**Irving John Good**, whose name was originally Isidore Jacob Gudak, was known to his friends and colleagues as Jack. His parents were Morris Edward Good (1885-1958) (also known by the Yiddish name Moshe Oved) and Sophia Polikoff. Morris Good was born in Poland at a time when it was part of the Russian Empire, He learnt the trade of a watchmaker before emigrating to England at the age of seventeen. In London he set up an antique jewellery shop. He wrote the autobiography

*Visions and Jewels*(London, 1952) under the name Moshe Oved and is described by his publisher as an:-

Sophia Polikoff was born in Russia and came to London at age eight with her parents. Morris and Sophia met in London and married. Jack Good showed remarkable mathematical talents when he was still a child. For example when he was nearly ten years old he caught diphtheria and was confined to bed for around six weeks. He occupied himself calculating the square root of 2 to about 12 decimal places. Quickly realising that the calculation would never terminate, he found rational approximations from Pell's equation... author, actor, jeweller, Zionist[and]also the founder of the Ben Uri Gallery in London.

*x*

^{2}- 2

*y*

^{2}= 1, discovering this method for himself. He then was able to prove that √2 could not be rational by discovering for himself the standard 'reductio ad absurdum' proof.

Jack attended Haberdashers' Aske's school in Hampstead from 1928 to 1935. He was encouraged by his mathematics teachers to go well beyond what they were teaching the other boys. At the age of thirteen he did the puzzles in Henry Dudeney's book *Amusements in Mathematics* and as a result of one of the problems, he discovered for himself the principle of induction. He continued to read mathematics books, for example when he was older he read Joseph Edward's *Differential Calculus with Applications and Numerous Examples* and G H Hardy's *Pure Mathematics*. Good entered Jesus College, Cambridge where he attended lectures by A E Ingham, J C Burkill, F P White and G H Hardy. His undergraduate tutor was L A Pars who Good described in [1] saying:-

As an undergraduate Good was extremely shy so, he said [1]:-He was a splendid mathematician and an excellent tutor. He later became the master of the college and he wrote a very well received book on classical mechanics published after my time. He always had slick proofs, but they were too slick. He taught us quite a bit about complex variables.

He played a lot of chess, particularly with John Francis O'Donovan who played board one for Ireland. Good won the Cambridgeshire Chess Championship in 1939. After being awarded a B.A. in 1938 he went on to do research, supervised first by A S Besicovitch and then by G H Hardy. He was awarded a Smith's Prize in 1940 for his essay on fractional dimensions of sets of simple continued fractions, and he received his doctorate in 1941 for his thesis... I didn't enjoy life as much as I might have done at that age.

*The topological concept of partial dimension based on the ideas of Henri Lebesgue*. His first publications appeared at this time:

*The approximate local monotony of measurable functions*(1940),

*The fractional dimensional theory of continued fractions*(1941), and

*Note on the summation of a classical divergent series*(1941).

By this time World War II had been in progress for some considerable time but as a mathematician Good had not been called up but put on the reserve list. He was given the choice of working on radar or at Bletchley Park and chose to work at Bletchley. One of those who interviewed him for the position at Bletchley was Hugh Alexander, who had twice been British Chess Champion and who Good knew, having played chess with him. Hugh Alexander became his boss at Bletchley when he took up his appointment in May 1941. Good said [1]:-

At first Good worked in Hut 8 headed by Alan Turing but after a year Turing moved to a different area and Hugh Alexander became head of Hut 8. In 1943 Good moved to the "Newmanry", working under Max Newman. He also worked with Donald Michie and later was joined by Henry Whitehead [5]:-Bletchley Park liked chess players - they believed that chess players and mathematicians had an aptitude for cryptanalysis. Hugh's closest friend was probably Stuart Milner-Barry, another chess master. I met him at a chess match a week or two before going to Bletchley, and I asked "Are you working on German ciphers?" and he said "No, my address is Room47, Foreign Office," but when I arrived at Bletchley Park I found, sure enough, that he was working on German ciphers.

Good described other aspects of life at Bletchley Park [1]:-In1943Good was one of a small group who helped specify Mark2of a large-scale(classified)binary electronic digital computer called Colossus. Colossus was the first large-scale digital computer but not quite general-purpose. ... The main users at first were Newman, Michie and Good, and some months later about twenty mathematicians. Good was a main user and produced more than half the theory for the use of the Colossi.

After the war ended, Good was appointed as a lecturer in Pure Mathematics at the University of Manchester working for Max Newman who had been appointed as professor. While at Manchester, Good was also involved in the project to develop an electronic computer. He remained there until 1948 when he took up a position at GCHQ (Government Communications Headquarters). During his years at Manchester he began publishing again having published little while working at Bletchley Park. For example:I played chess and Go and had intellectual and mathematical discussions with Alan Turing and David Rees. For example, Turing and I discussed the possibility of machine intelligence and automatic chess.

*On the regularity of a general method of summation*(1946),

*Normal recurring decimals*(1946),

*A note on positive determinants*(1947), and (with Gerd Edzard Harry Reuter)

*Bounded integral transforms*(1948).

One reason for Good leaving Manchester was that his book *Probability and the Weighing of Evidence* had been rejected by a publisher. However it was accepted by Charles Griffin & Co., Ltd., London and appeared in 1950. A H Copeland begins his review of the book as follows:-

He ends with the comment:-This book is concerned with the foundations of the theory of probability and an analysis of scientific inductive reasoning. The author produces rather convincing evidence to support the contention that every application of the theory of probability is based on some probability which is subjectively estimated. He develops a subjective theory in which probability is interpreted as a degree of belief depending on the state of mind of the individual doing the believing. For a body of beliefs to be reasonable the degrees of belief are required to possess a partial order which is invariant under changes in state of mind. Moreover when an individual assigns numerical probabilities to his beliefs then the partial order of the degrees of belief must agree with the numerical order of probabilities.

Good worked at GCHQ from 1948 until 1959. During this time he spent the summer of 1955 as a Visiting Research Associate Professor at Princeton and during 1958-59 worked for a few weeks as a consultant to IBM. During his years at GCHQ Good continued to publish papers such asThis book is interesting reading whether or not one agrees with the author. It is a significant contribution to a field in which contributions are badly needed.

*Rational decisions*(1952) which, Jimmie Savage writes:-

Examples of his many papers published during this time are:... deals first with the relationship between the theory of probability and the theory of rational behaviour. A method is then suggested for encouraging people to make accurate probability estimates, a connection with the theory of information being mentioned.

*Conjectures concerning the Mersenne numbers*(1955),

*The likelihood ratio test for Markoff chains*(1955), and

*Some terminology and notation in information theory*(1956). He continued to play chess at a very high level being runner-up in the West of England Chess Championship of 1958. In 1959 Good resigned from his position at GCHQ when offered a chair at the University of Chicago. However, he changed his mind, deciding not to accept the position. After working for IBM in the United States for a few weeks, he was then appointed to the Admiralty Research Laboratory in England where he worked for three years until 1962. At that time he returned to the United States and worked at the Communications Research Division of the Institute for Defense Analysis for two years. During this time he worked on writing the book

*The estimation of probabilities. An essay on modern Bayesian methods*which was published in 1965. Jimmie Savage wrote an interesting review which tells us much about Good's approach to probability:-

In 1964 Good returned to England, and he also returned to the academic life, when he accepted an offer from John Hammersley of a three-year appointment as a Senior Research Fellow at Trinity College, Oxford, and at the Atlas Computer Laboratory. The fellowship was funded by the Science Research Council. During this time he publishedThis book reviews, from the author's provocative and unusual viewpoints, the general and pervasive statistical problem of estimating probabilities from data too scant to permit the usual large-sample employment of success ratios and standard deviations inferred from them. Perhaps problems of estimating a single probability from scant data could be discounted as academic. But, as the author emphasizes, science and ordinary life constantly confront us with such problems as estimation of the probability associated with one of thousands or millions of entries in a contingency table on the basis of data, which, though absolutely extensive, may involve far fewer observations than there are entries in the table. Such problems are hard, but are generally felt to be important and not insurmountable. They can, however, be met only by exploring some sort of hypotheses of analogy or regularity. There are many different hypotheses that might be entertained, depending on the particular structure and context of the problem. The author has heretofore, and in this monograph, made a great diversity of attacks on scant-data problems, which he sees as not only important in themselves, but as a challenge to one's philosophy of statistics. His own present philosophy is an eclectic one in which subjective, logical, and frequency probabilities are all held to be real and pertinent to statistics. It emphasizes the impossibility of perfection: "In our theories, we rightly search for unification, but real life is both complicated and short, and we make no mockery of honest adhockery"

*Speculations Concerning the First Ultraintelligent Machine*(1965). In this he wrote:-

At the end of the three-year fellowship at Oxford, in July 1967, he took up an appointment as Research Professor of Statistics at Virginia Polytechnic Institute and State University. He said [1]:-Let an ultraintelligent machine be defined as a machine that can far surpass all the intellectual activities of any man however clever. Since the design of machines is one of these intellectual activities, an ultraintelligent machine could design even better machines; there would then unquestionably be an 'intelligence explosion,' and the intelligence of man would be left far behind. Thus the first ultraintelligent machine is the last invention that man need ever make.

In November 1969 he was made a University Distinguished Professor and remained at Virginia Polytechnic Institute throughout the rest of his career. We mention two further books: (jointly with David B Osteyee)I arrived in Blacksburg in the seventh hour of the seventh day of the seventh month of year seven of the seventh decade, and I was put in apartment7of block7of Terrace View Apartments, all by chance. I seem to have had more than my fair share of coincidences.

*Information, weight of evidence, the singularity between probability measures and signal detection*(1974) and

*Good thinking. The foundations of probability and its applications*(1983). This last mentioned book collects together many of Good's more philosophical and less mathematical papers. In a review, D V Lindley writes:-

Good retired and was made Professor Emeritus in July 1994. Throughout his career he received a large number of honours for his remarkable contributions. We list only a few: Outstanding Educators of America (1970), Horsley Prize (Virginia Academy of Science) (shared with his student R A Gaskins) (1972), Member of New York Academy of Sciences (1974), Fellow of American Academy of Arts and Sciences (8 May 1985), Honorary Member of the International Statistical Institute (1990), Fellow of Virginia Academy of Science (1993), International Order of Merit (1993), Computer Pioneer Award (Medal) from the Computer Society of IEEE (1998), American Medal of Honor (2002), Congressional Medal of Excellence (2004), and Honorary Fellow of the Royal Statistical Society (2004).In the introduction the author says: "clear reasoning about many important practical and philosophical questions is impossible except in terms of probability". The thinking of the title is concerned with the relationship between mathematics and the real world; it is about "applicable philosophy". Mathematicians, quite properly, concentrate on the manipulative aspect of the subject. The author, in this book, is interested in how the results of those manipulations in the probability calculus can be used in practical circumstances. Thus, to a mathematician, a probability is a real number: the author is concerned with how that number can be found. Those who agree with the quote that begins this paragraph will find this book important. Here we have a marvellous opportunity to read what one of the major thinkers on probability has to say about many topics.

Good died of natural causes at the age of 92. A memorial service was held on 19 April 2009 at the Blacksburg Jewish Community Center in Blacksburg, Virginia.

**Article by:** *J J O'Connor* and *E F Robertson*

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