Mind, Mechanism & MathematicsI
THE TURING CENTENARY RESEARCH PROJECT:
First Workshop, Milan, June 30, 2013
The Project Team:
is Professor of Mathematics at the University of Leeds.
Author and editor of numerous books,
including Computability Theory,
New Computational Paradigms, Computability in Context,
and Alan Turing - His Work and Impact, he
he is a leading advocate of multidisciplinary research at the
interface between incomputability and real world computability.
Chair of the Turing Centenary Advisory Committee, which
coordinated the international Turing Centenary celebrations,
he is President of the Association Computability in Europe, which is
responsible for the largest computability-themed international conference
series, and chairs the Editorial Board of its Springer book
series Theory and Applications of Computability. He is a
member of the Board of Advisors of the John Templeton Foundation.
Turing Research Fellows and
Mark Braverman is an assistant professor of computer science at
Princeton University. His main interests lie in the theory of
computation and its applications in the natural and social sciences.
He received his PhD from the University of Toronto in 2008 under the
supervision of Stephen Cook. [Turing Fellow]
Stephanie Dick is a doctoral candidate in the Department of History of Science at
Harvard University, under the supervision of Prof. Peter Galison. Her
dissertation is a historical exploration of attempts to automate
mathematical reasoning and theorem proving after the advent of
digital computing. She aims to recover and reconstruct the interactions
between theory, practice, and technology in historical understandings
of mind, computing, and mathematics. [Turing Fellow]
Gamez (University of Sussex)
David Gamez holds PhDs in both philosophy and computer science. His
cross-disciplinary research uses philosophy and neural modelling to
explore how a science of the mind can be developed based on mathematical
theories. Between 2009 and 2012 he was at Imperial College London,
where he worked on brain-inspired neural networks and robotics and
investigated new algorithms for making predictions about consciousness.
He is currently a Research Fellow at the Sackler Centre for
Consciousness Science, University of Sussex. [Turing Fellow]
Paul Grant (University of Cambridge)
Paul Grant is a postdoctoral fellow in the Department of Plant Sciences at
the University of Cambridge. He currently works on synthetic
pattern-forming genetic circuits in microbes but his background is in
developmental biology, having written his doctoral dissertation at the
University of Washington on neuron migration in zebrafish. He is originally
from Northern New York and maybe plans to pursue an academic career
back in the States. [Turing Fellow]
Noam Greenberg (Victoria University of Wellington)
I am a mathematician, interested in mathematical logic and in
particular in computability and set theory. I am interested in how
diverse notions such as randomness or algebraic structure, or
generalisations to large domains, influence our understanding of the
notion of computation. I work at Victoria University of Wellington,
New Zealand; I received a Ph.D. from Cornell University and a B.Sc.
from the Hebrew University of Jerusalem. [Turing Fellow]
Kuyper (Radboud University Nijmegen)
My name is Rutger Kuyper and I am a PhD candidate at the RadboudUniversity Nijmegen
in the Netherlands. I am currently researching probability logic, in which mathematical logic (the subfield of
mathematics in which logical reasoning is investigated) is combined
with probabilistic, inductive reasoning. [Turing Scholar]
I recently finished a Ph.D under the supervision of Theodore Slaman at
UC Berkeley. I am currently a postdoctoral fellow at Caltech. [Turing Scholar]
Simon Martiel (University of Nice-Sophia
I am starting a PhD in the University of Nice-Sophia Antipolis in
France supervised by Bruno Martin and Pablo Arrighi. I am interested in the
links between abstract notions in Computational Model Theory such as
model complexity or intrinsic universality and Theoretical Physics, trying
to answer the question: How does nature compute itself? [Turing Scholar]
Rebecca Schulman is an assistant professor of chemical and biomolecular
engineering and computer science at Johns Hopkins University. Her group is working to
learn how to program molecules to assemble complex patterns and structures that can learn and
adapt. [Turing Fellow]
The Big Questions
work and scientific legacy inhabits a world at the interface between
the physical universe and the abstractions of mathematics;
between the computational, the predictable, and the uncertainties
manifest in human creativity, and in the emergence of form from
randomness and chaotic environments; and a world at the
interface between science, reason, reductionism and the mysteries
of intuition, incomputability, and the failures of foundational thinking.
There are many deep and intractable questions arise in such a
context. We list below four main
around which the research project has been built:
- The Mathematics of Emergence:
The Mysteries of Morphogenesis
Does emergence exist as something new in nature,
and is it something which transcends classical models of computation?
(Develop a sufficiently fundamental model of emergence to
answer this question.)
To what extent do existing models of natural processes accommodate
(Characterize the mathematical characteristics of such models,
and the extent of their representational and recursive capacities.)
One of the less familiar but most innovative of Turing's
contributions was his successful capturing of emergence of various
patterns in nature in mathematical equations. This key innovation
now forms the basis of an active and important research field
There are still general questions concerning both the computational
content of emergence, of the unifying features extractable from
different contexts, and the nature of good mathematical models
applicable in different environments. John Holland, in his popular
1998 book on Emergence: From Chaos to Order
sets out the extent of our ignorance of the precise mathematical
character of emergence:
"It is unlikely that a topic as complicated as emergence will submit
meekly to a concise definition, and I have no such definition to offer."
Issues of self-organization, emergence of form from chaotic or
turbulent environments, non-locality in physics, and the
relevance of classical models of computability such as Turing
machines, cellular automata, neural nets, among others, are all
in need of further investigation. It is hoped that a more coherent
and mathematically-based viewpoint will shed light on some of the
issues arising under the following three headings.
- Possibility of Building a Brain:
Intelligent Machines, Practice and Theory
Is there a successful mathematical model of intelligent thought?
(Develop more basic models, capable of pinning down the exact
relationship between the human brain and standard paradigms
from classical computability and computer science.)
Is intelligent thought essentially experimental and inexact?
Are mistakes an essential aspect of intelligence?
What is the extent of application of the notion of a 'virtual
machine'? Is human thinking essentially embodied, and the
Turing paradigm of a universal machine inapplicable in this context?
Turing's seminal role in artificial intelligence - for instance,
his formulation of the Turing
Test and his interest in connectionist models - has given rise to
many different and contrasting approaches to building intelligent
machines. There is a basic dichotomy between experimental, ad hoc
approaches and more logic-based theoretical ones. There is a
general agreement that we need to understand more about the
relationship between these, and to obtain a better theoretical
grasp of the practicalities, and of the underlying theoretical
obstacles. The extent of the challenges facing researchers is
summed up by Rodney Brooks, in Nature in 2001:
"neither AI nor Alife has produced artifacts that could be
confused with a living organism for more than an instant."
The question of embodiment of intelligent computation is a key one,
and is related to the problems surrounding the character of
mental causation. As the Brown University philosopher Jaegwon Kim
puts it, in relation to the brain, in his book Physicalism,
or Something Near Enough (2005):
"... the problem of mental causation is solvable only if
mentality is physically reducible; however, phenomenal consciousness
resists physical reduction, putting its causal efficacy in peril."
Turing himself refers to the role of mistakes in human intelligence
(a feature without a positive role in current machine-hosted
"if a machine is expected to be infallible, it cannot also be
intelligent. There are several theorems which say almost exactly that"
(talk to the London Mathematical Society, February 20,
1947, quoted by Andrew Hodges in Alan Turing - the enigma,
One will be looking for new thinking in this area, or more insightful
unifying of existing knowledge. This is an area in which it is easy
to produce speculation detached from clearly formulated models, and
there will be an emphasis on rigor, clarity and mathematical content.
- Nature of Information:
Complexity, Randomness, Hiddenness of Information
To what extent is complexity practically computable?
(Elucidate the distinctions and identities pertaining to the
numerous computational complexity classes.)
How much mathematical randomness occurs in nature?
(Characterize quantum 'randomness',
and answer questions in the mathematical theory relating to
information theory, cryptology and physics.)
This is an area more concerned with practical issues of computation,
and the extraction of information from less favorable contexts.
The focus is more on what is possible in the way of information
processing. Cryptology, where this topic has its origins, has
advanced hugely since the time of Turing and his Bletchley Park
contemporaries, and there is a rich and developing mathematical
theory. There are basic issues of complexity,
also going back to the time of Gödel
and Turing, where progress depends on solutions to fundamental
own interest in computational efficiency, and by implication
complexity of programs, goes back to his work with the
at Bletchley Park in the 1940s. In her article on
Computing over the Reals: Where Turing Meets Newton
(Notices of the AMS, 2004), Lenore Blum traces Turing's
early contribution to the study of complexity in the context of
numerical analysis in Rounding-off Errors in Matrix Processes
in The Quarterly Journal of Mechanics and Applied Mathematics,
vol. I, 1948, pp. 287-308, where she quotes Turing:
"It is convenient to have a measure of the amount of work involved in a
computing process, even though it be a very crude one ...
We might, for instance, count the number of additions, subtractions,
multiplications, divisions, recordings of numbers, ... "
There are connections to the fast-developing body of work
concerned with randomness and hidden information, and to
questions of the extent to which randomness occurs in
nature, particularly at the quantum level.
- How should we compute?
New Models of Logic and Computation
Working at the interface between science and mathematics,
how do classical models from logic and computability theory
model real-world processes?
(Develop the mathematical theories of relative computability,
and solve long-standing open problems relevant to
higher-order features of the real universe.)
What are the inadequacies of classical models, and what are
the new computational models which capture more of the essentials
of natural and physical computation?
(Develop new computational models, and identify the extent of
their real-world application.)
This theme would concern the extraction of mathematical models
of computation from nature, and the investigation of their
theoretical properties, a key concern of Turing himself.
Even for those who adhere to classical models, there are still
questions relating to modeling and underlying theory.
Here is David Deutsch quoted in the New Scientist
"I am sure we will have [conscious computers], I expect they will
be purely classical, and I expect that it will be a long time
in the future. Significant advances in our philosophical
understanding of what consciousness is, will be needed."
So topics can range from classical Turing analyses of the incomputable,
to new computational paradigms in bio-informatics, to quantum or
relativistic models from physics. Contributions relating to a
wide spectrum of different computational models, both new and
established ones, would be welcome. However, both modeling and
theory should be motivated by the drive to say something new and
fundamental about the nature of computation in the real universe.
The real contexts within which the models originate can also be
very diverse, but a unifying viewpoint will be encouraged.