NOTE 25 Dec 2015
This is a developing version of a paper that was frozen in December 2015, as an
invited chapter to appear in The Incomputable edited
by S Barry Cooper and Mariya Soskova, to be published by Springer. A preprint
was frozen in December 2015, but this document, and others on this web site,
will continue growing, as the subject is so vast, and there are still so many
gaps in our understanding (or mine, anyway).
The frozen preprint is at:
http://www.cs.bham.ac.uk/research/projects/cogaff/incomputable-kits-sloman.pdf
A closely related paper:
TENTATIVE NON-MATHEMATICAL THOUGHTS
ON ENTROPY, EVOLUTION, AND CONSTRUCTION-KITS
(Entropy, Evolution and Lionel Penrose's Droguli)
can be found at
http://www.cs.bham.ac.uk/research/projects/cogaff/misc/entropy-evolution.html
Construction kits for biological evolution
(Including evolution of minds and mathematical abilities.)
The scientific/metaphysical explanatory role of construction kits:
fundamental and derived kits;
concrete, abstract and hybrid kits.
(CHANGING DRAFT: Stored copies will soon be out of date.)
Aaron Sloman
School of Computer Science, University of Birmingham.
The Turing-inspired Meta-Morphogenesis (M-M) project asks:
How can a cloud of dust give birth to a planet
full of living things as diverse as
life on Earth?
A Protoplanetary Dust Cloud?
[NASA artist's impression of a protoplanetary disk, from WikiMedia]
Part of my answer is a theory of construction-kits, including
construction-kits produced by biological evolution and its products.
This paper presents some preliminary, incomplete, ideas about types
of construction-kit and their roles in biological evolution.
Additional topics are included or linked at the main M-M web site:
http://www.cs.bham.ac.uk/research/projects/cogaff/misc/meta-morphogenesis.html
JUMP TO TABLE OF CONTENTS
Formats of this document
This paper is available in two forms (HTML -- primary) and (PDF -- derived):
http://www.cs.bham.ac.uk/research/projects/cogaff/misc/construction-kits.html
http://www.cs.bham.ac.uk/research/projects/cogaff/misc/construction-kits.pdf
(May be out of date.)
A version of this document was posted to slideshare.net on 5 Jan 2015, then
later updated in on 11th Sept 2015, making this document available in flash
format. That version will not necessarily be updated whenever the html/pdf
versions are. (Not all the links work in the slideshare version.) See
http://www.slideshare.net/asloman/construction-kits
This is part of the Meta-Morphogenesis project:
http://www.cs.bham.ac.uk/research/projects/cogaff/misc/meta-morphogenesis.html
An older version of this paper was prepared as a set of reconfigurable notes for
a
tutorial (presented at the ESSENCE Summer School, Edinburgh Informatics, August
2015), and is available at
http://www.cs.bham.ac.uk/research/projects/cogaff/misc/essence-kits-tut.html
A partial index of discussion notes here is in
http://www.cs.bham.ac.uk/research/projects/cogaff/misc/AREADME.html
DRAFT: LIABLE TO CHANGE.
Saved copies will grow out of date.
Last Revised: Dec 25, 2015
ABSTRACT
This is part of the Turing-inspired Meta-Morphogenesis project, which aims to
identify transitions in information-processing since the earliest
proto-organisms, in order to provide new understanding of varieties of
biological intelligence, including the mathematical intelligence that produced
Euclid's Elements. (Explaining evolution of mathematicians is much harder
than explaining evolution of consciousness!) Transitions depend on
"construction-kits", including the initial "Fundamental Construction Kit"
(FCK) based on physics and Derived Construction Kits (DCKs) produced by
evolution, development, learning and culture. Some construction kits (e.g. Lego,
Meccano, plasticine, sand) are concrete: using physical components and
relationships. Others (e.g. grammars, proof systems and programming languages)
are abstract: producing abstract entities, e.g. sentences, proofs, and new
abstract construction kits. Mixtures of the two are hybrid kits. Some are
meta-construction kits: able to create, modify or combine construction kits.
Construction kits are generative: they explain sets of possible construction
processes, and possible products, with mathematical properties and limitations
that are mathematical consequences of properties of the kit and its environment.
Evolution and development both make new construction kits possible.
Study of the FCK and DCKs can lead us to new answers to old questions, e.g.
about the nature of mathematics, language, mind, science, and life, exposing
deep connections between science and metaphysics. Showing how the FCK makes its
derivatives, including all the processes and products of natural selection,
possible is a challenge for science and philosophy. This is a long-term research
programme with a good chance of being progressive in the sense of Lakatos.
Later, this may explain how to overcome serious current limitations of
AI (artificial intelligence), robotics, neuroscience and psychology.
Note:
My ideas have probably been influenced in more ways than I recognise by
my colleague
Margaret Boden, whose work has linked AI/Cognitive Science to Biology over
several decades, notably in her magnum opus published in 2006
by OUP, Mind As Machine: A history of Cognitive Science (Vols 1-2).
I have also profited from thoughts and writings of
Noam Chomsky,
Max Clowes,
Paul Davies,
Richard Dawkins,
Daniel Dennett,
Tibor Ganti,
Brian Goodwin,
Annette Karmiloff-Smith,
Erwin Schrödinger,
John McCarthy,
Marvin Minsky,
Jean Piaget,
John Maynard Smith,
and many colleagues and students.
Background note in a separate document, 1 Mar 2015
Contents
Abstract (above)
Background note in a separate document (above)
(Everything else is below this list of contents.)
1
1 Background: What is science? Beyond Popper and Lakatos
2 Fundamental and Derived Construction Kits (FCK, DCKs)
2.1 Combinatorics of construction processes
2.2 Construction Kit Ontologies
2.3 Construction kits built during development (epigenesis)
2.4 The variety of biological construction kits
2.5 Increasingly varied mathematical structures
2.6 Thermodynamic issues
2.7 Scaffolding in construction kits
2.8 Biological construction kits
3 Concrete (physical), abstract and hybrid construction kits
3.1 Kits providing external sensors and motors
3.2 Mechanisms for storing, transforming and using information
3.3 Mechanisms for controlling position, motion and timing
3.4 Combining construction kits
3.5 Combining abstract construction kits
4 Construction kits generate possibilities and impossibilities
4.1 Construction kits for making information-users
4.2 Different roles for information
4.3 Motivational mechanisms
5 Mathematics: Some constructions exclude or necessitate others
5.1 Proof-like features of evolution
5.2 Euclid's construction kit
5.3 Mathematical discoveries based on exploring construction kits
5.4 Evolution's (blind) mathematical discoveries
6 Varieties of Derived Construction Kit
6.1 A new type of research project
6.2 Construction-kits for biological information-processing
6.3 Representational blind spots of many scientists
6.4 Representing rewards, preferences, values
7 Computational/Information-processing construction-kits
7.1 Infinite, or potentially infinite, generative power
8 Types and levels of explanation of possibilities
9 Alan Turing's Construction kits
9.1 Beyond Turing machines: chemistry
9.2 Using properties of a construction-kit to explain possibilities
9.3 Bounded and unbounded construction kits
10 Conclusion: Construction kits for Meta-Morphogenesis
1 Background: What is science? Beyond Popper and Lakatos
How is it possible for very varied forms of life to evolve from lifeless
matter, including a mathematical species able to make the discoveries
presented in Euclid's Elements?1
Explaining evolution of mathematical insight is much harder than explaining
evolution of consciousness! (Even insects must be conscious of aspects of their
surroundings.) An outline answer is based on
construction kits that make other things (including new construction kits)
possible. The need for science to include theories that explain how something is
possible has not been widely acknowledged. Explaining how X is possible (e.g.
how humans playing chess can produce a certain board configuration) need not
provide a basis for predicting when X will be realised, so the theory used
cannot be falsified by non-occurrence. Popper, [1934] labelled such theories
"non-scientific" - at best metaphysics.
His falsifiability criterion has been blindly followed by many
scientists who ignore the history of science. E.g. the ancient atomic theory of
matter was not falsifiable, but was an early example of a deep scientific
theory. Later, Popper shifted his ground, e.g. in
Popper, [1978], and expressed great admiration for Darwin's theory of
Natural Selection, despite its unfalsifiability.
Lakatos () extended Popper's philosophy of science,
showing how to evaluate competing scientific research programmes over time,
according to their progress. He offered criteria for distinguishing
"progressive" from "degenerating" research programmes, on the basis of their
patterns of development, e.g. whether they systematically generate questions
that lead to new empirical discoveries, and new applications.
It is not clear to me whether he understood that his distinction could also be
applied to theories explaining how something is possible.
Chapter 2 of
Sloman, [1978]2
modified the ideas of Popper and Lakatos to accommodate scientific theories
about what is possible, e.g. types of plant, types of animal, types of
reproduction, types of consciousness, types of thinking, types of learning,
types of communication, types of molecule, types of chemical interaction, and
types of biological information-processing. It presented criteria for evaluating
theories of what is possible and how they are possible, including theories that
straddle science and metaphysics. Insisting on sharp boundaries between science
and metaphysics harms both. Each can be pursued with rigour and openness to
specific kinds of criticism. A separate paper3
includes a section entitled "Why allowing non-falsifiable theories doesn't make
science soft and mushy", and discusses the general concept of "explaining
possibilities", its importance in science, the criteria for evaluating such
explanations, and how this notion conflicts with the falsifiability requirement
for scientific theories. Further examples are in Sloman, [1996a]. The extremely
ambitious Turing-inspired Meta-Morphogenesis project, proposed in
Sloman, [2013b]4
depends on these ideas, and will be a test of their fruitfulness, in a
combination of metaphysics and science.
This paper, straddling science and metaphysics asks: How is it possible for
natural selection, starting on a lifeless planet, to produce billions of
enormously varied organisms, in environments of many kinds, including
mathematicians able to discover and prove geometrical theorems? An outline
answer is presented in terms of construction kits: the Fundamental
(physical) Construction Kit (the FCK), and a variety of "concrete",
"abstract" and "hybrid" Derived Construction Kits (DCKs). that together are
conjectured to explain how evolution is possible, including evolution of
mathematicians. The FCK and its relations to DCKs are crudely depicted later in
Figures 1 and 2. Inspired by ideas in Kant, [1781],
construction kits are also offered as providing
Biological/Evolutionary foundations for core parts of mathematics, including
parts used by evolution (but not consciously, of course) long
before there were human mathematicians.
Note on "Making Possible":
"X makes Y possible" as used here does
not imply that if X does not exist then Y is
impossible, only that one route to existence of Y is via X.
Other things can also make Y possible, e.g., an alternative
construction kit. So "makes possible" is a relation of sufficiency, not
necessity. The exception is the case where X is the FCK - the Fundamental
Construction Kit - since all concrete constructions must start from it (in
this universe?). If Y is abstract, there need not be
something like the FCK from which it must be derived. The space of abstract
construction kits may not have a fixed "root". However, the abstract
construction kits that can be thought about by physically implemented thinkers
may be constrained by a future replacement for the Church-Turing thesis, based
on later versions of ideas presented here. Although my questions about
explaining possibilities arise in the overlap between philosophy and science
Sloman, [1978,Ch.2], I am not aware of any philosophical work that explicitly
addresses the theses discussed here, though there seem to be examples of
potential overlap, e.g. Bennett, [2011,Wilson, [2015].
2 Fundamental and Derived Construction Kits (FCK, DCKs)
Natural selection alone cannot explain how evolution happens, for it must have
options to select from. What sorts of mechanisms can produce options that differ
so much in so many ways, allowing
evolution to produce microbes, fungi, oaks, elephants, octopuses, crows, new
niches, ecosystems, cultures, etc.? Various sorts of construction kit, including
evolved/derived construction kits, help to explain the emergence of new options.
What explains the possibility of these construction kits? Ultimately features of
fundamental physics including those emphasised by Schrödinger (),
discussed below. Why did it take so much longer for evolution to produce baboons
than bacteria? Not merely because baboons are more complex, but also because
evolution had to produce more complex construction kits, to make baboon-building
possible.
Construction-kits are the "hidden heroes" of evolution. Life as we know it
requires construction kits supporting construction of machines with many
capabilities, including growing many types of material, many types of mechanism,
many types of highly functional bodies, immune systems, digestive systems,
repair mechanisms, reproductive machinery, and even mathematicians!
A kit needs more than basic materials. If all the atoms required for making a
loaf of bread could somehow be put into a container, no loaf could emerge. Not
even the best bread making machine, with paddle and heater, could produce bread
from atoms, since that requires atoms pre-assembled into the right amounts of
flour, sugar, yeast, water, etc. Only different, separate, histories
can produce the molecules and multi-molecule components, e.g. grains of yeast
or flour. Likewise, no modern fish, reptile, bird, or mammal could be created
simply by bringing together enough atoms of all the required sorts; and no
machine, not even an intelligent human designer, could assemble a functioning
airliner, computer, or skyscraper directly from the required atoms. Why not,
and what are the alternatives? We first state the problem of constructing very
complex working machines in very general terms and indicate some of the variety
of strategies produced by evolution, followed later by conjectured features of a
very complex, but still incomplete, explanatory story.
2.1 Combinatorics of construction processes
Reliable construction
of a living entity requires: types of matter, machines that
manipulate matter, physical assembly processes, stores of energy for use during
construction, and usually
information e.g. about which components to assemble at
each stage, how to assemble them, and how to decide in what order to do so. This
requires, at every stage, at least: (i) components available for remaining
stages, (ii) mechanisms capable of assembling the components, (iii) mechanisms
able to decide what should happen next, whether selection is random or
information-based.
If there are N types of basic component and a task requires an object of type O
composed of K basic components, the size of an blind exhaustive search for a
sequence of types of basic components to assemble an O is up to NK sequences,
a number that rapidly grows astronomically large as K increases. If, instead of
starting from the N types of basic components, construction uses M types
of pre-assembled components, each containing P basic components, then an O
will require only K/P pre-assembled parts. The search space for a route to O is
reduced in size to M(K/P). Compare assembling an essay of length 10,000
characters (a) by systematically trying elements of a set of about 30 possible
characters (including punctuation and spaces) with (b) choosing from a set of
1000 useful words and phrases, of average length 50 characters. In the first
case each choice has 30 options but 10,000 choices are required. In the second
case there are 1000 options per choice, but far fewer stages: 200 instead of
10,000 stages. So the size of the (exhaustive) search space is reduced from
3010000, a number with 14,773 digits, to about 1000200, a number with
only 602 digits: a very much smaller number. So trying only good pre-built
substructures at each stage of a construction process, can make a huge reduction
to the search space for solutions of a given size - possibly eliminating
some solutions.
So, learning from experience by storing useful subsequences can achieve dramatic
reductions, analogous to a house designer moving from thinking about how to
assemble atoms, to thinking about assembling molecules, then bricks, planks,
tiles, then pre-manufactured house sections. The reduced search space contains
fewer samples from the original possibilities, but the original space has a much
larger proportion of useless options. As sizes of pre-designed components
increase so does the variety of pre-designed options to choose from at each
step, though far, far, fewer search steps are required for a working solution: a
very much shorter evolutionary process. The cost may be exclusion of some design
options.
This indicates intuitively, but very crudely, how using increasingly large,
already tested useful part-solutions can enormously reduce the search for viable
solutions. The technique is familiar to many programmers, in the use of
"memo-functions" ("memoization") to reduce computation time, e.g. computing
fibonacci numbers. The family of computational search techniques known as
"Genetic
Programming"5
uses related ideas. The use of "crossover" in evolution (and in
Genetic Algorithms), allows parts of each parent's design specification to be
used in new combinations.
In biological evolution, instead of previous solutions being stored for
future re-use, information about how to build components of previous
solutions, is stored in genomes. Evolution, the Great
Blind Mathematician discovered memoization long before we did. A closely related
strategy is to record fragments that cannot be useful in certain types of
problem, to prevent wasteful attempts to use such fragments. Expert
mathematicians learn from experience which options are useless (e.g. dividing by
zero). This could be described as "negative-memoization". Are innate
aversions examples of evolution doing something like that?
Without prior information about useful components and combinations of pre-built
components, random assembly processes can be used. If mechanisms are available
for recording larger structures that have been found to be useful or useless,
the search space for new designs can be shrunk. By doing the searching and
experimentation using information about how to build things rather than
directly recombining the built physical structures themselves, evolution reduces
the problem of recording what has been learnt.
Figure 1
Figure 1: This is a crude representation of the Fundamental
Construction Kit (FCK) (on left) and (on right) a collection of trajectories
from the FCK through the space of possible trajectories to increasingly complex
mechanisms.
The Fundamental Construction Kit (FCK) provided by the physical universe made
possible all the forms of life that have so far evolved on earth, and also
possible but still unrealised forms of life, in possible types of
physical environment. Fig. 1 shows how a common initial construction
kit can generate many possible trajectories, in which components of the kit are
assembled to produce new instances (living or non-living). The space of possible
trajectories for combining basic constituents is enormous, but routes can be
shortened and search spaces shrunk by building derived construction kits (DCKs),
that assemble larger structures in fewer steps6,
as indicated in Fig. 2.
Figure 2
Figure 2: Further transitions: a
fundamental construction kit (FCK) on left gives rise to new evolved "derived"
construction kits, such as the DCK on the right, from which new trajectories can
begin, rapidly producing new more complex designs, e.g. organisms with new
morphologies and new information-processing mechanisms. The shapes and colours
(crudely) indicate qualitative differences between components of old and new
construction kits, and related trajectories. A DCK trajectory uses larger
components and is therefore much shorter than the equivalent FCK trajectory.
The history of technology, science and engineering includes many transitions in
which new construction kits are derived from old ones. That includes the science
and technology of digital computation, where new advances used an enormous
variety of discoveries and inventions, from punched cards (used in Jacquard
looms) through many types of electronic device, many types of programming
language, many types of external interface (not available on Turing machines!),
many types of operating system, many types of network connection, many types of
virtual machine, and many applications. Particular inventions were generalised,
using mathematical abstractions, to patterns that could be reused in new
contexts. The production of new applications frequently involved
production of new tools for building more complex applications.
Natural selection did all this on an even larger scale, with far more variety,
probably discovering many obscure problems and solutions still unknown to us.
(An educational moral: teaching only what has been found most useful can discard
future routes to possible major new advances - like depleting a gene pool.)
Biological construction kits derived from the FCK can combine to form new
Derived Construction Kits (DCKs), some specified in genomes, and (very much
later) some discovered or designed by individuals (e.g. during epigenesis
Sect. 2.3), or by groups, for example new languages. Compared
with derivation from the FCK, the rough calculations above show how DCKs can
enormously speed up searching for new complex entities with new properties and
behaviours. See Fig. 2.
New DCKs that evolve in different species in different locations, may have
overlapping functionality, based on different mechanisms: a form of
convergent evolution. E.g., mechanisms enabling elephants to learn to use
trunk, eyes, and brain to manipulate food may share features with those enabling
primates to learn to use hands, eyes, and brains to manipulate food. In both
cases competences evolve in response to structurally similar affordances in the
environment. This extends ideas in [Gibson, 1979] to include affordances for a
species, or collection of species.7
2.2 Construction Kit Ontologies
A construction kit (and its products) can exist without being described. However
scientists need to use various forms of language in order to describe the
entities they observe or postulate in explanations, and to formulate new
questions to be answered. So a physicist studying the FCK will need one or more
construction kits for defining concepts, formulating questions, formulating
theories and conjectures, constructing models, etc. Part of the process of
science is extending construction kits for theory formation. Something
similar must be done by natural selection: extending useful genetic information
structures that store specifications for useful components.
This relates to claims that have been made about requirements for control
systems and for scientific theories. For example, if a system is to be capable
of distinguishing N different situations and responding differently to them it
must be capable of being in at least N different states (recognition+control
states). This is a variant of Ashby's "Law of Requisite Variety"
Ashby, [1956].
Several thinkers have discussed representational requirements for scientific
theories, or for specifications of designs. Chomsky, [1965] distinguished
requirements for theories of language, which he labelled observational
adequacy (covering the variety of observed uses of a particular language),
descriptive adequacy (covering the intuitively understood principles that
account for the scope of a particular language) and explanatory adequacy
(providing a basis for explaining how any language can be acquired on the basis
of data available to the learner). These labels were vaguely echoed in
McCarthy and Hayes, [1969] who described a form of representation as being
metaphysically adequate if it can express anything that can be the case,
epistemologically adequate if it can express anything that can be known by
humans and future robots, and heuristically adequate if it supports
efficient modes of reasoning and problem-solving. (I have simplified all these
proposals.)
Requirements can also be specified for powers of various sorts of biological
construction kits. The fundamental construction kit (FCK) must have the power to
make any form of life that ever existed or will exist possible: if necessary
using huge search spaces. DCKs may meet different requirements, e.g. each
supporting fewer types of life form, but enabling those life forms to be
"discovered" in a reasonable time by natural selection, and reproduced
(relatively) rapidly. Early DCKs may support the simplest organisms that
reproduce by making copies of themselves [Ganti, 2003]. At later stages of
evolution, DCKs are needed that allow construction of organisms that change
their properties during development and change their control mechanisms
appropriately as they grow Thompson, [1917]. This requires the ability to
produce individuals whose features are parametrised with parameters that
change over time. More sophisticated DCKs must be able to produce species that
modify their knowledge and their behaviours not merely as required to
accommodate their own growth but also to cope with changing physical
environments, new predators, new prey and new shared knowledge. A special case
of this is having genetic mechanisms able to support development of a wide
enough range of linguistic competences to match any type of human language,
developed in any social or geographical context. However, the phenomenon is far
more general than language development, as discussed in the next section.
2.3 Construction kits built during development (epigenesis)
Some new construction kits are products of evolution of a species and are
initially shared only between a few members of the species (barring genetic
abnormalities), alongside cross-species construction kits shared between
species, such as those used in mechanisms of reproduction and growth in related
species. Evolution also discovered the benefits of "meta-construction-kits":
mechanisms that allow members of a species to build new construction kits during
their own development.
Examples include mechanisms for learning that are initially generic mechanisms
shared across individuals, and developed by individuals on the basis of their
own previously encountered learning experiences, which may be different in
different environments for members of the same species. Human language learning
is a striking example: things learnt at earlier stages make new things learnable
that might not be learnable by an individual transferred from a different
environment, part way through learning a different language. This contrast
between genetically specified and individually built capabilities for learning
and development was labelled a difference between "pre-configured" and
"meta-configured" competences in Chappell and Sloman, [2007], summarised in
Fig. 3.
The meta-configured competences are partly specified in the genome but that
specification is combined with information abstracted from individual
experiences.
Mathematical development in humans seems to be a
special case of growth of such meta-configured competences.
Related ideas are in Karmiloff-Smith, [1992].
Figure 3: A construction kit gives rise to very
different individuals if the genome interacts with the environment in
increasingly complex ways during development. Precocial species use only the
downward routes on the left, producing preconfigured competences. Competences of
altricial species, using staggered development, may be far more varied. Results
of using earlier competences interact with the genome,
producing meta-configured competences on the right.
Construction kits used for assembly of new organisms that start as a seed or an
egg enable many different processes in which components are assembled in
parallel, using abilities of the different sub-processes to constrain one
another. Nobody knows the full variety of ways in which parallel construction
processes can exercise mutual control in developing organisms. One implication
is that there are not simple correlations between genes and organism features.
Explaining the many ways in which a genome can
orchestrate parallel processes of growth, development, formation of connections,
etc. is a huge challenge. A framework allowing abstract specifications in a
genome to interact with details of the environment in instantiating complex
designs is illustrated schematically in Fig. 3. An example
might be the proposal in Popper, [1976] that newly evolved desires of
individual organisms (e.g. desires to reach fruit in taller trees) could
indirectly and gradually, across generations, influence selection of physical
characteristics (e.g. longer necks, abilities to jump higher) that improve
success-rates of actions triggered by those desires.
Various kinds of creativity, including mathematical creativity, might result
from such transitions. This generalises Waddington's "epigenetic landscape"
metaphor
Waddington, [1957], by allowing individual members of a species to
partially construct and repeatedly modify their own epigenetic landscapes
instead of merely following paths in a landscape that is common to the species.
Mechanisms that increase developmental variability may also make new
developmental defects possible (e.g. autism?)8.
2.4 The variety of biological construction kits
As products of physical construction kits become more complex, with more ways of
contributing to needs of organisms, and directly or indirectly to reproductive
fitness, they require increasingly sophisticated control mechanisms. New sorts
of control often use new types of information, so new construction kits for
building types of information-processing mechanism are needed. The simplest
organisms use only a few types of (mainly chemical) sensor providing information
about internal states and the immediate external physical environment, and have
very few behavioural options. They acquire, use and replace fragments of
information, using the same forms of information throughout their life, to
control deployment of a fixed repertoire of capabilities. More complex organisms
acquire information about enduring spatial locations in extended terrain
including static and changing routes between static and changing
resources and dangers. They need to construct and use far more complex (internal
or external) information stores about their environment, and, in some cases,
"meta-semantic" information about information-processing, in themselves and in
others, e.g. conspecifics, predators and prey.
What forms can such information take? Many controlled systems have states that
can be represented by a fixed set of physical measures, often referred to as
"variables", representing states of sensors, output signals,
and internal states of various sorts. Relationships between such
state-components are often represented mathematically by equations, including
differential equations, and constraints (e.g. inequalities)
specifying restricted, possibly time-varying, ranges of values for the
variables, or magnitude relations between the variables. A system with N
variables (including derivatives) has a state of a fixed dimension, N. The only
way to record new information in such systems is in static or dynamic values for
numeric variables - changing "state vectors" and possibly alterations in the
equations. A typical example is
Powers, [1973], inspired by Wiener, [1961] and Ashby, [1952].
There are many well understood special cases, such as simple forms of
homeostatic control using negative feedback. Neural net based controllers
often use large numbers of variables clustered into strongly
interacting sub-groups, groups of groups, etc.
For many structures and processes, a set of numerical values and rates of change
linked by equations (including differential equations) expressing their changing
relationships is an adequate form of representation, but not for all, as implied
by the discussion of types of adequacy in Section 2.2.
That's why
chemists use structural formulae, e.g. diagrams showing different sorts of
bonds between atoms and collections of diagrams showing how bonds change in
chemical reactions. Linguists, programmers, computer scientists, architects,
structural engineers, map-makers, map-users, mathematicians studying geometry
and topology, composers, and many others, work in domains where structural
diagrams, logical expressions, grammars, programming languages, plan formalisms,
and other non-numerical notations express information about structures and
processes that is not usefully expressed in terms of collections of numbers and
equations linking numbers.9
Of course, any information that can be expressed in 2-D written or printed
notation such as grammatical rules, parse trees, logical proofs and computer
programs, can also be converted into a large array of numbers by taking a
photograph and digitising it. Although such processes are useful for
storing or transmitting documents, they add so much irrelevant numerical detail
that the original functions, such as use in checking whether an inference is
valid, or manipulating a grammatical structure by transforming an active
sentence to a passive one, or determining whether two sentences have the same
grammatical subject, or removing a bug from a program, or checking whether a
geometric construction proves a theorem, become inaccessible until the
original non-numerical structures are extracted.
Similarly, collections of numerical values will not always adequately represent
information that is biologically useful for animal decision making, problem
solving, motive formation, learning, etc. Moreover, biological sensors are poor
at acquiring or representing very precise information, and neural states often
lack reliability and stability. (Such flaws can be partly compensated for by
using many neurons per numerical value and averaging.) More importantly, the
biological functions, e.g. of visual systems, may have little use for absolute
measures, if their functions are based on relational information, such as
that A is closer to B than to C, A is biting B, A is keeping B and C apart, A
can fit through the gap between B and C, the joint between A and B is non-rigid,
A cannot enter B unless it is reoriented, and many more. As Schrödinger, [1944]
pointed out, topological structures of molecules can reliably encode a wide
variety of types of genetic information, and may also turn out to be useful for
recording other forms of structural information. Do brains employ them? There
are problems about how such information can be acquired, derived, stored, used,
etc. Chomsky, [1965] pointed out that using inappropriate structures in models
may divert attention from important biological phenomena that need to be
explained-see Sect. 2.2, above. Max Clowes, who introduced me
to AI in 1969 made similar points about research in vision around that
time.10 So
subtasks for this project include identifying biologically important types of
non-numerical (e.g. relational) information content and ways in which such
information can be stored, transmitted, manipulated, and used. We also need to
explain how mechanisms performing such tasks can be built from the FCK,
using appropriate DCKs.
2.5 Increasingly varied mathematical structures
Electronic computers made many new forms of control possible, including use of
logic, linguistic formalisms, planning, learning, problem solving, vision,
theorem proving, teaching, map-making, automated circuit design, program
verification, and many more. The world wide web is an extreme case of a control
system made up of millions of constantly changing simpler control systems,
interacting in parallel with each other and with millions of display devices,
sensors, mechanical controllers, humans, and many other things. The types of
control mechanism in computer-based systems11
now extend far beyond the numerical sorts familiar to control engineers.
Organisms also need multiple control systems, not all numerical. A partially
constructed percept, thought, question, plan or terrain description has parts
and relationships, to which new components and relationships can be added and
others removed, as construction proceeds, and errors are corrected, building
structures with changing complexity - unlike a fixed-size
collection of variables
assigned changing values. Non-numerical types of mathematics are needed for
describing or explaining such systems, including topology, geometry, graph
theory, set theory, logic, formal grammars, and theory of computation. A full
understanding of mechanisms and processes of evolution and development may need
new branches of mathematics, including mathematics of non-numerical structural
processes, such as chemical change, or changing "grammars" for internal
records of complex structured information. The importance of non-numerical
information structures has been understood by many mathematicians, logicians,
linguists, computer scientists and engineers, but many scientists
still focus only on numerical structures and processes - sometimes seeking to
remedy their failures by using statistical methods, which in restricted contexts
can be spectacularly successful, as shown by recent AI successes, whose
limitations I have criticised elsewhere.12
The FCK need not be able to produce all biological structures and processes
directly, in situations without life, but it must be rich enough to support
successive generations of increasingly powerful DCKs that together suffice to
generate all possible biological organisms evolved so far, and their behavioural
and information-processing abilities. Moreover, the FCK, or DCKs derived from
it, must include abilities to acquire, manipulate, store, and use information
structures in DCKs that can build increasingly complex machines that encode
information, including non-numerical information. Since the
1950s we have also increasingly
discovered the need for new virtual machines as
well as physical machines
[Sloman, 2010,Sloman, 2013a].
Large scale physical processes usually involve a great deal of variability and
unpredictability (e.g. weather patterns), and sub-microscopic indeterminacy is a
key feature of quantum physics, yet, as Schrödinger, [1944] observed, life
depends on very complex objects built from very large numbers of small scale
structures (molecules) that can preserve their precise chemical structure,
despite continual thermal buffetting and other disturbances. Unlike non-living
natural structures, important molecules involved in reproduction and other
biological functions are copied repeatedly, predictably transformed with great
precision, and used to create very large numbers of new molecules required for
life, with great, but not absolute, precision. This is non-statistical
structure preservation, which would have been incomprehensible without quantum
mechanics, as explained by Schrödinger.
That feature of the FCK resembles "structure-constraining" properties of
construction kits such as
Meccano, TinkerToy and Lego13
that support structures with more or less complex, discretely varied topologies,
or kits built from digital electronic components, that also provide extremely
reliable preservation and transformations of precise structures, in contrast
with sand, water, mud, treacle, plasticene, and similar materials. Fortunate
children learn how structure-based kits differ from more or less amorphous
construction kits that produce relatively flexible or plastic structures with
non-rigid behaviours - as do many large-scale natural phenomena, such as snow
drifts, oceans, or weather systems.
Schrödinger's 1944 book stressed that quantum mechanisms can explain the
structural stability of individual molecules and explained how a set of atoms in
different arrangements can form discrete stable structures with very different
properties (e.g. in propane and isopropane only the location of the single
oxygen atom differs, but that alters both the topology and the chemical
properties of the molecule).14
He also pointed out the relationship between number of discrete changeable
elements and information capacity, anticipating Shannon, [1948]. Some complex
molecules with quantum-based structural stability are simultaneously capable of
continuous deformations, e.g. folding, twisting, coming together, moving
apart, etc., all essential for the role of DNA and other molecules in
reproduction, and many other biochemical processes. This combination of discrete
topological structure (forms of connectivity) used for storing very precise
information for extended periods and non-discrete spatial flexibility used in
assembling, replicating and extracting information from large structures, is
unlike anything found in digital computers, although it can to some extent be
approximated in digital computer models of molecular processes.
Highly deterministic, very small scale, discrete interactions between very
complex, multi-stable, enduring molecular structures, combined with continuous
deformations (folding, etc.) that alter opportunities for the discrete
interactions, may have hitherto unnoticed roles in brain functions, in addition
to their profound importance for reproduction, and growth. Much recent AI and
neuroscience uses statistical properties of complex systems with many
continuous scalar
quantities changing randomly in parallel, unlike symbolic mechanisms used in
logical and symbolic AI, though the latter are still far too restricted to model
animal minds.
The Meta-Morphogenesis project has extended a set of examples
studied four decades earlier (e.g. in Sloman, [1978]) of types of mathematical
discovery and reasoning that use perceived possibilities and
impossibilities for change in geometrical and topological structures. Further
work along these lines may help to reveal biological mechanisms that enabled the
great discoveries by Euclid and his predecessors that are still unmatched by AI
theorem provers (discussed in Section 5).
2.6 Thermodynamic issues
The question sometimes arises whether formation of life from non-living matter
violates the second law of thermodynamics, because life increases the amount of
order or structure in the physical matter on the planet, reducing entropy.
The standard answer is
that the law is applicable only to closed systems, and the earth is not a closed
system, since it is constantly affected by solar and other forms of radiation,
asteroid impacts, and other external influences. The law implies only that our
planet could not have generated life forms without energy from non-living
sources, e.g. the sun
(though future technologies may reduce or remove such dependence). Some of the
ways in which pre-existing dispositions can harness external sources of energy
to increase local structure are discussed in a collection of thoughts on
entropy, evolution, and construction-kits:
http://www.cs.bham.ac.uk/research/projects/cogaff/misc/entropy-evolution.html15
Our discussion so far suggests that the FCK has two sorts of components: (a) a
generic framework including space-time and generic constraints on what can
happen in that framework, and (b) components that can be non-uniformly and
dynamically distributed in the framework. The combination makes possible
formation of galaxies, stars, clouds of dust, planets, asteroids, and many other
lifeless entities, as well as supporting forms of life based on derived
construction kits (DCKs) that exist only in special conditions. Some local
conditions e.g. extremely high pressures, temperatures, and gravitational
fields, (among others) can mask some parts of the FCK, i.e. prevent them from
functioning. So, even if all sub-atomic particles required for earthly life
exist at the centre of the sun, local factors can rule out earth-like
life forms. Moreover, if the earth had been formed from a cloud of
particles containing no carbon, oxygen, nitrogen, iron, etc, then no DCK able to
support life as we know it could have emerged, since that requires a region of
space-time with a specific manifestation of the FCK, embedded in a larger region
that can contribute additional energy (e.g. solar radiation) and possibly other
resources.
As the earth formed, new physical conditions created new DCKs that made the
earliest life forms possible. Ganti, [2003], usefully summarised in
Korthof, [2003] and Fernando, [2008], presents an analysis of requirements for a
minimal life form, the "chemoton", with self-maintenance and reproductive
capabilities. Perhaps still unknown DCKs made possible formation of pre-biotic
chemical structures, and also the environments in which a chemoton-like
entity could survive and reproduce. Later, conditions changed in ways that
supported more complex life forms, e.g. oxygen-breathing forms. Perhaps
attempts to identify the first life form in order to show how it could be
produced by the FCK are misguided, because several important pre-life
construction kits were necessary: i.e. several DCKs made possible by conditions
on earth were necessary for precursors. Some of the components of the DCKs may
have been more complex than their living products, including components
providing scaffolding for constructing life forms, rather than materials.
2.7 Scaffolding in construction kits
An important feature of some construction kits is that they contain parts that
are used during assembly of products of the kit, but are not included in the
products. For example, meccano kits come with spanners and screwdrivers, used
for manipulating screws and nuts during assembly and disassembly, though they
are not normally included in the models constructed. Similarly kits for making
paper dolls and their
clothing16 may include
pencils and scissors, used for preparing patterns and cutting them out. But the
pencils and scissors are not parts of the dolls or their clothing. When houses
are built many items are used that are not part of the completed house,
including tools and scaffolding frameworks to support incomplete structures. A
loose analogy can be made with the structures used by climbing plants, e.g.
rock-faces, trees, or frames provided by humans: these are essential for the
plants to grow to the heights they need but are not parts of the plant. More
subtly, rooted plants that grow vertically make considerable use of the soil
penetrated by their roots to provide not only nutrients but also the stability
that makes tall stalks or trunks possible, including in some cases the ability
to resist strong winds most of the time. The soil forms part of the scaffolding.
A mammal uses parts of its mother as temporary scaffolding while developing in
the womb, and continues to use the mother during suckling and later when fed
portions of prey caught by parents. Other species use eggs with protective
shells and food stores. Plants that depend on insects for fertilization can be
thought of as using scaffolding in a more general sense.
This concept of scaffolding may be crucial for research into origins of life. As
far as I know nobody has found candidate non-living chemical substances made
available by the FCK that have the ability spontaneously to assemble themselves
into primitive life forms. It is possible that the search is doomed to fail
because there never were such substances: if the earliest life forms required
not only materials but also scaffolding - e.g. in the form of complex molecules
that did not form parts of the earliest organisms but played an essential causal
role in assembly processes, bringing together the chemicals needed by the
simplest organisms. Evolution might then have produced new organisms without
that reliance on the original scaffolding. The scaffolding mechanisms might
later have ceased to exist on earth, e.g. because they were consumed and wiped
out by the new life forms, or because physical conditions changed that prevented
them forming but did not destroy the newly independent organisms. A similar
suggestion is made in Mathis et al, [2015] - and for all I know has been
made elsewhere. So it is quite possible that many evolutionary transitions,
including transitions in information processing, our main concern, depended on
forms of scaffolding that later did not survive and were no longer needed to
maintain what they had helped to produce. So research into evolution of
information processing, our main goal, is inherently partly speculative.
2.8 Biological construction kits
How did the FCK generate complex life forms? Is the
Darwin-Wallace theory of natural selection the whole answer, as suggested
in [Bell, 2008]?
"Living complexity cannot be explained except through selection and does
not require any other category of explanation whatsoever."
No: the explanation must include both selection mechanisms and
generative mechanisms, without which selection processes will not have a supply
of new viable options. Moreover, insofar as environments providing
opportunities, challenges and threats are part of the selection process, the
construction kits used by evolution include mechanisms not intrinsically
concerned with life, e.g. volcanoes, earthquakes, asteroid impacts, lunar and
solar tides, and many more.
The idea of evolution producing construction kits is not new, though they are
often referred to as "toolkits".
Coates et al, [2014] ask whether there is "a genetic toolkit for
multicellularity" used by complex life-forms. Toolkits and construction kits
normally have users (e.g. humans or other animals), whereas the
construction kits we have been discussing (FCKs and DCKs) do not all need
separate users.
Both generative mechanisms and selection mechanisms change during evolution.
Natural selection (blindly) uses the initial enabling mechanisms provided by
physics and chemistry not only to produce new organisms, but also to produce new
richer DCKs, including increasingly complex information-processing mechanisms.
Since the mid 1900s, spectacular changes have also occurred in human-designed
computing mechanisms, including new forms of hardware, new forms of virtual
machinery, and networked social systems all unimagined by early hardware
designers. Similar changes during evolution produced new biological construction
kits, e.g. grammars, planners, geometrical constructors, not well understood by
thinkers familiar only with physics, chemistry and numerical mathematics.
Biological DCKs produce not only a huge variety of physical forms, and physical
behaviours, but also forms of information-processing required for
increasingly complex control problems, as organisms become more
complex and more intelligent in coping with their environments, including
interacting with predators, prey, mates, offspring, conspecifics, etc. In
humans, that includes abilities to form scientific theories and discover and
prove theorems in topology and geometry, some of which are also used unwittingly
in practical activities.17
I suspect many animals come close to this in their systematic but
unconscious abilities to perform complex actions that use mathematical features
of environments. Abilities used unconsciously in building nests or in hunting
and consuming prey may overlap with topological and geometrical competences of
human mathematicians. (See Section 6.2). E.g.
search for videos of weaver birds building nests.
3 Concrete (physical), abstract and hybrid construction kits
Products of a construction kit may be concrete, i.e. physical, or abstract, like
a theorem, a sentence, or a symphony; or hybrid, e.g. a written presentation of
a theorem or poem.
Concrete kits: Construction kits for children include physical parts that
can be combined in various ways to produce new physical objects that are not
only larger than the initial components but have new shapes and new behaviours.
Those are concrete construction kits. The FCK is (arguably?) a concrete
construction kit. Lego, Meccano, twigs, mud, and stones, can all be used in
construction kits whose constructs are physical objects occupying space and
time: concrete construction kits.
Abstract kits:
There are also non-spatial abstract construction kits,
for example components of languages, such as vocabulary and grammar, or methods
of construction of arguments or proofs. Physical representations of such
things, however, can occupy space and/or time, e.g. a spoken or written
sentence, a diagram, or a proof presented on paper. Using an abstract
construction kit, e.g. doing mental arithmetic, or composing poetry in your
head, requires use of one or more physical construction kits, directly or
indirectly implementing features of the abstract kit.
There are (deeply confused) fashions emphasising "embodied cognition" and
"symbol grounding" (previously known as "concept empiricism" and demolished
by Immanuel Kant and 20th Century philosophers of science). These fashions
disregard many examples of thinking, perceiving, reasoning and planning, that
require abstract construction kits. For example, planning a journey to a
conference does not require physically trying possible actions, like water
finding a route to the sea. Instead, you may use an abstract construction kit
able to represent possible options and ways of combining them. Being able
to talk requires use of a grammar specifying abstract structures that can be
assembled using a collection of grammatical relationships, to form new abstract
structures with new properties relevant to various tasks involving information.
Sentences allowed by a grammar for English are abstract objects that can be
instantiated physically in written text, printed text, spoken sounds, morse
code, etc.: so a grammar is an abstract construction kit whose constructs can
have concrete (physical) instances. The idea of a grammar is not restricted to
verbal forms: it can be extended to many complex structures, e.g.
grammars for sign languages, circuit diagrams, maps, proofs, architectural
layouts and
even molecules.
A grammar does not fully specify a language: a structurally related
semantic construction kit, is required for building possible meanings.
Use of a language depends on language users, for which more complex construction
kits are required, including products of evolution, development and learning.
Evolution of various types of language is discussed in
Sloman, [2008].
In computers, physical mechanisms implement abstract construction kits via
intermediate abstract kits - virtual machines; and
presumably also in brains.
Hybrid abstract+concrete kits: These are combinations, e.g. physical chess
board and chess pieces combined with the rules of chess, lines and circular arcs
on a physical surface instantiating Euclidean geometry, puzzles like the
mutilated chess-board puzzle, and many more. A particularly interesting hybrid
case is the use of physical objects (e.g. blocks) to instantiate arithmetic,
which may lead to the discovery of prime numbers when certain attempts at
rearrangement fail - and an explanation of the impossibility is found.18
In some hybrid construction kits, such as games like chess, the
concrete (physical) component may be redundant for some players: e.g.
chess experts who can play without physical pieces on a board.
But communication of moves needs physical mechanisms, as does the expert's
brain (in ways that are not yet understood). Related abstract structures, states
and processes can also be implemented in computers, which can now play chess
better than most humans, without replicating human brain mechanisms. In
contrast, physical components are indispensable in hybrid construction kits for
outdoor games, like cricket [Wilson, 2015]. (I don't expect to see good
robot cricketers soon.)
Physical computers, programming languages, operating systems and virtual
machines form hybrid construction kits that make things happen when they
run. Logical systems with axioms and inference rules can be thought of as
abstract kits supporting construction of logical proof-sequences, usually
combined with a physical notation for written proofs. Purely logical systems
cannot have physical causal powers whereas concrete instances can, e.g.
teaching a student, or programming a computer,
to distinguish valid and invalid proofs. Natural selection
"discovered" the power of hybrid construction kits using virtual machinery,
long before human engineers did. In particular, biological virtual machines used
by animal minds outperform current engineering designs in some ways, but they
also generate much confusion in the minds of philosophical individuals who are
aware that something more than purely physical machinery is at work, but don't
yet understand how to implement virtual machines in physical machines
[Sloman and Chrisley, 2003,Sloman, 2010,Sloman, 2013a].
Animal perception, learning, reasoning, and intelligent behaviour require
hybrid construction kits. Scientific study of such kits is still in
its infancy. Work done so far on the Meta-Morphogenesis project suggests that
natural selection "discovered" and used a staggering variety of types of
hybrid construction kit that were essential for reproduction, for developmental
processes (including physical development and learning), for performing complex
behaviours, and for social/cultural phenomena.
3.1 Kits providing external sensors and motors
Some construction kits can be used to make toys with moving parts, e.g. wheels
or grippers, that interact with the environment. A toy car may include a spring,
whose potential energy can be transformed into mechanical energy via gears,
axles and wheels in contact with external surfaces. Further interactions,
altering the direction of motion, may result from collisions with fixed or
mobile objects in the environment or some control device.
More sophisticated kits have sensors that provide information for internal
mechanisms that use information to select options for movement, etc. In
rats, whiskers gain information allowing frequent changes of direction and
speed of motion to avoid collisions, or to move towards a source of energy
when internal supplies are running low.
As noted in Sloman, [1978,Ch.6], the distinction between internal and
external components is often arbitrary - a fact frequently
rediscovered. For example, a music box may perform a tune under the control of
a rotating disc with holes or spikes. The disc can be thought of as part of the
music box, or as part of a changing environment.
If a toy train set has rails or tracks used to guide the motion of the train,
then the wheels can be thought of as sensing the environment and
causing changes of direction. This is partly like and partly unlike a toy
vehicle that uses an optical sensor linked to a steering mechanism, so that a
vehicle can follow a line painted on a surface. The railway track provides both
information and the forces required to change direction. A painted line,
however, provides only the information, and other parts of the vehicle have to
supply the energy to change direction, e.g. an internal battery that powers
sensors and motors. Evolution uses both sorts: e.g. wind blowing seeds away from
parent plants and a wolf following a scent trail left by its prey. An unseen
wall uses force to stop your forward motion in a dark room, whereas a visible,
or lightly touched, wall provides only information [Sloman, 2011].
3.2 Mechanisms for storing, transforming and using information
Often information is acquired, used, then lost because it is over-written, e.g.
sensor information in simple servo-control systems with "online intelligence",
where only the latest sensed state is used for deciding whether to speed
something up, change direction, etc. In more complex control systems, with
"offline intelligence" sensor information is saved, possibly combined with
previously stored information, and remains available for use on different
occasions for different purposes.19
In the "offline" case, the underlying
construction-kit needs to be able to support stores of information that grow
with time and can be used for different purposes at different times. A control
decision at one time may need items of information obtained at several different
times and places, for example information about properties of a material, where
it can be found, and how to transport it to where it is needed. Sensors used
online may become faulty or require adjustment. Evolution may provide mechanisms
for testing and adjusting. When used offline, stored information may need to be
checked for falsity caused by the environment changing, as opposed to sensor
faults. The offline/online use of visual information has caused much confusion
among researchers, including muddled attempts to interpret the difference in
terms of "what" and "where" information.20 Contrast
Sloman, [1983].
Ways of acquiring and using information have been discovered and modelled by AI
researchers, psychologists, neuroscientists, biologists and others. However,
evolution produced many more. Some of them require not just additional
storage space but very different sorts of information-processing architectures.
A range of possible architectures is discussed in
Sloman, [1978,Sloman, [1983,Sloman, [1993,Sloman, [2003,Sloman, [2006], whereas
AI engineers typically seek one architecture for a project. A complex biological
architecture may use sub-architectures that evolved at different times, meeting
different needs in different niches. In particular, I suspect there are
biological mechanisms for handling vast amounts of rapidly changing incoming
information (visual, auditory, tactile, haptic, proprioceptive, vestibular) by
using several different sorts of short-term storage plus processing subsystems,
operating on different time-scales in parallel, including a-modal information
structures.
This raises the question whether evolution produced "architecture kits" able
to combine evolved information-processing mechanisms in different ways, long
before software engineers discovered the need. Such a
kit could be particularly important for species that produce new subsystems,
or modify old ones, during individual development, e.g. during different phases
of learning by apes, elephants, and humans, as described in
Section 2.3, contradicting the common assumption that a
computational architecture must remain fixed.21
3.3 Mechanisms for controlling position, motion and timing
All concrete construction kits (and some hybrid kits) share a deep common
feature insofar as their components, their constructs and their construction
processes involve space and time, both during assembly and while working. Those
behaviours include both relative motion of parts, e.g. wheels rotating, joints
changing angles, and also motion of the whole object relative to other objects,
e.g. an ape grasping a berry. A consequence of spatiality is that objects
built from different construction kits can interact, by changing their spatial
relationships (e.g. if one object enters, encircles or grasps another), applying
forces transmitted through space, and using spatial sensors to gain information
used in control. Products of different kits can interact in varied ways, e.g.
one being used to assemble or manipulate another, or one providing energy or
information for the other. Contrast the problems of getting software components
available on a computer to interact sensibly: merely locating them in the same
virtual or physical machine will not suffice. Some rule-based systems
are composed of condition-action rules, managed by an interpreter that
constantly checks for satisfaction of conditions. Newly added rules may then be
invoked simply because their conditions become satisfied, though "conflict
resolution" mechanisms may be required if the conditions of more than one rule
are satisfied.22
New concrete kits can be formed by combining two or
more kits. In some cases this will require modification of a kit, e.g.
combining Lego and Meccano by adding pieces with Lego studs or holes alongside
Meccano sized screw holes. In other cases mere spatial proximity and contact
suffices, e.g. when one construction kit is used to build a platform and others
to assemble a house on it. Products of different biological
construction kits may also use
complex mixtures of juxtaposition and adaptation.
Objects that exist in space/time often need timing mechanisms. Organisms use
"biological clocks" operating on different time-scales controlling repetitive
processes, including daily cycles, heart-beats, breathing, and wing or limb
movements required for locomotion. More subtly there are adjustable speeds and
adjustable rates of change: e.g. a bird in flight
approaching a perch; an animal running to escape a predator and having to
decelerate as it approaches a tree it needs to climb; a hand moving to grasp a
stationary or moving object, with motion controlled by varying coordinated
changes of joint angles at waist, shoulder, elbow and finger joints so as to
bring the grasping points on the hand into suitable locations relative to the
intended grasping points on the object. (This can be very difficult for robots,
when grasping novel objects in novel situations: if they use
ontologies that are too simple.)
There are also biological mechanisms
for controlling or varying rates of production of
chemicals (e.g. hormones).
So biological construction kits need many mechanisms able to measure time
intervals and to control rates of repetition or rates of change of parts of the
organism. These kits may be combined with other sorts of construction kit that
combine temporal and spatial control, e.g. changing speed and direction,
3.4 Combining construction kits
At the molecular level there is now a vast, and rapidly growing, amount of
biological research
on interacting construction kits, for example interactions between
different parts of the reproductive mechanism during development of a fertilised
egg, interactions between invasive viral or bacterial structures and a host
organism, and interactions with chemicals produced in medical research
laboratories. In computers the ways of combining
different toolkits include the application of functions to arguments, although
both functions and their arguments can be far more complex than the cases
most people encounter in learning arithmetic. A function could
be a compiler, its arguments could be arbitrarily complex programs in a high
level programming language, and the outputs of the function might be either a
report on syntactic errors in the input program, or a machine code program ready
to run.
Applying functions to arguments is very different from assembling structures in
space time, where inputs to the process form parts of the output. If computers
are connected via digital to analog interfaces, linking them to surrounding
matter, or if they are mounted on machines that allow them to move around in
space and interact, that adds a kind of richness that goes beyond application of
functions to arguments.
The additional richness is present in the modes of interaction of
chemical structures that include both digital (on/off chemical bonds) and
continuous changes in relationships, as discussed in
Turing, [1952], the paper on chemistry-based morphogenesis
that inspired this Meta-Morphogenesis project [Sloman, 2013b].
3.5 Combining abstract construction kits
Section 2.1 showed how a new DCK using
combinations of old components can make some new developments very much quicker
to reach - fewer steps are required, and the search space for a sequence
of steps to a solution may be dramatically reduced. Combining concrete
construction kits uses space-time occupancy. Combining abstract
construction kits is less straightforward. Sets of letters and numerals are
combined to form labels for chess board squares, e.g. "a2", "c5", etc. A
human language and a musical notation can form a hybrid system for writing
songs. A computer operating system (e.g. Linux) can be combined with programming
languages (e.g. Lisp, Java). In organisms, as in computers, products of
different kits may share information, e.g. information for sensing,
predicting, explaining or controlling, including information about information
[Sloman, 2011]. Engineers combining different kinds of functionality
find it useful to design re-usable information-processing architectures
that provide frameworks for combining different mechanisms and information
stores[21], especially in large projects where different teams
work on sensors, learning, motor systems, reasoning systems, motivational
systems, various kinds of metacognition, etc., using specialised tools.
The toolkit mentioned in Note 22 is an example
framework.
It is often necessary to support different sorts of virtual machinery
interacting simultaneously with one another and with internal and external
physical environments, during perception and motion. This may require new
general frameworks for assembling complex information-processing
architectures, accommodating multiple interacting virtual machines, with
different modifications developed at different times
[Minsky, 1987,Minsky, 2006] [Sloman, 2003].
Self-extension is a topic for further research.23
Creation of new construction kits may start by simply recording parts of
successful assemblies, or better still parametrised parts, so that they can
easily be reproduced in modified forms - e.g. as
required for organisms that change
size and shape while developing. Eventually, parametrised stored designs may be
combined to form a "meta-construction kit" able to extend, modify or
combine previously created construction kits as human engineers have recently
learnt to do in debugging toolkits.
Evolution needs to be able to create new meta-construction kits
using natural selection. Natural selection, the great creator/meta-creator, is
now spectacularly aided and abetted by its products, especially humans!
4 Construction kits generate possibilities and impossibilities
Explanations of how things are possible (Sect. 1)
can refer to construction kits, either
manufactured, e.g. Meccano and Lego, or composed of naturally occurring
components, e.g. boulders, mud, or sand. (Not all construction kits have sharp
boundaries.) Each kit makes possible certain types of construct, instances of
which can be built by assembling parts from the kit. Some construction
kits use products of products of biological evolution, e.g. birds'
nests assembled from twigs or leaves.
In some kits, features of components, such as shape, are inherited by
constructed objects. E.g. objects composed only of Lego bricks joined in the
"standard" way have external surfaces that are divisible into faces parallel
to the surfaces of the first brick used. However, if two Lego bricks are joined
at a corner only, using only one stud and one socket, it is possible to have
continuous relative rotation (because studs and sockets are circular), violating
that constraint, as Ron Chrisley pointed out in a conversation. This illustrates
the fact that constructed objects can have "emergent" features none of the
components have, e.g. a hinge is a non-rigid object that can be made from two
rigid objects with aligned holes through which a screw is passed.
So, a construction kit that makes some things possible and others impossible can
be extended so as to remove some of the impossibilities, e.g. by adding a hinge
to Lego, or adding new parts from which hinges can be assembled.
4.1 Construction kits for making information-users
Not everything that can play a role in acquisition, storage or transfer of
information has information-processing capabilities. Consider a lump of
plasticine or damp clay that can be deformed under pressure, then retains the
deformation. If a coin is pressed against it the lump will change its shape.
Entities with information-processing capabilities (e.g. archaeologists) can use
the depression as a source of information about the coin. But the deformed lump
of material is not an information user. If the depression is used to control a
process, e.g. making copies of the coin, or to help a historian years later,
then the deformed material is used as a source of information about the coin.
The fact that some part of a brain is changed by perceptual processes in an
organism does not imply that that portion of the brain is an information user.
It may play a role analogous to the lump of clay, or a footprint in soil.
Additional mechanisms are required if the information is to be used:
different mechanisms for different types of use. A photocopier acquires
information from a sheet of paper, but all it can do with the information is
produce a replica (possibly after slight modifications such as changes in
contrast, intensity or magnification). Different mechanisms are required for
recognising text, correcting spelling, analysing the structure of an image,
interpreting it as a picture of a 3-D scene, using information about the scene
to guide a robot, building a copy of the scene, or answering a question about
which changes are possible. Thinking up ways of using the impression as a source
of information about the coin is left as an exercise for the reader.
Biological construction kits for producing information-processing mechanisms
evolved at different times. Sloman, [1993] discusses the diversity of uses of
information from biological sensors, including sharing of sensor information
between different uses, either concurrently or sequentially. Some of the
mechanisms use intermediaries, such as sound or light, to gain information about
the source or reflector of the sound or light; used in taking decisions, e.g.
whether to flee, or used in controlling actions such as grasping or walking past
a source of information.
Some mechanisms that use information seem to be
direct products of biological evolution, such as mechanisms that control reflex
protective blinking. Others are grown in individuals by epigenetic mechanisms
influenced by context, as explained in Section 2.3.
For example, humans in different cultures start with a
generic language construction kit (sometimes misleadingly labelled a "universal
grammar") which is extended and modified to produce locally useful language
understanding/generating mechanisms. Language-specific mechanisms, such as
mechanisms for acquiring, producing, understanding and correcting textual
information evolved long after mechanisms able to use visual information for
avoiding obstacles or grasping objects, shared between far more types of animal.
In some species there may be diversity in the construction kits produced by
individual genomes, leading to even greater diversity in adults, if they develop
in different physical and cultural environments using epigenetic mechanisms
discussed above.
4.2 Different roles for information
Despite huge diversity in biological construction-kits and the mechanisms in
individual organisms, some themes recur, such as functions of different sorts of
information in control: e.g. information about how things actually are or might
be ("belief-like" information contents), information about how things need to
be or might need to be for the individual information user ("desire-like"
information contents), and information about how to achieve or avoid certain
states ("procedural" information contents). Each type has different subtypes,
across species, across members of a species and across developmental stages in
an individual. How a biological construction kit supports the requirements
depends on the environment, the animal's sensors, its needs, the local
opportunities, and the individual's history. Different mechanisms performing
such a function may share a common evolutionary precursor after which they
diverged. Moreover, mechanisms with similar functions can evolve independently
- convergent evolution.
Information relating to targets and how to achieve or maintain them is
control information: the most basic type of biological information, from which
all others are derived.
A simple case is a thermostatic control, discussed in McCarthy, [1979]. It has
two sorts of information: (a) a target temperature (desire-like information) (b)
current temperature (belief-like information). A discrepancy between them causes
the thermostat to select between turning a heater on, or off, or doing nothing.
This very simple homeostatic mechanism uses information and a source of energy
to achieve or maintain a state. There are very many variants on this schema,
according to the type of target (e.g. a measured state or some complex
relationship) the type of control (on, off, or variable, with single or multiple
effectors), and the mechanisms by which control actions are selected, which may
be modified by learning, and may use simple actions or complex plans.
As Gibson, [1966] pointed out, acquisition of information often requires
cooperation between processes of sensing and acting. Saccades are visual actions
that constantly select new information samples from the environment (or the
optic cone). Uses of the information vary widely according to context, e.g.
controlling grasping, controlling preparation for a jump, controlling avoidance
actions, or sampling text to be read. A particular sensor can therefore be
shared between many control subsystems [Sloman, 1993], and the significance
of the sensor state will depend partly on which subsystems are connected to the
sensor at the time, and partly on which other mechanisms receive information from
the sensor (which may change dynamically - a possible cause of some types of
"change blindness").
The study of varieties of use of information in organisms is exploding, and
includes many mechanisms on molecular scales as well as many intermediate levels
of informed control, including sub-cellular levels (e.g. metabolism),
physiological processes of breathing, temperature maintenance, digestion, blood
circulation, control of locomotion, feeding and mating of large animals and
coordination in communities, such as collaborative foraging in insects and
trading systems of humans. Slime moulds include spectacular examples in which
modes of acquisition and use of information change dramatically.24
Figure 4
Figure 4: Between the simplest and most sophisticated organisms there are many
intermediate forms with very different information processing
requirements and capabilities.
The earliest organisms must have acquired and used information about things
inside themselves and in their immediate vicinity, e.g. using chemical detectors
in an enclosing membrane. Later, evolution extended those capabilities in
dramatic ways (crudely indicated in Fig. 4). In the simplest
cases, local information is used immediately to select between alternative
possible actions, as in a heating control, or trail-following mechanism. Uses of
motion in haptic and tactile sensing and use of saccades, changing vergence, and
other movements in visual perception, all use the interplay between sensing and
doing, in "online intelligence". But there are cases ignored by Gibson and
anti-cognitivists, namely organisms that exhibit "offline intelligence", using
perceptual information for tasks other than controlling immediate reactions, for
example, reasoning about remote future possibilities or attempting to explain
something observed, or working out that bending a straight piece of wire will
enable a basket of food to be lifted out of a tube as illustrated in
Fig. 4 [Weir et al, 2002]. Doing that requires use of
previously acquired information about the environment including particular
information about individual objects and their locations or states, general
information about learnt laws or correlations and information about what is and
is not possible (Note [12]).
An information-bearing structure (e.g. the impression of a foot or the shape of a
rock) can provide very different information to different information-users, or
to the same individual at different times, depending on (a) what kinds of
sensors they have, (b) what sorts of information-processing (storing, analysing,
comparing, combining, synthesizing, retrieving, deriving, using...) mechanisms
they have, (c) what sorts of needs or goals they can serve by using various
sorts of information (knowingly or not), and (d) what information they already
have. So, from the fact that changes in some portion of a brain correlate with
changes in some aspect of the environment we cannot conclude much about what
information about the environment the brain acquires and uses or how it does
that - since typically that will depend on context.
4.3 Motivational mechanisms
It is often assumed that every information user, U, constantly tries to achieve
rewards or avoid punishments (negative rewards), and that each new item of
information, I, will make some actions more likely for U, and others less
likely, on the basis of what U has previously learnt about which actions
increase positive rewards or decrease negative rewards under conditions
indicated by I. But animals are not restricted to acting on motives selected
by them on the basis of expected rewards. They may also have motive
generators that are simply triggered as "internal reflexes" just as evolution
produces phototropic reactions in plants without giving plants any ability to
anticipate benefits to be gained from light. Some reflexes, instead of directly
triggering behaviour, trigger new motives, which may or may not lead
to behaviour, depending on the importance of other competing motives. For
example, a kind person watching someone fall may acquire a motive to rush to
help - not acted on if competing motives are too strong. It is widely believed
that all motivation is reward-based. But a new motive triggered by an internal
reflex need not be associated with some reward. It may be "architecture-based
motivation" rather than "reward-based motivation" Sloman, [2009].
Triggering of architecture-based motives in playful intelligent young animals
can produce kinds of delayed learning that the individuals could not possibly
anticipate [Karmiloff-Smith, 1992].
Unforeseeable biological benefits of automatically triggered motives include
acquisition of new information by sampling properties of the environment. The
new information may not be immediately usable, but in combination with
information acquired later and genetic tendencies activated later, as indicated
in Fig. 3, it may turn out to be important, during hunting,
caring for young, or learning a language. A toddler may have no conception of
the later potential uses of information gained in play, though the ancestors of
that individual may have benefited from the presence of the information
gathering reflexes. In humans this seems to be crucial for mathematical
development.
During evolution, and also during individual development, the sensor mechanisms,
the types of information-processing, and the uses to which various types of
information are put, become more diverse and more complex, while the
information-processing architectures allow more of the processes to occur in
parallel (e.g. competing, collaborating, invoking, extending, recording,
controlling, redirecting, enriching, training, abstracting, refuting, or
terminating). Without understanding how the architecture grows, which
information-processing functions it supports, and how they diversify and
interact, we are likely to reach wrong conclusions about biological functions of
the parts: e.g. over-simplifying the functions of sensory subsystems, or
over-simplifying the variety of concurrent control mechanisms producing
behaviours. Moreover, the architectural knowledge about how such a systems
works, like information about the architecture of a computer operating system,
may not be expressible in sets of equations, or statistical learning mechanisms
and relationships. (Ideas about architectures for human information-processing
can be found in
Simon, [1967,Minsky, [1987,Minsky, [2006,Laird et al, [1987,Sloman, [2003,Sun, [2006], among many
others.)
Construction kits for building information-processing architectures, with
multiple sensors and motor subsystems, in complex and varied environments, differ
widely in the designs they can produce. Understanding that variety is not helped
by disputes about which architecture is best. A more complete discussion would
need to survey the design options and relate them to actual choices made by
evolution or by individuals interacting with their environments.
5 Mathematics: Some constructions exclude or necessitate others
Physical construction kits (e.g. Lego, plasticine, or a
combination of paper, scissors and paste) have parts and materials with
physical properties (e.g. rigidity, strength, flexibility, elasticity,
adhesion, etc.), possible relationships between parts and possible processes
that can occur when the parts are in those relationships (e.g. rotation,
bending, twisting and elastic or inelastic resistance to deformation).
Features of a physical construction kit - including the shapes and materials of
the basic components, ways in which the parts can be assembled into larger
wholes, kinds of relationships between parts and the processes that can
occur involving them - explain the possibility of entities that can be constructed
and the possibility of processes, including processes of
construction and behaviours of constructs.
Construction kits can also explain necessity and impossibility. A construction
kit with a large initial set of generative powers can be used to build a
structure realising some of the kit's possibilities, in which some further
possibilities are excluded, namely all extensions that do not include what has
so far been constructed. If a Meccano construction has two parts in a
substructure that fixes them a certain distance apart, then no extension
can include a new part that is wider than that distance in all
dimensions and is in the gap. Some extensions to the part-built structure that
were previously possible become impossible unless something is undone. That
example involves a limit produced by a gap size. There are many more examples of
impossibilities that arise from features of the construction kit.
Figure
Figure 5: The sequence demonstrates how the three-cornered shape has
the consequence that summing the three angles necessarily produces half a
rotation (180 degrees). Since the position, size, orientation, and precise shape
of the triangle can be varied without affecting the possibility of constructing
the sequence, this is a proof that generalises to any planar triangle. It
nowhere mentions Euclid's parallel axiom, used by "standard" proofs. This
unpublished proof was reported to me by Mary Pardoe, a former student who became
a mathematics teacher, in the early 1970s.
Euclidean geometry includes a construction kit that enables construction of
closed planar polygons (triangles, quadrilaterals, pentagons,
etc.), with interior angles whose sizes can be summed.
If the polygon has three sides, i.e. it is a triangle, then the interior
angles must add up to exactly half a rotation. Why?
In this case, no physical properties of a structure (e.g. rigidity or
impenetrability of materials) are involved, only spatial relationships. Figure
5 provides one way to answer the question, unlike the
standard proofs, which use parallel lines. It presents a proof, found by Mary
Pardoe, that internal angles of a planar triangle sum to a straight line, or 180
degrees.
Most humans are able to look at a physical situation, or a diagram representing
a class of physical situations, and reason about constraints on a class of
possibilities sharing a common feature. This may have evolved from earlier
abilities to reason about changing affordances in the environment
[Gibson, 1979]. Current AI perceptual and reasoning systems still lack most of
these abilities, and neuroscience cannot yet explain what's going on (as opposed
to where it's going on?). (See Note [12]).
These illustrate mathematical properties of construction kits (partly analogous
to mathematical properties of formal deductive systems and AI problem solving
systems). As parts (or instances of parts) of the FCK are combined, structural
relations between components of the kit have two opposed sorts of consequences:
they make some further structures possible (e.g. constructing a circle
that passes through all the vertices of the triangle), and other structures
impossible (e.g. relocating the corners of the triangle so that the angles add
up to 370 degrees). These possibilities and impossibilities are necessary
consequences of previous selection steps. The examples illustrate how a
construction kit with mathematical relationships can provide the basis for
necessary truths and necessary falsehoods in some constructions (as argued in
Sloman, [1962,Chap 7]).25
Being able
to think about and reason about alterations in some limited portion of the
environment is a very common requirement for intelligent action
[Sloman, 1996a]. It seems to be partly shared with other intelligent species,
e.g. squirrels, nest-builders, elephants, apes, etc. Since our examples of
making things possible or impossible, or changing ranges of possibilities, are
examples of causation (mathematical causation), this also provides the basis for
a Kantian notion of causation based on mathematical necessity [Kant, 1781],
so that not all uses of the notion of "cause" are Humean (i.e. based on
correlations), even if some are. Compare
Section 5.3.26
Neuroscientific theories about information-processing in brains currently
seem to me to
omit the processes involved in such mathematical discoveries, so
AI researchers influenced too much by neuroscience may fail
to replicate important brain functions. Progress
may require major conceptual advances regarding what the problems
are and what sorts of answers are relevant.
We now consider ways in which evolution itself can be understood as discovering
mathematical proofs - proofs of possibilities.
5.1 Proof-like features of evolution
A subset of the FCK produced fortuitously as a side effect of formation
of the earth, supported (a) primitive life forms and (b) processes of evolution
that produced more and more complex forms of life, including new, more complex,
derived, DCKs. New products of natural selection can make more complex products
more reachable, as with toy construction kits, and mathematical proofs.
However starting from those parts will make some designs unreachable
except by disassembling some parts.
Moreover, there is not just one sequence: different evolutionary
lineages evolving in parallel can produce different DCKs. According to the
"Symbiogenesis" theory, different DCKs produced independently can sometimes
merge to support new forms of life combining different evolutionary
strands.27 Creation
of new DCKs in parallel evolutionary streams with combinable products can hugely
reduce part of the search space for complex designs, at the cost of excluding
parts of the search space reachable from the FCK. For example, use of DCKs in
the human genome may speed up development of language and typical human
cognitive competences, while excluding the possibility of "evolving back" to
microbe forms that might be the only survivors after a cataclysm.
5.2 Euclid's construction kit
An old example of great significance for science, mathematics, and
philosophy, is
the construction kit specified in Euclidean geometry, starting with points,
lines, surfaces, and volumes, and methods of constructing new, more complex,
geometrical configurations using a straight edge for drawing straight lines in a
plane surface, and a pair of compasses for drawing circular arcs. This
construction kit makes it possible to bisect, but not trisect an arbitrary
planar angle. A slight extension, the "Neusis construction", known to
Archimedes, allows line segments to be translated and rotated in a plane while
preserving their length, and certain incidence relations. This
allows arbitrary angles to be trisected!
(See
http://www.cs.bham.ac.uk/research/projects/cogaff/misc/trisect.html)
The ability of
humans to discover such things must depend on evolved information-processing
capabilities of brains that are as yet unknown and not yet replicated
in AI reasoning systems.
The idea of a space of possibilities generated by a physical construction kit
may be easier for most people to understand than the comparison with generative
powers of grammars, formal systems, or geometric constructions, though the two
are related, since grammars and mathematical systems are abstract
construction kits that can be parts of hybrid construction kits.
Concrete construction kits corresponding to grammars can be built out of
physical structures: for example a collection of small squares with letters and
punctuation marks, and some blanks, can be used to form sequences that
correspond to the words in a lexicon. A cursive ("joined up") script requires
a more complex physical construction kit. Human sign-languages are far more
demanding, since they involve multiple body parts moving concurrently.
[Expand the following:]
Some challenges for construction kits used by evolution, and also challenges for
artificial intelligence and philosophy, arise from the need to explain both (a) how
natural selection makes use of mathematical properties of construction kits
related to geometry and topology, in producing organisms with spatial
structures and spatial competences, and also (b) how various subsets of those
organisms (e.g. nest-building birds) developed
specific topological and geometrical reasoning abilities
used in controlling actions or solving problems, and finally (c) how at least one
species developed abilities to reflect on the nature of those competences and
eventually, through unknown processes of individual development and social
interaction, using unknown representational and reasoning mechanisms, managed to
produce the rich, deep and highly organised body of knowledge published as
Euclid's Elements[1].
There are important aspects of those mathematical competences that as far as I
know have not yet been replicated in Artificial Intelligence or
Robotics28.
Is it possible that currently understood forms of digital computation are
inadequate for the tasks, whereas chemistry-based information-processing systems
used in brains are richer, because they combine both discrete and continuous
operations, as discussed in Section 2.5? (That's not a
rhetorical question: I don't know the answer.)
5.3 Mathematical discoveries based on exploring construction kits
Some mathematical discoveries result from observation of naturally occurring
physical construction kits and noticing how constraints on modes of composition
of components generate constraints on resulting constructs. E.g. straight line
segments on a surface can be joined end to end, to enclose a finite region,
but that is impossible with only two lines, as noted in Kant, [1781].
Likewise flat surfaces can be combined to enclose a volume,
such as a tetrahedron or cube, but it is impossible for only three flat surfaces
to enclose a finite space. It is not clear how humans detect such
impossibilities: no amount of trying and failing can establish impossibility.
Kant had no access to a 20th century formal axiomatisation of Euclidean
geometry. What he, and before him Euclid, Archimedes and others had were
products of evolution. What products?
Many mathematical domains (perhaps all of them) can be thought of as sets of
possibilities generated by construction kits. Physicists and engineers deal
with hybrid concrete and abstract construction kits. The space of possible
construction kits is also an example, though as far as I know this
domain has not been explored systematically by mathematicians, though
many special cases have.
In order to understand biological evolution on this planet we need to understand
the sorts of construction kits made possible by the existence of the physical
universe, and in particular the variety of construction kits inherent in the
physics and chemistry of the materials of which our planet was formed, along
with the influences of its environment (e.g. solar radiation, asteroid impacts).
An open research question is whether a construction kit capable of
producing all the non-living structures on the planet would also suffice for
evolution of all the forms of life on this planet, or whether life and evolution
have additional requirements, e.g. cosmic radiation?
5.4 Evolution's (blind) mathematical discoveries
Some construction kits and their products have
mathematical properties, so life and mathematics
are closely connected.
More complex relationships
arise after evolution of mathematical meta-cognitive mechanisms.
On the way to achieving those results, natural selection often works as "a blind
theorem-prover". The theorems are mainly about new possible structures,
processes, organisms, ecosystems, etc. The proofs that they are possible are
implicit in the evolutionary trajectories that lead to occurrences.
Proofs are often thought of as abstract entities that can be represented
physically in different ways (using different formalisms) for communication,
persuasion (including self-persuasion), predicting, explaining and planning. A
physical sequence produced unintentionally, e.g. by natural selection, or by
growth in a plant, that leads to a new sort of entity is a proof that some
construction kit makes that sort of entity possible. The evolutionary or
developmental trail, like a geometric construction,
answers the question: "how is that sort of thing possible?" So
biological evolution can be construed as a "blind theorem prover", despite
there being no intention, or explicit recognition,
regarding the proof. Proofs of impossibility (or
necessity) raise more complex issues, to be discussed elsewhere.
These observations seem to support a new kind of "Biological-evolutionary"
foundation for mathematics, that is closely related to Immanuel Kant's
philosophy of mathematics in his Critique of Pure Reason (1781), and my
attempt to defend his ideas in Sloman, [1962]. This answers questions like
"How is it possible for things that make mathematical discoveries to exist?",
an example of explaining a possibility (See Note 3). Attempting to
go too directly from hypothesized properties of the primordial construction kit
to explaining advanced capabilities such as human self-awareness, without
specifying all the relevant construction kits, including required temporary
scaffolding (Sect. 2.7)
will fail, because short-cuts omit essential details
of both the problems and the solutions, like mathematical proofs with gaps.
Many of the "mathematical discoveries" (or inventions?) produced
(blindly) by evolution, depend on mathematical properties of physical
structures or processes or problem types, whether they are specific solutions to
particular problems (e.g. use of negative feedback control loops), or new
construction-kit components that are usable across a very wide range of
different species (e.g. the use of a powerful "genetic code", the use of various
kinds of learning from experience, the use of new forms of representation for
information, use of new physical morphologies to support sensing, or locomotion,
or consumption of nutrients etc.)
These mathematical "discoveries" started happening long before there were any
humans doing mathematics (which refutes
claims that humans create mathematics). Many
of the discoveries were concerned with what is possible, either absolutely or
under certain conditions, or for a particular sort of construction-kit. Other
discoveries, closer to what are conventionally thought of as mathematical
discoveries, are concerned with limitations on what is possible, i.e. necessary
truths. Some discoveries are concerned with probabilities derived from
statistical learning, but I think the relative importance of statistical
learning in biology has been vastly over-rated because of misinterpretations of
evidence. (To be discussed elsewhere.) In particular, the discovery that
something important is possible does not require collection of statistics: A
single instance suffices. And no amount of statistical evidence can show that
something is impossible: structural constraints need to be understood. For human
evolution, a particularly important subtype of mathematical discovery was
unwitting discovery and use of mathematical (e.g. topological) structures in the
environment, a discovery process that starts in human children before they are
aware of what they are doing, and in some species without any use of language
for communication. Examples are discussed in the "Toddler Theorems" document
referenced in Note. 23.
6 Varieties of Derived Construction Kit
DCKs may differ (a) at different evolutionary stages
within a lineage, (b) across lineages (e.g. in different coexisting organisms),
and (c) during development of an individual that starts as a single cell and
develops mechanisms that support different kinds of growth, development and
information processing, at different stages of development
(Section 2.3). New construction kits can also be produced by
cultures or ecosystems (e.g. human languages) and applied sciences (e.g.
bioengineering, computer systems engineering). New
cases build on what was previously available. Sometimes separately evolved DCKs
are combined, for instance in symbiosis, sexual reproduction, and individual
creative learning.
What sort of kit makes it possible for a child to acquire competence in any one
of the thousands of different human languages (spoken or signed) in the first
few years of life? Children do not merely learn pre-existing languages: they
construct languages that are new for them, constrained by the need to
communicate with conspecifics, as shown dramatically by Nicaraguan deaf children
who developed a sign language going beyond what their teachers understood
[Senghas, 2005].
There are also languages that might have developed but have not (yet). Evolution
of human spoken language may have gone from purely internal languages needed for
perception, intention, etc., through collaborative actions then signed
communication, then spoken communication, as argued in Sloman, [2008].
If language acquisition were mainly a matter of learning from
expert users, human languages could not have existed, since initially there
were no expert users to learn from, and learning could not get started. This
argument applies to any competence thought to be based entirely
on learning from experts, including mathematical expertise. So data-mining in
samples of expert behaviours will never produce AI systems with human
competences - only inferior subsets at best, though some narrowly focused
machines based on very large data-sets or massive computational power may
outperform humans (e.g. IBM's Deep Blue chess machine and WATSON).29
The history of computing since the earliest calculators illustrates changes that
can occur when new construction kits are developed. There were not only changes
of size, speed and memory capacity: there have also been profound qualitative
changes, in new layers of virtual machinery, such as new sorts of mutually
interacting causal loops linking virtual machine control states with portions of
external environments, as in use of GPS-based navigation. Long before that,
evolved virtual machines provided semantic contents referring
to non-physical structures and processes, e.g. mathematical problems, rules of
games, and mental contents referring to possible future mental contents ("What
will I see if...?") including contents of other minds.
I claim, but will not argue here, that some new machines cannot be fully
described in the language of the FCK even though they are fully
implemented in physical reality. (See Section 2.2
on ontologies.) We now understand many key components and many modes
of composition that provide platforms on which human-designed layers of
computation can be constructed, including subsystems closely but not rigidly
coupled to the environment (e.g. a hand-held video camera).
Several different "basic" abstract construction kits have been proposed as
sufficient for the forms of (discrete) computation required by mathematicians:
namely Turing machines, Post's production systems, Church's
Lambda Calculus, and several more, each capable of generating the others.
The Church-Turing thesis claims that each is sufficient for all forms of
computation.30
There has been an enormous amount of research in computer science, and computer
systems engineering, on forms of computation that can be built from such
components. One interpretation of the Church-Turing thesis is that these
construction kits generate all possible forms of information-processing.
But it is not at all obvious that those discrete mechanisms suffice for all
biological forms of information-processing. For example, chemistry-based forms
of computation include both discrete mechanisms (e.g. forming or releasing
chemical bonds) of the sort Schrödinger discussed, and continuous
process, e.g. folding and twisting used in reproduction.
Ganti, [2003] shows how a chemical construction-kit
can support forms of biological information-processing that don't depend only on
external energy sources (a feature shared with battery-powered computers),
and also supports growth and reproduction using internal mechanisms, which
human-made computers cannot do (yet).
There seem to be
many different sorts of construction-kit that allow different sorts
of information-processing to be supported, including some that we
don't yet understand. In particular, the physical/chemical mechanisms that
support the construction of both physical structures and information-processing
mechanisms in living organisms may have abilities not available in digital
computers.31
6.1 A new type of research project
Most biological processes and associated materials and mechanisms are not well
understood, though knowledge is increasing rapidly. As far as I know, very few
of the derived construction kits have been identified and studied, and I am not
aware of any systematic attempt to identify features of the FCK that suffice to
explain the possibility of all known evolved biological DCKs. Researchers in
fundamental physics or cosmology do not normally attempt to ensure that their
theories explain the many materials and process types that have been explored by
natural selection and its products, in addition to known facts about physics and
chemistry.
Schrödinger () pointed out that a theory of
the physical basis of life should explain such phenomena, though he could not
have appreciated some of the requirements for sophisticated forms
of information-processing, because, at the time he wrote, scientists and
engineers had not learnt what we now know.
Curiously, although he mentioned the need to explain the occurrence of
metamorphosis in organisms, the example he mentioned was the transformation from
a tadpole to a frog. He could have mentioned more spectacular examples, such as
the transformation from a caterpillar to a butterfly via an
intermediate stage as a chemical soup in an outer case, from which the
butterfly later emerges.32
Penrose, [1994] attempted to show how features of quantum physics explain
obscure features of human consciousness, especially mathematical consciousness,
but he ignored all the intermediate products of biological evolution on which
animal mental functions build. Human mathematics, at least the ancient
mathematics done before the advent of modern algebra and logic, seems to build
on animal abilities, such as abilities to see various types of affordance.
The use of diagrams and spatial models by Penrose may be an example.
It is unlikely that there are very abstract human mathematical abilities that
somehow grow directly out of quantum mechanical aspects of the FCK, without
depending on the mostly unknown layers of perceptual, learning, motivational,
planning, and reasoning competences produced by billions of years of evolution.
20th century biologists understood some of the achievements of the FCK in
meeting physical and chemical requirements of various forms of life, though they
used different terminology from mine, e.g. Haldane.33 However, the task can
never be finished, since the process of construction of new derived biological
construction kits may continue indefinitely, producing new kits with components
and modes of composition that allow production of increasingly complex types of
structure and behaviour in organisms. That idea is familiar to
computer scientists and engineers since thousands of new sorts
of computational construction kit (new programming languages, new operating
systems, new virtual machines, new development toolkits) have been developed
from old ones in the last half century, making possible new kinds of computing
system that could not previously be built from the original computing machinery
without introducing new intermediate layers, including new virtual machines that
are able to detect and record their own operations, a capability that is often
essential for debugging and extending computing systems.
Sloman, [2013a] discusses the importance of virtual machines in extending
what information-processing systems can do, and the properties they can have.
6.2 Construction-kits for biological information-processing
It seems that the earliest evolved DCKs supported evolution of new
physical/chemical mechanisms, making it useful to develop
information-based control mechanisms used to select between available
competences and tune them - using previous results of perception, learning,
motive formation, planning, and decision making. In some organisms, implicit
mathematical discovery processes, enabled production of competences used in
generic understanding of sensory information, e.g. locating perceived objects
and events in space, synthesis of separate information fragments into coherent
wholes, and control systems using mechanisms for motive generation, plan
construction, selection and control of behaviour, and prediction. Many of
evolution's mathematical discoveries were "compiled" into designs producing
useful behaviours, e.g. use of negative feedback loops controlling temperature,
osmotic pressure and other states, use of geometric constraints by bees whose
cooperative behaviours produce hexagonal cells in honeycombs, and use of new
ontologies for separating situations requiring different behaviours, e.g.
manipulating different materials, or hunting different kinds of prey.
Later still, construction kits used by evolution produced meta-cognitive
mechanisms enabling individuals to notice and reflect on their own
discoveries (enabling some of them to notice and remove flaws in their
reasoning). In some cases those meta-cognitive capabilities allowed individuals
to communicate discoveries to others, discuss them, and organise them into
complex highly structured bodies of shared knowledge, such as Euclid's
Elements (Note 1). I don't think anyone knows how long all of
this took, what the detailed evolutionary changes were, and how the required
mechanisms of
perception, motivation, intention formation, reasoning and planning
evolved. Explaining how that could happen, and what it tells us about the nature
of mathematics and biological/evolutionary foundations for mathematical
knowledge is a long term goal of the Meta-Morphogenesis project. That includes
seeking unnoticed overlaps between the human competences discovered by
meta-cognitive mechanisms, and similar competences in animals that lack the
metacognition, like young humans making and using mathematical discoveries, on
which they are unable to reflect because the required architecture has not yet
developed, and similar discoveries in other intelligent species.
This could stimulate new research in robotics aimed at replicating the
developmental processes.
Most of these naturally occurring mathematical abilities have not yet been
replicated in Artificial Intelligence systems or robots, unlike logical,
arithmetical, and algebraic competences that are relatively new to humans and
(paradoxically?) easier to replicate on computers. Examples of topological
reasoning about equivalence classes of closed curves not yet modelled in
computers (as far as I know) are referenced in Note 31. Even the
ability to reason about alternative ways of putting a shirt on a child
(Note 17) is still lacking. It is not clear whether the difficulty
of replicating such mathematical reasoning processes is due to the need for a
kind of construction-kit that digital computers (e.g. Turing machines) cannot
support, or due to our lack of imagination in using computers to replicate some
of the products of biological evolution - or both! Perhaps there are
important forms of representation or types of information-processing
architecture still waiting to be discovered by AI researchers. Alternatively the
gaps may be connected with properties of chemistry-based information-processing
mechanisms combining discrete and continuous interactions, or other physical
properties that cannot be replicated exactly (or even approximately) in familiar
forms of computation. (This topic requires more detailed mathematical analysis.)
6.3 Representational blind spots of many scientists
Although I cannot follow all the details of writings of physicists, I think it
is clear that most of the debates regarding what should go into a fundamental
theory of matter ignore most of the biological demands on such a theory. For
example, presentations on dynamics of physical systems make deep use of branches
of mathematics concerned with numerical values, and the ways in which different
measurable or hypothesized physical values do or do not co-vary, as expressed in
(probabilistic or non-probabilistic) equations of various sorts.
But the biological functions of complex physiological structures, especially
structures that change in complexity, don't necessarily have those forms.
Biological mechanisms include: digestive mechanisms, mechanisms for transporting
chemicals, mechanisms for detecting and repairing damage or infection,
mechanisms for storing re-usable information about an extended structured
environment, mechanisms for creating, storing and using complex percepts,
thoughts, questions, values, preferences, desires, intentions and plans,
including plans for cooperative behaviours, and mechanisms that transform
themselves into new mechanisms with new structures and functions.
Forms of mathematics used by physicists are not necessarily useful
for studying such biological
mechanisms. Logic, grammars and map-like representations
are sometimes more appropriate, though I think little is actually known about
the variety of forms of representation (i.e. encodings of information) used in
human and animal minds and brains. We may need entirely new forms of mathematics
for biology, and therefore for specifying what physicists need to explain.
Many physicists, engineers and mathematicians who move into neuroscience assume
that states and processes in brains need to be expressed as collections of
numerical measures and their derivatives plus equations linking them, a form of
representation that is well supported by widely used tools such as Matlab, but
is not necessarily best suited for the majority of mental contents, and probably
not even well suited for chemical processes where structures form and interact
with multiple changing geometrical and topological relationships - one of the
reasons for the invention of symbolic chemical notations (now being extended in
computer models of changing, interacting molecular structures).
6.4 Representing rewards, preferences, values
It is often assumed that all intelligent decision making uses positive or
negative scalar reward or utility values that are comparable across options
[Luce and Raiffa, 1957]. But careful attention to consumer magazines, political
debates, and the varieties of indecision that face humans in real life shows
that reality is far more complex. For example, many preferences are expressed in
rules about how to choose between certain options. Furthermore preferences can
be highly sensitive to changes in context. A crude example is the change in
preference for type of car after having children. Analysis of examples in
consumer reports led to the conclusion that "better" is a complex,
polymorphic, logical concept with a rich structure that cannot be reduced to use
of comparisons of numerical values [Sloman, 1969,Sloman, 1970]. Instead of a
linear reward or utility metric, choices for intelligent individuals, or for
natural selection, involve a complex partial ordering network, with
"annotated" links between nodes (e.g. "better" qualified by conditions:
"better for", "better if"...). In the Birmingham CogAff project
[Sloman, 2003], those ideas informed computational
models of simple agents with complex choices to be made under varying
conditions, but the project merely scratched the surface, as reported in
[Beaudoin and Sloman, 1993,Beaudoin, 1994,Wright et al, 1996,Wright, 1997]. Most AI/Cognitive Science
models use much shallower notions of motivation.
Despite all the sophistication of modern psychology and neuroscience, I
believe they currently lack the conceptual resources required to describe
either functions of brains in dealing with these matters, including
forms of development and learning required, or the mechanisms
implementing those
functions. In particular, we lack deep explanatory theories about
mechanisms that led to: mathematical discoveries over thousands of years,
including mechanisms producing conjectures, proofs, counter-examples,
proof-revisions, new scientific theories, new works of art and new styles of
art. In part that's because models considered so far lack both sufficiently rich
forms of information-processing (computation), and sufficiently deep
methodologies for identifying what needs to be explained. There are other
unexplained phenomena concerned with artistic creation and enjoyment, but that
will not be pursued here.
7 Computational/Information-processing construction-kits
Since the mid 20th century we have been learning about abstract
construction-kits whose products are machines that can be used for increasingly
complex tasks. Such construction kits include programming languages, operating
systems, software development tools and environments, and network-technology
that allows ever more complex information-processing machines to be constructed
by combining simpler ones. A crucial, but poorly understood, feature
of that history is the growing use of construction-kits based on virtual
machinery, mentioned in Section 2.
A complete account of the role of construction kits in biological evolution
would need to include an explanation of how the fundamental construction kit
(FCK) provided by the physical universe could be used by evolution to produce an
increasing variety of types of virtual machinery as well as increasingly
varied physical structures and mechanisms.
7.1 Infinite, or potentially infinite, generative power
A construction kit implicitly specifies a large, in some cases infinite, set of
possibilities, though as an instance of the kit is constructed each addition of
a new component or feature changes the set of possibilities accessible in
later steps of that construction process.
For example, as you construct a sentence or phrase in a language, at each state
in the construction there are alternative possible additions (not necessarily at
the end) and each of those additions will alter the set of possible further
additions consistent with the vocabulary and grammar of the language.
When use of language is embedded in a larger activity, such as composing a poem,
that context can modify the constraints that are relevant.
Chemistry does something like that for types of molecule, types of process
involving molecular changes, and types of structure made of multiple molecules.
Quantum mechanics added important constraints to 19th century chemistry,
including both the possibility of highly stable structures (resistant to thermal
buffetting)
and also locks and keys as in catalysis. All of that is essential for
life as we know it, and also for forms of information-processing produced by
evolution (mostly not yet charted).
Research in fundamental physics is a search for the construction kit that has
the generative power to accommodate all the possible forms of matter, structure,
process, causation, that exist in our universe. However, physicists generally
seek only to ensure that their construction kits are capable of accounting for
phenomena observed in the physical sciences. Normally they do not assemble
features of living matter, or processes of evolution, development, or learning,
found in living organisms and try to ensure that their fundamental theories can
account for those features also. There are notable exceptions, such as
Schrödinger and others, but most physicists who discuss physics and life (in
my experience) ignore most of the details of life, including the variety of
forms it can take, the variety of environments coped with, the different ways in
which individual organisms cope and change, the ways in which products of
evolution become more complex and more diverse over time, and the many kinds of
information-processing and control in individuals, in colonies (e.g. ant
colonies), societies, and ecosystems.
If cosmologists and other theoretical physicists attempted to take note of a
wide range of biological phenomena (including the phenomena discussed here in
connection with the Meta-Morphogenesis project) I suspect that they would find
considerable explanatory gaps between current physical theories and the
diversity of phenomena of life - not because there is something about life that
goes beyond what science can explain, but because we do not yet have a
sufficiently rich theory of the constitution of the universe (or the Fundamental
Construct Kit). In part that could be a consequence of the forms of mathematics
known to physicists. (The challenge of
Anderson, [1972] is also relevant: see Section 10, below.)
If that is true it may take many years of research to find out what's missing
from current physics. Collecting phenomena that need to be explained, and trying
as hard as possible to construct detailed explanations of those phenomena
is one way to make progress: it may help us to pin-point gaps in our theories
and stimulate development of new, more powerful, theories, in something like the
profound ways in which our understanding of possible forms of computation has
been extended by unending attempts to put computation to new uses. Collecting
examples of such challenges helps us assemble tests to be passed by future
proposed theories: collections of possibilities that a deep physical theory
needs to be able to explain.
Perhaps the most tendentious proposal here is that an expanded physical theory,
instead of being expressed mainly in terms of equations relating measures, will
need a formalism better suited to specification of a construction kit, perhaps
sharing features of grammars, programming languages, partial orderings,
topological relationships, architectural specifications, and the structural
descriptions in chemistry - all of which will need to make use of appropriate
kinds of mathematics for drawing out implications of the theories, including
explanations of possibilities, both observed and unobserved, such as possible
future forms of intelligence. Theories of utility measures may need to be
replaced, or enhanced with new theories of how benefits, evaluations,
comparisons and preferences, can be expressed Sloman, [1969]. We must also
avoid assuming optimality. Evolution produces designs as diverse as microbes,
cockroaches, elephants and orchids, none of which is optimal or rational in any
simple sense, yet many of them survive and sometimes proliferate, because they
are lucky, at least for a while. Likewise human decisions, policies,
preferences, cultures, etc.
8 Types and levels of explanation of possibilities
Suppose someone uses a Meccano kit to construct a toy crane, with a jib that can
be moved up and down by turning a handle, and a rotating platform on a fixed
base, that allows the direction of the jib to be changed. What's the difference
between explaining how that is possible and how it was done? First of all, if
nobody actually builds such a crane then there is no actual crane-building to be
explained: yet, insofar as the Meccano kit makes such cranes possible it
makes sense to ask how it is possible. This has several types of answer,
including answers at different levels of abstraction, with varying generality
and economy of specification.
More generally, the question "How is it possible to create X using construction
kit Y?" or, simply, "How is X possible?" has several types of answer,
including answers at different levels of abstraction, with varying generality.
I'll assume that a particular construction kit is referred to either explicitly
or implicitly. The following is not intended to be an exhaustive survey of the
possible types of answer: merely as a first experimental foray, preparing the
ground for future work:
1 Structural conformity: The first type of answer, structural conformity
(grammaticality) merely identifies the parts and relationships between parts
that are supported by the kit, showing that X (e.g. a crane of the sort in
question) could be composed of such parts arranged in such relationships. An
architect's drawings for a building, specifying materials, components, and their
spatial and functional relations would provide such an explanation of how a
proposed building is possible, including, perhaps, answering questions about how
the construction would make the building resistant to very high winds, or to
earthquakes up to a specified strength. This can be compared with showing that a
sentence is acceptable in a language with a well-defined grammar, by showing how
the sentence would be parsed (analysed) in accordance with the grammar of that
language. A parse tree (or graph) also shows how the sentence can be built up
piecemeal from words and other grammatical units, by assembling various
sub-structures and, using them to build larger structures. Compare using a
chemical diagram to show how a collection of atoms can make up a particular
molecule, e.g. the ring structure of C6H6 (Benzene).
Some structures are specified in terms of piece-wise relations, where the whole
structure cannot possibly exist, because the relations cannot hold
simultaneously, e.g. X is above Y, Y is above Z, Z is above X. It is
possible to depict such objects, e.g. in pictures of impossible objects by
Reutersvard, Escher, Penrose, and others.34
Some logicians and computer scientists
have attempted to design languages in which specifications of impossible
entities are necessarily syntactically ill-formed. This leads to impoverished
languages with restricted practical uses, e.g. strongly typed programming
languages. For some purposes less restricted languages, needing greater care in
use, are preferable, including human languages [Sloman, 1971].
2 Process possibility:
The second type of answer demonstrates constructability by
describing a sequence of spatial trajectories by which such a collection of
parts could be assembled. This may include processes of assembly of temporary
scaffolding (Sect. 2.7)
to hold parts in place before the connections have been made that make
them self-supporting or before the final supporting structures have been built
(as often happens in large engineering projects, such as bridge construction).
Many different possible trajectories can lead to the same result. Describing (or
demonstrating) any such trajectory explains both how that construction process
is possible, and how the end result is possible.
In some cases a complex object has type 1 possibility although not type 2. For
example, from a construction kit containing several rings it is possible to
assemble a pile of three rings, but not possible to assemble a chain
of three rings even though each of the parts of the chain is exactly like the
parts of the pile.
3 Process abstraction: Some possibilities are described at a level of
abstraction that ignores detailed routes through space, and covers many
possible alternatives. For example, instead of specifying precise trajectories
for parts as they are assembled, an explanation can specify the initial and
final state of each trajectory, where each state-pair may be shared by a vast,
or even infinite collection, of different possible trajectories producing the
same end state, e.g. in a continuous space.
In some cases the possible trajectories for a moved component are all
continuously deformable into one another (i.e. they are topologically
equivalent): for example the many spatial routes by which a cup could be moved
from a location where it rests on a table to a location where it rests on a
saucer on the table, without leaving the volume of space above the table. Those
trajectories form a continuum of possibilities that is too rich to be captured
by a parametrised equation for a line, with a number of variables. If
trajectories include passing through holes, or leaving and entering the room via
different doors or windows then the different possible trajectories will not all
be continuously deformable into one another: there are different equivalence
classes of trajectories sharing common start and end states, for example, the
different ways of threading a shoe lace with the same end result.
The ability to abstract away from detailed differences between trajectories
sharing start and end points, thereby implicitly recognizing invariant features
of an infinite collection of possibilities, is an important aspect of animal
intelligence that I don't think has been generally understood. Many researchers
assume that intelligence involves finding optimal solutions. So they
design mechanisms that search using an optimisation process, ignoring the
possibility of mechanisms that can find sets of possible solutions (e.g. routes)
initially considered as a class of equivalent options, leaving questions
about optimal assembly to be settled later, if needed. These remarks are closely
related to the origins of abilities to reason about geometry and
topology.35
4 Grouping:
Another form of abstraction is related to the difference between
1 and 2. If there is a sub-sequence of assembly processes, whose order
makes no difference to the end result, they can be grouped to form an unordered
"composite" move, containing an unordered set of moves. If N components are
moved from initial to final states in a sequence of N moves, and it makes no
difference in what order they are moved, merely specifying the set of N
possibilities without regard for order collapses N factorial sets of possible
sequences into one composite move. If N is 15, that will collapse 1307674368000
different sequences into one.
Sometimes a subset of moves can be made in parallel. E.g. someone with two hands
can move two or more objects at a time, in transferring a collection of items
from one place to another. Parallelism is particularly important in many
biological processes where different processes occurring in parallel constrain
one another so as to ensure that instead of all the possible states that could
occur by moving or assembling components separately, only those end states occur
that are consistent with parallel constructions. In more complex cases the end
state may depend on the relative speeds of sub-processes and also continuously
changing spatial relationships.
This is important in epigenesis, since all forms of development from a single
cell to a multi-celled structure depend on many mutually constraining processes
occurring in parallel.
For some construction kits certain constructs made of a collection of
sub-assemblies may require different sub-assemblies to be constructed in
parallel, if completing some too soon may make the required final configuration
unachievable. For example, rings being completed before being joined could
prevent formation of a chain.
5 Iterative or recursive abstraction: Some process types involve
unspecified numbers of parts or steps, although each instance of the type has a
definite number, for example a process of moving chairs by repeatedly carrying a
chair to the next room until there are no chairs left to be carried, or building
a tower from a collection of bricks, where the number of bricks can be varied. A
specification that abstracts from the number can use a notion like "repeat
until", or a recursive specification: a very old idea in mathematics, such as
Euclid's algorithm for finding the highest common factor of two numbers.
Production of such a generic specification can demonstrate a large variety of
possibilities inherent in a construction-kit in an extremely powerful and
economical way. Many new forms of abstraction of this type have been discovered
by computer scientists developing programming languages, for operating not only
on numbers but many other structures, e.g. trees and graphs.
Evolution may also have "discovered" many cases, long before humans
existed, by taking advantage of mathematical structures inherent in the
construction-kits available and the trajectories by which parts can be assembled
into larger wholes. This may be one of the ways in which evolution produced
powerful new genomes, and re-usable genome components that allowed many
different biological assembly processes to result from a single discovery, or a
few discoveries, at a high enough level of abstraction.
Some related abstractions may have resulted from parametrisation: processes by
which details are removed from specifications in genomes and left to be
provided by the context of development of individual organisms, including the
physical or social environment. (See Section 2.3 on
epigenesis.)
6 Self-assembly:
If, unlike construction of a toy Meccano crane or a sentence or a
sorting process, the process to be explained is a self-assembly process, like
many biological processes, then the explanation of how the assembly is possible
will not merely have to specify trajectories through space by which the parts
become assembled, but also
-
What causes each of the movements (e.g. what manipulators are required)
-
Where the energy required comes from (an internal store, or external supply?)
- Whether the process involves pre-specified information about required steps or
required end states, and if so what mechanisms can use that information to
control the assembly process.
- How that prior information structure (e.g. specification of a goal
state to be
achieved, or plan specifying actions to be taken) came to exist, e.g. whether it
was in the genome as a result of previous evolutionary transitions, or whether
it was constructed by some planning or problem-solving mechanism in an
individual, or whether it was provided by a communication from an external
source.
- How these abilities can be acquired or improved by learning
or reasoning processes, or random variation (if they can).
7 Use of explicit intentions and plans: None of the explanation-types
above presupposes that the possibility being explained has ever been represented
explicitly by the machines or organisms involved. Explaining the possibility of
some structure or process that results from intentions or plans would require
specifying pre-existing information about the end state and in some cases also
intermediate states, namely information that existed before the process began -
information that can be used to control the process (e.g. intentions,
instructions, or sub-goals, and preferences that help with selections between
options). It seems that some of the reproductive mechanisms that depend on
parental care make use of mechanisms that generate intentions and possibly also
plans in carers, for instance intentions to bring food to an infant, intentions
to build nests, intentions to carry an infant to a new nest, intention to
migrate to another continent when temperature drops, and many more. Use of
intentions that can be carried out in multiple ways selected according to
circumstances rather than automatically triggered reflexes could cover a far
wider variety of cases, but would require provision of greater intelligence in
individuals.
Sometimes an explanation of possibility prior to construction is important for
engineering projects where something new is proposed and critics believe that
the object in question could not exist, or could not be brought into existence
using available known materials and techniques. The designer might answer
sceptical critics by combining answers of any of the above types, depending on
the reasons for the scepticism.
Concluding comment on explanations of possibilities:
Those are all examples of components of explanations of assembly processes,
including self-assembly. In biological reproduction, growth, repair,
development, and learning there are far more subdivisions to be considered, some
of them already studied piecemeal in a variety of disciplines. In the case of
human development, and to a lesser extent development in other species, there
are many additional sub-cases involving construction kits both for creating
information structures and creating information-processing mechanisms of many
kinds, including perception, learning, motive formation, motive comparison,
intention formation, plan construction, plan execution, language use, and many
more. A subset of cases, with further references can be found in
Sloman, [2006].
The different answers to "How is it possible to construct this type of object"
may be correct as far as they go, though some provide more detail than others.
More subtle cases of explanations of possibility include differences between
reproduction via egg-laying and reproduction via parturition, especially when
followed by caring for young. The latter allows a parent's influence to continue
during development, as does teaching of younger individuals by older ones. This
also allows development of cultures suited to different environments.
To conclude this rather messy section: the investigation of different types of
generality in modes of explanation for possibilities supported by a construction
kit is also relevant to modes of specification of new designs based on the kit.
Finding economical forms of abstraction may have many benefits, including
reducing search spaces when trying to find a new design and also providing a
generic design that covers a broad range of applications tailored to detailed
requirements. Of particular relevance in a biological context is the need for
designs that can be adjusted over time, e.g. during growth of an organism, or
shared across species with slightly different physical features or environments.
Many of the points made here are also related to changes in types of computer
programming language and software design specification languages. Evolution may
have beaten us to important ideas. That these levels of abstraction are possible
is a metaphysical feature of the universe, implied by the generality of the FCK.
9 Alan Turing's Construction kits
Turing, [1936] showed that a rather simple sort of machine, now known as a
Turing machine, could be used to specify an infinite set of constructions with
surprisingly rich mathematical features. The set of possibilities was infinite,
because a Turing machine is defined to have an infinite (or indefinitely
extendable) linear "tape" divided into discrete locations in
which symbols can be inserted.
A feature of a Turing machine that is not in most other construction kits
is that it can be set up and then started after which it will modify initial
structures and build new ones, possibly indefinitely, though in some cases the
machine will eventually halt.
Another type of construction kit with related properties is Conway's Game of
Life,36 a
construction kit that creates changing patterns in 2D regular arrays. Stephen
Wolfram has written a great deal about the diversity of constructions that can
be explored using such cellular automata. Neither a Turing machine nor a Conway
game has any external sensors: once started they run according to their stored
rules and the current (changing) state of the tape or grid-cells. In principle
either of them could be attached to external sensors that could produce changes
to the tape of a turing machine or the states of some of the cells in the Life
array. However any such extension would significantly alter the powers of the
machine, and theorems about what such a machine could or could not do would
change.
Modern computers use a variant of the Turing machine idea where each computer
has a finite memory but with the advantage of much more direct access between
the central computer mechanism and the locations in the memory (a von Neumann
architecture).
Increasingly, computers have also been provided with a variety of
external interfaces connected to sensors or motors so that while running they
can acquire information (e.g. from keyboards, buttons, joy-sticks, mice,
electronic piano keyboards, network connections, and many more) and can also
send signals to external devices. Theorems about disconnected Turing machines
may not apply to machines with rich two-way interfaces to an external
environment.
Turing machines and Game of Life machines can be described as
"self-propelling" because once set up they can be left to run according to the
general instructions they have and the initial configuration on the tape or in
the array. But they are not really self-propelling: they have to be implemented
in physical machines with an external power supply. In contrast,
Ganti, [2003] shows how the use of chemistry as a construction kit provides
"self-propulsion" for living things, though every now and again the chemicals need
to be replenished. A battery driven computer is a bit like that, but someone
else has to make the battery.
Living things make and maintain themselves, at least after being given a
kick-start by their parent or parents. They do need constant, or at least
frequent, external inputs, but, for the simplest organisms, those are only
chemicals in the environment, and energy either from chemicals or heat-energy
via radiation, conduction or convection. John McCarthy pointed out in a
conversation that some animals also use externally supplied mechanical energy,
e.g. rising air currents used by birds that soar. Unlike pollen-grains, spores,
etc. propagated by wind or water, the birds use internal information-processing
mechanisms to control how the wind energy is used, as does a human piloting a
glider.
9.1 Beyond Turing machines: chemistry
Turing also explored other sorts of construction kits, including types of neural
nets and extended versions of Turing machines with "oracles" added. Shortly
before his death (in 1954), he published Turing, [1952] in which he
explored a type of pattern-forming construction kit in which two chemical
substances can diffuse through the body of an expanding organism and interact
strongly wherever they meet. He showed that that sort of construction kit could
generate many of the types of surface physical structure observed on plants and
animals. I have been trying to show how that can be seen as a very simple
example of something far more general.
One of the important differences between types of construction kit mentioned
above is the difference between kits supporting only discrete changes (e.g. to a
first approximation Lego and Meccano (ignoring variable length strings and
variable angle joints) and kits supporting continuous variation, e.g. plasticine
and mud (ignoring, for now, the discreteness at the molecular level).
One of the implications of such differences is how they affect
abilities to search for solutions to problems. If only big changes in design are
possible the precise change needed to solve a problem may be inaccessible (as I
am sure many who have played with construction kits will have noticed - when
a partial construction produces a gap whose width does not exactly match the
width of any available pieces).
On the
other hand if the kit allows arbitrarily small changes it will, in principle,
permit exhaustive searches in some sub-spaces. The exhaustiveness comes at the
cost of a very much larger (infinite, or potentially infinite!) search-space.
That feature could be useless, unless
the space of requirements has a structure that allows approximate solutions to
be useful. In that case a mixture of big jumps to get close to a good solution,
followed by small jumps to home in on a (locally) optimal solution can be very
fruitful: a technique that has been used by Artificial Intelligence researchers,
called "simulated annealing".37
A recently published book Wagner, [2014] claims that the structure of the
search space generated by the molecules making up the genome increases the
chance of useful, approximate, solutions to important problems to be found with
relatively little searching (compared with other search spaces), after
which small random changes allow improvements to be found. I have not yet read
the book but it seems to illustrate the importance for evolution of the types of
construction-kit available.38
I have not yet had time to check whether the book discusses uses of abstraction
and the evolution of mathematical and meta-mathematical competences discussed
here. Nevertheless, it seems to be an (unwitting) contribution to the
Meta-Morphogenesis project. Recent work by Jeremy England at MIT39
may turn out also to be relevant.
..
9.2 Using properties of a construction-kit to explain possibilities
A formal axiomatic system can be seen as an abstract construction kit with
axioms and rules that support construction of proofs, ending in theorems. The
theorems are formulae that can occur at the end of a proof using only axioms and
inference rules in the system. The kit explains the possibility of some theorems
based on the axioms and rules. The non-theorems of an axiomatic system are
formulae for which no such proof exists. Proving that something is a
non-theorem can be difficult, and requires a proof in a meta-system.
Likewise, a physical construction kit can be used to demonstrate that some
complex physical objects can occur at the end of a construction process. In some
cases there are objects that are describable but cannot occur in a construction
using that kit: e.g. an object whose outer boundary is a surface that is
everywhere curved, cannot be produced in a construction based on Lego bricks or
a Meccano set, though one could occur in a construction based on plasticene, or
soap-film.
9.3 Bounded and unbounded construction kits
A rectangular grid of squares combined with the single digit numbers, 0,1,..,9
(strictly numerals representing numbers) allows construction of a set of
configurations in which numbers are inserted into the squares subject to various
constraints, e.g. whether some squares can be left blank, or whether certain
pairs of numbers can be adjacent, whether the same number can occur in more than
one square. For a given grid and a given set of constraints there will be a
finite set of possible configurations (although it may be a very large set).
If, in addition to insertion of a number, the "construction kit" allows extra
empty rows or columns to be added to the grid, no matter how large it is, then
the set of possible configurations becomes infinite.
Many types of infinite construction kits have been investigated by
mathematicians, logicians, linguists, computer scientists, musicians and other
artists.
Analysis of chemistry-based construction kits for information-processing systems
would range over a far larger class of possible systems than Turing machines (or
digital computers), because of the mixture of discrete and continuous changes
possible when molecules interact, e.g. moving together, moving apart, folding,
twisting, but also locking and unlocking - using catalysts [Kauffman, 1995]. I
don't know whether anyone has a deep theory of the scope and limits of
chemistry-based information-processing.
Recent discoveries indicate that some biological mechanisms use
quantum-mechanical features of the FCK that we do not yet fully understand,
providing forms of information-processing that are very different from what
current computers do. E.g. a presentation by Seth Lloyd, summarises quantum
phenomena used in deep sea photosynthesis, avian navigation, and odour
classification.40
This may turn out to be the tip of an iceberg of quantum-based
information-processing mechanisms.
There are some unsolved, very hard, partly ill-defined, problems about the
variety of functions of biological vision: e.g. simultaneously interpreting
a very large, varied and changing collection of visual fragments, perceived from
constantly varying viewpoints, e.g. as you walk through a garden with many
unfamiliar flowers, shrubs, bushes, etc. moving irregularly in a changing
breeze. Could some combination of quantum entanglement and non-local interaction
play a role in rapidly and simultaneously processing a large collection of
mutual constraints between multiple visual fragments? The ideas are not yet
ready for publication, but work in progress is recorded here:
http://www.cs.bham.ac.uk/research/projects/cogaff/misc/quantum-evolution.html.
Some related questions about perception of videos of fairly complex moving plant
structures are raised here:
http://www.cs.bham.ac.uk/research/projects/cogaff/misc/vision/plants/.
10 Conclusion: Construction kits for Meta-Morphogenesis
A useful survey of previous attempts to show how life and its products relate to
the physical world is in Keller, [2008,Keller, [2009], which concluded that attempts
so far have not been successful. Keller ends with the suggestion that the
traditional theory of dynamical systems is inadequate for dealing with
constructive processes and needs to be expanded to include "objects, their
internal properties, their construction, and their dynamics" i.e. a theory of
"Constructive dynamical systems". This paper outlines a project to do
that and more: including giving an account of branching layers of new
derived construction kits produced by evolution, development and other
processes. The physical world clearly provides a very powerful (chemistry-based)
fundamental construction kit that, together with natural selection processes and
processes within individuals as they develop, produced an enormous variety of
organisms on this planet, based on additional derived construction kits (DCKs),
including concrete, abstract and hybrid construction kits, and, most recently,
new sorts of construction kit used as toys or engineering resources.
The idea of a construction kit is offered as a new unifying concept for
philosophy of mathematics, philosophy of science, philosophy of biology,
philosophy of mind and metaphysics. The aim is to explain how it is possible for
minds to exist in a material world and to be produced by natural selection and
its products. Related questions arise about the nature of mathematics
and its role in life.
The ideas are still at an early stage of development and there are
probably many more distinctions to be made, and a need for a more formal,
mathematical presentation of properties of and relationships between
construction kits, including the ways in which new derived construction kits can
be related to their predecessors and their successors. The many new types of
computer-based virtual machinery produced by human engineers since around
1950 provide examples of non-reductive supervenience (as explained in
Sloman, [2013a]). They are also useful as relatively simple examples to be
compared with far more complex products of evolution.
In Esfeld et al, [in press] a distinction is made between two "principled" options for
the relationship between the basic constituents of the world and their
consequences. In the "Humean" option there is nothing but the distribution of
structures and processes over space and time, though there may be some
empirically discernible patterns in that distribution. The second option is
"modal realism", or "dispositionalism", according to which there is
something about the primitive stuff and its role in space-time that constrains
what can and cannot exist, and what types of process can or cannot occur. This
paper supports a "multi-layer" version of the modal realist option
(developing ideas in Sloman, [1962,Sloman, [1996a,Sloman, [2013a]).
I suspect that a more complete development of this form of modal realism can
contribute to answering the problem posed in Anderson's famous paper
[Anderson, 1972], namely how we should understand the relationships between
different levels of complexity in the universe (or in scientific theories). The
reductionist alternative claims that when the physics of elementary particles
(or some other fundamental physical level) has been fully understood, everything
else in the universe can be explained in terms of mathematically derivable
consequences of the basic physics. Anderson contrasts this with the
anti-reductionist view that different levels of complexity in the universe
require "entirely new laws, concepts and generalisations" so that, for
example, biology is not applied chemistry and psychology is not applied biology.
He writes: "Surely there are more levels of organization between human ethology
and DNA than there are between DNA and quantum electrodynamics, and each level
can require a whole new conceptual structure". However, the structural levels
are not merely in the concepts used by scientists, but actually in the world.
We still have much to learn about the powers of the fundamental construction kit
(FCK), including: (i) the details of how those powers came to be used for life
on earth, (ii) which sorts of derived construction kit (DCK) were required in
order to make more complex life forms possible, (iii) how those construction
kits support "blind" mathematical discovery by evolution, mathematical
competences in humans and other animals and eventually meta-mathematical
competences, then meta-meta-mathematical competences, at least in humans, (iv)
what possibilities the FCK has that have not yet been realised, (v) whether
and how some version of the FCK could be used to extend the intelligence of
current robots, and (vi) whether currently used Turing-equivalent forms of
computation have at least the same information-processing potentialities (e.g.
abilities to support all the biological information-processing mechanisms and
architectures), and (vii) if those forms of computation lack the potential, then
how are biological forms of information-processing different? Don't expect
complete answers soon.
In future, physicists wishing to show the superiority of their theories, should
attempt to demonstrate mathematically and experimentally that they can explain
more of the potential of the FCK to support varieties of construction kit
required for, and produced by, biological evolution than rival theories can.
Will that be cheaper than building bigger better colliders? Will it be harder?41
End Note
As I was finishing off this paper I came across a letter Turing wrote to W. Ross
Ashby in 1946 urging Ashby to use Turing's ACE computer to implement his ideas
about modelling brains. Turing expressed a view that seems to be
unfashionable among AI researchers at present (2015),
but accords with the aims of this
paper:
"In working on the ACE I am more interested in the possibility of producing
models of the actions of the brain than in the practical applications to
computing."
http://www.rossashby.info/letters/turing.html
It would be very interesting to know whether he had ever considered the question
whether digital computers might be incapable of accurately modelling
brains making deep use of chemical processes. He also wrote in Turing, [1950]
"In the nervous system chemical phenomena are at least as important as
electrical."
But he did not elaborate on the implications of that claim.42
Acknowledgements
The Meta-Morphogenesis project, including this paper, owes a great
debt to Barry Cooper who unfortunately died in October 2015. I had never met him
until we both contributed chapters to a book published in 2011 on Information
and Computation. Barry and I first met, by email, when we reviewed each others'
chapters. Later, out of the blue, he invited me to contribute to the Turing
centenary volume he was co-editing [Cooper and van Leeuwen, 2013]. I contributed three papers.
He then asked me for a contribution to Part 4 (on Emergence and Morphogenesis)
based on Turing's paper on morphogenesis published in 1952, two years before he
died. That got me wondering what Turing might have done if he had lived another
30-40 years. So I offered Barry a paper proposing "The Meta-Morphogenesis
Project" as an answer. He accepted it (as the final commentary paper in the
book) and ever since then I have been working full-time on the project. He later
encouraged me further by inviting me to give talks and to contribute a chapter
to this book. As a result we had several very enjoyable conversations. He
changed my life, by giving me a new research direction, which does not often
happen to 75-year old retired academics! (Now four years older.) I wish we could
continue our conversations.
I also owe much to the highly intelligent squirrels and magpies in our
garden, who have humbled me.
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Footnotes:
1
http://www.gutenberg.org/ebooks/21076
2http://www.cs.bham.ac.uk/research/projects/cogaff/crp/#chap2
3
http://www.cs.bham.ac.uk/research/projects/cogaff/misc/explaining-possibility.html
4Expanded in
http://www.cs.bham.ac.uk/research/projects/cogaff/misc/meta-morphogenesis.html
5https://en.wikipedia.org/wiki/Genetic_programming
6Assembly mechanisms are
part of the organism, as illustrated in a video of grass growing itself from
seed https://www.youtube.com/watch?v=JbiQtfr6AYk. In mammals with a
placenta, more of the assembly process is shared between mother and offspring.
7Implications for evolution of vision and language are discussed in
http://www.cs.bham.ac.uk/research/projects/cogaff/talks/\#talk111
8http://www.cs.bham.ac.uk/research/projects/cogaff/misc/autism.html
9Examples include:
https://en.wikipedia.org/wiki/Parse_tree
https://en.wikipedia.org/wiki/Structural_formula
https://en.wikipedia.org/wiki/Flowchart
https://en.wikipedia.org/wiki/Euclidean_geometry
https://en.wikipedia.org/wiki/Entity-relationship_model
https://en.wikipedia.org/wiki/Programming_language
10http://www.cs.bham.ac.uk/research/projects/cogaff/00-02.html\#71
11Often misleadingly labelled
"non-linear". http://en.wikipedia.org/wiki/Control_theory
http://en.wikipedia.org/wiki/Nonlinear_control
12E.g.
http://www.cs.bham.ac.uk/research/projects/cogaff/misc/impossible.html
13
https://en.wikipedia.org/wiki/Meccano,
https://en.wikipedia.org/wiki/Tinkertoy and
https://en.wikipedia.org/wiki/Lego
14E.g. see James Ashenhurst's tutorial:
http://www.masterorganicchemistry.com/2011/11/10/dont-be-futyl-learn-the-butyls/
15Partly inspired by memories of a talk by Lionel Penrose in Oxford around 1960
about devices he called droguli - singular drogulus.
Such naturally occurring multi-stable physical structures seem to me to render
redundant the apparatus proposed in Deacon, [2011] to explain how life
apparently goes against the second law of thermodynamics. See
https://en.wikipedia.org/wiki/Incomplete_Nature
16https://en.wikipedia.org/wiki/Paper_doll
17Such as putting a shirt on a child:
http://www.cs.bham.ac.uk/research/projects/cogaff/misc/shirt.html I think
Piaget noticed some of the requirements.
18http://www.cs.bham.ac.uk/research/projects/cogaff/misc/toddler-theorems.html\#primes
19Trehub, [1991] proposed an
architecture for vision that allows snapshots from visual saccades to be
integrated in a multi-layer fixation-independent visual memory.
20http://en.wikipedia.org/wiki/Two-streams_hypothesis
21The
BICA society aims to bring together researchers on biologically inspired
cognitive architectures. Some examples are here:
http://bicasociety.org/cogarch/
22Our SimAgent toolkit is
an example
http://www.cs.bham.ac.uk/research/projects/poplog/packages/simagent.html
[Sloman, 1996b].
23As discussed in connection with "toddler theorems" in
http://www.cs.bham.ac.uk/research/projects/cogaff/misc/toddler-theorems.html
Contributions from observant parents and child-minders are welcome. I think
deeper insights come from extended individual developmental trajectories
than from statistical snapshots of many individuals.
24
http://www.theguardian.com/cities/2014/feb/18/slime-mould-rail-road-transport-routes
25Such relationships between
possibilities provide a deeper, more natural, basis for understanding modality
(necessity, possibility, impossibility) than so called "possible world
semantics". I doubt that most normal humans who can think about possibilities
and impossibilities base that on thinking about truth in the whole world, past,
present and future, and in the set of alternative worlds.
26For more on Kantian vs Humean
causation see the presentations on different sorts of causal reasoning in humans
and other animals, by Chappell and Sloman at the Workshop on Natural and
Artificial Cognition (WONAC, Oxford, 2007):
http://www.cs.bham.ac.uk/research/projects/cogaff/talks/wonac
Varieties of causation that do not involve mathematical necessity, only
probabilities (Hume?) or propensities (Popper) will not be discussed here.
27http://en.wikipedia.org/wiki/Symbiogenesis
28Some
of them listed in
http://www.cs.bham.ac.uk/research/projects/cogaff/misc/mathstuff.html
29This
comparison needs further discussion. See
http://www.popsci.com/science/article/2012-12/fyi-which-computer-smarter-watson-or-deep-blue
30For more on this see:
http://en.wikipedia.org/wiki/Church-Turing_thesis
31Examples of human mathematical reasoning in geometry and
topology that have, until now, resisted replication on computers are presented
in
http://www.cs.bham.ac.uk/research/projects/cogaff/misc/torus.html
and
http://www.cs.bham.ac.uk/research/projects/cogaff/misc/triangle-sum.html
32http://en.wikipedia.org/wiki/Pupa
http://en.wikipedia.org/wiki/Holometabolism
33http://en.wikipedia.org/wiki/J.\_B.\_S.\_Haldane
34http://www.cs.bham.ac.uk/research/projects/cogaff/misc/impossible.html
35Illustrated in these discussion notes:
http://www.cs.bham.ac.uk/research/projects/cogaff/misc/changing-affordances.html
http://www.cs.bham.ac.uk/research/projects/cogaff/misc/triangle-theorem.html
http://www.cs.bham.ac.uk/research/projects/cogaff/misc/torus.html
36http://en.wikipedia.org/wiki/Conway.27s.Game.of.Life
37One of many online explanations is
http://www.theprojectspot.com/tutorial-post/simulated-annealing-algorithm-for-beginners/6
38An interview with the author is online
at
https://www.youtube.com/watch?v=wyQgCMZdv6E
39
https://www.quantamagazine.org/20140122-a-new-physics-theory-of-life/
40https://www.youtube.com/watch?v=wcXSpXyZVuY
41Here's a cartoon teasing particle physicists:
http://www.smbc-comics.com/?id=3554
42I think it will turn out that the ideas about "making possible" used here
are closely related to Alastair Wilson's ideas about grounding as "metaphysical
causation". Wilson, [2015].
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