Chapter 4 |
The Universities: |
Thus we observe in Babbage’s period a great flowering of astronomy and celestial mechanics. It was still too soon however to see a similar situation obtain in other branches of natural philosophy. As a consequence, great calculations were not attempted or, indeed, needed in other branches of physics. In fact, we may say as a general principle that large calculations will not be attempted in a given field until its practitioners can write down in mathematical form an unambiguous description of the phenomena in question.
This was not possible in Babbage’s time, for example, in the field of electricity, where Michael Faraday (1791–1867) was just then performing his beautiful experiments, which were to form the basis for Maxwell’s wonderful work. Moreover, it was not until the year of Babbage’s death that a chair of experimental physics was established at Cambridge. In 1871 James Clerk Maxwell (1831–1879) was called to the chair which was intended “especially for the cultivation and teaching of the subjects of Heat, Electricity, and Magnetism.” This was the first time that instruction in physics was provided at Cambridge. Prior to this physical research was done outside of the English university structure. In 1874 the Cavendish Laboratory was presented to the University by its Chancellor, the Duke of Devonshire, in honor of his kinsman, Henry Cavendish (1730–1810), the distinguished physicist who anticipated, among others, Coulomb and Faraday. Upon Maxwell’s untimely death in 1879, Lord Rayleigh succeeded him.
The university world of Babbage’s time was a very different one from our own, and it is a mistake to extrapolate back from our times to his without realizing the differences. Although Oxford and Cambridge were the only universities in England and Wales, until 1827 Roman Catholics, Protestant Dissenters, and Jews were excluded from them by an act of Parliament that was not revoked until 1856. Moreover, it was not until the year of Babbage’s death that restrictions were removed making it possible for such people to hold lay offices in either school. Thus both these schools exeluded some of die most able people from the scientific and political fields. During the eighteenth century annual admissions at Cambridge were under 200; in the 1820s, they rose to 400, in the 1870s to 600, and in the 1890s to 900. One should not however imagine that Oxford and Cambridge had completely changed. A. J. P. Taylor says that as late as the 1920s “the two systems of education catered for different classes and provided education, different in quality and content, for rulers and ruled.”1 To validate this point he states that “only one in a hundred [students] came from a working-class family.”
This exclusion was going on just at the time when the Industrial Revolution was making education ever more essential for all members of society. In 1823 George Birkbeck (1776–1841) founded his first Mechanics’ Institute in Scotland, and similar institutes spread into England under the patronage of Henry Brougham (1778–1868). These brought to the workingman the advantages of a technological training just when it was most needed in England. These schools are the place in which the engineers and mechanics learned their business—for an annual fee of one guinea. Most of these men were not middle-class; for example, Stephenson, the inventor of the locomotive, was a poor boy who taught himself to read when he was seventeen.
In 1827 Brougham founded University College in London, where those excluded from Oxford and Cambridge could obtain a real education without religious tests for students or faculty and without any theology in the curriculum. Science received an important role in this school. Then in competition the Church of England founded King’s College in London in 1828, and in 1836 these two London colleges were joined into the University of London. This new University admitted women for degrees in 1878 and has served as the model for all the universities founded in England since its establishment.
It is worthy of note that as late as 1860 Thomas Huxley (1825–1895) had to defend at Oxford Darwin’s and, more generally, science’s right to do research and to profess the results of this research without reference to the theological implications of these results. Trevelyan says:
Science was not yet a part of the established order. In the past, most investigation had been done by individuals like Priestley or Darwin working at their own expense and on their own account, and this era of private initiation was only gradually passing into the new era of endowed and organized research. In the fifties the Natural Science Tripos had been set up at Cambridge, largely owing to the intelligent patronage of the Chancellor of the University, the Prince Consort, whose royal presence was able to charm away opposition to new-fangled learning in just those academic quarters which were most obscurantist. In the sixties science was making itself felt as a power in the land. And so long as it was still struggling for freedom and recognition, with the word “evolution” inscribed on its banner militant, it could not fail to exert an influence favourable in a broad sense to liberal reform.2
Curiously, the Presbyterian Church in Scotland did not suppress free thought and was eager to provide education to all. Thus the university scene in Scotland was entirely different from that in England and Wales. In 1904 Kelvin, upon his installation as Chancellor of the University of Glasgow, was able to say: “The University of Adam Smith, James Watt, Thomas Reid was never stagnant. For two centuries and a quarter it has been very progressive. Nearly two centuries ago it had a laboratory of human anatomy. Seventy-five years ago it had the first chemical students’ laboratory. Sixty-five years ago it had the first Professorship of engineering in the British empire. Fifty years ago it had the first physical students’ laboratory—a deserted wine cellar of an old profesorial house, enlarged a few years later by the annexation of a deserted examination room. Thirty-four years ago … it acquired laboratories of physiology and zoology.”3
Clearly the creation of a chair in experimental physics and the calling of Maxwell to it was a most signal event in English education. It is not possible to conceive of a more remarkable figure than Maxwell to hold the Cavendish Chair. (He even edited Cavendish’s researches in electricity, which had been largely unpublished since doing this was of no interest to Cavendish.) We say more about him below. It was he who fitted up the Cavendish Laboratory and got experimental physics started in England. It was his successor, Lord Rayleigh (John William Strutt, 3d Baron Rayleigh, 1842–1919), who undertook to establish physics instruction in the thoroughgoing manner characteristic of all his activities. It was due to Rayleigh and his assistants that modern laboratory courses in heat, electricity, magnetism, matter, light, and sound were established on sound footings.
Neither Maxwell nor Rayleigh regarded the chair in physics as a sinecure as did Babbage his in mathematics. The holder of the professorship was expected to be in residence eighteen weeks during each academic year and to give at least forty lectures during this period.4 Contrast this with Babbage, who was never in residence and never lectured at the university.
Thus we see at least faintly the differences between the traditions in astronomy and in mathematical physics. In astronomy there was a longstanding tradition of large-scale calculation of very extensive tables to high accuracy and with great precision. In physics there was as yet little tradition for calculation, and in the future as it developed the need arose for rather smaller and more ad hoc—what are now called in slang “quick and dirty”—calculations. The physicist, to make a too-sweeping generalization, generally did not need the very large tables of the astronomer. Instead he needed much less accurate solutions of the differential equations of motion which described various phenomena. These differences were to cause the physicists to go off on an entirely different tack from the astronomers. The two groups did not come together until modern times with the advent of the modern, general-purpose computer.
In this connection it is interesting to read Maxwell’s views on calculation. He said in an address to the British Association:
I do not here refer to the fact that all quantities, as such, are subject to the rules of arithmetic and algebra, and are therefore capable of being submitted to those dry calculations which represent, to so many minds their only idea of mathematics.
The human mind is seldom satisfied, and is certainly never exercising its highest functions, when it is doing the work of a calculating machine. What the man of science, whether he is a mathematician or a physical inquirer, aims at is to acquire and develope clear ideas of the things he deals with. For this purpose he is willing to enter on long calculations, and to be for a season a calculating machine, if he can only at last make his ideas clearer.5
Before ending our brief description of the English university scene perhaps we can illustrate the problem by saying some words about George Boole (1815–1864), who was one generation after Babbage. Boole was born in Lincoln of lower middle-class parents and was in a stratum of society from which children did not go to university in spite of their religion. Instead, he, like Abraham Lincoln, six years his senior, educated himself. He trained himself in Latin and Greek, and his father, remarkably enough, was able to start him in mathematics. At age sixteen he needed to go to work to support his parents and took a job as an usher (an assistant teacher) in a school; in fact he taught in two schools for four years. During this period he taught himself French, German, and Italian in preparation for a career in the Church. Bell amusingly says: “In spite of all that has been said for and against God, it must be admitted even by his severest critics that he has a sense of humor. Seeing the ridiculousness of George Boole’s ever becoming a clergyman, he skilfully turned the young man’s eager ambition into less preposterous channels.”6
When he was twenty Boole opened his own school and taught himself mathematics—indeed, higher mathematics. He managed to read and to digest all the great masters of his time. This lonely study produced great results, which came about through the good offices of a Scot, D. F. Gregory, who was the editor of the Cambridge Mathematical Journal and was therefore able to bring Boole’s work before the mathematical world. The magnitude of his total contribution may be judged by Bertrand Russell’s summation: “Pure mathematics was discovered by George Boole in his work published in 1854.”
One of his important contributions, from our point of view, was to the so-called calculus of finite differences. This apparatus is the basic tool of the numerical analyst, and Boole’s accomplishments in helping to build it are certainly far from negligible.
But most important is his contribution to formal logics. In 1848 he published a little book entitled The Mathematical Analysis of Logic, which was to be the prelude to his great work in 1854: An Investigation of the Laws of Thought, on which are founded the Mathematical Theories of Logic and Probabilities.7 In these works he rendered logics into a precise and mathematical form. He set down postulates or axioms for logics for the first time—just as Euclid and others had done for geometry. Further, he gave to the whole subject an algebraic treatment. He said in his first chapter:
1. The design of the following treatise is to investigate the fundamental laws of those operations of the mind by which reasoning is performed: to give expression to them in the symbolical language of a Calculus, and upon this foundation to establish the science of Logic and instruct its method; … and finally to collect from the various elements of truth brought to view in the course of these inquiries some probable intimations concerning the nature and constitution of the human mind.
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3. But although certain parts of the design of this work have been entertained by others its general conception, its method, and to a considerable extent, its results, are believed to be original. For this reason I shall offer, in the present chapter, some preparatory statements and explanations, in order that the real aim of this treatise may be understood, and the treatment of its subject facilitated.
It is designed, in the first place, to investigate the fundamental laws of those operations of the mind by which reasoning is performed. It is unnecessary to enter here into any argument to prove that the operations of the mind are in a certain real sense subject to laws, and that a science of the mind is therefore possible. If these are questions which admit of doubt, that doubt is not to be met by an endeavour to settle the point of dispute a priori, but by directing the attention of the objector to the evidence of actual laws, by referring him to an actual science. And thus the solution of that doubt would belong not to the introduction to this treatise, but to the treatise itself….
4. Like all other sciences, that of the intellectual operations must primarily rest upon observations, —the subject of such observation being the very operations and processes of which we desire to determine the laws….
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6…. There is not only a close analogy between the operations of the mind in general reasoning and its operations in the particular science of Algebra, but there is to a considerable extent an exact agreement in the laws by which the two classes of operations are conducted….
Now the actual investigations of the following pages exhibit Logic, in its practical aspect, as a system of processes carried on by the aid of symbols having a definite interpretation, and subject to laws founded upon that interpretation alone. But at the same time they exhibit those laws as identical in form with the general symbols of algebra, with this single addition, viz., that the symbols of Logic are further subject to a special law (Chapter II), to which the symbols of quantity, as such, are not subject….8
The monumental work of Boole was to remain a curiosity for many years, and it was not until Whitehead and Russell wrote their great Principia Mathematica in 1910–13 that serious mathematicians took up formal logics. Since then the field has flowered into the stupendous achievements of Gödel and Cohen in our era.
In any case Boole’s contribution to logics made possible the works of subsequent logicians including Turing and von Neumann. As we shall see later, the work of von Neumann was essential to the modern computer. Even Babbage depended a great deal on Boole’s ideas—as well as those of de Morgan, Herschel, and Peacock—for his understanding of what mathematical operations really are. We mentioned before how Babbage understood the notion of a mathematical operation and the quantities upon which it operated. This was made possible for the first time in this period by this group of English algebraists.
Since Boole showed that logics can be reduced to very simple algebraic systems—known today as Boolean Algebras—it was possible for Babbage and his successors to design organs for a computer that could perform the necessary logical tasks. Thus our debt to this simple, quiet man, George Boole, is extraordinarily great and probably not adequately repaid. However, it is nice to know that he was befriended by de Morgan and that in 1849 he was called to a chair in mathematics of the newly-founded Queen’s College in Cork, Ireland.
His remark about a “special law to which the symbols of quantity are not subject” is very important: this law in effect is that x2 = x for every x in his system. Now in numerical terms this equation or law has as its only solution 0 and 1. This is why the binary system plays so vital a role in modern computers: their logical parts are in effect carrying out binary operations.
In Boole’s system 1 denotes the entire realm of discourse, the set of all objects being discussed, and 0 the empty set. There are two operations in this system which we may call + and ×, or we may say or and and. It is most fortunate for us that all logics can be comprehended in so simple a system, since otherwise the automation of computation would probably not have occurred—or at least not when it did. We need to say much more on this point and will do so a little later when it fits in more naturally against a larger background of understanding.
1 A. J. P. Taylor, English History, 1914–1945, Oxford History of England (New York and Oxford, 1965), p. 171.
2 Trevelyan, British History in the Nineteenth Century, p. 342.
3 Kelvin, Mathematical and Physical Papers, vol. VI (Cambridge, 1911), p. 374.
4 See the Introduction by R. Lindsay, p. viii, to Lord Rayleigh, The Theory of Sound (New York, 1945).
5 J. C. Maxwell, Address to the Mathematical and Physical Sections of the British Association, British Association Report, vol. XL (1870), in Collected Works, II, 215.
6 Bell, Men of Mathematics, p. 436.
7 Reprint edition, Laws of Thought (New York, 1953), the edition quoted here.
8 Boole, Laws of Thought, chap. 1.