Christmas Trilogy Part 2: Computing mathematical miracles.

In modern perception Charles Babbage has become reduced to narrow minded Victorian number cruncher whose only interest in life was producing mechanical computing machines to crunch ever more numbers. He has even been accused by the acolytes of St Ada of Lovelace of not understanding the real future purpose of those machines, knowledge of which had to be supplied by her saintliness. This rather dismal one-sided portrait of Babbage is very far from the truth Babbage being in reality a multi-talented man whose knowledge of the potentials embodied in the newly emerging machines of the nineteenth century was both very broad and deep. He even, within the context of a religious debate, conceived of the possibility of pre-programmed miracles a concept that he would demonstrate like a mechanical conjuror on early prototypes of his difference engine.

In the first half of the nineteenth century intellectual discourse in Victorian England was dominated by the concept of natural theology, particularly as presented by William Paley in his Natural Theology, or Evidences of the Existence and Attributes of the Deity collected from the Appearances of Nature published in 1802; Paley being today much loved by creationists and much derided by their opponents. The central argument of natural theology is very simple, stating that one can deduce the existence of God through the (scientific) study of the natural world. Paley is famous for having used the watchmaker analogy, the natural world resembles a watch in its complexity and design therefore there must be a watchmaker. (I have a sneaking feeling that I’m going to get hammered by my philosophical friends for this very simplified presentation of natural theology).

Paley was by no means the only believer in natural philosopher in that age and Francis Egerton, 8^th Earl of Bridgewater, left a bequest of £8 000, a lot of money in those days, to pay one or more authors to write one or more treatises defending the principle of natural theology against the then modern scientific discoveries; the money to be administered by the Royal Society. The Royal Society decide to divide the money into eight portions of £1 000 and to commission eight treatises covering the full range of the then natural and moral sciences.

John Herschel was originally approached to write the treatise on astronomy but he declined on the grounds that it was wrong for a scholar to write for money! This volume was then offered to William Whewell, who having neither Herschel’s wealth nor his scruples eagerly accepted the task. Whewell duly wrote and published the Third Bridgewater Treatise, Astronomy and General Physics considered with reference to Natural Theology, which became the most successful and widely read of all of them, running to nine editions in his own lifetime. Whewell produced all of the argument brought earlier by Isaac Newton, who can be considered natural theological, for a God designed cosmos but adding all of the newer astronomical discoveries made since Newton’s times even including Herschel’s very recent work on double stars, showing how they too obeyed the law of gravity. Whewell’s cosmos was one governed by the laws of science as laid out by a scientific God; having established that God’s cosmos is governed by scientific laws Whewell then goes on to expound his philosophy of science. As he was soon to declare in his legendary three volume History of the Inductive Sciences (1st ed. 1837) and two volume The Philosophy of the Inductive Sciences Founded upon their History (1st ed. 1840) Whewell was a Baconian through and through who argued that the laws of science are obtained through induction. Not content to leave it at that he then went on to deny the ability of mathematics and deductions to discover new laws of nature.

Whewell, Herschel and Babbage had been close friends as students at Cambridge[1] and although all three of them were excellent mathematicians, who together as students had fought for the introduction of the continental analysis into Cambridge, it was Babbage who most considered himself to be a mathematician and who took umbrage at what he saw as a personal slight in Whewell’s dismissal of mathematicians in the process of scientific discovery. Never one to take insults lying down Babbage rose to the challenge and wrote and published his own Bridgewater Treatise, although he was not one of the eight chosen authors. Entitled The Ninth Bridgewater Treatise A Fragment by Charles Babbage, Esq. it contained Whewell’s offending passage on its title page:

“We may thus, with the greatest propriety, deny to the mechanical philosophers and mathematicians of recent times any authority with regard to their views of the administration of the universe; we have no reason whatever to expect from their speculations any help, when we ascend to the first cause and supreme ruler of the universe. But we might perhaps go farther, and assert that they.are in some respects less likely than men employed in other pursuits, to make any clear advance towards such a subject of speculation.”—Bridgewater Treatise, by the REV. WM. WHEWELL, p. 334.

This small book contains much of interest but what concerns us here is Chapter II, Argument in Favour of Design from the changing of Laws in Natural Events, which is a clever move by Babbage the computing expert to score points over Whewell.

Not in his Bridgewater Treatise, but in his reviews of the two volumes Charles Lyell’s Principles of Geology from 1831and 1832 Whewell addressed a problem that was central to the problems of natural philosophy caused by the recent scientific developments, evolution. Although Darwin’s own theory of evolution still lay some decades in the future evolution as a scientific fact was becoming more and more established as the geologists and palaeontologists found and examined more and more fossils of extinct species. If God had created the world and all that was in it, how come the geological record clearly displayed the disappearance and appearance of different species over the ages. Whewell’s solution was to invoke a caretaker God who popped in from time to time introducing new species to replace those that had died out these interventions being in the form of miracles. It is here that Babbage set out to demonstrate the superiority of a mathematical computing God.

Babbage argued by analogy, he describes the possibility of a computer programme (not the terminology that Babbage uses by the way) that generates the natural numbers 1, 2, 3, 4, … up to and including 100,000,001 but then instead of producing the number 100,000,002 as expected jumps to 100,010,002, continuing the series 100,030,003; 100,060,004; 100,100,005; 100,150,006; 100,210,007 … and so forth. Babbage states that the law generating the series has changed at the jump. The expected numbers being exceeded by the series 10,000, 30,000, 60,000, 100,000, 150,000 … and so on this being the series of triangular numbers 1, 3, 6, 10, 15, … multiplied by 10,000.

Babbage goes on to explain that the operator does not need to interfere with the calculating engine (he is of course thinking of his own Difference Engine) at this point but can pre-programme it from the beginning to make the change at the given juncture.

Unlike Whewell’s God who has to intervene in his own laws of nature with miracles to explain the presence of new species in the geological record Babbage’s mathematical God can pre-programme his laws of nature to change at the required point in time thus pre-programming his miracles at the point of creation.

Babbage actually programmed one of the calculating units of his Difference Engine to perform a miracle of the type described here, which he then demonstrated to guests at the soirees he held at his home in London. It was one of these demonstrations that so impressed the seventeen year old Ada Byron in 1833 and drew her into Babbage’s sphere of influence.

Babbage was so pleased with his mathematical miracles that he included another account of them in his autobiography, Passages from the Life of a Philosopher originally published in London in 1864.

Some readers might note a strong similarity between Babbage’s argument, sketched here, for a divine pre-programmed replacement of species and the arguments of those modern Christians who accept the theory of evolution but state that this is God’s method of creating the world.

 

 

 

 

[1] Laura J. Snyder’s The Philosophical Breakfast Club, Broadway Books New York, 2011 is an excellent account of that friendship that I strongly recommend.