Chance in a Deterministic World - York University · zDefined a miracle as a violation of a law of nature through divine intervention. zCast great doubt on all the miracles of Christianity

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11Math 1700 – Chance versus Determinism

Chance in a Deterministic World

The Rise of Statistical Reasoning


Scientific laws describe nature precisely…

F=maG=g(M1M2/d2)

e = mc2

… or, at least they did until quantum mechanics.


In the 20th century, scientific laws began to incorporate indeterminism

Here, P is the probability of a particle being between x1 and x2.

Above is a derivation from Schrödinger’s wave equation for quantum mechanics.

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Back in the 19th century…

To be a “scientist” was to be committed to a fully determined world.What, then, was chance?


Chance and probability

My research interest:To explore what the notion of chance can mean in the context of a deterministic view of nature, and what a statement about the probability of an event is really saying.


My starting place:

John Venn, 1834-1923In 1866, Venn published The Logic of Chance, which explored the very questions that interested me.

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Gonville & Caius College, Cambridge

Venn was a Fellow of Gonville & Caius College, Cambridge, and its president for the last 20 years of his life.His archives are located in the college. His vast book collection is in the library of Cambridge University.


Sabbaticals in Cambridge

In September, 2004, my wife and I moved to Cambridge, England, for a year of immersion in 19th century ideas of chance and probability. We returned in January 2008 for another six months.I plan a third trip in January 2010.


The first mathematical treatment of indeterminism:Error theory in astronomy

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Human beings as “astronomical observations”

Adolphe Quetelet (1796-1874), a Belgian who eventually became the Royal Astronomer of BelgiumQuetelet went to Paris to study mathematics and learned the principles of Laplace’s error theory of astronomical observations.He applied these mathematical ideas to statistical measurements of human data.

Adolphe Quetelet, 1796-1874


L’homme moyen – The Average ManQuetelet viewed individual human beings as “errors”from God’s ideal form. Variations in human data, e.g. chest circumference, plotted at right, formed the same kind of bell curve as astronomical measurements of star positions.The central value, the average, was therefore the best measure of the ideal.

Adolphe Quetelet, 1796-1874


Statistics of human behaviour –The Science of Society

Quetelet sought to find the “true values” of all sorts of human data, both physical characteristics and behaviour, by statistical analysis.

Examples of empirically established facts:Constant suicide rates in major citiesPeak age for murder: 25 years old

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Historical determinism

Quetelet’s findings were taken to their logical conclusion by a young British whiz kid named Henry Thomas Buckle (1821-1862).Buckle, who was fluent in dozens of languages and was one of Europe’s strongest chess players in his day, had the idea that he could explain history scientifically, using Quetelet’s findings. Henry Thomas Buckle,

1821-1862


Historical determinism, 2

Buckle undertook to “explain” human history by arguing that human beings’ actions are completely determined by the unalterable laws of behaviour that Quetelet was discovering. Individual actions varied, he claimed, but the overall result was predetermined because as a group, people are predictable.

History of Civilization in England


Historical determinism, 3

Buckle’s book was all the rage in mid-19th century England.It appeared in two volumes, the first in 1857 and the second in 1861.For comparison, Darwin’s On the Origin of Species appeared in 1859.

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Predictable human behaviour

Among the data that Buckle found in Quetelet’s writings that he viewed as proof of the lack of free will:

Number of marriages related to the price of cornNearly constant number of unaddressed letters year after year


No free will

Science, applied to the natural world, was laying down fully deterministic laws that applied to every object in the universe.Buckle viewed Quetelet’s work as proof that such laws applied to human behaviour as well. He wrote:

“Thus it is that, in the ordinary march of society, an increasing perception of the regularity of nature destroys the doctrine of Chance, and replaces it by that of Necessary Connexion.”


Back to John Venn

Venn matriculated at Gonville & Caius in 1853 and took a 1st class degree in mathematics in 1857, graduating as 6th

wrangler. A “wrangler” is someone who graduates with first class honours in mathematics from Cambridge University.

Venn was top of his class at Gonville & Caius College and 6th overall at Cambridge University.

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John Venn, Churchman

John Venn came from a family of clerics. His father was a prominent evangelical priest, the leader of the Clapham Sect of the Church of England.

The Venns were the oldest continuous family of university graduates and Church of England clerics: father & son for 9 generations.

As soon as Venn graduated from Cambridge with his B.A., he entered the Church and took holy orders.

Venn became a deacon, then a priest. He held curacies at various churches in the London area for 5 years.


John Venn, Layman

In 1862, Venn returned to Cambridge and became a college lecturer in the moral sciences, eventually specializing in logic.

Problems reconciling church doctrine with his emerging new views.

Four years after that, in 1866, he published The Logic of Chance.


The Logic of Chance

The Logic of Chance: An Essay on the Foundations and Province of the Theory of Probability with Especial Reference to Its Application to Moral and Social Science.First edition, 1866. Later editions, 1876, 1888

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The Logic of Chance

Venn was prompted to think about chance, and to doubt Evangelical doctrines, by reading some of the radical literature of the 1850s, especially

Henry Thomas Buckle’s History of Civilization

John Stuart Mill’s System of Logic


British empiricism in the 19th

century

J. S. Mill’s Logicdiscussed the logic of inference from incomplete data.

Pitted against Continental idealism.


The meaning of the probability of an event, as understood in Venn’s time

Two chief interpretations:

Degree of Belief: measure of belief that the event will occur in the next trial. This was originally articulated by Leibniz and was the dominant view in continental Europe

Frequency: Ratio of occurrences of the event in some defined universe of trials. This fit well with British empiricism in philosophy and was the favoured view in the British Isles.

A subject of sharp debate.

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The Frequency Interpretation

Venn maintained that the frequency interpretation was the only intelligible meaning of probability.

Probability was seen as the frequency of occurrence of a particular outcome in a finite set of events, or the limit of the frequency in a potentially infinite set of events.


What is so bad about the Degree of Belief Interpretation?

Why did Venn take the trouble to write a long, detailed treatise to show that it was an untenable view?

What else rode along with this concept?


Degree of Belief

The view that the probability of an event is the rational expectation that one should have of its occurrence.

E.g., on the next available occasion: example, a coin toss.

Or, of a single, unique event: example, guilt or innocence of an accused.

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Miracles

What is the bearing of probability on the claims to the truth of miracles?

David Hume’s “Essay on Miracles,” 1748.

Defined a miracle as a violation of a law of nature through divine intervention.

Cast great doubt on all the miracles of Christianity.


Venn’s way out

“It appears to me…that the science of Probability has really little or no direct bearing on the credibility of miracles….” (LOC, p. 302)

Only induction from experience of frequency is a rational measure.

Beliefs depend on personal prepossessions (faith) and will vary by individual.


The Hulsean Lectures

3 years after the publication of The Logic of Chance, Venn gave the Hulsean Lectures at Cambridge – “upon the evidences of Christianity or upon Scriptural difficulties.”

Venn’s lectures: On Some of the Characteristics of Belief, Scientific and Religious.

Ostensibly advice to undergraduates on how to cope with conflicting claims of science and religion.

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Venn’s muddle, and his escape

Scientific beliefs: Deterministic laws of nature.

Religious beliefs: Omnipotence of God, and the power to suspend the laws of nature.

These beliefs will vary from person to person. Therefore they cannot be objective measured. Therefore assessing (objective) probability by (subjective) degree of belief is doomed.


Separating science and religion

Removing belief from probability lets it remain an objective assessment, uncluttered by individual convictions.

This also conveniently removes the necessity of addressing the complications of statistical, historical determinism and free will.


Statistical inference versus questions of causation

Venn’s pure view of probability as (the limiting) frequency nicely paves the way for statistical inference from sampling, without the necessity of assessing causation.

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What about chance?

In The Logic of Chance, Venn dodges the issue of what chance is really all about.But in 1878, he published a long discursive article, “The Foundations of Chance,” in the Princeton Review.

Here, finally, he gets round to chance.


Sort of…

Venn’s illustrative example of chance: the casinos in Monaco.This is, Venn says, “pure chance”:“It is chance, pure and simple, without admixture of fraud on the part of the conductors, or skill on the part of the public. No imputation of any kind of unfairness seems ever to be brought against the managers, even by the most desperate or despondent of the ruined. It is fate, or some such impersonal agent…that they abuse when they find themselves reduced to their last five-franc piece. There is, moreover, no opening for skill or acuteness on the part of the players, whatever they may persuade themselves on this subject…. No more typical instance of the nature and working of ‘chance’ could easily be chosen.”


The great insight! Fanfare….

Casino operators can count on regular profits even though individual results are unpredictable.Venn remarks:“…the clear and adequate appreciation of this kind of uniformity may be regarded as the great logical achievement of modern times. It is like the discovery of a new continent, since it is not merely the extension of old methods over adjacent territories, but rather the discovery of a new method. It is the acquisition, for purposes of inference, of a region where no possibility of inference was formerly known to exist.”

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The momentous discovery: what probability measures.

“Regularity in the long-run combined with perfect irregularity in the details.”Chance, then, seems to apply only to the details.


The momentous clarification:

Everything that happens is fully determined. Therefore there isn’t any such thing as chance.

“If then we are asked whether there is such a thing as chance in the world or not, we should say certainly not.”

But nevertheless, there is “perfect irregularity” in the details.


Randomness vs. pseudo-randomness

Venn confused randomness, which exhibits no pattern at all with “pseudo-randomness” which exhibits no recognizable pattern.Here is Venn’s example of a random sequence of numbers:

2653589793238462643383279502884197 1693993751058209749445923078164062 86208998628034825342117067982148

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The “random” sequence is not random at all

The previous example was not random at all. It is fully determined by the ratio of the diameter to the circumference of a circle.The example given is the first 100 digits of π after the recognizable 3.14159

3.1415926535897932384626433832795028 841971693993751058209749445923078164 06286208998628034825342117067982148


The perfect muddle

In 1897, Cambridge University announced a new chair in philosophy, the Professor of Logic and Mental Philosophy.

Venn was the odds-on favourite to get the appointment, but it went to someone else.Venn immediately dropped all work on logic and probability and spent the remaining 26 years of his life writing historical works on his college and the university.He missed all the uproar over chance and indeterminism in physics in the 20th century.


What Venn is remembered for

Though John Venn grappled with chance versus determinism he failed to resolve the incompatibility of the two concepts for science.He is remembered now only for an idea he had on how to represent the truth value of logical propositions by a series of interlocking diagrams.

Venn diagrams

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Statistical inference versus questions of causation

Venn’s pure view of probability as (the limiting) frequency nicely paves the way for statistical inference from sampling, without the necessity of assessing causation. Statistics was free to evolve without the baggage of a pre-established interpretation.


Quetelet’s “Average Man” revisited

Quetelet took the average value as the best estimate of the ideal: God’s plan. Any deviation from the average was an aberration.Venn’s interpretation, that the data is simply a frequency pattern, gives no special place to the average, nor to the deviations.

This gave room for a different interpretation.


Francis Galton, 1822-1911

Francis Galton, a cousin of Charles Darwin, was also interested in human statistics, but took greatest interest not in the averages, but the extremes.Venn’s work gave no special significance to the extreme deviations, but only emphasized their rarity.

Francis Galton, 1822-1911

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For Galton, the deviations were of more interest than the central value.

Galton took great interest in extraordinary people and made a study of the extent to which genius is an inheritable trait, with a view to trying to understand evolution and if possible to control it.


Francis Ysidro Edgeworth, 1845-1926

Another cousin of Darwin’s (and Galton’s) was Francis YsidroEdgeworth, known primarily for contributions to mathematical economics.He corresponded with both Venn and Galton about statistical inference, and adopted Venn’s frequency interpretation of probability. He tried to quantify the pleasure principle as a fundamental unit for psychology and economics. For Edgeworth, statistics provided the most reliable description of a group of people.

Francis Ysidro Edgeworth, 1845-1926


Karl Pearson, 1857-1936

Karl Pearson was one of the great innovators of statistics in the second half of the 19th century.Pearson, when serving as lecturer in geometry at Gresham College in London, undertook to give a series of lectures on the Laws of Chance.

Karl Pearson, 1857-1936

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Pearson’s Gresham College Lectures

He cited both Venn and Edgeworth in his lectures.In 1893, due to illness, Pearson had to find people to substitute for him in his lectures. He asked Venn to lecture on a topic of his choice.Venn chose the topic “Curves of Error”—preserving Quetelet’s sense of data representing errors from a pre-ordained model. Pearson agreed, but suggested that Venn change the title to “On Frequency Curves” which would be less confusing to his non-technical audience.

This underscored the permanent change in thinking that Venn brought about. Statistical data was seen just as frequencies, not as errors from a “correct”answer.

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Chance in a Deterministic World - York University · zDefined a miracle as a violation of a law of nature through divine intervention. zCast great doubt on all the miracles of Christianity