(Comments and criticism welcome: Send to a.sloman[at]cs.bham.ac.uk)

Notes on Mathematics, Metaphysics, Evolution
The importance of chemistry for biological emergence

Expanded version of notes written after the FraMEPhys Levels of Explanation Workshop (https://twitter.com/framephys?lang=en) at the University of Birmingham 3rd April 2019.

Aaron Sloman
Honorary Professor of Artificial Intelligence and Cognitive Science
School of Computer Science, University of Birmingham


Levels of Explanation Workshop Details here:

Unfortunately I was able to attend only the last two talks, by Eleanor Knox, "From Abstraction to Explanation to Levels: some thoughts", and Alex Franklin, "How do levels emerge?".

This discussion paper attempts to summarise some of the important aspects of chemistry that are essential for the metaphysical creativity of biological evolution and which were not mentioned in the talks -- unless I missed something! This extends my previous notes on Mathematics, Metaphysics, Evolution following a FraMEPhys workshop in January 2019, http://www.cs.bham.ac.uk/research/projects/cogaff/misc/evo-framephys.html (also pdf) and themes from a talk to philosophy students in Zurich in March 2019 http://www.cs.bham.ac.uk/research/projects/cogaff/misc/zurich-talk.html
     How can a physical universe produce mathematical minds?
     And why are they so hard to replicate in current AI systems?

Additional changes were made to this document in April and May 2018, aiming to clarify what I am proposing, and its implications for philosophy, science and engineering (design of intelligent machines).

(DRAFT: Liable to change)

Contents below
Installed: 14 Apr 2019
Last updated: 22 Apr 2019; 27 Apr 2019;18 May 2019
This paper is
This is an extension of the Meta-Morphogenesis project, pointing out some of the connections between products of biological evolution and varieties of dynamic metaphysical grounding.

A partial index of discussion notes in this directory is in

Closely related:
-- Immanuel Kant's views on mathematics (1781)
http://www.cs.bham.ac.uk/research/projects/cogaff/misc/kant-maths.html or pdf
-- Alan Turing (1938) on mathematical intuition vs mathematical ingenuity.
He thought only the latter could be implemented on digital computers Turing(1938) -- discussed in:
http://www.cs.bham.ac.uk/research/projects/cogaff/misc/turing-intuition.html or pdf
-- The multiple roles of compositionality in biology
http://www.cs.bham.ac.uk/research/projects/cogaff/misc/sloman-compositionality.html (or pdf)
-- Jane Austen's concept of information (Not Claude Shannon's)
http://www.cs.bham.ac.uk/research/projects/cogaff/misc/austen-info.html (or pdf)
-- The Meta-Morphogegenesis project
http://www.cs.bham.ac.uk/research/projects/cogaff/misc/meta-morphogenesis.html (Also pdf)

This paper is a contribution to discussion of the chemical basis of biological emergence in a physical universe, partly inspired by Schrödinger's discussion in What is life? (1944) and Turing's 1952 paper on chemistry-based morphogenesis.

I haven't found a short way to summarise the network of ideas, so I apologise for length, and repetition of parts of related documents. I'll welcome comments, criticisms, suggestions or corrections.


Background (above)
The Chemical Basis of Emergence in a Physical Universe
Turing on mathematical intuition vs mathematical ingenuity
Virtual machinery running on brains

The Chemical Basis of Emergence in a Physical Universe

Biological evolution provides strikingly creative examples of emergence which would not have been possible without chemistry, for reasons identified by Schrödinger in What is life? (1944).

I suggest that is essentially because quantum mechanisms (e.g. bond formation and catalysis) allow molecular structures to have discrete changes between highly stable structures alongside continuous changes such as moving together or apart, twisting and folding.

Some of that creativity includes production of new kinds of virtual machinery required for information processing by individuals and groups of individuals, about which more below. (Most of what philosophers have written about virtual machines seems to be based on ignorance of some of their important features, mentioned below.)

Those sorts of creativity involve discovery and use of (pre-existing) parametrisable mathematical structures that are instantiated in multi-layered genomes (designs for organism types) and used to generate some of their products, including foraging behaviours using chemical trails, termite cathedrals, flight control mechanisms, human languages, human mathematical discoveries and practical applications of mathematics.

Some genome instantiation is done in cooperation with the environment: in humans and other intelligent species, important genome instantiation phases are staggered (delayed while information is acquired from the environment using previously instantiated genome layers e.g. in development of linguistic, spatial manipulation and and mathematical capabilities).

This blurs the boundary between machine and environment, and the results are totally different from the common view of computation as what happens in machines running fixed previously constructed programs.

Using the creativity provided by these mechanisms, evolution produced increasingly rich types of emergence, including emergence of ancient mathematical minds, and before that many animals with (unreflective) spatial intelligence.

Computer based AI systems have so far failed to replicate those forms of spatial intelligence. They cannot be replicated either using logic based theorem provers (which need to be given axioms and definitions to start from) or statistics based neurally inspired deep-learning systems, because spatial intelligence includes noticing impossibility and necessary consequences, and neither impossibility nor necessity can be derived from statistical evidence: they are not extremes of probability.


Examples include squirrels defeating "squirrel proof" bird feeders (search on youtube for examples) and this 17.5 month toddler recognizing and satisfying fairly complex 3D topological and geometrical possibilities simultaneously at her first attempt (although she had previous experience of pushing things through holes in rigid plastic or metal toys):
     4.5 minute video -- if short of time skip till the baby appears!
(Observant parents can find many more examples!)

I have many examples that require some of the kinds of thinking that could have led to the discoveries organised by Euclid in his Elements.

Something not in Euclid, but illustrating the point: Try this:

You probably find it obvious that a planar cut can be used to slice exactly one vertex off any convex polyhedron to produce a new convex polyhedron. (What brain mechanisms could support such knowledge?)
What effect will that cut have on the total numbers of vertices, edges and faces on the remaining polyhedron?

How do you know that no other result is possible?

(This is a simplified, disguised, special case of a well known topological theorem.)

At present, I don't think anyone knows how brains do that sort of thing. And nothing in current AI can replicate it either.

In particular, known neural mechanisms derive probabilities, and cannot find proofs of necessity or impossibility.

There are theorem provers that can derive geometrical or topological results, but only because they start from a collection of axioms expressed in a logical notation (e.g. Hilbert's axiomatisation of Euclid's Elements).

But the ancient mathematicians did not need that: rather their axioms were products of spatial intelligence, not arbitrarily adopted starting points.

(Sadly geometry is no longer taught as a standard part of education in this country -- seriously depriving young minds.)

To explain such mathematical capabilities we need to find designs for suitable virtual machines capable of running on brain mechanisms, and possibly also on future computers.

The latter may be impossible on digital computers, if some mathematical discoveries, or some forms of practical spatial reasoning, require a mixture of discrete and continuous processing: digital computers provide only the former -- they can run digital programs to simulate 2D and 3D continuous motion, but perceiving and understanding such motion is a different matter.

Turing on mathematical intuition vs mathematical ingenuity

I suspect this is connected with Alan Turing's unexplained remark in his thesis, that computers can emulate human mathematical ingenuity but not mathematical intuition. He didn't say why not, though I suspect it's related to his reasons for working on chemical processes over a decade later.

[ I was unaware of Turing's remark till last year. Discussed at length here:
http://www.cs.bham.ac.uk/research/projects/cogaff/misc/turing-intuition.html (also pdf) ]

I suspect that remark is related to his (unstated) motivation for switching attention to chemistry based morphogenesis shortly before he died.

For anyone who does not already know Turing's amazing 1952 paper "The chemical basis of morphogenesis", it is nicely summarised by Philip Ball:

"Forging patterns and making waves from biology to geology: a commentary on Turing (1952) `The chemical basis of morphogenesis'", in Royal Society Philosophical Transactions B
Related video interview: https://www.youtube.com/watch?v=6ed54_95kP4

Returning to emergence: we need to understand how biological information processing mechanisms such as brains can produce known forms of spatial reasoning about possibility, impossibility and necessity (among many other 'spatial' processes, e.g. perceiving, controlling actions, planning future actions, explaining successes and failures, learning, acquiring new concepts, communicating etc.)

Virtual machinery running on brains

We have learnt during the last 70-80 years that whereas some information processing systems can be fully implemented in mechanical devices -- e.g. an ancient mechanical calculator able to add, subtract, multiply or divide -- more sophisticated general purpose reasoning mechanisms need virtual machines running on physical machines supporting very complex rapidly changeable information processing mechanisms.

Virtual machinery can support much richer processes than physical machinery can, including very rapid creation and changes of complex structures, e.g. syntactic and semantic structures created by your brain while you read my tortuous prose.

Most philosophers seem not to have noticed key features of that engineering progress. Some of the important points about (emergent) computer-based virtual machinery were made in this paper, criticising common philosophical accounts of functionalist philosophies of mind:

Corey Maley and Gualtiero Piccinini, Get the Latest Upgrade: Functionalism 6.3.1, Philosophia Scientiae, 17 (2) 2013, pp. 1--15,
A summary, with comments on important gaps in their paper is in this messy document (in need of re-writing): http://www.cs.bham.ac.uk/research/projects/cogaff/misc/vm-functionalism.html#related

One of the most important points about virtual machinery often ignored by philosophers, is that virtual machines running in a physical computer (or collection of physical computers) are typically not fully implemented in the hardware in that physical machine or network, insofar as the virtual machine's functionality depends on causal links with other things.

For the same reason your mind (one of the most sophisticated types of virtual machine that we currently know of) is not, and cannot possibly be, fully implemented in your brain, or any other unchanging bounded portion of the physical universe. For example, your mind will, in future, refer to things that did not don't yet exist, including newborn humans and new kinds of technology. It remains the same mind after those changes, but the future implementation involves both new structures in your brain and new structures and relations in the physical environment, including, perhaps, technology that has not yet been thought of.

The physical components of the implementation substrate can change over time in both biological and human-created running virtual machines. For example email systems use a very complex virtual machine composed of many different virtual machines running on hardware scattered all over the planet, and constantly changing over time, e.g. as parts are added, removed or replaced, and switched on or off at different times. Likewise animal control systems implemented in brains, nerve networks and underlying physiological mechanisms supplying energy, maintaining functionality (e.g. repairing damage) etc.

The need for such virtual machines was discovered by biological evolution millions of years before human engineers made related discoveries during the second half of the 20th century. But there are still many things we have not learnt about these amazing products of evolution (also products of their own products). So we don't know how to replicate their functionality in computer-based machines, and it may be impossible to do so if brains make essential use of mechanisms implemented in a mixture of discrete and continuous process provided by sub-neural chemistry.

Why Chemistry?

The importance of chemistry as a layer of mechanism between physics and biology seems to have been largely ignored by (analytical?) philosophers of physics and philosophical physicists -- especially the fact that chemistry-based information processing may be intrinsically richer than the types of electronics-based information processing that now support artificial computation, including AI. I once wrote (but cannot remember where) that the fact that computers are digital (i.e. based on discrete mechanisms) is not a serious limitation because a continuous process can be approximated as closely as desired by a discrete process. But that is a mistake for at least two reasons: at any choice of resolution for discrete approximation, the representation of a wildly (infinitely) fluctuating continuous process will mostly be wrong.

That's because chemical processes combine both discrete state changes (bonds forming or being released especially by catalysts) and continuous changes (folding, twisting, moving closer/further etc.)

(A few neuroscientists have begun exploring the implications regarding limitations of network models of brain processing. E.g.
Patrick C. Trettenbrein, (2016), The Demise of the Synapse As the Locus of Memory: A Looming Paradigm Shift?, Frontiers in Systems Neuroscience 88

Schrödinger noticed some of the implications of that combination of discreteness and continuity in 1944, although he was writing before AI had been born (apart from the little known speculations of Ada Lovelace a century earlier) so he said nothing directly relevant to brains or minds.

Note that discreteness can emerge from continuity (e.g. repeatedly folding a flat sheet of paper to produce multiple distinct regions) whereas continuity cannot emerge in a discrete universe.

Contrast (classical) mathematics: the infinite sequences of zeros and ones form a mathematical continuum.

I suspect some physicists would argue (mistakenly?) that the continuity in chemical processes is an illusion because fundamental physics is discrete?
[David Deutsch seems to think something like that. The Beginning of Infinity: Explanations That Transform the World, Allen Lane & Penguin Books, 2011, page 458.]

Particle-based physical theories may allow continuous motion, but without any "locking" to produce persistent structures that can change continuously. (E.g. work on causal sets dosn't allow that -- unless I've misunderstood it?)

Even if my description is not accurate there remains a collection of deep differences between

-- physical processes using formation and release of chemical bonds and

-- sub-chemical interacting particles, which cannot directly produce interacting partly rigid partly flexible structures like birds nests, and skeleton based mechanisms in vertebrates.

Fundamental physics is used to explain the formation of enduring structures such as galaxies and stars. But the formation of relatively rigid entities like the solid portions of planets and asteroids requires chemical bonds holding things together while acted on by strong physical forces.

A more precise formulation would also include flexible structures forming (e.g.) nets with partly fixed topology and changing geometry.

Evolution's use of (Austen/not-Shannon) information
There is an important additional point that's only implicit so far:

The main metaphysical creativity of biological evolution, including its non-reductive emergence, depends on increasingly sophisticated uses of information, not in Shannon's (syntactic) sense, but in the much older sense of "information", used for example by Jane Austen in her novels, as explained in Jane Austen's concept of information.

The evolution of increasingly sophisticated kinds of (mostly virtual?) machinery producing and using information in that sense has products that resist the kinds of reduction that I think were discussed in the two talks I heard at the workshop.

E.g. such virtual machines can refer to entities, including past and future states and processes and even mythical entities, beyond the boundaries of the physical mechanisms in which the machines are implemented.

[This can be seen as a development of P.F. Strawson's claim, in Individuals -- an essay in Descriptive Metaphysics (1959) that some individuals have both P and M (physical and mental) properties -- though as far as I know he knew nothing about AI or computation.]

Unfortunately, in my experience, the crucial features of computer-based virtual machinery are not understood by most philosophers (even though they now use examples every day). So it's hard for them to think about the power added by chemistry-based virtual machinery.

The Maley/Piccinini account mentioned earlier is more complete than most of what I hear/or read from philosophers, but still incomplete, alas. (As far as I can tell, it has not been updated since the paper was published.)

Even Daniel Dennett gets some important things wrong (e.g. when he compares virtual machines with centres of gravity) though some of what he writes seems to agree with what I am claiming. However, he doesn't seem to have any interest in explaining how ancient mathematical discoveries were possible.

Evolution's creativity depends in part on use of multi-layered genomes where different layers are expressed at different times, and later layers are *parametrised*, i.e. they get parameters from collected consequences of earlier gene expression.

These layers control both the developing physical mechanisms in individual organisms and their developing information-processing capabilities. But the layers are products of previous evolutionary history. This idea is explained in more detail in this discussion of meta-configured genomes:

Thus the metaphysical creativity of a particular genome can depend on the number of generations it has been involved in, and how varied the environments of previous generations have been -- including variety in the products of earlier instances of the same genome. (I.e. the kinds of cultural evolution supported by the genome.) It can also be affected by similar evolutionary processes in other species in the environment, including food, predators, competitors, and symbionts of many kinds.

As a result of all these mechanisms and processes, the increasing abilities of humans to change the planet is in part a result of how many layers of gene expression they currently have and the time gaps that allow genes expressed during earlier phases of development to gather information for later genes to make use of. This is very different from standard models of learning.

(I think it is related to Annette Karmiloff-Smith's notion of "Representational Redescription" in her 1992 book Beyond Modularity.)

I regard Alistair Wilson's claim "grounding is metaphysical causation" as one way of summarising the huge variety of examples of metaphysically novel products of evolutionary and developmental processes.

This depends essentially on the fact that he allows grounding to be a temporally extended process, which I think is unusual among metaphysicians.

This is "work in progress": part of an extended project with many strands, explored in a collection of online papers growing in parallel: The Meta-Morphogenesis project: http://www.cs.bham.ac.uk/research/projects/cogaff/misc/meta-morphogenesis.html

Comments, criticisms, suggestions, questions all welcome.

Closely related documents:

Immanuel Kant's views on mathematics (1781)
http://www.cs.bham.ac.uk/research/projects/cogaff/misc/kant-maths.html or pdf

Alan Turing (1938) on mathematical intuition vs mathematical ingenuity.
He thought only the latter could be implemented on digital computers Turing(1938) -- discussed in:
http://www.cs.bham.ac.uk/research/projects/cogaff/misc/turing-intuition.html or pdf

Jane Austen's concept of information (Not Claude Shannon's)

The multiple roles of compositionality in biology
http://www.cs.bham.ac.uk/research/projects/cogaff/misc/sloman-compositionality.html (or pdf)

Copied from another document -- still to be pruned

A partial index of discussion notes in this directory is in

Jordana Cepelewicz, 2016 How Does a Mathematician's Brain Differ from That of a Mere Mortal? Scientific American Online April 12, 2016

Jackie Chappell and Aaron Sloman (2007a). Natural and artificial meta-configured altricial information-processing systems. (2007a) International Journal of Unconventional Computing, 3(3), 211-239. http://www.cs.bham.ac.uk/research/projects/cogaff/07.html#717

Jackie Chappell and Aaron Sloman, (2007b) Two ways of understanding causation: Humean and Kantian,
Contributions to WONAC: International Workshop on Natural and Artificial Cognition Pembroke College, Oxford, June 25-26, 2007, http://www.cs.bham.ac.uk/research/projects/cogaff/talks/wonac

M.B. Clowes, 1973, Man the creative machine: A perspective from Artificial Intelligence research, in The Limits of Human Nature, Ed. J. Benthall, Allen Lane, London.

Chris Christensen (2013) Review of Biographies of Alan Turing, Cryptologia, 37:4, 356-367,

Kenneth Craik, 1943, The Nature of Explanation, Cambridge University Press, London, New York
Craik drew attention to previously unnoticed problems about biological information processing in intelligent animals. For a draft incomplete discussion of his contribution, see

David Deutsch, 1997 The Fabric of Reality,
Allen Lane and Penguin Books

David Deutsch, 2011 The Beginning of Infinity: Explanations That Transform the World,
Allen Lane and Penguin Books, London.

Euclid and John Casey (2007) The First Six Books of the Elements of Euclid, Project Gutenberg, Salt Lake City, Third Edition, Revised and enlarged. Dublin: Hodges, Figgis, \& Co., Grafton-St. London: Longmans, Green, \& Co. 1885,

H. Gelernter, 1964, Realization of a geometry-theorem proving machine, reprinted in Computers and Thought, Eds. Edward A. Feigenbaum and Julian Feldman, McGraw-Hill, New York, pp. 134-152,

Robert Geretschlager, 1995. Euclidean Constructions and the Geometry of Origami, Mathematics Magazine, 68, 5, pp. 357--371, Mathematical Association of America, http://www.jstor.org/stable/2690924

James J. Gibson, 1979 The Ecological Approach to Visual Perception, Houghton Mifflin, Boston, MA,

Ira Goldstein, 1973, Elementary Geometry Theorem Proving MIT AI Memo 280, April 1973

Yacin Hamami and John Mumma, 2013, Prolegomena to a Cognitive Investigation of Euclidean Diagrammatic Reasoning, in Journ Log Lang Inf 22, pp 421-448

David Hilbert, 1899, The Foundations of Geometry,, available at Project Gutenberg, Salt Lake City, http://www.gutenberg.org/ebooks/17384 2005, Translated 1902 by E.J. Townsend, from 1899 German edition,

Andrew Hodges, 1999. Turing. New York: Routledge.

T. Ida and J. Fleuriot, Eds., Proc. 9th Int. Workshop on Automated Deduction in Geometry (ADG 2012), Edinburgh, September, 2012, University of Edinburgh, Informatics Research Report,

Immanuel Kant's Critique of Pure Reason (1781)
has relevant ideas and questions, but he lacked our present understanding of information processing (which is still too limited)

Imre Lakatos, Proofs and Refutations,
Cambridge University Press, 1976,

John McCarthy and Patrick J. Hayes, 1969, "Some philosophical problems from the standpoint of AI", Machine Intelligence 4, Eds. B. Meltzer and D. Michie, pp. 463--502, Edinburgh University Press,

Kenneth Manders (1998) The Euclidean Diagram, reprinted 2008 in The Philosophy of Mathematical Practice, OUP, pp.80--133 Ed Paolo Mancosu,

Kenneth Manders (2008) "Diagram-Based Geometric Practice", In Paolo Mancosu (ed.), The Philosophy of Mathematical Practice. OUP, pp.65--79

D. Mareschal and M. S. C. Thomas, "How computational models help explain the origins of reasoning," in IEEE Computational Intelligence Magazine, vol. 1, no. 3, pp. 32-40, Aug. 2006. doi: 10.1109/MCI.2006.1672986

Noboru Matsuda and Kurt Vanlehn (2004), GRAMY: A Geometry Theorem Prover Capable of Construction, Journal of Automated Reasoning Vol 32 (3--33) Kluwer Academic Publishers. Netherlands.

David Mumford, 2016, Grammar isn't merely part of language, Online Blog,

Tuck Newport, Brains and Computers: Amino Acids versus Transistors,
2015, Kindle,
Discusses implications of href="#von-Neumann-brain">von Neumann 1958,

Jean Piaget, (1952). The Child's Conception of Number. London: Routledge & Kegan Paul.

Piaget, 1981, 1983 Jean Piaget's last two (closely related) books written with collaborators are relevant, though I don't think he had good explanatory theories.

Possibility and Necessity
Vol 1. The role of possibility in cognitive development (1981)
Vol 2. The role of necessity in cognitive development (1983)
University of Minnesota Press, Tr. by Helga Feider from French in 1987

(Like Kant, Piaget had deep observations but lacked an understanding of information processing mechanisms, required for explanatory theories.)

Gualtiero Piccinini (2003), Alan Turing and the Mathematical Objection, Minds and Machines
Feb, 2003, Kluwer Academic

L. J. Rips, A. Bloomfield and J. Asmuth, 2008, From Numerical Concepts to Concepts of Number, The Behavioral and Brain Sciences, Vol 31, no 6, pp. 623--642,

Carl J. Posy (Ed), 1992 Kant's Philosophy of Mathematics-Modern Essays Synthese Library Vol 219 Kluwer Academic Publishers

Erwin Schrödinger (1944) What is life? CUP, Cambridge,
An annotated transcript of part of this book is here
http://www.cs.bham.ac.uk/research/projects/cogaff/misc/schrodinger-life.html (or PDF)

Dana Scott, 2014, Geometry without points. (Video lecture, 23 June 2014,University of Edinburgh)

Frege on the Foundation of Geometry in Intuition Journal for the History of Analytical Philosophy Vol 3, No 6. pp 1-23,

Siemann, J., & Petermann, F. (2018). Innate or Acquired? - Disentangling Number Sense and Early Number Competencies. Frontiers in psychology, 9, 571. doi:10.3389/fpsyg.2018.00571

Sloman, A. (1962). Knowing and Understanding: Relations between meaning and truth, meaning and necessary truth, meaning and synthetic necessary truth (DPhil Thesis), Oxford University. (Transcribed version online.)

Aaron Sloman, 1965, "Necessary", "A Priori" and "Analytic", Analysis, Vol 26, No 1, pp. 12--16.

A. Sloman, 1971, "Interactions between philosophy and AI: The role of intuition and non-logical reasoning in intelligence", in Proc 2nd IJCAI, pp. 209--226, London. William Kaufmann. Reprinted in Artificial Intelligence, vol 2, 3-4, pp 209-225, 1971.
A slightly expanded version was published as chapter 7 of Sloman 1978, available here.

A. Sloman, 1978 The Computer Revolution in Philosophy,
Harvester Press (and Humanities Press), Hassocks, Sussex.
Free, partly revised, edition online:

A. Sloman, (1978b). What About Their Internal Languages? Commentary on three articles by Premack, D., Woodruff, G., by Griffin, D.R., and by Savage-Rumbaugh, E.S., Rumbaugh, D.R., Boysen, S. in BBS Journal 1978, 1 (4). Behavioral and Brain Sciences, 1(4), 515.

Aaron Sloman (2012-...), The Meta-Morphogenesis (Self-Informing Universe) Project (begun 2012, with several progress reports, but still work in progress).

Aaron Sloman (2014), Tentative non-mathematical thoughts on entropy, evolution, and construction-kits (Discussion Note)

Aaron Sloman, 2015-18, Some (possibly) new considerations regarding impossible objects, (Their significance for mathematical cognition, current serious limitations of AI vision systems, and philosophy of mind, i.e. contents of consciousness), Online research presentation,

Aaron Sloman, 2013--2018, Jane Austen's concept of information (Not Claude Shannon's)
Online technical report, University of Birmingham,

Aaron Sloman, 2016, Natural Vision and Mathematics: Seeing Impossibilities, in Proceedings of Second Workshop on: Bridging the Gap between Human and Automated Reasoning, IJCAI 2016, pp.86--101, Eds. Ulrich Furbach and Claudia Schon, July, 9, New York,

A. Sloman (with help from Jackie Chappell), 2017-8, The Meta-Configured Genome (unpublished)

A. Sloman, 2018a, A Super-Turing (Multi) Membrane Machine for Geometers Part 1
(Also for toddlers, and other intelligent animals)
PART 1: Philosophical and biological background

A. Sloman, 2018b A Super-Turing (Multi) Membrane Machine for Geometers Part 2
(Also for toddlers, and other intelligent animals)
PART 2: Towards a specification for mechanisms

Aaron Sloman, 2018c,
Biologically Evolved Forms of Compositionality
Structural relations and constraints vs Statistical correlations and probabilities
Expanded version of paper accepted for First Symposium on Compositional Structures (SYCO 1)
Sept 2018 School of Computer Science, University of Birmingham, UK

Wikipedia contributors, Tarski's axioms for geometry Wikipedia, The Free Encyclopedia,
[Accessed 6-November-2018]

Trettenbrein, Patrick C., 2016, The Demise of the Synapse As the Locus of Memory: A Looming Paradigm Shift?, Frontiers in Systems Neuroscience, Vol 88,

A. M. Turing, (1950) Computing machinery and intelligence,
Mind, 59, pp. 433--460, 1950,
(reprinted in many collections, e.g. E.A. Feigenbaum and J. Feldman (eds)
Computers and Thought McGraw-Hill, New York, 1963, 11--35),
WARNING: some of the online and published copies of this paper have errors,
including claiming that computers will have 109 rather than 109 bits
of memory. Anyone who blindly copies that error cannot be trusted as a commentator.

A. M. Turing, (1952), 'The Chemical Basis Of Morphogenesis', in
Phil. Trans. R. Soc. London B 237, 237, pp. 37--72.
(Also reprinted(with commentaries) in S. B. Cooper and J. van Leeuwen, EDs (2013)).

A useful summary of Turing's 1952 paper for non-mathematicians is:
Philip Ball, 2015, Forging patterns and making waves from biology to geology: a commentary on Turing (1952) `The chemical basis of morphogenesis', Royal Society Philosophical Transactions B,

John von Neumann, 1958 The Computer and the Brain (Silliman Memorial Lectures), Yale University Press. 3rd Edition, with Foreword by Ray Kurzweill. Originally published 1958.

Wikipedia contributors, 2018, Mathematics of paper folding Wikipedia, The Free Encyclopedia,

Alastair Wilson, 2017, Metaphysical Causation, Nous

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