## 1. Outline of Life




Alan Turing's short and extraordinary life has attracted wide interest.
It has inspired his mother's memoir (E. S. Turing 1959), a detailed
biography (Hodges 1983), a play and television film (Whitemore 1986),
and various other works of fiction and art. 




There are many reasons for this interest, but one is that in every
sphere of his life and work he made unexpected connections between
apparently unrelated areas. His central contribution to science and
philosophy came through his treating the subject of symbolic logic as a
new branch of applied mathematics, giving it a physical and engineering
content. Unwilling or unable to remain within any standard role or
department of thought, Alan Turing continued a life full of
incongruity. Though a shy, boyish, man, he had a pivotal role in world
history through his role in Second World War cryptology. Though the
founder of the dominant technology of the twentieth century, he
variously impressed, charmed or disturbed people with his unworldly
innocence and his dislike of moral or intellectual compromise.




Alan Mathison Turing was born in London, 23 June 1912, to
upper-middle-class British parents. His schooling was of a traditional
kind, dominated by the British imperial system, but from earliest life
his fascination with the scientific impulse--expressed by him as
finding the 'commonest in nature'--found him at odds
with authority. His scepticism, and disrespect for worldly values, were
never tamed and became ever more confidently eccentric. His moody
humour swung between gloom and vivacity. His life was also notable as
that of a gay man with strong emotions and a growing insistence on his
identity.




His first true home was at King's College, Cambridge University,
noted for its progressive intellectual life centred on J. M. Keynes.
Turing studied mathematics with increasing distinction and was elected
a Fellow of the college in 1935. This appointment was followed by a
remarkable and sudden debut in an area where he was an unknown
figure: that of mathematical logic. The paper "On Computable
Numbers..." (Turing 1936-7) was his first and perhaps greatest
triumph. It gave a definition of computation and an absolute limitation
on what computation could achieve, which makes it the founding work of
modern computer science. It led him to Princeton for more advanced work
in logic and other branches of mathematics. He had the opportunity to
remain in the United States, but chose to return to Britain in 1938,
and was immediately recruited for the British communications war.




From 1939 to 1945 Turing was almost totally engaged in the mastery
of the German enciphering machine, Enigma, and other cryptological
investigations at now-famous Bletchley Park, the British government's
wartime communications headquarters. Turing made a unique logical
contribution to the decryption of the Enigma and became the chief
scientific figure, with a particular responsibility for reading the
U-boat communications. As such he became a top-level figure in
Anglo-American liaison, and also gained exposure to the most advanced
electronic technology of the day.




Combining his ideas from mathematical logic, his experience in
cryptology, and some practical electronic knowledge, his ambition, at
the end of the war in Europe, was to create an electronic computer in
the full modern sense. His plans, commissioned by the National Physical
Laboratory, London, were overshadowed by the more powerfully supported
American projects. Turing also laboured under the disadvantage that his
wartime achievements remained totally secret. His ideas led the field
in 1946, but this was little recognised. Frustrated in his work, he
emerged as a powerful marathon runner, and almost qualified for the
British team in the 1948 Olympic games.




Turing's motivations were scientific rather than industrial or
commercial, and he soon returned to the theoretical limitations of
computation, this time focussing on the comparison of the power of
computation and the power of the human brain. His contention was that
the computer, when properly programmed, could rival the brain. It
founded the 'Artificial Intelligence' program of coming
decades.




In 1948 he moved to Manchester University, where he partly fulfilled
the expectations placed upon him to plan software for the pioneer
computer development there, but still remained a free-ranging thinker.
It was here that his famous 1950 paper, "Computing Machinery and
Intelligence," (Turing 1950b) was written. In 1951 he was elected a
Fellow of the Royal Society for his 1936 achievement, yet at the same
time he was striking into entirely new territory with a mathematical
theory of biological morphogenesis (Turing 1952).




This work was interrupted by Alan Turing's arrest in February 1952
for his sexual affair with a young Manchester man, and he was obliged,
to escape imprisonment, to undergo the injection of oestrogen intended
to negate his sexual drive. He was disqualified from continuing secret
cryptological work. His general libertarian attitude was enhanced
rather than suppressed by the criminal trial, and his intellectual
individuality also remained as lively as ever. While remaining formally
a Reader in the Theory of Computing, he not only embarked on more
ambitious applications of his biological theory, but advanced new ideas
for fundamental physics.




For this reason his death, on 7 June 1954, at his home in Wilmslow,
Cheshire, came as a general surprise. In hindsight it is obvious that
Turing's unique status in Anglo-American secret communication work
meant that there were pressures on him of which his contemporaries were
unaware; there was certainly another 'security' conflict
with government in 1953 (Hodges 1983, p. 483). Some commentators, e.g.
Dawson (1985), have argued that assassination should not be ruled out.
But he had spoken of suicide, and his death, which was by cyanide
poisoning, was most likely by his own hand, contrived so as to allow
those who wished to do so to believe it a result of his penchant for
chemistry experiments. The symbolism of its dramatic element--a
partly eaten apple--has continued to haunt the intellectual Eden
from which Alan Turing was expelled.


## 2. The Turing Machine and Computability




Alan Turing drew much between 1928 and 1933 from the work of the
mathematical physicist and populariser A. S. Eddington, from J. von
Neumann's account of the foundations of quantum mechanics, and then
from Bertrand Russell's mathematical logic. Meanwhile, his lasting
fascination with the problems of mind and matter was heightened by
emotional elements in his own life (Hodges 1983, p. 63). In 1934 he
graduated with an outstanding degree in mathematics from Cambridge
University, followed by a successful dissertation in probability theory
which won him a Fellowship of King's College, Cambridge, in 1935. This
was the background to his learning, also in 1935, of the problem which
was to make his name. 




It was from the lectures of the topologist M. H. A. (Max) Newman in
that year that he learnt of Godel's 1931 proof of the formal
incompleteness of logical systems rich enough to include arithmetic,
and of the outstanding problem in the foundations of mathematics as
posed by Hilbert: the "Entscheidungsproblem" (decision problem). Was
there a method by which it could be decided, for any given mathematical
proposition, whether or not it was provable?




The principal difficulty of this question lay in giving an
unassailably correct and general definition of what was meant by such
expressions as 'definite method' or 'effective
procedure.' Turing worked on this alone for a year until April
1936; independence and isolation was to be both his strength, in
formulating original ideas, and his weakness, when it came to promoting
and implementing them.




The word 'mechanical' had often been used of the
formalist approach lying behind Hilbert's problem, and Turing seized on
the concept of the *machine.*Turing's solution lay in defining
what was soon to be named the *Turing machine.* With this he
defined the concept of 'the mechanical' in terms of simple
atomic operations. The Turing machine formalism was modelled on the
teleprinter, slightly enlarged in scope to allow a paper tape that
could move in both directions and a 'head' that could read,
erase and print new symbols, rather than only read and punch permanent
holes.




The Turing machine is 'theoretical,' in the sense that
it is not intended actually to be engineered (there being no point in
doing so), although it is essential that its atomic components (the
paper tape, movement to left and right, testing for the presence of a
symbol) are such as *could* actually be implemented. The whole
point of the formalism is to reduce the concept of 'method'
to simple operations that can unquestionably be
'effected.'




Nevertheless Turing's purpose was to embody the most general
mechanical process as carried out by a *human being.* His
analysis began not with any existing computing machines, but with the
picture of a child's exercise book marked off in squares. From the
beginning, the Turing machine concept aimed to capture what the
*human mind* can do when carrying out a procedure.




In speaking of 'the' Turing machine it should be made
clear that there are *infinitely many* Turing machines, each
corresponding to a different method or procedure, by virtue of having a
different 'table of behaviour.' Nowadays it is almost
impossible to avoid imagery which did not exist in 1936: that of the
computer. In modern terms, the 'table of behaviour' of a
Turing machine is equivalent to a computer program.




If a Turing machine corresponds to a computer program, what is the
analogy of the computer? It is what Turing described as a
*universal* machine (Turing 1936, p. 241). Again, there are
*infinitely many* universal Turing machines, forming a subset of
Turing machines; they are those machines with 'tables of
behaviour' complex enough to read the tables of other Turing
machines, and then do what those machines would have done. If this
seems strange, note the modern parallel that any computer can be
simulated by software on another computer. The way that tables can read
and simulate the effect of other tables is crucial to Turing's theory,
going far beyond Babbage's ideas of a hundred years earlier. It also
shows why Turing's ideas go to the heart of the modern computer, in
which it is essential that programs are themselves a form of data which
can be manipulated by other programs. But the reader must always
remember that in 1936 there were no such computers; indeed the modern
computer arose *out of* the formulation of 'behaving
mechanically' that Turing found in this work.




Turing's machine formulation allowed the precise definition of the
*computable:* namely, as what can be done by a Turing machine
acting alone. More exactly, computable operations are those which can
be effected by what Turing called *automatic* machines. The
crucial point here is that the action of an automatic Turing machine is
totally determined by its 'table of behaviour'. (Turing
also allowed for 'choice machines' which call for human
inputs, rather than being totally determined.) Turing then proposed
that this definition of 'computable' captured precisely
what was intended by such words as 'definite method, procedure,
mechanical process' in stating the
*Entscheidungsproblem.*




In applying his machine concept to the
*Entscheidungsproblem,* Turing took the step of defining
*computable numbers.* These are those real numbers, considered
as infinite decimals, say, which it is possible for a Turing machine,
starting with an empty tape, to print out. For example, the Turing
machine which simply prints the digit 1 and moves to the right, then
repeats that action for ever, can thereby compute the number
.1111111... A more complicated Turing machine can compute the
infinite decimal expansion of p.




Turing machines, like computer programs, are countable; indeed they
can be ordered in a complete list by a kind of alphabetical ordering of
their 'tables of behaviour'. Turing did this by encoding
the tables into 'description numbers' which can then be
ordered in magnitude. Amongst this list, a subset of them (those with
'satisfactory' description numbers) are the machines which
have the effect of printing out infinite decimals. It is readily shown,
using a 'diagonal' argument first used by Cantor and
familiar from the discoveries of Russell and Godel, that there can
be no Turing machine with the property of deciding whether a
description number is satisfactory or not. The argument can be
presented as follows. Suppose that such a Turing machine exists. Then
it is possible to construct a new Turing machine which works out in
turn the Nth digit from the Nth machine possessing a satisfactory
description number. This new machine then prints an Nth digit differing
from that digit. As the machine proceeds, it prints out an infinite
decimal, and therefore has a 'satisfactory' description
number. Yet this number must by construction differ from the outputs of
every Turing machine with a satisfactory description number. This is a
contradiction, so the hypothesis must be false (Turing 1936, p. 246).
From this, Turing was able to answer Hilbert's
*Entscheidungsproblem* in the negative: there can be no such
general method.




Turing's proof can be recast in many ways, but the core idea depends
on the *self-reference* involved in a machine operating on
symbols, which is itself described by symbols and so can operate on its
own description. Indeed, the self-referential aspect of the theory can
be highlighted by a different form of the proof, which Turing preferred
(Turing 1936, p. 247). Suppose that such a machine for deciding
satisfactoriness does exist; then apply it to its own description
number. A contradiction can readily be obtained. However, the
'diagonal' method has the advantage of bringing out the
following: that a real number may be *defined* unambiguously,
yet be *uncomputable.* It is a non-trivial discovery that
whereas some infinite decimals (e.g. p) may be encapsulated in a
finite table, other infinite decimals (in fact, almost all) cannot.
Likewise there are decision problems such as 'is this number
prime?' in which infinitely many answers are wrapped up in a
finite recipe, while there are others (again, almost all) which are
not, and must be regarded as requiring infinitely many different
methods. 'Is this a provable proposition?' belongs to the
latter category.




This is what Turing established, and into the bargain the remarkable
fact that anything that *is* computable can in fact be computed
by *one* machine, a universal Turing machine.




It was vital to Turing's work that he justified the definition by
showing that it encompassed the most general idea of
'method'. For if it did not, the
*Entscheidungproblem* remained open: there might be some more
powerful type of method than was encompassed by Turing computability.
One justification lay in showing that the definition included many
processes a mathematician would consider to be natural in computation
(Turing 1936, p. 254). Another argument involved a human calculator
following written instruction notes. (Turing 1936, p. 253). But in a
bolder argument, the one he placed first, he considered an
'intuitive' argument appealing to the *states of
mind* of a human computer. (Turing 1936, p. 249). The entry of
'mind' into his argument was highly significant, but at
this stage it was only a mind following a rule.




To summarise: Turing found, and justified on very general and
far-reaching grounds, a precise mathematical formulation of the
conception of a general process or method. His work, as presented to
Newman in April 1936, argued that his formulation of
'computability' encompassed 'the possible processes
which can be carried out in computing a number.' (Turing 1936,
p. 232). This opened up new fields of discovery both in practical
computation, and in the discussion of human mental processes. However,
although Turing had worked as what Newman called 'a confirmed
solitary' (Hodges 1983, p 113), he soon learned that he was not
alone in what Gandy (1988) has called 'the confluence of ideas in
1936.'




The Princeton logician Alonzo Church had slightly outpaced Turing in
finding a satisfactory definition of what he called 'effective
calculability.' Church's definition required the logical
formalism of the *lambda-calculus.* This meant that from the
outset Turing's achievement merged with and superseded the formulation
of Church's Thesis, namely the assertion that the
lambda-calculus formalism correctly embodied the concept of effective
process or method. Very rapidly it was shown that the mathematical
scope of Turing computability coincided with Church's definition (and
also with the scope of the *general recursive functions* defined
by Godel). Turing wrote his own statement (Turing 1939, p. 166) of
the conclusions that had been reached in 1938; it is in the Ph.D.
thesis that he wrote under Church's supervision, and so this statement
is the nearest we have to a joint statement of the 'Church-Turing
thesis':



> A function is said to be 'effectively
> calculable' if its values can be found by some purely mechanical
> process. Although it is fairly easy to get an intuitive grasp of this
> idea, it is nevertheless desirable to have some more definite,
> mathematically expressible definition. Such a definition was first
> given by Godel at Princeton in 1934... These functions were
> described as 'general recursive' by Godel...
> Another definition of effective calculability has been given by
> Church... who identifies it with lambda-definability. The author
> [i.e. Turing] has recently suggested a definition corresponding more
> closely to the intuitive idea... It was stated above that 'a
> function is effectively calculable if its values can be found by a
> purely mechanical process.' We may take this statement literally,
> understanding by a purely mechanical process one which could be carried
> out by a machine. It is possible to give a mathematical description, in
> a certain normal form, of the structures of these machines. The
> development of these ideas leads to the author's definition of a
> computable function, and to an identification of computability with
> effective calculability. It is not difficult, though somewhat
> laborious, to prove that these three definitions are
> equivalent.




Church accepted that Turing's definition gave a compelling, intuitive
reason for why Church's thesis was true. The recent exposition by
Davis (2000) emphasises that Godel also was convinced by Turing's
argument that an absolute concept had been identified (Godel
1946). The situation has not changed since 1937. (For further
comment, see the article on the
 Church-Turing Thesis. 
The recent selection of Turing's papers edited by Copeland (2004),
and the review of Hodges (2006), continue this discussion.)




Turing himself did little to evangelise his formulation in the world
of mathematical logic and early computer science. The textbooks of
Davis (1958) and Minsky (1967) did more. Nowadays Turing computability
is often reformulated (e.g. in terms of 'register
machines'). However, computer simulations (e.g.,
 Turing's World,
 from Stanford) have brought Turing's
original imagery to life.




Turing's work also opened new areas for decidability questions
within pure mathematics. From the 1970s, Turing machines also took on
new life in the development of *complexity theory,* and as such
underpin one of the most important research areas in computer science.
This development exemplifies the lasting value of Turing's special
quality of giving concrete illustration to abstract concepts.


## 3. The Logical and the Physical




As put by Gandy (1988), Turing's paper was 'a paradigm of
philosophical analysis,' refining a vague notion into a precise
definition. But it was more than being an analysis *within* the
world of mathematical logic: in Turing's thought the question that
constantly recurs both theoretically and practically is the
relationship of the logical Turing machine to the physical world. 




'Effective' means *doing,* not merely imagining
or postulating. At this stage neither Turing nor any other logician
made a serious investigation into the physics of such
'doing.' But Turing's image of a teleprinter-like machine
does inescapably refer to something that could actually be physically
'done.' His concept is a distillation of the idea that one
can only 'do' one simple action, or finite number of simple
actions, at a time. How 'physical' a concept is it?




The tape never holds more than a finite number of marked squares at
any point in a computation. Thus it can be thought of as being finite,
but always capable of further extension as required. Obviously this
unbounded extendibility is unphysical, but the definition is still of
practical use: it means that anything done on a finite tape, however
large, is computable. (Turing himself took such a finitistic approach
when explaining the practical relevance of computability in his 1950
paper.) One aspect of Turing's formulation, however, involves absolute
finiteness: the table of behaviour of a Turing machine must be finite,
since Turing allows only a finite number of
'configurations' of a Turing machine, and only a finite
repertoire of symbols which can be marked on the tape. This is
essentially equivalent to allowing only computer programs with finite
lengths of code.




'Calculable by finite means' was Turing's
characterisation of computability, which he justified with the argument
that 'the human memory is necessarily limited.' (Turing
1936, p. 231). The whole point of his definition lies in encoding
infinite potential effects, (e.g. the printing of an infinite decimal)
into finite 'tables of behaviour'. There would be no point
in allowing machines with infinite 'tables of behaviour'.
It is obvious, for instance, that any real number could be printed by
such a 'machine', by letting the Nth configuration be
'programmed' to print the Nth digit, for example. Such a
'machine' could likewise store any countable number of
statements about all possible mathematical expressions, and so make the
*Entscheidungsproblem* trivial.




Church (1937), when reviewing Turing's paper while Turing was in
Princeton under his supervision, actually gave a bolder
characterisation of the Turing machine as an *arbitrary finite
machine.*



> The author [i.e. Turing] proposes as a criterion that an
> infinite sequence of digits 0 and 1 be "computable" that it shall be
> possible to devise a computing machine, occupying a finite space and
> with working parts of finite size, which will write down the sequence
> to any desired number of terms if allowed to run for a sufficiently
> long time. As a matter of convenience, certain further restrictions are
> imposed on the character of the machine, but these are of such a nature
> as obviously to cause no loss of generality--in particular, a
> human calculator, provided with pencil and paper and explicit
> instructions, can be regarded as a kind of Turing machine.




Church (1940) repeated this characterisation. Turing neither
endorsed it nor said anything to contradict it, leaving the general
concept of 'machine' itself undefined. The work of Gandy
(1980) did more to justify this characterisation, by refining the
statement of what is meant by 'a machine.' His results
support Church's statement; they also argue strongly for the view that
natural attempts to extend the notion of computability lead to
trivialisation: if Gandy's conditions on a 'machine' are
significantly weakened then every real number becomes calculable (Gandy
1980, p. 130ff.). (For a different interpretation of Church's
statement, see the article on the
 Church-Turing Thesis.)




Turing did not explicitly discuss the question of the *speed*
of his elementary actions. It is left implicit in his discussion, by
his use of the word 'never,' that it is not possible for
infinitely many steps to be performed in a finite time. Others have
explored the effect of abandoning this restriction. Davies (2001), for
instance, describes a 'machine' with an infinite number of
parts, requiring components of arbitrarily small size, running at
arbitrarily high speeds. Such a 'machine' could perform
uncomputable tasks. Davies emphasises that such a machine cannot be
built in our own physical world, but argues that it could be
constructed in a universe with different physics. To the extent that it
rules out such 'machines', the Church-Turing thesis must
have at least some physical content.




True physics is quantum-mechanical, and this implies a different
idea of matter and action from Turing's purely classical picture. It is
perhaps odd that Turing did not point this out in this period, since he
was well versed in quantum physics. Instead, the analysis and practical
development of quantum computing was left to the 1980s. Quantum
computation, using the evolution of wave-functions rather than
classical machine states, is the most important way in which Turing
machine model has been challenged. The standard formulation of quantum
computing (Deutsch 1985, following Feynman 1982) does not predict
anything beyond computable effects, although within the realm of the
computable, quantum computations may be very much more efficient than
classical computations. It is possible that a deeper understanding of
quantum mechanical physics may further change the picture of what can
be physically 'done.'


## 4. The Uncomputable




Turing turned to the exploration of the *uncomputable* for his
Princeton Ph.D. thesis (1938), which then appeared as *Systems of
Logic based on Ordinals* (Turing 1939). 




It is generally the view, as expressed by Feferman (1988), that this
work was a diversion from the main thrust of his work. But from another
angle, as expressed in (Hodges 1997), one can see Turing's development
as turning naturally from considering the mind when following a rule,
to the action of the mind when *not* following a rule. In
particular this 1938 work considered the mind when seeing the truth of
one of Godel's true but formally unprovable propositions, and
hence going beyond rules based on the axioms of the system. As Turing
expressed it (Turing 1939, p. 198), there are 'formulae, seen
intuitively to be correct, but which the Godel theorem shows are
unprovable in the original system.' Turing's theory of
'ordinal logics' was an attempt to 'avoid as far as
possible the effects of Godel's theorem' by studying the
effect of adding Godel sentences as new axioms to create stronger
and stronger logics. It did not reach a definitive conclusion.




In his investigation, Turing introduced the idea of an
'oracle' capable of performing, as if by magic, an
uncomputable operation. Turing's oracle cannot be considered as some
'black box' component of a new class of machines, to be put
on a par with the primitive operations of reading single symbols, as
has been suggested by (Copeland 1998). An oracle is *infinitely more
powerful* than anything a modern computer can do, and nothing like
an elementary component of a computer. Turing defined
'oracle-machines' as Turing machines with an additional
configuration in which they 'call the oracle' so as to take
an uncomputable step. But these oracle-machines are *not purely
mechanical.* They are only partially mechanical, like Turing's
choice-machines. Indeed the *whole point* of the oracle-machine
is to explore the realm of what *cannot* be done by purely
mechanical processes. Turing emphasised (Turing 1939, p. 173):



> We shall not go any further into the nature of this oracle
> apart from saying that it cannot be a machine.




Turing's oracle can be seen simply as a mathematical tool, useful
for exploring the mathematics of the uncomputable. The idea of an
oracle allows the formulation of questions of *relative* rather
than absolute computability. Thus Turing opened new fields of
investigation in mathematical logic. However, there is also a possible
interpretation in terms of human cognitive capacity. On this
interpretation, the oracle is related to the 'intuition'
involved in seeing the truth of a Godel statement. M. H. A.
Newman, who introduced Turing to mathematical logic and continued to
collaborate with him, wrote in (Newman 1955) that the oracle resembles
a mathematician 'having an idea', as opposed to using a
mechanical method. However, Turing's oracle cannot actually be
*identified* with a human mental faculty. It is too powerful: it
immediately supplies the answer as to whether any given Turing machine
is 'satisfactory,' something no human being could do. On
the other hand, anyone hoping to see mental 'intuition'
captured completely by an oracle, must face the difficulty that Turing
showed how his argument for the incompleteness of Turing machines could
be applied with equal force to oracle-machines (Turing 1939, p. 173).
This point has been emphasised by Penrose (1994, p. 380). Newman's
comment might better be taken to refer to the different oracle
suggested later on (Turing 1939, p. 200), which has the property of
recognising 'ordinal formulae.' One can only safely say
that Turing's interest at this time in uncomputable operations appears
in the *general setting* of studying the mental
'intuition' of truths which are not established by
following mechanical processes (Turing 1939, p. 214ff.).




In Turing's presentation, intuition is in practice present in every
part of a mathematician's thought, but when mathematical proof is
formalised, intuition has an explicit manifestation in those steps
where the mathematician sees the truth of a formally unprovable
statement. Turing did not offer any suggestion as to what he considered
the brain was physically doing in a moment of such
'intuition'; indeed the word 'brain' did not
appear in his writing in this era. This question is of interest because
of the views of Penrose (1989, 1990, 1994, 1996) on just this issue:
Penrose holds that the ability of the mind to see formally unprovable
truths shows that there must be uncomputable physical operations in the
brain. It should be noted that there is widespread disagreement about
whether the human mind is really seeing the truth of a Godel
sentence; see for instance the discussion in (Penrose 1990) and the
reviews following it. However Turing's writing at this period accepted
without criticism the concept of intuitive recognition of the
truth.




It was also at this period that Turing met Wittgenstein, and there
is a full record of their 1939 discussions on the foundations of
mathematics in (Diamond 1976). To the disappointment of many, there is
no record of any discussions between them, verbal or written, on the
problem of Mind.




In 1939 Turing's various energetic investigations were broken off
for war work. This did, however, have the positive feature of leading
Turing to turn his universal machine into the practical form of the
modern digital computer.


## 5. Building a Universal Machine




When apprised in 1936 of Turing's idea for a universal machine,
Turing's contemporary and friend, the economist David Champernowne,
reacted by saying that such a thing was impractical; it would need
'the Albert Hall.' If built from relays as then employed in
telephone exchanges, that might indeed have been so, and Turing made no
attempt at it. However, in 1937 Turing did work with relays on a
smaller machine with a special cryptological function (Hodges 1983, p.
138). World history then led Turing to his unique role in the Enigma
problem, to his becoming the chief figure in the mechanisation of
logical procedures, and to his being introduced to ever faster and more
ambitious technology as the war continued. 




After 1942, Turing learnt that electronic components offered the
speed, storage capacity and logical functions required to be effective
as 'tapes' and instruction tables. So from 1945, Turing
tried to use electronics to turn his universal machine into practical
reality. Turing rapidly composed a detailed plan for a modern
stored-program computer: that is, a computer in which data and
instructions are stored and manipulated alike. Turing's ideas led the
field, although his report of 1946 postdated von Neumann's more famous
EDVAC report (von Neumann 1945). It can however be argued, as does
Davis (2000), that von Neumann gained his fundamental insight into the
computer through his pre-war familiarity with Turing's logical work. At
the time, however, these basic principles were not much discussed. The
difficulty of engineering the electronic hardware dominated
everything.




It therefore escaped observers that Turing was ahead of von Neumann
and everyone else on the future of software, or as he called it, the
'construction of instruction tables.' Turing (1946) foresaw
at once:



> Instruction tables will have to be made up by
> mathematicians with computing experiences and perhaps a certain
> puzzle-solving ability. There will probably be a great deal of work to
> be done, for every known process has got to be translated into
> instruction table form at some stage. 
> 
> 
> 
> The process of constructing instruction tables should be very
> fascinating. There need be no real danger of it ever becoming a drudge,
> for any processes that are quite mechanical may be turned over to the
> machine itself.
> 
> 
> 




These remarks, reflecting the universality of the computer, and its
ability to manipulate its own instructions, correctly described the
future trajectory of the computer industry. However, Turing had in mind
something greater: 'building a brain.' 


## 6. Building a Brain




The provocative words 'building a brain' from the outset
announced the relationship of Turing's technical computer engineering
to a philosophy of Mind. Even in 1936, Turing had given an
interpretation of computability in terms of 'states of
mind'. His war work had shown the astounding power of the
computable in mechanising expert human procedures and judgments. From
1941 onwards, Turing had also discussed the mechanisation of
chess-playing and other 'intelligent' activities with his
colleagues at Bletchley Park (Hodges 1983, p. 213). But more
profoundly, it appears that Turing emerged in 1945 with a conviction
that computable operations were sufficient to embrace *all*
mental functions performed by the brain. As will become clear from the
ensuing discussion, the uncomputable 'intuition' of 1938
disappeared from Turing's thought, and was replaced by new ideas all
lying within the realm of the computable. This change shows even in the
technical prospectus of (Turing 1946), where Turing referred to the
possibility of making a machine calculate chess moves, and then
continued: 



> This ... raises the question 'Can a machine play
> chess?' It could fairly easily be made to play a rather bad game.
> It would be bad because chess requires intelligence. We stated ...
> that the machine should be treated as entirely without intelligence.
> There are indications however that it is possible to make the machine
> display intelligence at the risk of its making occasional serious
> mistakes. By following up this aspect the machine could probably be
> made to play very good chess.




The puzzling reference to 'mistakes' is made clear by a
talk Turing gave a year later (Turing 1947), in which the issue of
mistakes is linked to the issue of the significance of seeing the truth
of formally unprovable statements.



> ...I would say that fair play must be given to the
> machine. Instead of it giving no answer we could arrange that it gives
> occasional wrong answers. But the human mathematician would likewise
> make blunders when trying out new techniques... In other words
> then, if a machine is expected to be infallible, it cannot also be
> intelligent. There are several mathematical theorems which say almost
> exactly that. But these theorems say nothing about how much
> intelligence may be displayed if a machine makes no pretence at
> infallibility.




Turing's post-war view was that mathematicians make mistakes, and so
do not in fact see the truth infallibly. Once the possibility of
mistakes is admitted, Godel's theorem become irrelevant.
Mathematicians and computers alike apply computable processes to the
problem of judging the correctness of assertions; both will therefore
sometimes err, since seeing the truth is known not to be a computable
operation, but there is no reason why the computer need do worse than
the mathematician. This argument is still very much alive. For
instance, Davis (2000) endorses Turing's view and attacks Penrose
(1989, 1990, 1994, 1996) who argues against the significance of human
error on the grounds of a Platonist account of mathematics.




Turing also pursued more constructively the question of how
computers could be made to perform operations which did not appear to
be 'mechanical' (to use common parlance). His guiding
principle was that it should be possible to simulate the operation of
human brains. In an unpublished report (Turing 1948), Turing explained
that the question was that of how to simulate 'initiative'
in addition to 'discipline'--comparable to the need
for 'intuition' as well as mechanical ingenuity expressed
in his pre-war work. He announced ideas for how to achieve this: he
thought 'initiative' could arise from systems where the
algorithm applied is not consciously designed, but is arrived at by
some other means. Thus, he now seemed to think that the mind when
*not* actually following any conscious rule or plan, was
nevertheless carrying out some computable process.




He suggested a range of ideas for systems which could be said to
modify their own programs. These ideas included nets of logical
components ('unorganised machines') whose properties could
be 'trained' into a desired function. Thus, as expressed
by (Ince 1989), he predicted neural networks. However, Turing's nets
did not have the 'layered' structure of the neural
networks that were to be developed from the 1950s onwards. By the
expression 'genetical or evolutionary search', he also
anticipated the 'genetic algorithms' which since the late
1980s have been developed as a less closely structured approach to
self-modifying programs. Turing's proposals were not well developed in
1948, and at a time when electronic computers were only barely in
operation, could not have been. Copeland and Proudfoot (1996) have
drawn fresh attention to Turing's connectionist ideas, which
have since been tried out (Teuscher 2001).




It is important to note that Turing identified his prototype neural
networks and genetic algorithms as *computable.* This has to be
emphasised since the word 'nonalgorithmic' is often now
confusingly employed for computer operations that are not explicitly
planned. Indeed, his ambition was explicit: he himself wanted to
implement them as programs on a computer. Using the term Universal
Practical Computing Machine for what is now called a digital computer,
he wrote in (Turing 1948):



> It should be easy to make a model of any particular machine
> that one wishes to work on within such a UPCM instead of having to work
> with a paper machine as at present. If one also decided on quite
> definite 'teaching policies' these could also be programmed
> into the machine. One would then allow the whole system to run for an
> appreciable period, and then break in as a kind of 'inspector of
> schools' and see what progress had been made. One might also be
> able to make some progress with unorganised
> machines...




The upshot of this line of thought is that all mental operations are
*computable* and hence realisable on a universal machine: the
computer. Turing advanced this view with increasing confidence in the
late 1940s, perfectly aware that it represented what he enjoyed calling
'heresy' to the believers in minds or souls beyond material
description.




Turing was not a mechanical thinker, or a stickler for convention;
far from it. Of all people, he knew the nature of originality and
individual independence. Even in tackling the U-boat Enigma problem,
for instance, he declared that he did so because no-one else was
looking at it and he could have it to himself. Far from being trained
or organised into this problem, he took it on despite the prevailing
wisdom in 1939 that it was too difficult to attempt. His arrival at a
thesis of 'machine intelligence' was not the outcome of
some dull or restricted mentality, or a lack of appreciation of
individual human creativity.


## 7. Machine Intelligence




Turing relished the paradox of 'Machine Intelligence': an
apparent contradiction in terms. It is likely that he was already
savouring this theme in 1941, when he read a theological book by the
author Dorothy Sayers (Sayers 1941). In (Turing 1948) he quoted from
this work to illustrate his full awareness that in common parlance
'mechanical' was used to to mean 'devoid of
intelligence.' Giving a date which no doubt had his highly
sophisticated Enigma-breaking machines secretly in mind, he wrote that
'up to 1940' only very limited machinery had been used, and
this 'encouraged the belief that machinery was necessarily
limited to extremely straightforward, possibly even to repetitious,
jobs.' His object was to dispel these connotations. 




In 1950, Turing wrote on the first page of his Manual for users of
the Manchester University computer (Turing 1950a):



> Electronic computers are intended to carry out any definite
> rule of thumb process which could have been done by a human operator
> working in a disciplined but unintelligent manner.




This is, of course, just the 1936 universal Turing machine, now in
electronic form. On the other hand, he also wrote in the more famous
paper of that year (Turing 1950b, p. 460)



> We may hope that machines will eventually compete with men
> in all purely intellectual fields.




How could the *intelligent* arise from operations which were
themselves totally *routine and 
mindless*--'entirely without intelligence'? This is the core of the
problem Turing faced, and the same problem faces Artificial
Intelligence research today. Turing's underlying argument was that the
human brain must somehow be organised for intelligence, and that the
organisation of the brain must be realisable as a finite discrete-state
machine. The implications of this view were exposed to a wider circle
in his famous paper, "Computing Machinery and Intelligence," which
appeared in *Mind* in October 1950.




The appearance of this paper, Turing's first foray into a journal of
philosophy, was stimulated by his discussions at Manchester University
with Michael Polanyi. It also reflects the general sympathy of Gilbert
Ryle, editor of *Mind,* with Turing's point of view.




Turing's 1950 paper was intended for a wide readership, and its fresh and
direct approach has made it one of the most frequently cited and republished
papers in modern philosophical literature. Not surprisingly,
the paper has attracted many critiques. Not all commentators note the
careful explication of computability which opens the paper, with an
emphasis on the concept of the universal machine. This explains why if
mental function can be achieved by any finite discrete state machine,
then the same effect can be achieved by programming a computer (Turing
1950b, p. 442). (Note, however, that Turing makes no claim that the
nervous system should resemble a digital computer in its structure.)
Turing's treatment has a severely finitistic flavour: his argument is
that the relevant action of the brain is not only computable, but
realisable as a totally finite machine, i.e. as a Turing machine that
does not use any 'tape' at all. In his account, the full
range of computable functions, defined in terms of Turing machines that
use an infinite tape, only appears as being of 'special
theoretical interest.' (Of uncomputable functions there is, *a
fortiori,* no mention.) Turing uses the finiteness of the nervous
system to give an estimate of about 109 bits
of storage required for a limited simulation of intelligence (Turing
1950b, p. 455).




The wit and drama of Turing's 'imitation game' has
attracted more fame than his careful groundwork. Turing's argument was
designed to bypass discussions of the nature of thought, mind, and
consciousness, and to give a criterion in terms of external observation
alone. His justification for this was that one only judges that other
human beings are thinking by external observation, and he applied a
principle of 'fair play for machines' to argue that the
same should hold for machine intelligence. He dramatised this viewpoint
by a thought-experiment (which nowadays can readily be tried out). A
human being and a programmed computer compete to convince an impartial
judge, using textual messages alone, as to which is the human being.
If the computer wins, it must be credited with intelligence.




Turing introduced his 'game' confusingly with a poor
analogy: a party game in which a man pretends to be a woman. His loose
wording (Turing 1950b, p. 434) has led some writers wrongly to suppose
that Turing proposed an 'imitation game' in which a machine
has to imitate a man imitating a woman. Others, like Lassegue
(1998), place much weight on this game of gender pretence and its real
or imaginary connotations. In fact, the whole point of the
'test' setting, with its remote text-message link, was to
*separate* intelligence from other human faculties and
properties. But it may fairly be said that this confusion reflects
Turing's richly ambitious concept of what is involved in human
'intelligence'. It might also be said to illustrate his own
human intelligence, in particular a delight in the Wildean reversal of
roles, perhaps reflecting, as in Wilde, his homosexual identity. His
friends knew an Alan Turing in whom intelligence, humour and sex were
often intermingled.




Turing was in fact sensitive to the difficulty of separating
'intelligence' from other aspects of human senses and
actions; he described ideas for robots with sensory attachments and
raised questions as to whether they might enjoy strawberries and cream
or feel racial kinship. In contrast, he paid scant attention to the
questions of authenticity and deception implicit in his test,
essentially because he wished to by-pass questions about the reality of
consciousness. A subtle aspect of one of his imagined
'intelligent' conversations (Turing 1950b, p. 434) is where
the computer imitates human intelligence by giving the *wrong
answer* to a simple arithmetic problem. But in Turing's setting we
are not supposed to ask whether the computer 'consciously'
deceives by giving the impression of innumerate humanity, nor why it
should wish to do so. There is a certain lack of seriousness in this
approach. Turing took on a second-rank target in countering the
published views of the brain surgeon G. Jefferson, as regards the
objectivity of consciousness. Wittgenstein's views on Mind would have
made a more serious point of departure.




Turing's imitation principle perhaps also assumes (like
'intelligence tests' of that epoch) too much of a shared
language and culture for his imagined interrogations. Neither does it
address the possibility that there may be kinds of thought, by animals
or extra-terrestrial intelligences, which are not amenable to
communication.




A more positive feature of the paper lies in its constructive
program for research, culminating in Turing's ideas for 'learning
machines' and educating 'child' machines (Turing
1950b, p. 454). It is generally thought (e.g. in Dreyfus and Dreyfus
1990) that there was always an antagonism between programming and the
'connectionist' approach of neural networks. But Turing
never expressed such a dichotomy, writing that both approaches should
be tried. Donald Michie, the British AI research pioneer profoundly
influenced by early discussions with Turing, has called this suggestion
'Alan Turing's Buried Treasure', in an allusion to a
bizarre wartime episode in which Michie was himself involved (Hodges
1983, p. 345). The question is still highly pertinent.




It is also a commonly expressed view that Artificial Intelligence
ideas only occurred to pioneers in the 1950s *after* the success
of computers in large arithmetical calculations. It is hard to see why
Turing's work, which was rooted from the outset in the question of
mechanising Mind, has been so much overlooked. But through his failure
to publish and promote work such as that in (Turing 1948) he largely
lost recognition and influence.




It is also curious that Turing's best-known paper should appear in a
journal of philosophy, for it may well be said that Turing, always
committed to materialist explanation, was not really a philosopher at
all. Turing was a mathematician, and what he had to offer philosophy
lay in illuminating its field with what had been discovered in
mathematics and physics. In the 1950 paper this was surprisingly
cursory, apart from his groundwork on the concept of computability. His
emphasis on the sufficiency of the computable to explain the action of
the mind was stated more as a hypothesis, even a manifesto, than argued
in detail. Of his hypothesis he wrote (Turing 1950b, p. 442):



> ...I believe that at the end of the century the use of
> words and general educated opinion will have altered so much that one
> will be able to speak of machines thinking without expecting to be
> contradicted. I believe further that no useful purpose is served by
> concealing these beliefs. The popular view that scientists proceed
> inexorably from established fact to established fact, never being
> influenced by any unproved conjecture, is quite mistaken. Provided it
> is made clear which are proved facts and which are conjecture, no harm
> can result. Conjectures are of great importance since they suggest
> useful lines of research.




Penrose (1994, p.21), probing into Turing's conjecture, has
presented it as 'Turing's thesis' thus:



> It seems likely that he viewed physical action in general--which would include the action of a human brain--to be
> always reducible to some kind of Turing-machine action.




The statement that all physical action is in effect computable goes
beyond Turing's explicit words, but is a fair characterisation of the
implicit assumptions behind the 1950 paper. Turing's consideration of
'The Argument from Continuity in the Nervous System,' in
particular, simply asserts that the physical system of the brain can be
approximated as closely as is desired by a computer program (Turing
1950b, p. 451). Certainly there is nothing in Turing's work in the
1945-50 period to contradict Penrose's interpretation. The more
technical precursor papers (Turing 1947, 1948) include wide-ranging
comments on physical processes, but make no reference to the
possibility of physical effects being uncomputable.




In particular, a section of (Turing 1948) is devoted to a general
classification of 'machines.' The period between 1937 and
1948 had given Turing much more experience of actual machinery than he
had in 1936, and his post-war remarks reflected this in a down-to-earth
manner. Turing distinguished 'controlling' from
'active' machinery, the latter being illustrated by
'a bulldozer'. Naturally it is the former--in modern
terms 'information-based machinery'--with which
Turing's analysis is concerned. It is noteworthy that in 1948 as in
1936, despite his knowledge of physics, Turing made no mention of how
quantum mechanics might affect the concept of
'controlling'. His concept of 'controlling'
remained entirely within the classical framework of the Turing machine
(which he called a Logical Computing Machine in this paper.)




The same section of (Turing 1948) also drew the distinction between
*discrete* and *continuous* machinery, illustrating the
latter with 'the telephone' as a continuous, controlling
machine. He made light of the difficulty of reducing continuous physics
to the discrete model of the Turing machine, and though citing
'the brain' as a continuous machine, stated that it could
probably be treated as if discrete. He gave no indication that physical
continuity threatened the paramount role of computability. In fact, his
thrust in (Turing 1947) was to promote the digital computer as *more
powerful* than analog machines such as the differential analyser.
When he discussed this comparison, he gave the following informal
version of the Church-Turing thesis:



> One of my conclusions was that the idea of a 'rule of
> thumb' process and a 'machine process' were
> synonymous. The expression 'machine process' of course
> means one which could be carried out by the type of machine I was
> considering [i.e. Turing machines]




Turing gave no hint that the discreteness of the Turing machine
constituted a real limitation, or that the non-discrete processes of
analog machines might be of any deep significance.




Turing also introduced the idea of 'random elements' but
his examples (using the digits of p) showed that he considered
*pseudo-random* sequences (i.e. computable sequences with
suitable 'random' properties) quite adequate for his
discussion. He made no suggestion that randomness implied something
uncomputable, and indeed gave no definition of the term
'random'. This is perhaps surprising in view of the fact
that his work in pure mathematics, logic and cryptography all gave him
considerable motivation to approach this question at a serious
level.


## 8. Unfinished Work




From 1950 Turing worked on a new mathematical theory of morphogenesis,
based on showing the consequences of non-linear equations for chemical
reaction and diffusion (Turing 1952). He was a pioneer in using a
computer for such work. Some writers have referred to this theory as
founding Artificial Life (A-life), but this is a misleading
description, apt only to the extent that the theory was intended, as
Turing saw it, to counter the Argument from Design. A-life since the
1980s has concerned itself with using computers to explore the logical
consequences of evolutionary theory without worrying about specific
physiological forms. Morphogenesis is complementary, being concerned to
show which physiological pathways are feasible for evolution to
exploit. Turing's work was developed by others in the 1970s and is now
regarded as central to this field. 




It may well be that Turing's interest in morphogenesis went back to
a primordial childhood wonder at the appearance of plants and flowers.
But in another late development, Turing went back to other stimuli of
his youth. For in 1951 Turing did consider the problem, hitherto
avoided, of setting computability in the context of quantum-mechanical
physics. In a BBC radio talk of that year (Turing 1951) he discussed
the basic groundwork of his 1950 paper, but this time dealing rather
less certainly with the argument from Godel's theorem, and this
time also referring to the quantum-mechanical physics underlying the
brain. Turing described the universal machine property, applying it to
the brain, but said that its applicability required that the machine
whose behaviour is to be imitated



> ...should be of the sort whose behaviour is in
> principle predictable by calculation. We certainly do not know how any
> such calculation should be done, and it was even argued by Sir Arthur
> Eddington that on account of the indeterminacy principle in quantum
> mechanics no such prediction is even theoretically
> possible.




Copeland (1999) has rightly drawn attention to this sentence in his
preface to his edition of the 1951 talk. However, Copeland's critical
context suggests some connection with Turing's 'oracle.'
There is is in fact no mention of oracles here (nor anywhere in
Turing's post-war discussion of mind and machine.) Turing here is
discussing the possibility that, when seen as as a
*quantum-mechanical machine* rather than a classical machine,
the Turing machine model is inadequate. The correct connection to draw
is not with Turing's 1938 work on ordinal logics, but with his
knowledge of quantum mechanics from Eddington and von Neumann in his
youth. Indeed, in an early speculation, influenced by Eddington, Turing
had suggested that quantum mechanical physics could yield the basis of
free-will (Hodges 1983, p. 63). Von Neumann's axioms of quantum
mechanics involve two processes: unitary evolution of the wave
function, which is predictable, and the measurement or reduction
operation, which introduces unpredictability. Turing's reference to
unpredictability must therefore refer to the reduction process. The
essential difficulty is that still to this day there is no agreed or
compelling theory of when or how reduction actually occurs. (It should
be noted that 'quantum computing,' in the standard modern
sense, is based on the predictability of the unitary evolution, and
does not, as yet, go into the question of how reduction occurs.) It
seems that this single sentence indicates the beginning of a new field
of investigation for Turing, this time into the foundations of quantum
mechanics. In 1953 Turing wrote to his friend and student Robin Gandy
that he was 'trying to invent a new Quantum Mechanics but it
won't really work.'




At Turing's death in June 1954, Gandy reported in a letter to Newman
on what he knew of Turing's current work (Gandy 1954). He wrote of
Turing having discussed a problem in understanding the reduction
process, in the form of



> ...'the Turing Paradox'; it is easy to
> show using standard theory that if a system start in an eigenstate of
> some observable, and measurements are made of that observable N times a
> second, then, even if the state is not a stationary one, the
> probability that the system will be in the same state after, say, 1
> second, tends to one as N tends to infinity; i.e. that continual
> observation will prevent motion. Alan and I tackled one or two
> theoretical physicists with this, and they rather pooh-poohed it by
> saying that continual observation is not possible. But there is nothing
> in the standard books (e.g., Dirac's) to this effect, so that at least the
> paradox shows up an inadequacy of Quantum Theory as usually
> presented.




Turing's investigations take on added significance in view of the
assertion of Penrose (1989, 1990, 1994, 1996) that the reduction
process must involve something uncomputable. Probably Turing was aiming
at the opposite idea, of finding a theory of the reduction process that
would be predictive and computable, and so plug the gap in his
hypothesis that the action of the brain is computable. However Turing
and Penrose are alike in seeing this as an important question affecting
the assumption that all mental action is computable; in this they both
differ from the mainstream view in which the question is accorded
little significance.




Alan Turing's last postcards to Robin Gandy, in March 1954, headed
'Messages from the Unseen World' in allusion to Eddington,
hinted at new ideas in the fundamental physics of relativity and
particle physics (Hodges 1983, p. 512). They illustrate the wealth of
ideas with which he was concerned at that last point in his life, but
which apart from these hints are entirely lost. A review of such lost
ideas is given in (Hodges 2004), as part of a larger volume on
Turing's legacy (Teuscher 2004). 


## 9. Alan Turing: the Unknown Mind




It is a pity that Turing did not write more about his ethical
philosophy and world outlook. As a student he was an admirer of Bernard
Shaw's plays of ideas, and to friends would openly voice both the
hilarities and frustrations of his many difficult situations. Yet the
nearest he came to serious personal writing, apart from occasional
comments in private letters, was in penning a short story about his
1952 crisis (Hodges 1983, p. 448). His last two years were particularly
full of Shavian drama and Wildean irony. In one letter (to his friend
Norman Routledge; the letter is now in the Turing Archive at King's
College, Cambridge) he wrote: 



> Turing believes machines think
>   
> 
>  Turing lies with men
>   
> 
>  Therefore machines do not think




The syllogistic allusion to Socrates is unmistakeable, and his
demise, with cyanide rather than hemlock, may have signalled something
similar. A parallel figure in World War II, Robert Oppenheimer,
suffered the loss of his reputation during the same week that Turing
died. Both combined the purest scientific work and the most effective
application of science in war. Alan Turing was even more directly on
the receiving end of science, when his sexual mind was treated as a
machine, against his protesting consciousness and will. But amidst all
this human drama, he left little to say about what he really thought of
himself and his relationship to the world of human events.




Alan Turing did not fit easily with any of the intellectual movements
of his time, aesthetic, technocratic or marxist. In the 1950s,
commentators struggled to find discreet words to categorise him: as
'a scientific Shelley,' as possessing great 'moral
integrity'. Until the 1970s, the reality of his life was
unmentionable. He is still hard to place within twentieth-century
thought. He exalted the science that according to existentialists had
robbed life of meaning. The most original figure, the most insistent
on personal freedom, he held originality and will to be susceptible to
mechanisation. The mind of Alan Turing continues to be an
enigma.




But it is an enigma to which the twenty-first century seems
increasingly drawn. The year of his centenary, 2012, witnessed
numerous conferences, publications, and cultural events in his
honor. Some reasons for this explosion of interest are obvious. One is
that the question of the power and limitations of computation now
arises in virtually every sphere of human activity. Another is that
issues of sexual orientation have taken on a new importance in modern
democracies. More subtly, the interdisciplinary breadth of Turing's
work is now better appreciated. A landmark of the centenary period was
the publication of *Alan Turing, his work and impact*
(eds. Cooper and van Leeuwen, 2013), which made available almost all
aspects of Turing's scientific oeuvre, with a wealth of modern
commentary. In this new climate, fresh attention has been paid to
Turing's lesser-known work, and new light shed upon his
achievements. He has emerged from obscurity to become one of the most
intensely studied figures in modern science.