The difficult legacy of Turing’s wager
The difficult legacy of Turing's wager
Andrew Thwaites 0
Andrew Soltan 0
Eric Wieser 0
Ian NimmoSmith 0
Action Editor: Jonathan David Victor 0
0 MRC Cognition and Brain Sciences Unit , Cambridge , UK
1 Andrew Thwaites
Describing the human brain in mathematical terms is an important ambition of neuroscience research, yet the challenges remain considerable. It was Alan Turing, writing in 1950, who first sought to demonstrate how timeconsuming such an undertaking would be. Through analogy to the computer program, Turing argued that arriving at a complete mathematical description of the mind would take well over a thousand years. In this opinion piece, we argue that  despite seventy years of progress in the field  his arguments remain both prescient and persuasive.
Computational neuroscience; Philosophy of neuroscience; Policy; Alan Turing

‘We are not’, Alan Turing once remarked, ‘interested in the fact
that the brain has the consistency of cold porridge’
(Turing
1952)
. For Turing, the key to understanding the brain lay not
in mapping its anatomy or measuring its density, but in
characterising its behaviour mathematically. His view is one
shared by many. Mathematical equations are expressive and
unambiguous — and arguably the only language suited to
describing so complex an object.
Recent medical innovations are a reminder of the practical
value of these equations. Hearing aids and prosthetic limbs
exploit algorithms that mimic the equations of the brain and
University of Cambridge, Cambridge, UK
nervous system
(Flesher et al. 2016; Moore 2003)
, as do novel
treatments for neurodegenerative disorders such as
Parkinson’s, dementia and lockedin syndrome
(Chou et al.
2015; Famm et al. 2013; Lebedev and Nicolelis 2006;
Vansteensel et al. 2016)
. Beyond medicine, mathematical
characterisations of the brain inform the field of artificial
intelligence and the design of novel computer components
(Hassabis 2012; The Economist 2013)
. As the demand for
such technology increases, so too does the need to characterise
ever more of the brain in mathematical terms.
Policymakers have reflected these aspirations in their
funding decisions. In 2013, the European Commission
awarded the Human Brain Project [HBP] €1 billion with
the explicit aim of constructing an accurate mathematical
model of the brain
(Markram 2012)
. Later that year, the US
government launched the BRAIN Initiative, aiming to
invest $4.5 billion over twelve years towards a study of
the brain in which modelling was intended to play a
substantial role
(National Institutes of Health 2014)
.
China and Japan have since announced projects with
similar aims
(Grillner et al. 2016)
, and there appears to
be every reason to hope that detailed mathematical
models of the healthy and diseased brain may one day
become viable and useful propositions.
Yet the difficulties involved are substantial. Turing himself,
a pioneer of computational theory, viewed the enterprise with
blunt scepticism and sought to highlight the challenges
involved with a practical illustration. He started by writing a
short computer program on his departmental workstation at
the University of Manchester. This program accepted a single
number, performed a series of unspecified calculations on it,
and returned a second number. It would be extremely difficult,
Turing argued, for anyone to guess these calculations from the
input and output numbers alone. Determining the calculations
taking place in the brain, he reasoned, must be harder still: not
only does the brain accept tensofthousands of inputs from
sensory receptors around the body, but the calculations these
inputs undergo are far more complicated than anything written
by a single programmer. Turing underscored his argument
with a wager: that it would take an investigator at least a
thousand years to guess the full set of calculations his
Manchester program employed. Guessing the full set of
calculations taking place in the brain, he noted, would appear
prohibitively timeconsuming
(Turing 1950)
.
If characterising the brain’s equations is so difficult, what
accounts for the recent surge in investment? There are two
reasons. The first is that researchers are now able to measure
physiological properties inside the brain, including blood
oxygenation levels and actionpotential spike rates
(Ulmer
and Jansen 2010; Smith et al. 1985)
. Turing’s wager
demonstrates why this development makes guessing the
calculations of the brain easier: if an investigator was
allowed to measure the electrical charge of different
components inside the Manchester computer while it was
running, this would constrain the number of equations that
the investigator need consider. The same logic holds true of
characterising the brain.
The second development is that modern supercomputers
are now able to take on the burden of testing equations
themselves. In 1950, an engineer trying to establish the program
running in Turing’s computer would have to hypothesise each
possible sequence of calculations, use these to predict what the
output would be, and then test if these predictions
corresponded with the observed output, repeating the process
until they found the correct calculation. Today, modern
supercomputers are able to test thousands of predictions a second,
vastly speeding up the search.
As neuroimaging resolutions and supercomputing speeds
have improved, the number of established brainrelated
equations has grown. Neuroscientists have employed a wide
variety of equationfinding strategies, each of which makes use of
different configurations of brain measurements and biological
assumptions [see
(Eliasmith and Trujillo 2014)
,
(de Garis et al.
2010)
for review]. The apparent pace of these advances has
led some commentators to argue that a complete catalogue of
the brain’s equations may be in reach, perhaps within the next
fifty years
(Kurzweil 2012; Markram 2012)
.
This is unfortunate, because such optimism is almost
certainly misplaced. This is largely because the wager is less
influenced by technological advances than it first appears.
Generating hypothetical equations, for example, is a
considerable bottleneck. Many things the mind does — such as face
recognition or language comprehension — are extremely
difficult to describe mathematically, and few equations that claim
to model such processes have been proposed. Although
supercomputers can be programmed to automatically generate,
and subsequently test, thousands of equations at a time in an
investigatory or heuristic manner (a process known as
‘machine learning’), this is of limited use when the processes
they model are intrinsically complicated. Novel equations
may come from the emerging fields of artificial intelligence
and deeplearning, but the task here is onerous: three of the
most requested search terms in the Kymata Atlas (a database
of equations maintained by the University of Cambridge) are
‘morality’, ‘consciousness’ and ‘ego’ — features of human
cognition that no one has yet been able to capture in an
equation.1
More challenging, however, is the original difficulty the
wager aimed to emphasise: that the number of possible
equations that an investigator would need to assess in order to fully
characterise the brain is unimaginably vast. To determine the
calculations behind the Manchester program, the investigator
would potentially need to test more candidate equations than
there are atoms in the observable universe; to do the same for
all — or even part of — the human brain would require the
testing of higher numbers still. It remains a stubborn truism
that, impressive as recent neuroscience advances have been,
they have revealed only a very small proportion of all possible
equations taking place. Neither improvements in
neuroimaging resolution, nor the current exponential acceleration in
computing power, are likely to increase this proportion
substantially in the foreseeable future.
These stumbling blocks do not indicate that attempts at
brain modelling should be abandoned; after all, the wager
does not argue that mathematical modelling of the brain is
impossible — only that it is timeconsuming. With this in
mind, which approaches are likely to be most effective?
Aside from the extensive use of supercomputing, two stand
out. First are those approaches that restrict their scrutiny to
modelling processes related to specific technological
applications or clinical conditions, which will, by their nature,
be more tractable than mapping the entire brain. Second
are those approaches that aim to establish the basic
underlying principles of cortical function, in the hope that this
will reduce the number of possibilities when determining
equations of a more complex nature. Such criteria
increasingly influence funding decisions: both the BRAIN and
Chinese initiatives favour approaches that demonstrate
clinical or applied objectives, or that attempt to reduce
the equation search space through research into basic
biological principles
(National Institutes of Health 2014;
Grillner et al. 2016)
. The European HBP, by contrast, with
its explicit aspirations of simulating a complete model of
the brain, was required to scale back these ambitions after
widespread criticism from the scientific community
(Marquardt 2015)
. This preference for the practical over
1 The availability of such equations is not clear cut. Cf., for instance, Tononi
and colleagues, who provide what they believe to be plausible mathematical
characterizations of consciousness
(Tononi 2004; Oizumi et al. 2014)
, and
their critics, who believe these characterizations to be untestable, and therefore
unscientific (see Cerullo (2015) for overview).
the comprehensive should be welcomed: given the lack of
constraints over what the totality of possible equations in
the brain might be, it makes no sense for today’s
neuroscience researcher, as with Turing’s investigator, to spend
time and resources reaching too far into the distance when
useful equations are closer at hand.
Underlying Turing’s wager is a plea for realism. So
much has been learnt about the brain over the last twenty
years that it is tempting to think that simply offering the
field of neuroscience more resources will remove the
limitations of the wager and the workings of the brain will be
laid bare. This is a mistake. In the endless search for the
mathematical basis of the mind, Turing reflected that there
was only one truth we can be sure of: ‘We certainly know
of no circumstances under which we could say BWe have
searched enough.^’
(Turing 1950)
.
Alan Turing (right) and colleagues working on the Ferranti Mark 1 computer at the University
of Manchester. Science Museum Collection (CCBYNCND).
The Turing Test and Turing’s Wager
Turing introduced both the Turing Test and Turing’s Wager in the same essay, ‘Computing
Machinery and Intelligence’, published in Mind in 1950. The Turing Test (as it became
known) defines the criteria by which a machine can be said to possess ‘intelligence’
equivalent to that of a human. Turing’s Wager (as we refer to it here) is an argument aiming to
demonstrate that characterising the brain in mathematical terms will take over a thousand
years.
Turing was optimistic about mankind’s ability to build a machine able to pass the Turing Test
(‘about sixty workers, working steadily through the fifty years might accomplish the job, if
nothing went into the wastepaper basket’). He was not, by contrast, optimistic about their
chances of beating Turing’s Wager — and with good reason: modelling a brain accurately is a
much harder problem than imitating it approximately. Indeed, the former is subproblem of
the latter: if an individual was able to overcome Turing’s Wager and come into possession of a
complete mathematical characterisation of the human brain, then that individual would, as a
consequence, be in possession of a machine able to pass the Turing Test.
Acknowledgements We are grateful to Henry Nicholls for comments
and suggestions during the writing of this paper, and to the Centre for
Speech, Language and the Brain for providing AT with office space.
Compliance with ethical standards
Conflict of interest The authors declare no conflict of Interest.
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