Friday, October 24, 2008
Sunday, October 19, 2008
Man is Nature's agent and interpreter; he does and understands only as much as he has observed of the order of nature in fact or by inference' he does not know and cannot do more.
Human knowledge and human power come to the same thing, because ignorance of cause frustrates effect. For Nature is conquered only by obedience; and that which in thought is a cause, is like a rule in practice.
All man can do to achieve results is to bring natural bodies together and take them apart; Nature does the rest internally.
The subtlety of nature far surpasses the subtlety of sense and intellect, so that men's fine meditations, speculations, and endless discussions are quite insane, except that there is no one who notices.
Francis Bacon (1620) Novum Organon
Sunday, October 12, 2008
This Ratio once discovered, and manifestly serving to a wise purpose, we conclude the Ratio itself, or if you will the Form of the Die, to be an Effect of Intelligence and Design.
Abraham De Moivre (1756) The Doctrine of Chances, 3rd ed.
1711: Dr. John Arbuthnot argues for "Divine Providence" on the basis of the statistical significance of the relative difference between the birth rates of boys and girls recorded in the London Bills of Mortality for the previous 82 years.
His reason: a chance process could not have produced this discrepancy.
Bernoulli's response: "if sex is likened to tossing a 35-sided die, with 18 faces labelled "male," and 17 labelled "female," then Arbuthnot's data are entirely consistent with the outcome of chance."
[see S. L. Zabell (1988) "Symmetry and Its Discontents"]
De Moivre makes his remark in response to Bernoulli. The issue, as Hacking ably demonstrates, arises during a metamorphosis of the concepts of chance and probability. The crucial issue was defining the class of phenomena to which the concept of probability properly applies.
Zabell emphasizes the role of symmetry here: Arbuthnot believes the lack of symmetry in birth rates indicates divine design; Bernoulli demonstrates the presence of symmetry; De Moivre questions the source of this symmetry.
But isn't De Moivre's request for a source of "the Die" just a request for a deeper symmetry? The "wise purpose" which a discrepancy in birth rates seems to "serve" indicates the need for a deeper explanation, hence the appeal to "Design." Unlike the discrepancy in birth rates, the outcomes of die rolls do not appear to favor (statistically) any wise purpose. The new asymmetry here is between stereotypical "chance" events, such as die rolls, oblivious to providence, and a seemingly analogous case, conception, which disanalogously manifests providence.
One goal of an effective theory of evolution is to remove this asymmetry, to account for the discrepancy in birth rates and the providential nature of this discrepancy, by appealing to mechanisms of a purely die-like oblivion. In other words, De Moivre's request for an explanation concerning the shape of the die must itself be answered with an account of the random (and manifestly non-providential) process producing that shape: dice within dice.
Whether De Moivre would be satisfied with such a response is irrelevant here. Furthermore, this is not explicitly a question concerning evolution or design. We can see its modern cousin in string theory's anthropic landscape - a desperate attempt to rule out alternate solutions to the physical equations of string theory by positing a multiverse and arbitrarily licensing consideration of only those universes which can provide for human life.
Effectively, assuming string theory, our universe is then the result of the throw of a 10500 sided die. Only one face on this die produces a universe hospitable to human life. Again, as with De Moivre, this asymmetry, this apparent providence demands an explanation which restores symmetry. The preferred move of modern string theorists, rather than positing divine intervention, is simply to posit 10500 dice, each thrown simultaneously (= "the cosmic landscape").
Their solution is every bit as unsatisfying and anti-scientific as De Moivre's, however. For De Moivre's solution is unsatisfying not because of its appeal to God, but because the appeal to God indicates the end of scientific inquiry: and you cannot provide me a suitable explanation. Likewise with the multiverse, for, by its very nature, the other "universes" of the multiverse are beyond empirical inquiry, and the positing of this monster licenses string theorists to declare victory for their theory, ending inquiry by appealing to that the properties of which cannot be investigated via the experimental method.
In the case of the economy, the situation is far worse. For the die throws which govern the economy (unlike those which govern the fundamental physical constants) are subject to intervention. We are dissatisfied with the result of rolling this 6-sided die, so we replace it with a 35-sided one, suitably labeled. Providence attributed to the hand of God in the case of birth rates is now attributed to the hand of Man.
But surely the problem here is one of calculation: it took careful analysis of 82 years of records to determine the operation of a single 35-sided die. If there is any analog in the economy, surely many different dice are being thrown at asynchronous intervals. But how can the underlying form of these die be discovered if, rather than observing and analyzing this complex phenomenon, we insist upon participating, toss our own homemade dice upon the table, or taking the die of our neighbor and shaving one of its sides?
But wait, you say: participation is demanded, how else can we avoid catastrophe? Here, perhaps, it is the trust to divine providence itself which is the scientific response, and not the egocentric hubris of the modern man.
Thursday, October 9, 2008
Axiomatically, mathematics is concerned solely with relations among undefined things. This property is well illustrated by the game of chess. It is impossible to "define" chess otherwise than by stating a set of rules. the conventional shape of the pieces may be described to some extent, but it will not always be obvious which piece is intended for "king." The chessboard and the pieces are helpful, but they can be dispensed with. the essential thing is to know how the pieces move and act. It is meaningless to talk about the "definition" or the "true nature" of a pawn or a king. Similarly, geometry does not care what a point and a straight line "really are." They remain undefined notions, and the axioms of geometry specify the relations among them: two points determine a line, etc. These are the rules, and there is nothing sacred about them. We change the axioms to study different forms of geometry, and the logical structure of the several non-Euclidean geometries is independent of their relation to reality. Physicists have studied the motion of bodies under laws of attraction different from Newton's, and such studies are meaningful even if Newton's law of attraction is accepted as true in nature.
Newton's notions of a field of force and of action at a distance and Maxwell's concept of electromagnetic "waves" were at first decried as "unthinkable" and "contrary to intuition." Modern technology and radio in the homes have popularized these notions to such an extent that they form part of the ordinary vocabulary. Similarly, the modern student has no appreciation of the modes of thinking, the prejudices, and other difficulties against which the theory of probability had to struggle when it was new. Nowadays newspapers report on samples of public opinion, and the magic of statistics embraces all phrases of life to the extent that young girls watch the statistics of their chances to get married. Thus everyone has acquired a feeling for the meaning of statements such as "the chances are three in five." Vague as it is, this intuition serves as background and guide for the first step. It will be developed as the theory progresses and acquaintance is made with more sophisticated applications.
The concepts of geometry and mechanics are in practice identified with certain physical objects, but the process is so flexible and variable that no general rules can be given. The notion of a rigid body is fundamental and useful, even though no physical object is rigid. Whether a given body can be treated as if it were rigid depends on the circumstances and the desired degree of approximation. Rubber is certainly not rigid, but in discussing the motion of automobiles text books treat the rubber tires as rigid bodies. Depending on the purpose of the theory, we disregard the atomic structure of matter and treat the sun now as a ball of continuous matter, now as a single mass point.
. . . The manner in which mathematical theories are applied does not depend on preconceived ideas; it is a purposeful technique depending on, and changing with, experience.
Feller, William (1950) An Introduction to Probability Theory and Its Applications, 2nd ed.