paper in 2004 which examined the philosophical and logical assumptions of Cox’s theorem (assumptions usually left implicit by its proponents), and argued that these are inappropriate for many (perhaps even most) domains with uncertainty.

M. Colyvan [2004]: The philosophical significance of Cox’s theorem. International Journal of Approximate Reasoning, 37: 71-85.

Colyvan’s work complements Glenn Shafer’s attack on the theorem, which noted that it assumes that belief should be represented by a real-valued function.

G. A. Shafer [2004]: Comments on “Constructing a logic of plausible inference: a guide to Cox’s theorem” by Kevin S. Van Horn. International Journal of Approximate Reasoning, 35: 97-105.

Although these papers are several years old, I mention them here for the record -  and because I still encounter invocations of Cox’s Theorem.

IME, most statisticians, like most economists, have little historical sense. This absence means they will not appreciate a nice irony: the person responsible for axiomatizing classical probability theory – Andrei Kolmogorov – is also one of the people responsible for axiomatizing intuitionistic logic, a version of classical logic which dispenses with the law of the excluded middle. One such axiomatization is called BHK Logic (for Brouwer, Heyting and Kolmogorov) in recognition.

Liking cladistics as I do, I thought it useful to list all the potential purposes of models and modeling.   The only discussion that considers this topic that I know is a brief discussion by game theorist Ariel Rubinstein in an appendix to a book on modeling rational behaviour (Rubinstein 1998).  Rubinstein considers several alternative purposes for economic modeling, but ignores many others.   My list is as follows (to be expanded and annotated in due course):

• 1. To better understand some real phenomena or existing system.   This is perhaps the most commonly perceived purpose of modeling, in the sciences and the social sciences.
• 2. To predict (some properties of) some real phenomena or existing system.  A model aiming to predict some domain may be successful without aiding our understanding  of the domain at all.  Isaac Newton’s model of the motion of planets, for example, was predictive but not explanatory.   I understand that physicist David Deutsch argues that predictive ability is not an end of scientific modeling but a means, since it is how we assess and compare alternative models of the same phenomena.    This is wrong on both counts:  prediction IS an end of much modeling activity (especially in business strategy and public policy domains), and it not the only means we use to assess models.  Indeed, for many modeling activities, calibration and prediction are problematic, and so predictive capability may not even be  possible as a form of model assessment.
• 3. To manage or control (some properties of) some real phenomena or existing system.
• 4. To better understand a model of some real phenomena or existing system.  Arguably, most of economic theorizing and modeling falls into this category, and Rubinstein’s preferred purpose is this type.   Macro-economic models, if they are calibrated at all, are calibrated against artificial, human-defined, variables such as employment, GDP and inflation, variables which may themselves bear a tenuous and dynamic relationship to any underlying economic reality.   Micro-economic models, if they are calibrated at all, are often calibrated with stylized facts, abstractions and simplifications of reality which economists have come to regard as representative of the domain in question.    In other words, economic models are not not usually calibrated against reality directly, but against other models of reality.  Similarly, large parts of contemporary mathematical physics (such as string theory and brane theory) have no access to any physical phenomena other than via the mathematical model itself:  our only means of apprehension of vibrating strings in inaccessible dimensions beyond the four we live in, for instance, is through the mathematics of string theory.    In this light, it seems nonsense to talk about the effectiveness, reasonable or otherwise, of mathematics in modeling reality, since how we could tell?
• 5. To predict (some properties of) a model of some real phenomena or existing system.
• 6. To better understand, predict or manage some intended (not-yet-existing) artificial system, so to guide its design and development.   Understanding a system that does  not yet exist is qualitatively different to understanding an existing domain or system, because the possibility of calibration is often absent and because the model may act to define the limits and possibilities of subsequent design actions on the artificial system.  The use of speech act theory (a model of natural human language) for the design of artificial machine-to-machine languages, or the use of economic game theory (a mathematical model of a stylized conceptual model of particular micro-economic realities) for the design of online auction sites are examples here.   The modeling activity can even be performative, helping to create the reality it may purport to describe, as in the case of the Black-Scholes model of options pricing.
• 7. To provide a locus for discussion between relevant stakeholders in some business or public policy domain.  Most large-scale business planning models have this purpose within companies, particularly when multiple partners are involved.  Likewise, models of major public policy issues, such as epidemics, have this function.  In many complex domains, such as those in public health, models provide a means to tame and domesticate the complexity of the domain.  This helps stakeholders to jointly consider concepts, data, dynamics, policy options, and assessment of potential consequences of policy options,  all of which may need to be socially constructed. 
• 8. To provide a means for identification, articulation and potentially resolution of trade-offs and their consequences in some business or public policy domain.   This is the case, for example, with models of public health risk assessment of chemicals or new products by environmental protection agencies, and models of epidemics deployed by government health authorities.
• 9. To enable rigorous and justified thinking about the assumptions and their relationships to one another in modeling some domain.   Business planning models usually serve this purpose.   They may be used to inform actions, both to eliminate or mitigate negative consequences and to enhance positive consequences, as in retroflexive decision making.
• 10. To enable a means of assessment of managerial competencies of the people undertaking the modeling activity. Investors in start-ups know that the business plans of the company founders are likely to be out of date very quickly.  The function of such business plans is not to model reality accurately, but to force rigorous thinking about the domain, and to provide a means by which potential investors can challenge the assumptions and thinking of management as way of probing the managerial competence of those managers.    Business planning can thus be seen to be a form of epideictic argument, where arguments are assessed on their form rather than their content, as I have argued here.
• 11. As a means of play, to enable the exercise of human intelligence, ingenuity and creativity, in developing and exploring the properties of models themselves.  This purpose is true of that human activity known as doing pure mathematics, and perhaps of most of that academic activity known as doing mathematical economics.   As I have argued before, mathematical economics is closer to theology than to the modeling undertaken in the natural sciences. I see nothing wrong with this being a purpose of modeling, although it would be nice if academic economists were honest enough to admit that their use of public funds was primarily in pursuit of private pleasures, and any wider social benefits from their modeling activities were incidental.
  

POSTSCRIPT (Added 2011-06-17):  I have just seen Joshua Epstein’s 2008 discussion of the purposes of modeling in science and social science.   Epstein lists 17 reasons to build explicit models (in his words, although I have added the label “0″ to his first reason):

0. Prediction
1. Explain (very different from predict)
2. Guide data collection
3. Illuminate core dynamics
4. Suggest dynamical analogies
5. Discover new questions
6. Promote a scientific habit of mind
7. Bound (bracket) outcomes to plausible ranges
8. Illuminate core uncertainties
9. Offer crisis options in near-real time. [Presumably, Epstein means "crisis-response options" here.]
10. Demonstrate tradeoffe/ suggest efficiencies
11. Challenge the robustness of prevailing theory through peturbations
12. Expose prevailing wisdom as imcompatible with available data
13. Train practitioners
14. Discipline the policy dialog
15. Educate the general public
16. Reveal the apparently simple (complex) to be complex (simple).

These are at a lower level than my list, and I believe some of his items are the consequences of purposes rather than purposes themselves, at least for honest modelers (eg, #11, #12, #16).

References:

Joshua M Epstein [2008]: Why model? Keynote address to the Second World Congress on Social Simulation, George Mason University, USA.  Available here (PDF).

Robert E Marks [2007]:  Validating simulation models: a general framework and four applied examples. Computational Economics, 30 (3): 265-290.

David F Midgley, Robert E Marks and D Kunchamwar [2007]:  The building and assurance of agent-based models: an example and challenge to the field. Journal of Business Research, 60 (8): 884-893.

Robert Rosen [1985]: Anticipatory Systems. Pergamon Press.

Ariel Rubinstein [1998]: Modeling Bounded Rationality. Cambridge, MA, USA: MIT Press.  Zeuthen Lecture Book Series.

Ariel Rubinstein [2006]: Dilemmas of an economic theorist. Econometrica, 74 (4): 865-883.

Music critic Alex Ross discusses John Cage’s music in a recent article in The New Yorker.    Ross goes some way before he trips up, using those dreaded  - and completely inappropriate – words “randomness” and “chance”:

Later in the forties, he [Cage] laid out “gamuts” – gridlike arrays of preset sounds – trying to go from one to the next without consciously shaping the outcome.  He read widely in South Asian and East Asian thought, his readings guided by the young Indian musician Gita Sarabhai and, later, by the Zen scholar Daisetz Suzuki.  Sarabhai supplied him with a pivotal formulation of music’s purpose:  “to sober and quiet the mind, thus rendering it susceptible to divine influences.”  Cage also looked to Meister Eckhart and Thomas Aquinas, finding another motto in Aquinas’s declaration that “art imitates nature in the manner of its operation.”

 . . .

In 1951, writing the closing movement of his Concerto for Prepared Piano, he finally let nature run its course, flipping coins and consulting the I Ching to determine which elements of his charts should come next.   “Music of Changes,” a forty-three-minute piece of solo piano, was written entirely in this manner, the labor-intensive process consuming most of a year.

As randomness took over, so did noise.  “Imaginary Landscape No. 4″ employs twelve radios, whose tuning, [page-break] volume, and tone are governed by chance operations.”  [pages 57-58]

That even such a sympathetic, literate, and erudite observer as Alex Ross should misconstrue what Cage was doing with the I Ching as based on chance events is disappointing.  But, as I’ve argued before about Cage’s music, the belief that the material world is all there is is so deeply entrenched in contemporary western culture that westerners seem rarely able to conceive of other ways of being.  Tossing coins may seem to be a chance operation to someone unversed in eastern philosophy, but was surely not to John Cage.   

References:

Alex Ross [2010]:  Searching for silence.  John Cage’s art of noise.   The New Yorker, 4 October 2010, pp. 52-61.

James Pritchett [1993]:  The Music of John Cage.  Cambridge, UK:  Cambridge University Press.

Here are other posts on music and art.

Technorati Tags: Alex Ross, I Ching, John Cage

In business strategy, good decision-making requires consideration of the consequences of potential actions, which in turn requires the consideration of the potential actions of other actors and stakeholders in response to the first set of actions.  These actors may include entities whose existence is not yet known or even suspected, for example, future competitors to a product whose launch creates a new product category.   Why, there’s even a whole branch of strategy analysis, devoted to scenario planning, a discipline that began in the military analysis of alternative post-nuclear worlds, and whose very essence involves the creation of imagined futures (for forecasting and prognosis) and/or imagined pasts (for diagnosis and analysis).   Every good air-crash investigation, medical diagnosis, and police homicide investigation, for instance, involves the creation of imagined alternative pasts, and often the creation of imaginary entities in those imagined pasts, whose fictional attributes we may explore at length.   Arguably, in one widespread view of the philosophy of mathematics, pure mathematicians do nothing but explore the attributes of entities without material existence.

And not just in business, medicine, the military, and the professions.   In computer software engineering, no new software system development is complete without due and rigorous consideration of the likely actions of users or other actors with and on the system, for example.   Users and actors here include those who are the intended target users of the system, as well as malevolent or whimsical or poorly-behaved or bug-ridden others, both human and virtual, not all of whom may even exist when the system is first developed or put into production.      If creative articulation and manipulation of imaginary futures (possible or impossible) is to be outlawed, not only would we have no literary fiction or much poetry, we’d also have few working software systems either.

Brian Dillon reviews a British touring exhibition of the art of John Cage, currently at the Baltic Mill Gateshead.

Two quibbles:  First, someone who compare’s Cage’s 4′ 33” to a blank gallery wall hasn’t actually listened to the piece.  If Dillon had compared it to a glass window in the gallery wall allowing a view of the outside of the gallery, then he would have made some sense.  But Cage’s composition is not about silence, or even pure sound, for either of which a blank gallery wall might be an appropriate visual representation.  The composition is about ambient sound, and about what sounds count as music in our culture.

Second, Dillon rightly mentions that the procedures used by Cage for musical composition from 1950 onwards (and later for poetry and visual art) were based on the Taoist I Ching.  But he wrongly describes these procedures as being based on “the philosophy of chance.”     Although widespread, this view is nonsense, accurate neither as to what Cage was doing, nor even as to what he may have thought he was doing.   Anyone subscribing to the Taoist philosophy underlying them understands the I Ching procedures as examplifying and manifesting hidden causal mechanisms, not chance.   The point of the underlying philosophy is that the random-looking events that result from the procedures express something unique, time-dependent, and personal to the specific person invoking the I Ching at the particular time they invoke it. So, to a Taoist, the resulting music or art is not “chance” or “random” or “aleatoric” at all, but profoundly deterministic, being the necessary consequential expression of deep, synchronistic, spiritual forces. I don’t know if Cage was himself a Taoist (I’m not sure that anyone does), but to an adherent of Taoist philosophy Cage’s own beliefs or attitudes are irrelevant to the workings of these forces.  I sense that Cage had sufficient understanding of Taoist and Zen ideas (Zen being the Japanese version of Taoism) to recognize this particular feature:  that to an adherent of the philosophy the beliefs of the invoker of the procedures are irrelevant.

In my experience, the idea that the I Ching is a deterministic process is a hard one for many modern westerners to understand, let alone to accept, so entrenched is the prevailing western view that the material realm is all there is.  This entrenched view is only historically recent in the west:  Isaac Newton, for example, was a believer in the existence of cosmic spiritual forces, and thought he had found the laws which governed their operation.    Obversely, many easterners in my experience have difficulty with notions of uncertainty and chance; if all events are subject to hidden causal forces, the concepts of randomness and of alternative possible futures make no sense.  My experience here includes making presentations and leading discussions on scenario analyses with senior managers of Asian multinationals.  

We are two birds swimming, each circling the pond, warily, neither understanding the other, neither flying away.

References:

Kyle Gann [2010]: No Such Thing as Silence.  John Cage’s 4′ 33”.  New Haven, CT, USA:  Yale University Press. 

James Pritchett [1993]:  The Music of John Cage.  Cambridge, UK:  Cambridge University Press.

Technorati Tags: John Cage, I Ching, Zen, Taoism

One wonders, fourthly, where are recorded all the prior distributions of beliefs which this theory requires investigators to articulate before doing research.  Surely someone must be writing them down, so that we consumers of science can know that our researchers are honest, and hold them to potential account.   That there is such a disconnect between what Bayesian theorists say researchers do and what those researchers demonstrably do should trouble anyone contemplating a choice of statistical paradigms, surely.   Finally, one wonders how a theory that requires non-zero probabilities be allocated to models of which the investigators have not yet heard or even which no one has yet articulated, for those models to be tested, passes muster at the statistical methodology corral.

To my mind, Bayesianism is a theory from some other world – infinite gambles, imagined prior distributions, models that disregard time or requirements for constructability,  unrealistic abstractions from actual scientific practice – not from our own.

So, how could the Bayesians make as much headway as they have these last six decades? Perhaps it is due to an inherent pragmatism of statisticians – using whatever techniques work, without much regard as to their underlying philosophy or incoherence therein.  Or perhaps the battle between the two schools of thought has simply been asymmetric:  the Bayesians being more determined to prevail (in my personal experience, to the point of cultism and personal vitriol) than the adherents of frequentism.  Greg Wilson’s 2001 PhD thesis explored this question, although without finding definitive answers.

Now,  Andrew Gelman and the indefatigable Cosma Shalizi have written a superb paper, entitled “Philosophy and the practice of Bayesian statistics”.  Their paper presents another possible reason for the rise of Bayesian methods:  that Bayesianism, when used in actual practice, is most often a form of hypothesis-testing, and thus not as untethered to reality as the pure theory would suggest.  Their abstract:

A substantial school in the philosophy of science identifies Bayesian inference with inductive inference and even rationality as such, and seems to be strengthened by the rise and practical success of Bayesian statistics. We argue that the most successful forms of Bayesian statistics do not actually support that particular philosophy but rather accord much better with sophisticated forms of hypothetico-deductivism. We examine the actual role played by prior distributions in Bayesian models, and the crucial aspects of model checking and model revision, which fall outside the scope of Bayesian confirmation theory. We draw on the literature on the consistency of Bayesian updating and also on our experience of applied work in social science.

Clarity about these matters should benefit not just philosophy of science, but also statistical practice. At best, the inductivist view has encouraged researchers to fit and compare models without checking them; at worst, theorists have actively discouraged practitioners from performing model checking because it does not fit into their framework.

References:

Andrew Gelman and Cosma Rohilla Shalizi [2010]:  Philosophy and the practice of Bayesian statistics.  Available from Arxiv.  Blog post here.

Technorati Tags: hypothesis-testing, statistics, Bayesianism, frequentism, inductive inference, hypothetico-deductivism

If Akerlof and Shiller’s grip on the history of economic thought is shaky, they also fail to grasp why Keynes rejected the idea that markets are self-stabilising. Throughout Animal Spirits they portray him as reintegrating psychology with economic theory. No doubt this was one of Keynes’s goals, but it is not his most fundamental revision of economic orthodoxy. Among his other accomplishments he was the author of A Treatise on Probability (1921), in which he tried to develop a theory of ‘rational degrees of belief’. By his own account he failed, and in his canonical General Theory of Employment, Interest and Money (1936) he concluded that there was no way anyone could make forecasts. Future interest rates and prices, new inventions and the likelihood of a European war cannot be predicted: there is no ‘basis on which to form any calculable probability whatever. We simply do not know!’ For Keynes, markets are unstable less because they are driven by emotion than because the future is unknowable. To suggest that the source of market volatility is unreason is to imply that if people were fully rational markets could be stable. But even if people were affectless calculating machines they would still be ignorant of the future, and markets would still be volatile. The root cause of market instability is the insuperable limitation of human knowledge.

Later economists have made much of a distinction between risk, which can be assessed in terms of quantifiable likelihood, and uncertainty, where probabilities cannot be attached to possible outcomes. The trouble is that when attempting to forecast the course of the economy we often cannot confidently distinguish between the two. Even our list of possible outcomes may turn out to have omitted the ones that are most important in shaping events. Such an omission was one of the factors that led Long-Term Capital Management, a highly leveraged hedge fund set up by two Nobel Prize winning economists, to fail in 1998-2000. The information used in applying the formula did not include the possibility of such events as the Asian financial crisis and Russia’s default on its sovereign debt, which destabilised global financial markets and helped destroy the fund. The orthodoxy that came unstuck with the collapse of LTCM was not faulty because it neglected the vagaries of human moods; its mistake was to think that the unknown future could be turned into a set of calculable risks and, in effect, conjured out of existence, which was impossible. Several centuries earlier, Pascal – one of the founders of probability theory – had come to the same conclusion, when in the Pensées he asks ironically: ‘Is it probable that probability brings certainty?’

The central flaw of the economic orthodoxy against which Keynes fought in the 1930s was to imagine that an insoluble problem – human ignorance of the future – had been solved. The error was repeated in the 1990s, when economists came to believe that complex mathematical formulae could tame uncertainty in the murky world of derivatives. Steeped in history as they were, this was a delusion that none of the classical economists entertained. It began to shape economics only towards the end of the 19th century, with the rise of Positivism, according to which the natural sciences are the only legitimate repository of human knowledge. It was the formative influence of this philosophy on the Chicago School that enabled the orthodoxy of the 1930s to re-emerge triumphant, and the result was an immense boost to the prestige of economics as a discipline. Economists could claim to be scientists, who with the aid of their mathematical magic could pierce the veil that conceals the future.

The hegemony of Positivism in economics obscured Keynes’s scepticism about probabilistic knowledge, his most important contribution to the discipline. G.L.S. Shackle set Keynes’s argument out systematically in his neglected masterpiece Epistemics and Economics: A Critique of Economic Doctrines (1972). Shackle is probably the only significant economist to have been influenced both by Keynes and by his arch-rival, F.A. Hayek. He knew both of them well, but argued that neither had digested the full implications for economics of our ignorance of the future. Hayek said that governments could never know enough to plan the economy successfully – a claim vindicated by the miserable record of central planning in Communist countries. At the same time, he attributed near omniscience to markets, and never doubted that if left to its own devices the economy would liquidate mistaken investments and return to equilibrium. Against this, Keynes had shown that there is no market mechanism that ensures revival; economic contraction can be self-reinforcing, and only government action can then create a way out.

Shackle took Keynes’s argument a step further, and showed that no economic policy can ensure economic stability indefinitely. ‘Keynesian’ policies are no exception to this rule. Deficit financing and monetary expansion may have worked well in the conditions that existed after the Second World War. It is not clear that they will be so effective today, when globalisation has brought a freedom of capital movements that did not exist then. The lesson of Shackle is that we must be resourceful in devising new remedies, while not losing sight of the fact that none of them works for long.

Akerlof and Shiller claim that their account of the role of psychology helps to explain the financial crisis. ‘Our theory of animal spirits,’ they say, ‘provides an answer to a conundrum: why did most of us utterly fail to foresee the current economic crisis? How can we understand this crisis when it seems to have come out of the blue with no cause?’ They are right that part of the answer lies in an intellectual default within economics, but they seem oblivious of the role of ideology in producing this default. The deformation of economics was not the result only of factors internal to the discipline, it was also part of the short-lived Western triumphalism that followed the end of the Cold War.

. . .

Keynes and the classical economists before him knew that there is no realm of market exchange that obeys laws of the kind that can be formulated in the natural sciences. Economics and politics are not separate branches of human activity, and economic life cannot be studied independently of social divisions and political conflicts among populations, along with their cultures and religions. Familiar to Keynes and most of the economists of his generation, these truisms have been forgotten, or rejected, by many economists today. The result is an economic imperialism that tries to explain every human activity in terms of a conception of rational action that does not work even when applied to the behaviour of markets.

Of course, there is a standard response to these observations, which is that unrealism in economic theories doesn’t matter. As developed by Milton Friedman, among others, this is in effect a version of instrumentalism, a tenable position in the philosophy of science. For instrumentalists, the goal of science is not a true representation of the world; it is to organise our observations into a theoretical framework that serves practical goals, such as prediction and control. But what practical goals have been served by the type of economics dominant over the past two decades? It has been useful neither in making predictions nor in responding to unforeseen developments.

Akerlof and Shiller intend their analysis to contribute to an intellectual reformation in economics, as a consequence of which the discipline will become more useful to policy-makers. It must be doubted, though, that the authors will succeed in persuading economists of the inadequacy of the conception of rational action. The profession is one of the few areas of human activity in which that conception is applicable. In its intra-academic varieties, at any rate, economics is insulated from the world not only by its narrow explanatory methodology but also because it rewards the mathematical modelling that resulted in nearly all of its members failing to anticipate the financial crisis. As institutionalised in universities, the notion of rational decision-making is self-perpetuating. Economics as currently practised may have only a slight grip on market behaviour, but it seems to be powerfully predictive of the behaviour of economists.”

Reference:

John Gray [2009]: We Simply Do Not Know!  London Review of Books,  31 (22): 13-14, 19 November 2009.  Review of: Animal Spirits: How Human Psychology Drives the Economy, and Why It Matters for Global Capitalism, by George Akerlof and Robert Shiller. (Princeton).

Technorati Tags: George Shackle, John Gray, George Akerlof, Robert Shiller

Here then is a list of mathematical ideas which have had great influence on my thinking, along with their progenitors.  Not all of these ideas have yet proved useful in any practical sense, either to me or to the world – but there is still lots of time.   Some of these theories are very beautiful, and it is their elegance and beauty and profundity to which I respond.  Others are counter-intuitive and thus thought-provoking, and I recall them for this reason.

• Euclid’s axiomatic treatment of (Euclidean) geometry
• The various laws of large numbers, first proven by Jacob Bernoulli (which give a rational justification for reasoning from samples to populations)
• The differential calculus of Isaac Newton and Gottfried Leibniz (the first formal treatment of change)
• The Identity of Leonhard Euler:  exp ( i * \pi) + 1 = 0, which mysteriously links two transcendental numbers (\pi and e), an imaginary number i (the square root of minus one) with the identity of the addition operation (zero) and the identity of the multiplication operation (1).
• The epsilon-delta arguments for the calculus of Augustin Louis Cauchy and Karl Weierstrauss
• The non-Euclidean geometries of Janos Bolyai, Nikolai Lobachevsky and Bernhard Riemann (which showed that 2-dimensional (or plane) geometry would be different if the surface it was done on was curved rather than flat – the arrival of post-modernism in mathematics)
• The diagonalization proof of Gregor Cantor that the Real numbers are not countable (showing that there is more than one type of infinity) (a proof-method later adopted by Godel, mentioned below)
• The axioms for the natural numbers of Guiseppe Peano
• The space-filling curves of Guiseppe Peano and others (mapping the unit interval continuously to the unit square)
• The axiomatic treatments of geometry of Mario Pieri and David Hilbert (releasing pure mathematics from any necessary connection to the real-world)
• The algebraic topology of Henri Poincare and many others (associating algebraic structures to topological spaces)
• The paradox of set theory of Bertrand Russell (asking whether the set of all sets contains itself)
• The Fixed Point Theorem of Jan Brouwer (which, inter alia, has been used to prove that certain purely-artificial mathematical constructs called economies under some conditions contain equilibria)
• The theory of measure and integration of Henri Lebesgue
• The constructivism of Jan Brouwer (which taught us to think differently about mathematical knowledge)
• The statistical decision theory of Jerzy Neyman and Egon Pearson (which enabled us to bound the potential errors of statistical inference)
• The axioms for probability theory of Andrey Kolmogorov (which formalized one common method for representing uncertainty)
• The BHK axioms for intuitionistic logic, associated to the names of Jan Brouwer, Arend Heyting and Andrey Kolmogorov (which enabled the formal treatment of intuitionism)
• The incompleteness theorems of Kurt Godel (which identified some limits to mathematical knowledge)
• The theory of categories of Sam Eilenberg and Saunders Mac Lane (using pure mathematics to model what pure mathematicians do, and enabling concise, abstract and elegant presentations of mathematical knowledge)
• Possible-worlds semantics for modal logics (due to many people, but often named for Saul Kripke)
• The topos theory of Alexander Grothendieck (generalizing the category of sets)
• The proof by Paul Cohen of the logical independence of the Axiom of Choice from the Zermelo-Fraenkel axioms of Set Theory (which establishes Choice as one truly weird axiom!)
• The non-standard analysis of Abraham Robinson and the synthetic geometry of Anders Kock (which formalize infinitesimal arithmetic)
• The non-probabilistic representations of uncertainty of Arthur Dempster, Glenn Shafer and others (which provide formal representations of uncertainty without the weaknesses of probability theory)
• The information geometry of Shunichi Amari, Ole Barndorff-Nielsen, Nikolai Chentsov, Bradley Efron, and others (showing that the methods of statistical inference are not just ad hoc procedures)
• The robust statistical methods of Peter Huber and others 
• The proof by Andrew Wiles of The Theorem Formerly Known as Fermat’s Last (which proof I don’t yet follow).

Some of these ideas are among the most sublime and beautiful thoughts of humankind.  Not having an education which has equipped one to appreciate these ideas would be like being tone-deaf.

Technorati Tags: mathematics, statistics, differential calculus, epsilon-delta, non-Euclidean geometries, diagonalization proof, space-filling curves, algebraic topology, Fixed Point Theorem, constructivism, statistical decision theory, statistical inference, probability theory, BHK axioms, intuitionistic logic, intuitionism, incompleteness, categories, Possible-worlds, modal logics, topos theory, Axiom of Choice, non-standard analysis, synthetic geometry, uncertainty, information geometry

A recent issue of the TLS ran a review by Terence Hawkes of the biography by Michael Holzman of Jim Angleton, head of counter-intelligence at the CIA.  Holzman’s book, although mostly written from secondary sources, is a fine summary of Angleton’s life and career.  It is marred, however, by (a) Holzman’s annoying (academic) habit of quoting something or somebody and  then repeating, verbatim, key words from that very quotation in the following paragraph, as if we readers were idiots, unable to read for ourselves or contemplate an idea for longer than a paragraph.  And, (b) by a casual sloppiness about dates.  Call me old-fashioned, but I think a historian should not simply say “in  May that  year”  when the last mention of the specific year was some tens of pages and several anecdotes or set-pieces back.   No doubt Holzman always knows which of the 71 years of Angleton’s life and the various ones before or since he is currently referring to, but this is rarely obvious to the reader of this book, even to a careful reader.   In view of the subject matter and Holzman’s theme (that Angleton’s training in so-called practical criticism was invaluable to his career in counter-intelligence), one has to wonder if such sloppiness is deliberate.

Holzman also does not tell us much about the actual theory and practice of counter-intelligence, despite the title and the claims he makes up front.   In particular, his treatment of the Nosenko case is misleading, partly he believes the official CIA line and because he does not refer to the most recent publication on the case, namely the book by Bagley. Hawkes seems to have followed Holzman in his garden-path-up-straying.

Unlike literary criticism, espionage is not only about what to believe, it is also about what to do.  It may be the case that Yuri Nosenko was a genuine Soviet defector, as Holzman claims CIA eventually came to believe.  Others closely involved in the case, such as retired CIA agent Tennent Bagley (2007) have argued compellingly that Nosenko was in fact a KGB plant, not a genuine defector.

Whether or not Nosenko was a genuine defector, and whether or not CIA leadership believed him to be a genuine defector, CIA would also need to concern itself with what impact a revelation of their beliefs would have on KGB, as I have argued before, and thus on what proposition to seek to have KGB believe about CIA’s beliefs in the matter.   If CIA were seen by KGB to accept Nosenko’s testimony (inconsistent and incomplete, by his own admission) too quickly, KGB may not accept as genuine any CIA profession of belief in his bona fides.  So, some delay and equivocation in decision-making was called for.  If CIA professed to believe that Nosenko was a plant or allowed KGB to conclude that CIA believed Nosenko to be a plant, then CIA risked signalling to KGB that they (CIA) were also rejecting all the testimony he arrived in the west with, which included detailed protestations of KGB non-involvement in the assassination of President John F. Kennedy.    Whether or not CIA believed that KGB were involved in that assassination, they may or may not have wished to let KGB know what they believed, at least at that particular moment.  In any case, perhaps a clever (and cunning) CIA would seek to have KGB believe that Nosenko was believed, in order to see how the game played itself out.

 So, one possible course of action for CIA was to signal to KGB that they accepted Nosenko as a genuine defector, but to signal also that they came to this decision only slowly and painfully.   How better to do this than to interrogate the man at length and (allegedly) harshly, and then, after years of apparent indecision and multiple internal investigations (some of which may even have been genuine), decide to accept him publicly as a true defector.   This public acceptance – consultancy fees, letters, flags, medals, and all – even now, four decades later, may have absolutely no connection whatever with what CIA leadership really believed then or, indeed, what they believe now.

It’s not only litcrit that gets an outing in these events.  If any philosopher reading this wonders about the practical usefulness of dynamic epistemic modal logic, wonder no more.

References:

Tennent H. Bagley [2007]:  Spy Wars.  New Haven, CT, USA:  Yale University Press.

Terence Hawkes [2009]: “William Empson’s Influence on the CIA.”  Times Literary Supplement, 2009-06-10.

Michael Holzman [2008]:  James Jesus Angleton, the CIA and the Craft of Counterintelligence.  Boston, MA, USA: University of Massachusetts Press.

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