(Including evolution of minds and mathematical abilities.)
The scientific/metaphysical explanatory role of construction kits
(CHANGING DRAFT: Stored copies will soon be out of date.)
NB: Last modified on 17 Apr 2015 (Major reorganisation)
This paper is
From time to time a PDF version will be generated from the html, available as:
http://www.cs.bham.ac.uk/research/projects/cogaff/misc/construction-kits.pdf (May be out of date.)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, making
this available in flash format, updated occasionally. That version will not
necessarily be updated whenever this one is. (Not all the links work in the
slideshare version.) See
This is part of the Meta-Morphogenesis project:
NOTE: An earlier paper posted to this location on "Construction kits as
explanations of possibilities", defending Chapter 2 of
The Computer Revolution in
Philosophy (1978) against criticisms by reviewers has now been moved to a
A partial index of discussion notes here is in
I'll present a partial outline answer to the question: How was it possible for the known and unknown varieties of life to evolve from lifeless matter, including some varieties that are able to make the mathematical discoveries assembled in Euclid's Elements?1 The answer proposed here is based on construction kits, both fundamental and derived.
Karl Popper,  distinguished scientific and non-scientific statements. He required the former to be empirically falsifiable, otherwise they were metaphysical. Unfortunately this 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. Popper (unlike many of his admirers, who use falsifiability as a criterion for scientific content) acknowledged that some unfalsifiable "metaphysical" theories could be precursors of scientific theories. But labelling them "metaphysics" rather than "science" is misleading, because of their importance for science.2
Imre Lakatos (1980) extended Popper's philosophy of science, proposing ways of evaluating competing scientific research programmes, based on their progress over time. He offered criteria for distinguishing "progressive" from "degenerating" research programmes over time, on the basis of their different patterns of development, and specified criteria for deciding which of two research programmes is better, though it is not always possible to decide while both are being developed. The history of science shows that what appears to be a decisive victory (like Thomas Young's evidence of diffraction of light, apparently refuting Newton's particle theory of light) may later be overturned (e.g. by evidence for the dual wave-particle nature of light).
Chapter 2 of Sloman, 3 extended the ideas of Popper and Lakatos to accommodate scientific theories concerned with what is possible, e.g. types of plant, types of animal, types of reproduction, types of thinking, types of learning, types of verbal communication, types of molecule, types of chemical interaction, and types of biological information-processing -- the focus of the Meta-Morphogenesis project first presented in Sloman, [2013b]4 . A separate paper5 discusses in more detail 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].
Insisting on sharp boundaries between science and metaphysics harms both. Each can be pursued with rigour and openness to specific kinds of criticism. Some of the criteria for evaluating theories of what is possible, including theories that straddle science and metaphysics, were presented in sections 2.5.4-5 of Sloman, [1978,Ch. 2], and in the section entitled "Why allowing non-falsifiable theories doesn't make science soft and mushy" in the companion paper to this (Note 5). The ideas presented here are offered as a contribution to metaphysics as well as science -- but incomplete science, leaving much to be done. There are also connections with Kant's philosophy of mathematics, extended here to biological/evolutionary foundations of mathematics (BEFM). Possible engineering applications include addressing current gaps between AI and animal intelligence.
Life requires construction kits supporting construction of machines with many capabilities, including growing highly functional bodies, immune systems, digestive systems, repair mechanisms, and reproductive machinery. The requirements for life include information-processing (e.g. deciding what to repair) as well as physical construction (assembling matter). The Fundamental Construction Kit (FCK) provided by the physical universe when our planet came into existence was sufficient to make possible all the forms of life that have so far evolved on earth, meeting challenges that drove selection of new life forms. The FCK also makes possible many unrealised but possible forms of life, in possible but unrealised types of physical environment. How does it make all these things possible?
Figure FCK: Fundamental Construction Kit and possible trajectories
Think of time and increasing complexity going approximately from left to right.
Fig. FCK, above indicates crudely how a common initial construction kit (FCK, on the left) could explain many possible trajectories in which components of the kit are assembled to produce new instances of possible living and non-living physical forms using increasingly complex mechanisms.
Products of a construction kit can have mathematical features that are useful, e.g. negative feedback. So evolution produces mathematical competences implicit in biological mechanisms. As explained (sketchily) below, this may lead later to explicit mathematical and meta-mathematical competences in some species, eventually providing new biological/evolutionary foundations for mathematics.
The history of technology, science and engineering includes many transitions in which new construction kits were derived from old ones. That includes the science and technology of digital computation, where new advances used (among other things):
Products of evolutionary trajectories from the FCK may combine to form Derived Construction Kits (DCKs) (some specified in genomes, and some designed or discovered by individuals, or groups), that speed up construction of more complex entities with new types of properties and behaviours, as crudely indicated in Fig. DCK, below. In cases of convergent evolution, new DCKs evolve in different species in different locations, with overlapping functionality, using different mechanisms. A DCK producing mechanisms enabling elephants to learn to use trunk, eyes, and brain to manipulate food may share features with a DCK enabling primates to acquire abilities to use hands, eyes, and brains to manipulate food. Both competences, apparently using related mathematical control structures, evolved after the last common ancestor.
Figure DCK: Derived Construction Kit and new possible trajectories
Figure DCK: Derived Construction Kit and new possible trajectories
Biological evolution seems to have produced many branching lineages of increasingly complex re-usable construction kits, adding new, more complex, types of physical and chemical process (e.g. new forms of reproduction), and increasingly complex forms of information-processing.
Details of human-designed forms of computation look very different from evolved biological layers of machinery for assembling complex information-processing systems from simpler ones. But there may be deep similarities of function, including use of virtual machinery, discussed below. Over time, human designers use their evolved mechanisms, to produce larger, more complex, and more powerful systems, with the aid of increasingly complex tools for designing, building, testing and debugging. Likewise evolution.
Some new biological construction kits allow creation of new physical materials with new properties -- e.g. different weight/strength ratios, different kinds of flexibility and elasticity, different sorts of permeability, different ways of storing, releasing and using energy, different ways of producing motion, different forms of reproduction, and many more, all making use of new chemical mechanisms, including products of "biological nano-engineering".
Different life-forms (microbes, fungi, slime moulds, plants of many sizes and shapes, invertebrate and vertebrate animals of many kinds) have produced different sorts of physical materials used in constructing bodies, or extensions of bodies such as webs, cocoons and egg-shells. Examples include the cellulose and lignin structures that help to provide the strength of large plant structures that grow upwards out of soil, the materials in animals that produce rigid or semi-rigid structures (bones, shells, teeth, cartilage), the materials used in flexible structures with high tensile strength (e.g. tendons, vines), materials used in absorbing nutrients, oxygen, or water from the environment, materials transported between body parts, for different purposes (nutrients, waste matter, hormones, information, e.g. about stress or damage), materials concerned with storage and transfer or deployment of energy, for heat, for applying forces, for mobility, for reproduction, and many more.
Note on Making Possible:
The assertion "X makes Y possible" does not imply that if X does not exist then Y is impossible. All that is claimed is that one route to existence of Y is via existence of X. If X is built, that makes construction of Y easier. However, other things than X can make Y possible, for instance, an alternative construction kit. So "makes possible" should be interpreted in our discussion as 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 that. If X and Y are abstract, it is not clear that there is something like the FCK to which they must be traceable. The space of abstract construction kits may not have a fixed "root" kit. However, the abstract construction kits that can be thought about by physically implemented thinkers may be more constrained.
Note on 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. 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 the construction kit for theory formation, which includes extending the language used. Some of the later theories about DCKs (including theories about virtual machines in computer systems engineering) may include concepts that are not definable in terms of the concepts used in theories about the FCK, even though everything created using the DCK is fully implemented in the FCK. For more on this see Sloman, [2013a].
The simplest microbes use only a few (usually chemical) sensors 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 types of internal information throughout their life.
More complex organisms acquire and use information about enduring spatial locations in extended terrain whose contents include static and changing resources and dangers, e.g. noxious substances or lurking predators. Some can construct and use complex (internal or external) information stores about their environment.
Some of them also acquire and use information about information-processing, in themselves and in others, e.g. conspecifics, predators and prey. What features of construction kits support these developments?
Some controlled systems have states represented by a fixed set of physical measures, often referred to as "variables" and "constants", representing states of sensors, output signals, and internal states of various sorts. Relationships between state-components are represented mathematically by equations, including differential equations, and possibly also constraints (e.g. inequalities) specifying restricted, possibly time-varying, ranges of values for the variables. Such a system with N variables has a state of a fixed dimension, N.
The only way to store new information in such systems is in static or dynamic values for the variables -- changing "state vectors". A typical example is Powers, , inspired by Wiener,  and Ashby, . There are many well understood special cases of this pattern, such as simple forms of homeostatic control using negative feedback. Neural net controllers may be very much more complex, with variables typically clustered into strongly interacting sub-groups, and perhaps groups of groups, etc.
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.6
This may turn out to be the tip of an iceberg of quantum-based information-processing mechanisms (e.g. Hameroff and Penrose, ).
The world wide web can be thought of as 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. So the types of control mechanism in computer-based systems now extend far beyond the sorts familiar to control engineers, and studied in control theory.7
Many different sorts of control system may be required in the life of a single organism, e.g. between an egg being fertilised and the death of the organism.
Not all natural control functions are 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 the construction proceeds and the product (percept, thought, plan, map) becomes more complex -- unlike a fixed size collection of changing numerical values.
Different branches of numerical and non-numerical mathematics are suited to the problem of designing or understanding such systems, including graph theory, lattice theory, knot theory, category theory, set theory, logic, mathematical linguistics and others.
For a full understanding of mechanisms and processes of evolution and development, new branches of mathematics are likely to be needed, including mathematics relevant to complex non-numerical structural changes, such as revising a grammar for internal records of complex structured information.
Traditional vector- and equation-based control theories, even with probabilistic extensions, are not general enough for intelligent control systems that build and use sentences, problem descriptions, changing ontologies, explanatory theories, plans of varying complexity, new types of learning mechanism, systems of motives, values, social rules, and rule-based games, among other things.
A fixed set of equations cannot adequately represent steady growth of increasingly complex molecular structures [Anderson, 1972]. Evolution, like human mathematicians and computer scientists millions of years later, built construction kits and information structures able to cope with structures and processes of changing complexity, unlike models and mechanisms based only on fixed sets of variables linked by equations -- unable to represent either the structure of the meaning of a complex sentence, such as this one, or what can exist on a skyscraper construction-site, or many other perceived processes, including waves breaking on a rocky seashore, an intricately choreographed ballet, or a symphony.
It is unlikely that all the required forms of information, all the forms of control, and all the types of physical mechanism required for implementation are already understood by scientists and engineers. Yet the FCK along with the DCKs produced directly or indirectly by natural selection must be sufficiently general to model and explain everything that has evolved so far, and the things they have created and will create in future.
The huge variety of types of construction kit cannot be surveyed here. Instead of a complete theory: this paper merely presents a first-draft research framework within which gaps in our understanding can be discovered and in some cases filled, possibly over several decades, or even centuries. In particular, this first draft specifies some features of old and new construction kits, in the hope that additional research will extend the answers.
The planet on its own could not generate all those life forms. Energy from solar radiation is crucial for life on earth (though future technologies may remove that dependence). Other external influences that were important for the particular forms of life that evolved on earth included asteroid impacts, and cosmic radiation.8
Before our solar system formed, the fundamental construction-kit was potentially available everywhere in the universe, making possible the formation of galaxies, stars, clouds of dust, planets, asteroids, and many other lifeless entities, as well as supporting all forms of life, possibly through derived construction kits (DCKs) that exist only in special conditions. Local conditions e.g. extremely high pressures, temperatures, gravitational fields, distribution of kinds of matter, etc. can locally mask some parts of the FCK or prevent them from functioning.
According to some physical theories, every physical particle is (or can be) spread out over large areas, or possibly over the whole universe: nevertheless there must be differences in what exists in different places, for different processes can occur in different places. So the contents of the FCK are not necessarily distributed uniformly throughout the universe and some developments based on the FCK are impossible in certain parts of the universe lacking the required matter, or other pre-requisites.
The FCK must in some sense be available at the centre of the sun, but that does not mean that animal life or plant life can exist there. Likewise if the cloud of dust from which the earth is thought to have formed had been composed mostly of grains of sand, then no DCK capable of supporting life as we know it could have emerged, since earth-life depends on the presence of carbon, oxygen, hydrogen, iron, and many other elements.
As the earth formed, the new physical conditions created new DCKs that made the earliest life forms possible. Ganti,  presents a deep analysis of requirements for a DCK that supports primitive life forms. That DCK (building on the FCK) made possible both the formation of pre-biotic chemical structures and very simple life forms, and also the environments in which they could survive and reproduce. But there's more to life than primitive life forms!
There is a huge variety of types of construction kit, that cannot all be surveyed here. This work is still in its infancy and only very shallow discussions using a small number sub-cases can be offered here.
Construction kits that will not be discussed here but should be in a more complete investigation include internet-based virtual construction kits such as Minecraft (https://minecraft.net/) currently used by millions of people. Other sorts of virtual machinery will be mentioned later.
In some cases, properties of components, such as shape, are inherited by constructed objects. E.g. objects composed only of Lego bricks joined in the "standard" way all have external surfaces that are divisible into faces parallel to the surfaces of the first brick used. However, as Ron Chrisley pointed out to me, when two Lego bricks are joined at a corner only, using only one stud and socket, it is possible to have continuous relative rotation (because studs and sockets are circular).
More generally, constructed objects can have features none of the components have, e.g. a hinge is a non-rigid object that can be made from rigid objects: two rigid objects with aligned holes through which a rod or screw is passed, creating a flexible object from non-flexible parts. A connected structure in a 2-D film cannot have a channel going right through it, whereas a 3-D structure can. There are many such examples of emergent novelty [Anderson, 1972]. I am not aware of any exhaustive taxonomy of ways of producing novel powers, structures and processes by combining old parts in new ways: apart from the implicit taxonomy in life forms.
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. Another option is to recruit something outside the kit, e.g. a gravitational field. Something like a seesaw can be made using gravity (part of the FCK) to keep one piece supporting another that behaves as if hinged at the centre.
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. There are also non-spatial abstract construction kits, whose products do not occupy space-time, 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, or orally. There are also hybrid concrete+abstract construction kits, such as the physical components of a chess set combined with abstract rules of chess, specifying legal moves, and conditions for winning and losing.
In some hybrid construction kits, such as chess, the physical pieces are not essential. For an expert, physical components of a chess set are dispensable: the abstract kit suffices for representing the abstract structures, states and processes, though communication of moves between players needs physical mechanisms, as does a player'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, which have different strengths and weaknesses.
The clay does not, in itself, have the ability to make use of those relationships, but if something else can inspect the clay it may be able to take decisions or answer questions about the things that have been pressed into it, including quite abstract questions, e.g. whether any two of the objects were similar in shape, or how they differ. But we must be careful not to jump to conclusions from uses we can make of physical differences, as may happen when scientists discover changes in brain states correlated with things for which we have labels.
Additional mechanisms are required if available information is to be used: What sort of mechanism will depend on what sort 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). Additional mechanisms are required for recognising text, correcting spelling, analysing the structure of an image, interpreting it as a picture of a 3-D scene, or using information about the scene to guide a robot, or build a copy of the scene.
Different sorts of construction kit are required for producing those mechanisms. In organisms, the kits have different evolutionary histories: for example, mechanisms for finding, understanding, and correcting text evolved long after mechanisms able to use visual information for avoiding obstacles or for grasping objects.
In some cases, the mechanisms that use information seem to be direct products of biological evolution, including blinking as a defense mechanism, and other reflexes. In other cases, the detailed mechanisms are developed by individuals using mechanisms produced by evolution: for example: individual humans in different cultures develop different language-understanding mechanisms, but presumably they use a generic language construction kit shared with other humans. After use of such a kit begins it may be modified in ways that support further learning or development of a specific type of language.
In some species, especially those using sexual reproduction, there may be considerable 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.
A simple case is a thermostat that turns a heater on or off, discussed in McCarthy, . It has two sorts of information: (a) about a target temperature set by a user (desire-like information) and (b) about current ambient temperature, provided by a sensor (belief-like information). The discrepancy between the two information items is used by the thermostat to select between turning a heater on, or off, or leaving it as it is. This is a very simple homeostatic mechanism, using information and a source of energy to maintain a state.
Many biological and human-designed control mechanisms acquire information through transducers and use the information in combination with energy sources, to produce, maintain or avoid various states of affairs. The causal role a physical state or change plays in controlling something else, e.g. controlling deployment of energy, altering direction of growth, selection of mode of analysis of information, among many others, can be described as providing information, in this case control information.
As Gibson,  pointed out, acquisition of information often requires cooperation between processes of sensing and acting. In animal vision, saccades are actions that constantly select new information samples from the environment (e.g. from the optic cone). The use of that information is very different in different contexts, e.g. controlling grasping, controlling preparation for a jump, controlling avoidance actions, or sampling portions of text while reading. A particular sensor can therefore be shared between many control subsystems [Sloman, 1993], and the significance of particular sensor inputs will depend partly on which subsystems are in control of the sensor at the time, partly on which others happen to receive information from the sensor (assuming channels can be turned on or off).
The study of varieties of use of information in organisms is exploding, and now includes many mechanisms on molecular scales within much larger organisms as well as many intermediate levels of informed control, including sub-cellular levels (e.g. metabolism), physiological processes of breathing, temperature maintenance, digestion of food, blood circulation, control of locomotion, feeding and mating of large animals and coordination across 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.9
The earliest evolved machines 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. 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 exemplify the interplay between sensing and doing, in "online intelligence".
But there are cases ignored by Gibson and by researchers opposed to cognitive theories, 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. Offline intelligence 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.
One information-bearing structure (e.g. the impression of a foot, the shape of a rock or even a neural state) can provide very different information to different information-users, depending at least on (a) what kinds of sensors (including internal sensors) they can use to get information from the structure, (b) what sorts of information-processing (storing, analysing, comparing, combining, synthesizing, retrieving, deriving, using...) mechanisms the users have, (c) what sorts of needs or goals they can serve by using various sorts of information (knowingly or not).
So, from the fact that changes in some portion of a brain are strongly correlated 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 - any more than discovering footprints in the sand where animals walk, tells us that a beach perceives animals.
However, this ignores some of the sophistication of evolution. Animals are not all restricted to acting on motives selected on the basis of rewards expected by the individual. They may also have motive construction mechanisms that are simply triggered as "internal reflexes" by certain states of affairs, without having any knowledge or expectations regarding beneficial consequences of achieving those motives, 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 construction of new motives, which may or may not lead to behaviour, depending on how important other competing behaviours are. For example, in a kind person, watching someone fall may trigger a motive to rush to help. But that motive may not generate action if competing motives are too strong.
Moreover, such a motive need not be selected because acting on it will produce some reward for the actor, contrary to the widely held view that all motivation is reward-based. Sloman,  labelled such reflex motive generation as "architecture-based motivation" in contrast with "reward-based motivation" where motives are selected on the basis of anticipated rewards. Behaviours apparently produced by architecture-based motivations can be observed in young children and the young of other playful intelligent animals. When watching such "idle" behaviours it may be tempting to invent hypothetical rewards but the assumption that expected rewards must always play a role in motive generation is just a prejudice. In some cases choosing between motives can take rewards into account, but moral principles or mere habits, may suffice instead.
One of the benefits of certain automatically triggered motives is that acting on them will sometimes produce new information, by sampling properties of the environment. That information may not be immediately usable, but in combination with other episodes of information storage may enable some later processes to analyse and reorganise the stored information. The individual need not have any conception of that later process when the information is acquired, though the ancestors of that individual may have benefited from the presence of the mechanisms of information gathering later used for information reorganisation (labelled "Representational Redescription" in Karmiloff-Smith, ).
During evolution, and in some species 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).
If we don't understand the architecture and the many information-processing functions it supports, and how they are related, and how they grow and diversify, 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. The architectural knowledge about how such a systems works may not be expressible in sets of differential equations, or statistical learning mechanisms and relationships. (For important but partial attempts to characterise some architectural roles in human information-processing see Minsky, [1987,Minsky, [2006,Laird et al, [1987,Sun, . Compare Sloman, . Earlier pioneering work was done by Herbert Simon, in "Motivational and emotional controls of cognition", reprinted in his Models of Thought, Yale University Press, pp. 29--38, 1967.)
The construction kits required for building information-processing architectures, with multiple sensors and multiple motor subsystems developing in complex and varied environments may differ in
This list is not meant to be complete: it merely illustrates the complexity and variety of challenges in understanding the construction kits required for producing theories or models of biological information-processing. Not all biological information-processing systems have all these capabilities. Some are required for all organisms, though their forms can vary. Others, including abilities to reason about hypothetical possibilities, and to modify ontologies used, must have been products of relatively recent evolution.
There is much we still do not know about the construction kits used in these processes, and what they are used for. The Meta-Morphogenesis project aims to investigate the huge variety of uses of information and how they evolved, partly in the expectation there will turn out to be many mechanisms and many uses that we have not noticed, that are essential for understanding, or replicating, the more complex control phenomena in living things, including brain functions.
I suspect that assumptions made by neuroscientists about the information-processing in brains omit some important types, and that AI researchers influenced by those assumptions therefore fail to replicate important functions of brains in their machines.
Progress in this investigation may require major conceptual advances regarding what the problems are and what sorts of answers are relevant. E.g. "Where in the brain are discoveries made?" "Where do emotions occur in the brain?" "Where in the brain is musical ability?" "Where do visual experiences (qualia) occur in the brain?" "Where does understanding occur when you read a sentence?" are all nonsensical questions. (Compare "Where exactly in the USA was the president elected?") But that does not mean there are no mental states and processes, including detection of changes in qualia! (E.g. when internal self-monitoring processes notice changes in visual or other sensory information when nothing perceived in the environment has changed.)
How did the FCK generate complex life forms? Is the Darwin-Wallace theory of natural selection the whole answer? Graham Bell wrote 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, in addition to evolved construction kits and their products.
The idea of evolution producing construction kits is not new, though they are often referred to as "toolkits". Coates et al,  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 external users. Ganti,  explained how chemistry supports self-sufficiency in very simple organisms.
Both generative mechanisms and selection mechanisms change during evolution (partly by influencing each other). 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 whose products are incomprehensible to thinkers familiar only with physics and chemistry, without any first had experience of designing and implementing virtual machinery.
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, such as putting a shirt on a child10.
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 7.2 below.)
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 a construction kit with concrete and abstract aspects, the subject of much research by physicists.
Despite the current (deeply confused) fashion emphasising embodied cognition, many examples of thinking, perceiving, reasoning and planning, 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 by exploring possible route-fragments. Instead an abstract construction kit representing possible options and ways of combining them can be used. Being able to talk requires use of a grammar specifying a 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 use of information. The sentences allowed by a grammar for English can be thought of as abstract objects that can be instantiated in written text, printed text, spoken sounds, morse code, semaphore, and other concrete forms: 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 kinds of complex structures, e.g. grammars for sign languages, circuit diagrams, maps, architectural layouts and even molecules. Human sign languages use different structures from spoken languages.
A grammar does not specify a language: a semantic construction kit, structurally related to the grammar, is required for building possible meanings for the language to express. Use of a language depends on language users, for which more complex construction kits are required, also products of evolution and learning. (Evolution of various types of language is discussed in Sloman, , which argues that internal languages must have evolved first, then sign languages.)
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 is found.
In computing technology, physical computers, programming languages, operating systems and virtual machines form hybrid construction kits that can make things happen when they run. A logical system with axioms and inference rules can be thought of as an abstract kit supporting construction of logical proof-sequences, usually combined with a physical notation for written proofs. A purely logical system cannot have physical causal powers whereas its concrete instances can, e.g. helping a student distinguish valid and invalid proofs.
Natural selection seems to have "discovered" the power of hybrid construction kits, especially the use of sophisticated virtual machinery, long before human engineers did. In particular, biological virtual machines used by animal minds are more powerful than current engineering designs [Sloman, 2010]. All examples of perception, learning, reasoning, and intelligent behaviour are based on hybrid construction kits, though scientific study of such kits is still in its infancy. This discussion merely scratches the surface of a huge multi-disciplinary research area. Work done so far on the Meta-Morphogenesis project project suggests that natural selection "discovered" and used a staggering variety of types of hybrid construction kits that were essential for reproduction, for developmental processes (including physical development and learning), for performing complex behaviours, and for social/cultural phenomena. Jablonka and Lamb,  seem to come close to making this point, though they use different terminology.
Some construction kits allow assembly of such toys. More sophisticated kits include sensors that can be used to provide information for an internal mechanism that uses the information to take decisions concerning deployment of available energy, for instance using light, sonar, or in the case of rats, using whiskers, to gain information that allows frequent changes of direction or speed of motion, e.g. in order to avoid collisions, or in order to move towards a source of electrical or chemical energy when internals supplies are running low. Some examples are provided in Braitenberg, , though he (or at least some of his admirers) unfortunately over-interpreted his vehicles as being capable of love, fear, etc.11
In some cases the distinction between internal and external components is arbitrary. For example, a musical box may perform a tune under the control of a rotating disc with holes or spikes that cause a tone to be produced when they reach a certain location, during the rotation. The disc can be thought of as part of the music box. It can also be thought of as part of a changing environment, in which case the devices that detect the holes or spikes are external sensors.
If a toy train set has rails or tracks used to guide the motion of the train as it moves, then the wheels of the train can be thought of as sensing the environment and causing changes of direction in the train. 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 the information about where to go and the forces required to change direction. The 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 perceived wall provides information, not force, causing deceleration [Sloman, 2011].
The offline/online use of visual information has caused much confusion among researchers, including attempts to interpret the difference in terms of "what" and "where" information.12 Compare Sloman, .
There are hugely varied ways of acquiring and using information, some of which have been discovered (or re-discovered) and modelled by AI researchers, psychologists, neuroscientists, biologists and others, though it seems that evolution has achieved a great deal more, not only in humans, but in other intelligent animals. Many of these achievements require not just additional storage space but very different sorts of information-processing architectures. A range of possible architectures is discussed in Sloman, [1993,Sloman, [2006,Sloman, . Some types use sub-architectures that evolved at different times, meeting different needs, in different biological niches [Sloman, 2000].
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 individuals 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 5. (The BICA society aims to bring together researchers on biologically inspired cognitive architectures.13)
A consequence of the common 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), by applying forces that are transmitted through space, and in other ways. Products of different kits can interact in more complex ways, e.g. one being used to manipulate another, or one providing energy or information for the other.
This contrasts starkly with the problems of getting software components available on a computer to interact: merely co-locating them in the same virtual machine on the same computer will not suffice. There are some rule-based systems 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 special "conflict resolution" mechanisms may be required if the conditions of more than one rule are satisfied.14
Spatial embedding of products allows new construction kits to be formed by combining two or more concrete kits. In some cases this will require modification of a kit, e.g. supporting combinations of 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 the platform. In organisms, products of different construction kits may use complex mixtures of juxtaposition and adaptation. As mentioned in section 2.1, there is evidence that some organisms can also make use of non-local quantum effects when complex mechanisms are made of interacting components, a topic requiring much further research.
Another consequence of the fact that objects exist in space/time is the need for timing mechanisms. Organisms use many "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 of motion or change, and adjustable rates of change. Examples: a bird in flight approaching a perch on which it is to land; an animal running towards a tree to escape a predator and having to decelerate as it approaches the tree to avoid a dangerous crash; 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 a suitable location relative to the selected grasping points on the object. (The last example is still very difficult for robots, when grasping novel objects in novel situations: partly because of designs that use only sensory-motor ontologies.)
There are also mechanisms for controlling or varying rates of production of chemicals (e.g. hormones).
So biological construction kits need many mechanisms with abilities to measure time intervals and to control rates of repetition or rates of change of parts of the organism. These construction kits may be combined with other sorts of construction kit that require temporal as well as spatial control, e.g. changing speed and direction of motion simultaneously. There are different requirements for controlling growth of fixed structures, e.g. trees growing branches, and for mobile animals.
In the realm of digital computation 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 simple cases most people encounter in learning about arithmetic. For example a function could be a compiler, its arguments could be arbitrarily complex programs in a high level programming language, and the output of the function in each case might be either a report on syntactic errors in the input program, or, if there are no errors, a machine code program to run on a particular type of computer.
The application of functions to arguments is a very different process from assembling structures in space time. In the latter case inputs to the process form parts of the output, which need not be the case with a mathematical or computational function. If computers are connected via digital to analog interfaces, linking them to other things, e.g. 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.
That additional richness is present in the modes of interaction of chemical structures which include both digital (on/off chemical bonds) and continuous changes in relationships, as discussed by Turing in his paper on the chemical basis of morphogenesis Turing,  (the paper that inspired the Meta-Morphogenesis project Sloman, [2013b]).
In living organisms, there are interactions between products of the same or different kits that involve information, e.g. use of information for sensing, predicting, explaining or controlling, including information about information Sloman, .
Researchers on systems combining many kinds of functionality have found it useful to design information-processing architectures that provide frameworks for combining different mechanisms and information stores. This is particularly important in large projects where different research groups are working on sensors, learning mechanisms, motor subsystems, reasoning systems, motivational systems, various kinds of meta-cognition, etc., with associated sets of tools supporting processes of design, implementation, testing, debugging. Our own SimAgent toolkit Sloman, [1996c], mentioned in Note 14 is one among very many.
Some of the common principles include the need to be able to support different sorts of virtual machines with causal interactions between them and the physical environment (including perception and physical actions), as explained in this tutorial overview: Sloman (2013).
In addition to design patterns for physical mechanisms, biological evolution also discovered re-usable frameworks for assembling complex information-processing architectures, accommodating multiple interacting virtual machines, with different modifications developed by different species -- including humans [Minsky, 1987,Minsky, 2006]. This is a topic for further research, which will provide new insights into complex mental states and processes, including forms of self-consciousness, varieties of affective states, and processes of cognitive development that help to explain mathematical development.15
Adding a new DCK can make some possible further developments quicker to reach - fewer additional steps are required than were originally required, and the total search space for a suitable sequence of steps to a solution may be considerably reduced. This is partly analogous to the role of previously proved theorems in a new proof. Using previous results can considerably shorten a proof, make it more comprehensible, and have a dramatic effect on the size of the search-space when searching for a proof. If the number of steps to a solution has been reduced by 10 and there are two options at every step, the search for a complete design may have been reduced by a factor of 210, i.e. 1024: reducing the remaining evolutionary search space required by a factor over a thousandfold - if a solution exists in the remaining search space. Evolutionary search spaces are very much larger, and in principle re-use of designs could have an even larger impact on search spaces. So, the ability to re-use modified versions of useful designs could dramatically reduce an evolutionary search space - if there is a solution in the remaining search space.
Creation of new construction kits may start by simply recording parts of successful assemblies, so that they can easily be reproduced. At later stages previous stores may be combined to form an appropriate "meta-construction kit" able to extend or modify or combine previously created construction kits. 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 and their products!
Some new construction kits are products of the process of evolution of a species and are shared between all members of the species (barring genetic abnormalities), alongside construction kits shared between species, such as those used in mechanisms of reproduction and growth in related species. But evolution has also discovered the benefits of what might be called "meta-construction-kits", namely mechanisms provided for members of a species that allow individuals to build new construction kits during their own development.
Examples include mechanisms for learning that are 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, having experienced 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, , summarised below in Fig. EVO-DEVO, below. Mathematical development in humans seems to be a special case of growth of meta-configured competences.
The 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. As far as I can tell, 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.
Turing's (1952) examples of diffusing chemicals causing patterns when they interact include only formation of superficial 2-D patterns. Explaining the different ways in which features of a genome can directly or indirectly orchestrate many parallel processes of growth, development, formation of connections, etc. is a far greater challenge.
A possible framework for allowing abstract specifications in the genome to interact with details of the environment in instantiating complex designs is illustrated schematically in Fig. EVO-DEVO. This generalises Waddington's "epigenetic landscape" metaphor Waddington, , by allowing individual members of a species to partially construct their own epigenetic landscapes instead of merely following paths in a landscape that is common to the species. Related ideas are in Karmiloff-Smith, .
Some of the implications of these ideas for attempts to understand genetic abnormalities such as autism are discussed in http://www.cs.bham.ac.uk/research/projects/cogaff/misc/autism.html
Features of a physical construction kit -- including the shapes and materials of the basic components, the ways in which the parts can be assembled into larger wholes, the kinds of relationships between parts and the processes that can occur involving them -- all contribute to explaining the possibility of entities that can be constructed from those components, and the possibility of processes, including both the processes of construction and the behaviours of the constructs.
Construction kits can also explain necessity and impossibility. A construction kit that has a very large set of generative powers initially 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. Some of the extensions that were possible before the last addition become impossible unless the last step is undone.
Interactions between structure and remaining possibilities:
Moreover, what has been done may make some further steps possible and others impossible: e.g. the size of a gap between two rigidly assembled components will make it impossible to extend the structure by placing some components in the gap: A beam of 20cm square cross section cannot fit in a 10cm gap. Narrower beams can fit in the gap, but the angles by which their orientations can vary will depend on their diameter, the diameter of the gap, and other spatial relations. the narrower or shorter a beam in the gap, is the wider the angle through which it can rotate in a plane through the gap. The wider the gap is the wider the angle through which a beam of a certain width can rotate, while the longer the gap is the narrower the angle of rotation possible in that plane. Examples are in Figure GAPS. Both human engineers and evolution can make use of similar, though usually more complex, mathematical relationships, in skeletal geometry for example.
Figure Triangle illustrates a different sort of example, where no physical properties of a structure (e.g. rigidity or impenetrability of materials) are involved, only spatial relationships. It presents a proof, found by Mary Pardoe, that internal angles of a triangle sum to a straight line, or 180 degrees.
Mary Pardoe's proof of the triangle sum theorem.
Unlike the "standard" proofs, this proof makes no explicit reference to Euclid's parallel axiom. The human mathematical ability 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 certain constraints 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, though that may change.
These are simple examples of the 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, and they make other structures impossible - and their absence or opposites, e.g. geometrical or topological properties, will then be necessary consequences of previous selection steps.
These examples illustrate how a construction kit with mathematical relationships can provide the basis for necessary truths and necessary falsehoods in some constructions (as in Sloman, [1962,Chap 7]). Such relationships between possibilities provide a deeper, more natural, basis for understanding modality (necessity, possibility, impossibility) than so called "possible world semantics". Being able to think about and reason about alterations in some limited portion of the environment is very common and a requirement for intelligent action [Sloman, 1996a]. In contrast being able to think about the whole world, past, present and future, and the set of alternative complete worlds is a far more demanding requirement. Moreover it is not clear how we could decide whether an individual language user has that capability.
Since our examples of making things possible or impossible, or changing ranges of possibilities, are examples of 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 6.3.16
Varieties of causation that do not involve mathematical necessity, only probabilities (Hume?) or propensities (Popper) will not be discussed here.
Moreover, there was not just one sequence of DCKs: 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.17
So 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. Likewise adding previously proved theorems to a set of axioms, for use as starting points for new proofs will reduce the search space for proofs of related theorems.
A different sort of geometry allows line segments to be translated and rotated in a plane while preserving their length. This is an assumption underlying the use of rulers for measuring length. Adding movable and rotatable line segments to Euclidean geometry allows an arbitrary angle to be divided into three equal parts, which is not possible in standard Euclidean geometry. See http://www.cs.bham.ac.uk/research/projects/cogaff/misc/trisect.html A related construction is possible using "Origami geometry".
The idea of spaces of possibilities generated by different sorts of physical construction kit may be easier for most people to understand than the comparison with generative powers of grammars or formal systems, though the two are closely connected, since grammars and axiom systems are both 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 can be used to form sequences that correspond to the words in a lexicon. Adding some blank squares and specifying rules of a grammar based on that lexicon, produces a new grammar that can be applied to sequences of squares, with blanks as word-separators, generating a set of possible physical sentences conforming to the grammar. The use of cursive ("joined up") script provides a more complex physical construction kit.
Some challenges for construction kits used by evolution, and also challenges for artificial intelligence and philosophy, arise from the need to explain both 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 how various subsets of those organisms developed specific topological and geometrical reasoning abilities used in controlling actions and solving problems, and finally 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 (Note 1).
There are important aspects of those mathematical competences that as far as I know have not yet been replicated in Artificial Intelligence or Robotics18. 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?
Many mathematical domains (perhaps all of them) can be thought of as sets of possibilities generated by construction kits of various kinds. 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 is not a domain that has been explored systematically by mathematicians, though many special cases have.
In order to understand how the sorts of biological evolution that occurred on this planet are possible 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 interesting research question is whether any 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. external influences such as cosmic radiation.
Insofar as construction kits have mathematical properties, life and mathematics are closely interconnected, as we have already seen. More complex relationships arise after evolution of mathematical meta-cognitive mechanisms.
Proofs are often thought of as abstract entities that can be represented physically in different ways (e.g. using different formalisms) for the purpose of communication or persuasion (including self-persuasion), predicting, explaining and planning. It can also be argued that 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 sort of (unwitting) proof that some construction kit makes that sort of entity possible. The evolutionary or developmental trail answers the question: how is that sort of thing possible? In that sense biological evolution can be construed as a "blind theorem prover", despite there being no intention behind 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, . This answers questions like "How is it possible for things that make mathematical discoveries to exist?", an example of explaining a possibility (See Note 5).
Those who try to go too directly from hypothesized properties of the primordial construction kit to explaining advanced capabilities such as human self-awareness (e.g. Schrödinger, [1944,Penrose, ) are likely to fail, because short-cuts will omit essential details of both the problems and the solutions, like mathematical proofs with gaps.
The success of 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" (discussed in more detail on the Meta-Morphogenesis web site19) started happening long before there were any humans doing mathematics (which refutes Wittgenstein's suggestion that mathematics is an anthropological phenomenon). 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 important discovery that something is possible does not require collection of statistics: A single instance suffices. And no amount of statistical evidence can show that something is impossible.
For human evolution, a particularly important subclass of mathematical discoveries has been unwitting discovery and use of mathematical 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 before uses of language for communication have developed. Examples are discussed in the "Toddler Theorems" document (Note 15).
There is also variety in construction kits produced by cultures or ecosystems, illustrated by human languages, applied sciences as in bioengineering, notations for logic, the theory of computation and computer systems engineering. All 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 young child to acquire competence in use of any one of the thousands of different human languages (whether spoken or signed) in the first few years of life? There is evidence that children do not merely learn an existing language: 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 many possible human languages that might have developed but have not (yet).
Evolutionary trajectories leading to human spoken language capabilities may have gone from internal languages through collaborative actions then signed communication, then spoken communication, as argued in Sloman, .
If language acquisition were solely, or mainly, a matter of learning from language users, human languages could never have existed, since initially there were no expert users to learn from, and the process could not get started. This argument applies to many competences that might be thought to be based entirely on learning from experts, including mathematical expertise. So AI systems based on data-mining in samples of expert behaviours will never produce AI systems with human competences -- only subsets at best.
The history of computing since the earliest calculators demonstrates some of the kinds of change that can arise when new construction kits are developed. The technological changes were not merely changes of size, speed and memory capacity: there have been profound qualitative changes, in part because development of new layers of virtual machinery produced new types of mechanism, including new sorts of mutually interacting causal loops linking virtual machine control states with portions of external environments, for instance use of GPS in navigation using a "satnav" device.
Some of the new powers, states and processes include semantic contents referring to non-physical structures and processes, e.g. mathematical problems, rules of games, and mental contents including past or possible future mental contents and contents of other minds. So the machines cannot be fully described in the language of the FCK even though they are fully implemented in physical reality. (See note on ontologies.)
We now understand some of the key components and modes of composition providing platforms on which human-designed layers of computation can be constructed, including subsystems closely but not rigidly coupled to the environment (e.g. using video cameras and propulsion by propellers, when coping with a cross-wind).
Several different sorts of "basic" abstract construction kits suffice to generate the forms of (discrete) computation so far studied. Those basic types include Turing machines, Post's production systems, Church's Lambda Calculus, and several more, each capable of implementing the others. 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.20
One interpretation of the Church-Turing thesis is that these construction kits generate all possible forms of information-processing -- a claim I question. It is not obvious that those discrete mechanisms suffice for all biological forms of information-processing. In contrast, use of a wholly or partly chemical basis allows forms of computation that include both discrete and continuous mechanisms that were essential for some forms of biological assembly and information-processing. In some cases the assembly processes (including continuous changes such as folding, twisting, coming together, moving apart), seem to be self-controlling because partial structures constrain later possibilities. But the ability to form and release chemical bonds also provides discrete control. Ganti,  shows how a chemical construction-kit supports forms of biological information-processing that don't depend only on external energy sources (a fact that's also true of battery-powered computers), and also supports growth and reproduction using internal mechanisms, which human-made computers cannot do (yet).
There may be many different sorts of construction-kit that allow different sorts of information-processing (computation) 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.21
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 explain the possibility of 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.
Schroedinger (1944) 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.22
Penrose,  attempted to show how features of quantum physics explain obscure features of human consciousness, especially mathematical consciousness, but ignores 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, for instance abilities to see various types of affordance. The use of diagrams and spatial models by Penrose could be an example of that. It seems to be 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 layers of perceptual, planning, and reasoning competences produced by billions of years of evolution. (I have not yet fully understood Penrose's claims, however.)
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.23 However, the task can never be finished, since the process of construction of new derived construction kits may continue indefinitely, producing new kits with components and modes of composition that allow production of more complex types of structure and more complex forms of behaviour in organisms.
That idea is familiar to computer scientists and computer systems engineers since thousands of new sorts of computational construction kit (new programming languages, new operating systems, new virtual machines) 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 machinery in extending what information-processing systems can do, and the properties they can have.
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.
Later still, construction kits used by evolution produced meta-cognitive mechanisms enabling individuals to notice and reflect on their own mathematical discoveries (enabling some of them to notice and remove flaws in their reasoning). In some cases those meta-cognitive capabilities allowed individuals to communicate their 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 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. For a draft discussion of evolution of mathematical mechanisms, see Note 24.
Many of these naturally occurring mathematical abilities have not yet been replicated in Artificial Intelligence systems or robots, unlike logical, arithmetical, and algebraic competences. Examples of topological reasoning about equivalence classes of closed curves not yet modelled in computers (as far as I know) are referenced in Note 21. Even the ability to reason about alternative ways of putting a shirt on a child (Note 10) 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 a mixture! 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. Compare Penrose, .)
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) differential 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 normally 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.
Despite all the sophistication of modern psychology and neuroscience, I don't believe they currently have 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 human mechanisms that led to: mathematical discoveries over thousands of years, including mechanisms producing new 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 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.
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.
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 buffeting) 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 mentioned above, such as Schrödinger and Penrose, 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, 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.
One of the issues some physicists have discussed is whether the formation of life from non-living matter requires violation of the second law of thermodynamics, because evolution increases the amount of order or structure in the physical matter on the planet. The standard answer is that the second law of thermodynamics 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. Some of the ways in which pre-existing dispositions can harness external sources of energy to increase local structure are discussed in a short collection of thoughts on entropy, evolution, and construction-kits: http://www.cs.bham.ac.uk/research/projects/cogaff/misc/entropy-evolution.html
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,  is also relevant: see Section 11, 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 may 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, including 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. 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 decision making.
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 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. 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 supports 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. There may be several different routes to the same end result.
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.25
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. If each move can be represented only by start and end states, as in 3, that will further reduce the space of alternatives.
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 5 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
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, 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, .
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 been 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.
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 Life26, 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 (from keyboards, buttons, joy-sticks, mice, electronic piano keyboards, or network connections) 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  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. 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.
(It is perhaps worth mentioning that one of the differences between 2-D and 3-D structures is that a connected 3-D structure can have an interior space and an exterior space with two or more distinct routes joining them (essential for an organism to consume food through one opening and excrete through a separate one), whereas in a 2-D space any structure with two holes (or a through-route) would not be fully connected. This severely limits the possibilities for 2-D life forms.)
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). 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".27
A recently published book Wagner,  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.28 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.
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.
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.
Connections between Meta-Morphogenesis and entropy are discussed in a related
work-in-progress document on issues involving entropy
changes in evolutionary trajectories:
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. A long-term 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. The idea is 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?
Here's a cartoon teasing particle physicists: http://www.smbc-comics.com/?id=3554
"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."
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, 
4 Extended in http://www.cs.bham.ac.uk/research/projects/cogaff/misc/meta-morphogenesis.html
The role of entropy is discussed briefly in:
Some examples are here:
The Birmingham SimAgent toolkit is an example
As discussed in connection with
"toddler theorems" in
(Contributions from observant parents and child-minders are welcome. Deep insights come from individual developmental trajectories rather than statistical patterns of development across individuals.)
For 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):
Some of them listed in
For more on this see:
Examples of human mathematical reasoning in geometry and
topology that have, until now, resisted replication on computers are presented
Illustrated in these discussion notes:
One of many online explanations is
An interview with the author (Wagner)is online