Progressive ideology generally asserts that there is a wide variety of elements of conceivable and feasible policy and technology that would significantly improve average quality of life. It may be further assumed that the pursuit of any of those elements or any combination of those elements can be expected to be fruitful. To some extent this assumption – what may be termed the “parallel progress assumption” – seems to be implicit in much of present-day progressive activism: the agenda is usually very eclectic and often somewhat vague on specifics. While this is partly a tactic that is aimed at maintaining wide appeal, there is also the implication that as long as general principles are agreed to, laying out a detailed workplan is not necessary since progress can be made on any of many items quite independently of each other. While I accept the progressive assumption (i.e., that progress is possible) it appears to me that the parallel progress assumption is incorrect.

First, technological progress, by itself, doesn’t hold much promise. The dramatic technological progress that occurred over the last four decades – especially in the fields of automation and telecommunications, but in many other fields as well – has yielded surprisingly little in terms of improved quality of life. Most notably, automation has translated into the elimination of jobs, forcing workers into a desperate search for employment yielding lower and lower returns. In general, since technological progress is the product of concentrated resources, it can be expected in general that the fruits of technological progress would flow to those who control the resources. Thus, until political democratization is achieved – making control of resources more evenly distributed – there is no reason to expect that technological progress would result in increased quality of life.

In terms of public policy, there are many proposals that could hold promise, but it seems that most of them would not be feasible or would be easily reversible, unless, again, changes in government structure would act as an enabling factor. Political elites seem to be able to block any significant progressive public policy changes, or erode those changes that do manage to get implemented. Public pressure rarely gets to the level that forces progress, and when it does, steady, patient, determined regressive pressures effectively reverse over time any gains.

One important possible exception to the infeasibility of progressive public policy is Basic Income Guarantee. The experience of the Alaska permanent fund dividend shows that once such a program is in place, it becomes entrenched due to the fact that it provides direct tangible benefits very large parts of the population. The fact that the policy is relatively clear cut and equalitarian makes it more difficult to weaken it through the usual mechanisms of ratcheting complication and differentiation.

BIG, however, is the exception, I believe. Most progress would depend on policy that requires constant micro-managing, so that as long as those who do the managing are not sympathetic to the progressive goals, it would be unlikely that such policy would be effective.

Thus, by and large, progress is unlikely until government reform – namely, democratization – is achieved. Democratization, it seems to me, depends crucially on two elements:

1. Sortition
2. Democratic media

Those two elements, I believe, are both necessary and sufficient (and mutually enabling) for democratization of government. Therefore progressive activism must focus its energies on those elements: disseminating those idea among the population and putting them at the top of any list of objectives. Until that happens, it can be expected that the initial success of grass root progressive movements such as the Indignados in Spain and Greece, the Tent protest in Israel and the Occupy movement in the U.S. and elsewhere, will ultimately fizzle out without making long term gains.

[1] Even there the detailed picture is mixed:

In 737 U.S. counties out of more than 3,000, life expectancies for women declined between 1997 and 2007. For life expectancy to decline in a developed nation is rare. Setbacks on this scale have not been seen in the U.S. since the Spanish influenza epidemic of 1918, according to demographers.

(Noam N. Levey, L.A. Times, June 15, 2011)

Proposition 2:

For every l, l >= ES, P(S ≥ l) is maximized when q₁ = ··· = q_n-mz = q = ES / (n – m_z), and q_{n – mz + 1} = ··· = q_n = 0, for some m_z.

That is, in the terminology of Proposition 1, for all non-trivial cases, m_o = 0, and thus the probability of deviation is maximized by some Bernoulli variable (rather than a shifted Bernoulli variable).

Proof:

Let ES < l. Let the parameters q₁, …, q_n be as described in Proposition 1, with m_o > 0. Let S’ = S – B₁ – B_{mo + 1} = S – 1 – B_{mo + 1}. Then S’ – m_o + 1 is a Binomial variable with parameters n’ = n – m_o – m_z – 1 and q whose expectation is ES – m_o – q = (ES – m_o) n’ / (n’ + 1).

The density of a Binomial variable with parameters n’ and q is unimodal with a mode smaller or equal to max((n’ + 1) q – 1, 0). Therefore, the density of S’ is unimodal with mode smaller or equal to the maximum of m_o – 1 and

m_o – 1 + ES’ (n’ + 1) / n’ – 1 = m_o – 1 + (ES – m_o) – 1 = ES – 2.

Since by assumption ES < l (and therefore m_o < l) the mode of the distribution of S’ is lower or equal to l – 2. Thus P(S’ = l – 2) > P(S’ = l – 1) and therefore, following the argument in the proof of Proposition 1, P(S” >= l) > P(S >= l), where S” = S’ + B’₁ + B’_{mo + 1} and B’₁ and B’_{mo + 1} are independent Bernoulli variables both with parameter 1/2. ¤

A similar argument proves

Proposition 3:

For every l, l > ES + 1, P(S ≥ l) is maximized when q₁ = ··· = q_n = ES / n.

That is, unless l ≤ ceil(ES), P(S >= l) is maximized by a Binomial variable with parameters n and ES / n. It turns out that the special case that motivated this investigation is indicative of the situation in a rather limited domain. For most cases, the same Bernoulli variable that maximizes the variance of the family under consideration also maximizes the probability of deviation.

Finally, the domain is limited but is not empty.

Proposition 4:

For l + 2 – 1 / l – (1 + 1 / l)^{l + 1} < ES, P(S_l >= l) > P(S_{l + 1} >= l), where S_l is a Binomial with parameters l and ES / l, and S_{l + 1} is a Binomial with parameters l + 1 and ES / (l + 1).

Proof:

By direct comparison of (ES / l)^l and (ES / (l + 1))^{l + 1} + (l + 1) (ES / (l + 1))^l (1 – ES / (l + 1)). ¤

A numerical investigation shows that indeed in the range l – 1 < ES < l, the probability of deviation is maximized by Binomial variables with parameters n and ES / n for some finite n. The diagrams below illustrate the situation (generated for l = 10):

The solid thin lines correspond to a sum of 10 IID Bernoulli variables, while the dashed and dotted lines correspond to sums of 11 and 20 variables, respectively. The thick solid lines correspond to a Poisson variable – the limiting distribution as the number of IID Bernoulli variables in the sum increases indefinitely.

It is worth noting that the same line of argument used above can be used to show that shifted Binomials maximize the density values of sums of independent non-identically distributed Bernoulli variables, i.e., P(S = l).

Clearly, if l ≤ ES, then the tightest bound is 1. This is attained by setting q₁ = ··· = q_l = 1.

This example shows that while the variance of S is maximized by setting q_i = ES / n, i = 1, ···, n, at least for some values, l, P(S ≥ l) is maximized by having the B_i not identically distributed.

Proposition 1:

For every l, P(S ≥ l) is maximized when q₁ = ··· = q_mo = 1, q_mo+1 = ··· = q_n-mz = q, and q_n-mz+1 = ··· = q_n = 0, for some m_o and m_z, and for q = (ES – m_o) / (n – m_o – m_z).

Proof:

Assume that 1 > q_i > q_j > 0. Let S’ = S – B_i – B_j. Then

P(S ≥ l) = P(S’ ≥ l) + p₁ (q_i + q_j – q_i q_j) + p₂ q_i q_j,

where p₁ = P(S’ = l – 1) and p₂ = P(S’ = l – 2). Thus, keeping q_i + q_j fixed, but varying the proportion between them, P(S ≥ l) is a linear function of q_i q_j. Unless p₁ = p₂, P(S ≥ l) will be increased by varying q_i and q_j – with a maximum either when they are equal or when one of them is zero or one.

Therefore, P(S ≥ l) cannot be at a maximum if there exist 1 > q_i > q_j > 0, unless p₁ = p₂. But in that case, the same probability can be achieved by setting the parameter of B_i to be q’_i = 0 (if q_i + q_j < 1) or q’_i = 1 (otherwise), and the parameter of B_j to be q’_j = q_i + q_j – q’_i. Therefore, in that case, there would exist a set of parameters, q’₁, …, q’_n, that would achieve that same probability but with fewer parameters that are not equal to zero or one. Thus, in the set of parameter settings maximizing P(S ≥ l), there exists a solution – namely the one which maximizes the number of parameters with extreme values (zeros and ones) – in which there is only one non-extreme value. ¤

The next step is to investigate which specific parameter setting correspond to various combinations of l and ES.

the use of violence and threats to intimidate or coerce, especially for political purposes,

clearly conflicts with accepted usage, for if it were believed then any military activity, or even police activity, would constitute a terrorist activity. Other definitions come closer to the intended meaning:

Terrorism is defined as political violence in an asymmetrical conflict that is designed to induce terror and psychic fear (sometimes indiscriminate) through the violent victimization and destruction of noncombatant targets (sometimes iconic symbols). [Attributed by Wikipedia to Carsten Bockstette.]

Obviously, military activity would often fall within those terms if it were not for the “asymmetrical” condition. Such activity, when considered legitimate by the speaker, is never referred to as “terrorism”. On the other hand, this condition makes it impossible to have terrorism by a military, which clearly contradicts common usage. Systematic campaigns of massacre by a military are often referred to as “terrorism”. Thus, the definition is inaccurate either with or without the “asymmetrical” condition.

It appears to me that the reason for the difficulty in finding an accurate definition is the idea that “terrorism” can be defined solely in terms of the physical activity of the agent in question or its effects on those affected by it. This, I think, is simply untrue. It seems that this idea ignores a crucial factor, which is the perception of legitimacy of aims and methods. This implies that a policy – defined in terms of the ways violence is used – would be described either as “terrorist” or in different terms, based on circumstances that have nothing to do with the policy itself or its foreseeable effects.

The perception of legitimacy – and thus the decision of whether to apply the term “terrorism” – is in fact dominated by the perception of intent. If the violent actor is perceived as using violence against noncombatants deliberately or without justification then the actor is a terrorist. The same acts, when perceived as being due to error or “necessity” would not be termed so.

Now, intent is of course extremely difficult to determine objectively. Almost any act can be described as being motivated by good intentions. In a bind, one can concede that the act itself was wrong, but still ascribe good intentions to the perpetrator. This last position is very common in American media as regarding military activity of the US or its allies, when things are such that it is no longer feasible to pretend that widespread targeting of civilians doesn’t occur. The perpetrator is then described as having been naive or errant, but not a terrorist. The terrorist attacks non-combatants because he is morally degraded, not because he must or because he legitimately, even if mistakenly, believes it serves a greater good.

In light of the above, I propose the following as a definition of terrorism:

Terrorism is defined as violent victimization of civilians, deliberately or through negligence, unless justified by unavoidable needs stemming from the genuine pursuit of legitimate or worthy objectives.

It may be inferred that

Insisting in general terms that one is very scrupulous in making sure civilians are not harmed is thus an important (by not sufficient, or even strictly necessary) step toward not being considered a terrorist. It is much more important than the particulars of any policy or event.

Whatever may be the advantages of this definition in terms of accurately capturing the sense in which the term “terrorism” is used, its drawback is clear: it explicates the notion that the term (as it is used) is inherently subjective, and thus completely useless as a tool for rational discussion. According to such a definition, the term is little more than a device of propaganda, reinforcing pre-conceived points of view. Taking seriously the quip about the subjectivity of the difference between a terrorist and a freedom fighter (or, even more problematically, between a terrorist and a soldier) may be inconvenient.

The object of our analysis is, not to provide a machine, or method of blind manipulation, which will furnish an infallible answer, but to provide ourselves with an organised and orderly method of thinking out particular problems; and, after we have reached a provisional conclusion by isolating the complicating factors one by one, we then have to go back on ourselves and allow, as well as we can, for the probable interactions of the factors amongst themselves. This is the nature of economic thinking. Any other way of applying our formal principles of thought (without which, however, we shall be lost in the wood) will lead us into error. It is a great fault of symbolic pseudo-mathematical methods of formalising a system of economic analysis, such as we shall set down in section vi of this chapter, that they expressly assume strict independence between the factors involved and lose all their cogency and authority if this hypothesis is disallowed; whereas, in ordinary discourse, where we are not blindly manipulating but know all the time what we are doing and what the words mean, we can keep “at the back of our heads” the necessary reserves and qualifications and the adjustments which we shall have to make later on, in a way in which we cannot keep complicated partial differentials “at the back” of several pages of algebra which assume that they all vanish. Too large a proportion of recent “mathematical” economics are mere concoctions, as imprecise as the initial assumptions they rest on, which allow the author to lose sight of the complexities and interdependencies of the real world in a maze of pretentious and unhelpful symbols.

There is much truth in the above, I think, and it is truth that applies not only to “economic thinking” but to any kind of thinking that relies on formalization. Statistical analysis is plagued with this kind of problems. Keynes does lay too much stress on the matter of interaction between factors. The problem with formal methods is not particularly with neglecting various effects – it is that they simply are false in various ways (neglecting various effects is only one of the sources of falsehoods). Informal methods have the same problem, of course – and in addition have problems associated with informality.

The special problem with formal methods is their privileged status in discourse. Formal methods are usually difficult for the uninitiated to analyze – pushing them out of the realm of contestability for a large section of the audience. The initiated, on the other hand, are often also indoctrinated into accepting certain assumptions that are convenient for their formal analysis. In addition, the initiated often have a vested interest in maintaining the privileged status of the formal analysis. Thus those who can contest often don’t. Another important source of privilege is that mathematical models enjoy a reputation for scientific rigor and practical power which makes out-of-hand dismissal a difficult position.

Thus the issue is not inherent in formalization. There is no particular problem in keeping things at ‘the back of our heads’ when manipulating formal symbols. People who carry out such manipulation don’t do so in a blind manner (I certainly don’t and I would be surprised to learn others do). Each manipulation step is motivated by an intuitive understanding of the meaning of the step and the implication of the result. The implications of assumptions on the analysis can be examined and alternatives can be considered. When this is not done, it is not done due to practical considerations, not because the method does not allow it. Such practical considerations are present when carrying out a word-argument as well.

The strength of informal methods, then, is that they are usually obviously contestable. People are used to examining informal arguments and contesting them, opening the possibility of rational critique. Of course, not too infrequently informal argumentation becomes hard to examine. A non-trivial book length argument can easily be outside the realm of examinability for most people. This is the situation with, say, Marxist theory, various philosophical schools, or Keynes’s own work. In such situations, there seems to be little advantage to the word-arguments over formalized arguments. (Contrary to the situation with mathematics, a blanket statement dismissing such work is socially acceptable, due to the commonplace and undeniable fact that word-arguments are not made truer by being longer or more complicated. But such out-of-hand dismissal is not any more of a rational decision than the uncritical acceptance of an argument.)

Formalization does have its uses. Its main advantage over words is that it eliminates ambiguity. Associated advantages are often brevity and facility in ascertaining consistency and identifying missing details, and similar effects. That is, it does a good job at providing “an organised and orderly method of thinking out particular problems.”

All this goes to explain why I think that having a mathematical model of Keynes’s General Theory is useful despite contravening Keynes’s own strictures in that same work. I have spent considerable time and effort understanding that work. I find Keynes’s thinking compelling in its method – it is careful, self-reflective and wide-ranging; it is philosophical in the most positive sense of the word. The book is well worth reading. Still, I believe the presentation suffers due to a lack of a general formal framework. Oddly, Keynes does not avoid mathematics altogether (as he mentions the passage above) – but his few mathematical sections focus largely (as far as I can tell) on minor or tangential points. They do not seem to illuminate the main argument but to obscure it.

The model I presented attempts to correct this situation. Unlike Hicks’s oft-referred-to paper, Mr. Keynes and the Classics (1937), this model describes Keynes’s main lines of thinking as they appear in the book (as I understand them, of course), rather than by trying to reconcile them with contemporary convention.

Data source: New York Times/CBS poll, April 15-20, 2011.

template <typename T>
void insert(std::list< std::pair<T,unsigned int> > &l, T v)
{
    unsigned int sp = 0;
    while (!l.empty() && l.back().first < v)
    {
        sp++;
        l.pop_back();
    }
    l.push_back(std::pair<T,unsigned int>(v,sp));
}

template <typename T>
void advance(std::list< std::pair<T,unsigned int> > &l)
{
    if (l.front().second > 0)
        l.front().second--;
    else
        l.pop_front();
}

template <typename T>
void max_in_window(T const *in, T *out, size_t n, size_t k)
{
    std::list< std::pair<T,unsigned int> > l;
    unsigned int i;
    for (i = 0; i < k - 1 && i < n; i++)
        insert(l,in[i]);

    for (; i < n; i++)
    {
        insert(l,in[i]);
        out[0] = l.front().first;
        out++;
        advance(l);
    }
}

Interestingly, an algorithm for the median in a sliding window will have runtime of at least O(n log k), since such an algorithm can be used to sort O(n / k) sequences of length O(k) each:

Let a₁, a₂, …, a_n be a sequence of numbers in a known interval, say (0, 1). Create a sequence of length 3n – 2 by padding the sequence with a prefix of (n – 1) 0‘s and a suffix of (n – 1) 1‘s. Now execute the sliding window median algorithm with the padded sequence as input, and with k = 2n – 1.

The i-th output element will be the median of a sequence that is made of the entire sequence a₁, a₂, …, a_n, n – i zeros, and i – 1 ones. That median is a_(i), the i-th smallest elements of the sequence a₁, a₂, …, a_n. Thus, the output of the algorithm will be the sorted sequence a₍₁₎, a₍₂₎, …, a_(n).