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Abstract

Background note in separate document, 1 Mar 2015

Contents

1  Background: What is science? Beyond Popper and Lakatos

The need for science to include theories that explain how something is possible has not been widely acknowledged. Explaining how X is possible (e.g. how humans playing chess can produce a certain board configuration) need not provide a basis for predicting when X will be realised, so the theory used cannot be falsified by non-occurrence. Popper, [1934] labelled such theories "non-scientific" - at best metaphysics. His falsifiability criterion has been blindly followed by many scientists who ignore the history of science. E.g. the ancient atomic theory of matter was not falsifiable, but was an early example of a deep scientific theory. Later, Popper shifted his ground, e.g. in [Popper 1978], and expressed great admiration for Darwin's theory of Natural Selection, despite its unfalsifiability.

Lakatos [1980] extended Popper's philosophy of science, showing how to evaluate competing scientific research programmes over time, according to their progress. He offered criteria for distinguishing "progressive" from "degenerating" research programmes, on the basis of their patterns of development, e.g. whether they systematically generate questions that lead to new empirical discoveries, and new applications. It is not clear to me whether he understood that his distinction could also be applied to theories explaining how something is possible.

Chapter 2 of [Sloman 1978]³ modified the ideas of Popper and Lakatos to accommodate scientific theories about what is possible, e.g. types of plant, types of animal, types of reproduction, types of consciousness, types of thinking, types of learning, types of communication, types of molecule, types of chemical interaction, and types of biological information-processing. It presented criteria for evaluating theories of what is possible and how they are possible, including theories that straddle science and metaphysics. Insisting on sharp boundaries between science and metaphysics harms both. Each can be pursued with rigour and openness to specific kinds of criticism. A separate paper⁴ includes a section entitled "Why allowing non-falsifiable theories doesn't make science soft and mushy", and discusses the general concept of "explaining possibilities", its importance in science, the criteria for evaluating such explanations, and how this notion conflicts with the falsifiability requirement for scientific theories. Further examples are in [Sloman 1996 1]. The extremely ambitious Turing-inspired Meta-Morphogenesis project, proposed in [Sloman 2013 2]⁵ depends on these ideas, and will be a test of their fruitfulness, in a combination of metaphysics and science.

This paper, straddling science and metaphysics asks: How is it possible for natural selection, starting on a lifeless planet, to produce billions of enormously varied organisms, living in environments of many kinds, including mathematicians able to discover and prove geometrical and topological theorems?

An outline answer is presented in terms of construction kits: the Fundamental (physical) Construction Kit (the FCK), and a variety of "concrete", "abstract" and "hybrid" Derived Construction Kits (DCKs). that together are conjectured to explain how evolution is possible, including evolution of mathematicians. The FCK and its relations to DCKs are crudely depicted below in Figures FCK and DCK. Inspired by ideas in [Kant 1781], construction kits are also offered as providing Biological/Evolutionary foundations for core parts of mathematics, including mathematical facts used by evolution long before there were human mathematicians. At a later stage evolution also produced meta-construction kits: construction kits that are able to create, modify or combine construction kits.

Note on "Making Possible": "X makes Y possible" as used here does not imply that if X does not exist then Y is impossible, only that one route to existence of Y is via X. Other things can also make Y possible, e.g., an alternative construction kit. So "makes possible" is a relation of sufficiency, not necessity. The exception is the case where X is the FCK - the Fundamental Construction Kit - since all concrete constructions must start from it (in this universe?). If Y is abstract, there need not be something like the FCK from which it must be derived. The space of abstract construction kits may not have a fixed "root". However, the abstract construction kits that can be thought about by physically implemented thinkers may be constrained by a future replacement for the Church-Turing thesis, based on later versions of ideas presented here.

Although my questions about explaining possibilities arise in the overlap between philosophy and science [Sloman 1978,Ch.2], I am not aware of any philosophical work that explicitly addresses the theses discussed here, though there seem to be examples of potential overlap, e.g. [Bennett 2011,Wilson 2015].

2  Fundamental and Derived Construction Kits (FCK, DCKs)

What makes all of this possible is the construction kit provided by fundamental physics, the Fundamental Construction Kit (FCK) about which we still have much to learn, even if modern physics has got beyond the stage lampooned in this SMBC cartoon:

As hinted by the cartoon, there is not yet agreement among physicists as to what exactly the FCK is, or what it can do. Perhaps important new insights into properties of the FCK will be among the long term outcomes of our attempts to show how the FCK can support all the DCKs required for developments across billions of years, and across no-one knows how many layers of complexity, to produce animals as intelligent as elephants, crows, squirrels, or even humans (or their successors).

Construction-kits are the "hidden heroes" of evolution. Life as we know it requires construction kits supporting construction of machines with many capabilities, including growing many types of material, many types of mechanism, many types of highly functional bodies, immune systems, digestive systems, repair mechanisms, reproductive machinery, information processing machinery (including digital, analogue and virtual machinery) and even producing mathematicians, such as Euclid and his predecessors!

A kit needs more than basic materials. If all the atoms required for making a loaf of bread could somehow be put into a container, no loaf could emerge. Not even the best bread making machine, with paddle and heater, could produce bread from atoms, since that requires atoms pre-assembled into the right amounts of flour, sugar, yeast, water, etc. Only different, separate, histories can produce the molecules and multi-molecule components, e.g. grains of yeast or flour.

Likewise, no modern fish, reptile, bird, or mammal could be created simply by bringing together enough atoms of all the required sorts; and no machine, not even an intelligent human designer, could assemble a functioning airliner, computer, or skyscraper directly from the required atoms. Why not, and what are the alternatives? We first state the problem of constructing very complex working machines in very general terms and indicate some of the variety of strategies produced by evolution, followed later by conjectured features of a very complex, but still incomplete, explanatory story.

2.1  Combinatorics of construction processes

The mechanisms involved in construction of an organism can be thought of as a construction kit, or collection of construction kits. Some components of the kit are parts of the organism and are used during the life of the mechanism, e.g. for growth and repair. Construction kits used for building information-processing mechanisms may continue being used and extended long after birth as discussed in the section on epigenesis below. All of the construction kits must ultimately come from the Fundamental Construction kit (FCK) provided by physics and chemistry.

If there are N types of basic component and a task requires an object of type O composed of K basic components, the size of a blind exhaustive search for a sequence of types of basic components to assemble an O is up to N^K sequences, a number that rapidly grows astronomically large as K increases. If, instead of starting from the N types of basic components, construction uses M types of pre-assembled components, each containing P basic components, then an O will require only K/P pre-assembled parts. The search space for a route to O is reduced in size to M^(K/P).

Compare assembling an essay of length 10,000 characters (a) by systematically trying elements of a set of about 30 possible characters (including punctuation and spaces) with (b) choosing from a set of 1000 useful words and phrases, of average length 50 characters. In the first case each choice has 30 options but 10,000 choices are required. In the second case there are 1000 options per choice, but far fewer stages: 200 instead of 10,000 stages. So the size of the (exhaustive) search space is reduced from 30¹⁰⁰⁰⁰, a number with 14,773 digits, to about 1000²⁰⁰, a number with only 602 digits: a very much smaller number. So trying only good pre-built substructures at each stage of a construction process, can make a huge reduction to the search space for solutions of a given size - possibly eliminating some solutions.

So, learning from experience by storing useful subsequences can achieve dramatic reductions, analogous to a house designer moving from thinking about how to assemble atoms, to thinking about assembling molecules, then bricks, planks, tiles, then pre-manufactured house sections. The reduced search space contains fewer samples from the original possibilities, but the original space is likely to have a much larger proportion of useless options. As sizes of pre-designed components increase so does the variety of pre-designed options to choose from at each step, though far, far, fewer search steps are required for a working solution, if one exists: a very much shorter evolutionary process. The cost may be exclusion of some design options.

This indicates intuitively, but very crudely, how using increasingly large, already tested useful part-solutions can enormously reduce the search for viable solutions. The technique is familiar to many programmers, for example in the use of "memo-functions" ("memoization") to reduce computation time, e.g. computing fibonacci numbers. Storing partial solutions to sub-problems while trying to solve a complex problem may allow later sub-problems to be solved more quickly: a technique known as "structure sharing", a technique that is often re-invented by AI researchers. The family of computational search techniques known as "Genetic Programming"⁶ uses related ideas. The use of "crossover" in evolution (and in Genetic Algorithms), is partly analogous insofar as it allows parts of each parent's design specification to be used in new combinations.

In biological evolution, instead of previous solutions being stored for future re-use, information about how to build components of previous solutions, is stored in genomes, insofar as information structures specifying designs (e.g. blue-prints for complex machines) are much cheaper and quicker to store, copy, modify and re-use than the physical instances of those designs (the machines themselves). Evolution, the Great Blind Mathematician discovered memoization and the usefulness of exploration in a space of specifications for working systems rather than the systems themselves, long before we did. A closely related strategy is to record fragments that cannot be useful in certain types of problem, to prevent wasteful attempts to use such fragments. Expert mathematicians learn from experience which options are useless (e.g. dividing by zero). This could be described as "negative-memoization". Are innate aversions examples of evolution doing something like that? In principle it might be possible for a genome to include warnings about not attempting certain combinations of design fragments, but I know of no evidence that that happens.

Without prior information about useful components and combinations of pre-built components, random assembly processes can be used. If mechanisms are available for recording larger structures that have been found to be useful or useless, the search space for new designs can be shrunk. By doing the searching and experimentation using information about how to build things rather than directly recombining the built physical structures themselves, evolution reduces the problem of recording what has been learnt.

Figure FCK
A crude representation of the Fundamental Construction Kit (FCK) (on left) and (on right) a collection of trajectories from the FCK through the space of possible trajectories to increasingly complex mechanisms.

The Fundamental Construction Kit (FCK) provided by the physical universe made possible all the forms of life that have so far evolved on earth, and also possible but still unrealised forms of life, in possible types of physical environment. Fig. 1 shows how a common initial construction kit can generate many possible trajectories, in which components of the kit are assembled to produce new instances (living or non-living). The space of possible trajectories for combining basic constituents is enormous, but routes can be shortened and search spaces shrunk by building derived construction kits (DCKs), that assemble larger structures in fewer steps⁷, as indicated in Fig. 2.

Further transitions: a fundamental construction kit (FCK) on left gives rise to new evolved "derived" construction kits, such as the DCK on the right, from which new trajectories can begin, rapidly producing new more complex designs, e.g. organisms with new morphologies and new information-processing mechanisms. The shapes and colours (crudely) indicate qualitative differences between components of old and new construction kits, and related trajectories. A DCK trajectory uses larger components and is therefore much shorter than the equivalent FCK trajectory.

Figure DCK
Further transitions: a fundamental construction kit (FCK) on left gives rise to new evolved "derived" construction kits, such as the DCK on the right, from which new trajectories can begin, rapidly producing new more complex designs, e.g. organisms with new morphologies and new information-processing mechanisms. The shapes and colours (crudely) indicate qualitative differences between components of old and new construction kits, and related trajectories. A DCK trajectory uses larger components and is therefore much shorter than the equivalent FCK trajectory.

The history of technology, science and engineering includes many transitions in which new construction kits are derived from old ones. That includes the science and technology of digital computation, where new advances used an enormous variety of discoveries and inventions, from punched cards (used in Jacquard looms) through many types of electronic device, many types of programming language, many types of external interface (not available on Turing machines!), many types of operating system, many types of network connection, many types of virtual machine, and many applications. Particular inventions were generalised, using mathematical abstractions, to patterns that could be reused in new contexts. The production of new applications frequently involved production of new tools for building more complex applications.

Natural selection did all this on an even larger scale, with far more variety, probably discovering many obscure problems and solutions still unknown to us. (An educational moral: teaching only what has been found most useful can discard future routes to possible major new advances - like depleting a gene pool.)

Biological construction kits derived from the FCK can combine to form new Derived Construction Kits (DCKs), some specified in genomes, and (very much later) some discovered or designed by individuals (e.g. during epigenesis Sect. 2.3), or by groups, for example new languages. Compared with derivation from the FCK, the rough calculations above show how DCKs can enormously speed up searching for new complex entities with new properties and behaviours. See Fig. 2.

New DCKs that evolve in different species in different locations, may have overlapping functionality, based on different mechanisms: a form of convergent evolution. E.g., mechanisms enabling elephants to learn to use trunk, eyes, and brain to manipulate food may share features with those enabling primates to learn to use hands, eyes, and brains to manipulate food. In both cases competences evolve in response to structurally similar affordances in the environment. This extends ideas in [Gibson 1979] to include affordances for a species, or collection of species.⁸

2.2  Construction Kit Ontologies

This relates to claims that have been made about requirements for control systems and for scientific theories. For example, if a system is to be capable of distinguishing N different situations and responding differently to them it must be capable of being in at least N different states (recognition+control states). This is a variant of Ashby's "Law of Requisite Variety" [Ashby 1956]. Several thinkers have discussed representational requirements for scientific theories, or for specifications of designs. [Chomsky 1965] distinguished requirements for theories of language, which he labelled observational adequacy (covering the variety of observed uses of a particular language), descriptive adequacy (covering the intuitively understood principles that account for the scope of a particular language) and explanatory adequacy (providing a basis for explaining how any language can be acquired on the basis of data available to the learner). These labels were vaguely echoed in [McCarthy  Hayes 1969] who described a form of representation as being metaphysically adequate if it can express anything that can be the case, epistemologically adequate if it can express anything that can be known by humans and future robots, and heuristically adequate if it supports efficient modes of reasoning and problem-solving. (I have simplified all these proposals.)

Requirements can also be specified for powers of various sorts of biological construction kits. The fundamental construction kit (FCK) must have the power to make any form of life that ever existed or will exist possible: if necessary using huge search spaces. DCKs may meet different requirements, e.g. each supporting fewer types of life form, but enabling those life forms to be "discovered" in a reasonable time by natural selection, and reproduced (relatively) rapidly. Early DCKs may support the simplest organisms that reproduce by making copies of themselves [Ganti 2003]. At later stages of evolution, DCKs are needed that allow construction of organisms that change their properties during development and change their control mechanisms appropriately as they grow [Thompson 1917]. This requires the ability to produce individuals whose features are parametrised with parameters that change over time. More sophisticated DCKs must be able to produce species that modify their knowledge and their behaviours not merely as required to accommodate their own growth but also to cope with changing physical environments, new predators, new prey and new shared knowledge. A special case of this is having genetic mechanisms able to support development of a wide enough range of linguistic competences to match any type of human language, developed in any social or geographical context. However, the phenomenon is far more general than language development, as discussed in the next section.

2.3  Construction kits built during development (epigenesis)

Examples include mechanisms for learning that are initially generic mechanisms shared across individuals, and developed by individuals on the basis of their own previously encountered learning experiences, which may be different in different environments for members of the same species. Human language learning is a striking example: things learnt at earlier stages make new things learnable that might not be learnable by an individual transferred from a different environment, part way through learning a different language. This contrast between genetically specified and individually built capabilities for learning and development was labelled a difference between "pre-configured" and "meta-configured" competences in [Chappell  Sloman 2007], summarised in Fig. 3. The meta-configured competences are partly specified in the genome but that specification is combined with information abstracted from individual experiences. Mathematical development in humans seems to be a special case of growth of such meta-configured competences. Related ideas are in [Karmiloff-Smith 1992].

Figure 3: Figure EVO-DEVO:
A construction kit gives rise to very different individuals if the genome interacts with the environment in increasingly complex ways during development, allowing enormously varied developmental trajectories based on the same genome. Precocial species use only the downward routes on the left, producing only "preconfigured" competences. Competences of altricial species, using staggered development, may be far more varied within a species. Results of using earlier competences interact with the genome, producing "meta-configured" competences shown on the right. This is a modified version of a figure in [Chappell  Sloman 2007].

Construction kits used for assembly of new organisms that start as a seed or an egg enable many different processes in which components are assembled in parallel, using abilities of the different sub-processes to constrain one another. Nobody knows the full variety of ways in which parallel construction processes can exercise mutual control in developing organisms. One implication is that there are not simple correlations between genes and organism features.

Explaining the many ways in which a genome can orchestrate parallel processes of growth, development, formation of connections, etc. is a huge challenge. A framework allowing abstract specifications in a genome to interact with details of the environment in instantiating complex designs is illustrated schematically in Fig. 3. An example might be the proposal in [Popper 1976] that newly evolved desires of individual organisms (e.g. desires to reach fruit in taller trees) could indirectly and gradually, across generations, influence selection of physical characteristics (e.g. longer necks, abilities to jump higher) that improve success-rates of actions triggered by those desires. Various kinds of creativity, including mathematical creativity, might result from such transitions. This generalises Waddington's "epigenetic landscape" metaphor [Waddington 1957], by allowing individual members of a species to partially construct and repeatedly modify their own epigenetic landscapes instead of merely following paths in a landscape that is common to the species. Mechanisms that increase developmental variability may also make new developmental defects possible (e.g. autism?)⁹.

2.4  The variety of biological construction kits

What forms can such information take? Many controlled systems have states that can be represented by a fixed set of physical measures, often referred to as "variables", representing states of sensors, output signals, and internal states of various sorts. Relationships between such state-components are often represented mathematically by equations, including differential equations, and constraints (e.g. inequalities) specifying restricted, possibly time-varying, ranges of values for the variables, or magnitude relations between the variables. A system with N variables (including derivatives) has a state of a fixed dimension, N. The only way to record new information in such systems is in static or dynamic values for numeric variables - changing "state vectors" and possibly alterations in the equations. A typical example is [Powers 1973], inspired by [Wiener 1961] and [Ashby 1952]. There are many well understood special cases, such as simple forms of homeostatic control using negative feedback. Neural net based controllers often use large numbers of variables clustered into strongly interacting sub-groups, groups of groups, etc.

For many structures and processes, a set of numerical values and rates of change linked by equations (including differential equations) expressing their changing relationships is an adequate form of representation, but not for all, as implied by the discussion of types of adequacy in Section 2.2. That's why chemists use structural formulae, e.g. diagrams showing different sorts of bonds between atoms and collections of diagrams showing how bonds change in chemical reactions. Linguists, programmers, computer scientists, architects, structural engineers, map-makers, map-users, mathematicians studying geometry and topology, composers, and many others, work in domains where structural diagrams, logical expressions, grammars, programming languages, plan formalisms, and other non-numerical notations express information about structures and processes that is not usefully expressed in terms of collections of numbers and equations linking numbers.¹⁰

Of course, any information that can be expressed in 2-D written or printed notation such as grammatical rules, parse trees, logical proofs and computer programs, can also be converted into a large array of numbers by taking a photograph and digitising it. Although such processes are useful for storing or transmitting documents, they add so much irrelevant numerical detail that the original functions, such as use in checking whether an inference is valid, or manipulating a grammatical structure by transforming an active sentence to a passive one, or determining whether two sentences have the same grammatical subject, or removing a bug from a program, or checking whether a geometric construction proves a theorem, become inaccessible until the original non-numerical structures are extracted.

Similarly, collections of numerical values will not always adequately represent information that is biologically useful for animal decision making, problem solving, motive formation, learning, etc. Moreover, biological sensors are poor at acquiring or representing very precise information, and neural states often lack reliability and stability. (Such flaws can be partly compensated for by using many neurons per numerical value and averaging.) More importantly, the biological functions, e.g. of visual systems, may have little use for absolute measures, if their functions are based on relational information, such as that A is closer to B than to C, A is biting B, A is keeping B and C apart, A can fit through the gap between B and C, the joint between A and B is non-rigid, A cannot enter B unless it is reoriented, and many more. As [Schrödinger 1944] pointed out, topological structures of molecules can reliably encode a wide variety of types of genetic information, and may also turn out to be useful for recording other forms of structural information. Do brains employ them? There are problems about how such information can be acquired, derived, stored, used, etc. [Chomsky 1965] pointed out that using inappropriate structures in models may divert attention from important biological phenomena that need to be explained-see Sect. 2.2, above. Max Clowes, who introduced me to AI in 1969 made similar points about research in vision around that time.¹¹ So subtasks for this project include identifying biologically important types of non-numerical (e.g. relational) information content and ways in which such information can be stored, transmitted, manipulated, and used. We also need to explain how mechanisms performing such tasks can be built from the FCK, using appropriate DCKs.

2.5  Increasingly varied mathematical structures

Organisms also need multiple control systems, not all numerical. A partially constructed percept, thought, question, plan or terrain description has parts and relationships, to which new components and relationships can be added and others removed, as construction proceeds, and errors are corrected, building structures with changing complexity - unlike a fixed-size collection of variables assigned changing values. Non-numerical types of mathematics are needed for describing or explaining such systems, including topology, geometry, graph theory, set theory, logic, formal grammars, and theory of computation. A full understanding of mechanisms and processes of evolution and development may need new branches of mathematics, including mathematics of non-numerical structural processes, such as chemical change, or changing "grammars" for internal records of complex structured information. The importance of non-numerical information structures has been understood by many mathematicians, logicians, linguists, computer scientists and engineers, but many scientists still focus only on numerical structures and processes - sometimes seeking to remedy their failures by using statistical methods, which in restricted contexts can be spectacularly successful, as shown by recent AI successes, whose limitations I have criticised elsewhere.¹³

The FCK need not be able to produce all biological structures and processes directly, in situations without life, but it must be rich enough to support successive generations of increasingly powerful DCKs that together suffice to generate all possible biological organisms evolved so far, and their behavioural and information-processing abilities. Moreover, the FCK, or DCKs derived from it, must include abilities to acquire, manipulate, store, and use information structures in DCKs that can build increasingly complex machines that encode information, including non-numerical information. Since the 1950s we have also increasingly discovered the need for new virtual machines as well as physical machines [Sloman 2010,Sloman 2013 1].

Large scale physical processes usually involve a great deal of variability and unpredictability (e.g. weather patterns), and sub-microscopic indeterminacy is a key feature of quantum physics, yet, as [Schrödinger 1944] observed, life depends on very complex objects built from very large numbers of small scale structures (molecules) that can preserve their precise chemical structure, despite continual thermal buffetting and other disturbances. Unlike non-living natural structures, important molecules involved in reproduction and other biological functions are copied repeatedly, predictably transformed with great precision, and used to create very large numbers of new molecules required for life, with great, but not absolute, precision. This is non-statistical structure preservation, which would have been incomprehensible without quantum mechanics, as explained by Schrödinger. That feature of the FCK resembles "structure-constraining" properties of construction kits such as Meccano, TinkerToy and Lego¹⁴ that support structures with more or less complex, discretely varied topologies, or kits built from digital electronic components, that also provide extremely reliable preservation and transformations of precise structures, in contrast with sand, water, mud, treacle, plasticene, and similar materials. Fortunate children learn how structure-based kits differ from more or less amorphous construction kits that produce relatively flexible or plastic structures with non-rigid behaviours - as do many large-scale natural phenomena, such as snow drifts, oceans, or weather systems.

Schrödinger's 1944 book stressed that quantum mechanisms can explain the structural stability of individual molecules and explained how a set of atoms in different arrangements can form discrete stable structures with very different properties (e.g. in propane and isopropane only the location of the single oxygen atom differs, but that alters both the topology and the chemical properties of the molecule).¹⁵ He also pointed out the relationship between number of discrete changeable elements and information capacity, anticipating [Shannon 1948]. Some complex molecules with quantum-based structural stability are simultaneously capable of continuous deformations, e.g. folding, twisting, coming together, moving apart, etc., all essential for the role of DNA and other molecules in reproduction, and many other biochemical processes. This combination of discrete topological structure (forms of connectivity) used for storing very precise information for extended periods and non-discrete spatial flexibility used in assembling, replicating and extracting information from large structures, is unlike anything found in digital computers, although it can to some extent be approximated in digital computer models of molecular processes.

Highly deterministic, very small scale, discrete interactions between very complex, multi-stable, enduring molecular structures, combined with continuous deformations (folding, etc.) that alter opportunities for the discrete interactions, may have hitherto unnoticed roles in brain functions, in addition to their profound importance for reproduction, and growth. Much recent AI and neuroscience uses statistical properties of complex systems with many continuous scalar quantities changing randomly in parallel, unlike symbolic mechanisms used in logical and symbolic AI, though the latter are still far too restricted to model animal minds. The Meta-Morphogenesis project has extended a set of examples studied four decades earlier (e.g. in [Sloman 1978]) of types of mathematical discovery and reasoning that use perceived possibilities and impossibilities for change in geometrical and topological structures. Further work along these lines may help to reveal biological mechanisms that enabled the great discoveries by Euclid and his predecessors that are still unmatched by AI theorem provers (discussed in Section 5).

2.6  Thermodynamic issues

Our discussion so far suggests that the FCK has two sorts of components: (a) a generic framework including space-time and generic constraints on what can happen in that framework, and (b) components that can be non-uniformly and dynamically distributed in the framework. The combination makes possible formation of galaxies, stars, clouds of dust, planets, asteroids, and many other lifeless entities, as well as supporting forms of life based on derived construction kits (DCKs) that exist only in special conditions. Some local conditions e.g. extremely high pressures, temperatures, and gravitational fields, (among others) can mask some parts of the FCK, i.e. prevent them from functioning. So, even if all sub-atomic particles required for earthly life exist at the centre of the sun, local factors can rule out earth-like life forms. Moreover, if the earth had been formed from a cloud of particles containing no carbon, oxygen, nitrogen, iron, etc, then no DCK able to support life as we know it could have emerged, since that requires a region of space-time with a specific manifestation of the FCK, embedded in a larger region that can contribute additional energy (e.g. solar radiation) and possibly other resources.

As the earth formed, new physical conditions created new DCKs that made the earliest life forms possible. [Ganti 2003], usefully summarised in [Korthof 2003] and [Fernando 2008], presents an analysis of requirements for a minimal life form, the "chemoton", with self-maintenance and reproductive capabilities. Perhaps still unknown DCKs made possible formation of pre-biotic chemical structures, and also the environments in which a chemoton-like entity could survive and reproduce. Later, conditions changed in ways that supported more complex life forms, e.g. oxygen-breathing forms. Perhaps attempts to identify the first life form in order to show how it could be produced by the FCK are misguided, because several important pre-life construction kits were necessary: i.e. several DCKs made possible by conditions on earth were necessary for precursors. Some of the components of the DCKs may have been more complex than their living products, including components providing scaffolding for constructing life forms, rather than materials.

2.7  Scaffolding in construction kits

This concept of scaffolding may be crucial for research into origins of life. As far as I know nobody has found candidate non-living chemical substances made available by the FCK that have the ability spontaneously to assemble themselves into primitive life forms. It is possible that the search is doomed to fail because there never were such substances: if the earliest life forms required not only materials but also scaffolding - e.g. in the form of complex molecules that did not form parts of the earliest organisms but played an essential causal role in assembly processes, bringing together the chemicals needed by the simplest organisms. Evolution might then have produced new organisms without that reliance on the original scaffolding. The scaffolding mechanisms might later have ceased to exist on earth, e.g. because they were consumed and wiped out by the new life forms, or because physical conditions changed that prevented them forming but did not destroy the newly independent organisms. A similar suggestion is made in [Mathis, Bhattacharya,  Walker. 2015] - and for all I know has been made elsewhere. So it is quite possible that many evolutionary transitions, including transitions in information processing, our main concern, depended on forms of scaffolding that later did not survive and were no longer needed to maintain what they had helped to produce. So research into evolution of information processing, our main goal, is inherently partly speculative.

2.8  Biological construction kits

How did the FCK generate complex life forms? Is the Darwin-Wallace theory of natural selection the whole answer, as suggested in [Bell 2008]? "Living complexity cannot be explained except through selection and does not require any other category of explanation whatsoever." No: the explanation must include both selection mechanisms and generative mechanisms, without which selection processes will not have a supply of new viable options. Moreover, insofar as environments providing opportunities, challenges and threats are part of the selection process, the construction kits used by evolution include mechanisms not intrinsically concerned with life, e.g. volcanoes, earthquakes, asteroid impacts, lunar and solar tides, and many more.

The idea of evolution producing construction kits is not new, though they are often referred to as "toolkits". [Coates, Umm-E-Aiman,  Charrier. 2014] ask whether there is "a genetic toolkit for multicellularity" used by complex life-forms. Toolkits and construction kits normally have users (e.g. humans or other animals), whereas the construction kits we have been discussing (FCKs and DCKs) do not all need separate users.

Both generative mechanisms and selection mechanisms change during evolution. Natural selection (blindly) uses the initial enabling mechanisms provided by physics and chemistry not only to produce new organisms, but also to produce new richer DCKs, including increasingly complex information-processing mechanisms. Since the mid 1900s, spectacular changes have also occurred in human-designed computing mechanisms, including new forms of hardware, new forms of virtual machinery, and networked social systems all unimagined by early hardware designers. Similar changes during evolution produced new biological construction kits, e.g. grammars, planners, geometrical constructors, not well understood by thinkers familiar only with physics, chemistry and numerical mathematics.

Biological DCKs produce not only a huge variety of physical forms, and physical behaviours, but also forms of information-processing required for increasingly complex control problems, as organisms become more complex and more intelligent in coping with their environments, including interacting with predators, prey, mates, offspring, conspecifics, etc. In humans, that includes abilities to form scientific theories and discover and prove theorems in topology and geometry, some of which are also used unwittingly in practical activities.¹⁸ I suspect many animals come close to this in their systematic but unconscious abilities to perform complex actions that use mathematical features of environments. Abilities used unconsciously in building nests or in hunting and consuming prey may overlap with topological and geometrical competences of human mathematicians. (See Section 6.2). E.g. search for videos of weaver birds building nests.

3  Concrete (physical), abstract and hybrid construction kits

There are (deeply confused) fashions emphasising "embodied cognition" and "symbol grounding" (previously known as "concept empiricism" and demolished by Immanuel Kant and 20th Century philosophers of science). These fashions disregard many examples of thinking, perceiving, reasoning and planning, that require abstract construction kits. For example, planning a journey to a conference does not require physically trying possible actions, like water finding a route to the sea. Instead, you may use an abstract construction kit able to represent possible options and ways of combining them. Being able to talk requires use of a grammar specifying abstract structures that can be assembled using a collection of grammatical relationships, to form new abstract structures with new properties relevant to various tasks involving information. Sentences allowed by a grammar for English are abstract objects that can be instantiated physically in written text, printed text, spoken sounds, morse code, etc.: so a grammar is an abstract construction kit whose constructs can have concrete (physical) instances. The idea of a grammar is not restricted to verbal forms: it can be extended to many complex structures, e.g. grammars for sign languages, circuit diagrams, maps, proofs, architectural layouts and even molecules.

A grammar does not fully specify a language: a structurally related semantic construction kit, is required for building possible meanings. Use of a language depends on language users, for which more complex construction kits are required, including products of evolution, development and learning. Evolution of various types of language is discussed in [Sloman 2008].

In computers, physical mechanisms implement abstract construction kits via intermediate abstract kits - virtual machines; and presumably also in brains.
Hybrid abstract+concrete kits: These are combinations, e.g. physical chess board and chess pieces combined with the rules of chess, lines and circular arcs on a physical surface instantiating Euclidean geometry, puzzles like the mutilated chess-board puzzle, and many more. A particularly interesting hybrid case is the use of physical objects (e.g. blocks) to instantiate arithmetic, which may lead to the discovery of prime numbers when certain attempts at rearrangement fail - and an explanation of the impossibility is found.¹⁹

In some hybrid construction kits, such as games like chess, the concrete (physical) component may be redundant for some players: e.g. chess experts who can play without physical pieces on a board. But communication of moves needs physical mechanisms, as does the expert's brain (in ways that are not yet understood). Related abstract structures, states and processes can also be implemented in computers, which can now play chess better than most humans, without replicating human brain mechanisms. In contrast, physical components are indispensable in hybrid construction kits for outdoor games, like cricket [Wilson 2015]. (I don't expect to see good robot cricketers soon.)

Physical computers, programming languages, operating systems and virtual machines form hybrid construction kits that make things happen when they run. Logical systems with axioms and inference rules can be thought of as abstract kits supporting construction of logical proof-sequences, usually combined with a physical notation for written proofs. Purely logical systems cannot have physical causal powers whereas concrete instances can, e.g. teaching a student, or programming a computer, to distinguish valid and invalid proofs. Natural selection "discovered" the power of hybrid construction kits using virtual machinery, long before human engineers did. In particular, biological virtual machines used by animal minds outperform current engineering designs in some ways, but they also generate much confusion in the minds of philosophical individuals who are aware that something more than purely physical machinery is at work, but don't yet understand how to implement virtual machines in physical machines [Sloman  Chrisley 2003,Sloman 2010,Sloman 2013 1].

Animal perception, learning, reasoning, and intelligent behaviour require hybrid construction kits. Scientific study of such kits is still in its infancy. Work done so far on the Meta-Morphogenesis project suggests that natural selection "discovered" and used a staggering variety of types of hybrid construction kit that were essential for reproduction, for developmental processes (including physical development and learning), for performing complex behaviours, and for social/cultural phenomena.

3.1  Kits providing external sensors and motors

As noted in [Sloman 1978,Ch.6], the distinction between internal and external components is often arbitrary - a fact frequently rediscovered. For example, a music box may perform a tune under the control of a rotating disc with holes or spikes. The disc can be thought of as part of the music box, or as part of a changing environment.

If a toy train set has rails or tracks used to guide the motion of the train, then the wheels can be thought of as sensing the environment and causing changes of direction. This is partly like and partly unlike a toy vehicle that uses an optical sensor linked to a steering mechanism, so that a vehicle can follow a line painted on a surface. The railway track provides both information and the forces required to change direction. A painted line, however, provides only the information, and other parts of the vehicle have to supply the energy to change direction, e.g. an internal battery that powers sensors and motors. Evolution uses both sorts: e.g. wind blowing seeds away from parent plants and a wolf following a scent trail left by its prey. An unseen wall uses force to stop your forward motion in a dark room, whereas a visible, or lightly touched, wall provides only information [Sloman 2011].

3.2  Mechanisms for storing, transforming and using information

In the "offline" case, the underlying construction-kit needs to be able to support stores of information that grow with time and can be used for different purposes at different times. A control decision at one time may need items of information obtained at several different times and places, for example information about properties of a material, where it can be found, and how to transport it to where it is needed. Sensors used online may become faulty or require adjustment. Evolution may provide mechanisms for testing and adjusting. When used offline, stored information may need to be checked for falsity caused by the environment changing, as opposed to sensor faults. The offline/online use of visual information has caused much confusion among researchers, including muddled attempts to interpret the difference in terms of "what" and "where" information.²¹ Contrast [Sloman 1983].

Ways of acquiring and using information have been discovered and modelled by AI researchers, psychologists, neuroscientists, biologists and others. However, evolution produced many more. Some of them require not just additional storage space but very different sorts of information-processing architectures. A range of possible architectures is discussed in [Sloman 1978], [Sloman 1983], [Sloman 1993], [Sloman 2003], [Sloman 2006], whereas AI engineers typically seek one architecture for a project. A complex biological architecture may use sub-architectures that evolved at different times, meeting different needs in different niches. In particular, I suspect there are biological mechanisms for handling vast amounts of rapidly changing incoming information (visual, auditory, tactile, haptic, proprioceptive, vestibular) by using several different sorts of short-term storage plus processing subsystems, operating on different time-scales in parallel, including a-modal information structures.

This raises the question whether evolution produced "architecture kits" able to combine evolved information-processing mechanisms in different ways, long before software engineers discovered the need. Such a kit could be particularly important for species that produce new subsystems, or modify old ones, during individual development, e.g. during different phases of learning by apes, elephants, and humans, as described in Section 2.3, contradicting the common assumption that a computational architecture must remain fixed.²²

3.3  Mechanisms for controlling position, motion and timing

New concrete kits can be formed by combining two or more kits. In some cases this will require modification of a kit, e.g. combining Lego and Meccano by adding pieces with Lego studs or holes alongside Meccano sized screw holes. In other cases mere spatial proximity and contact suffices, e.g. when one construction kit is used to build a platform and others to assemble a house on it. Products of different biological construction kits may also use complex mixtures of juxtaposition and adaptation.

Objects that exist in space/time often need timing mechanisms. Organisms use "biological clocks" operating on different time-scales controlling repetitive processes, including daily cycles, heart-beats, breathing, and wing or limb movements required for locomotion. More subtly there are adjustable speeds and adjustable rates of change: e.g. a bird in flight approaching a perch; an animal running to escape a predator and having to decelerate as it approaches a tree it needs to climb; a hand moving to grasp a stationary or moving object, with motion controlled by varying coordinated changes of joint angles at waist, shoulder, elbow and finger joints so as to bring the grasping points on the hand into suitable locations relative to the intended grasping points on the object. (This can be very difficult for robots, when grasping novel objects in novel situations: if they use ontologies that are too simple.) There are also biological mechanisms for controlling or varying rates of production of chemicals (e.g. hormones).

So biological construction kits need many mechanisms able to measure time intervals and to control rates of repetition or rates of change of parts of the organism. These kits may be combined with other sorts of construction kit that combine temporal and spatial control, e.g. changing speed and direction,

3.4  Combining construction kits

Applying functions to arguments is very different from assembling structures in space time, where inputs to the process form parts of the output. If computers are connected via digital to analog interfaces, linking them to surrounding matter, or if they are mounted on machines that allow them to move around in space and interact, that adds a kind of richness that goes beyond application of functions to arguments.

The additional richness is present in the modes of interaction of chemical structures that include both digital (on/off chemical bonds) and continuous changes in relationships, as discussed in [Turing 1952], the paper on chemistry-based morphogenesis that inspired this Meta-Morphogenesis project [Sloman 2013 2].

3.5  Combining abstract construction kits

Creation of new construction kits may start by simply recording parts of successful assemblies, or better still parametrised parts, so that they can easily be reproduced in modified forms - e.g. as required for organisms that change size and shape while developing. Eventually, parametrised stored designs may be combined to form a "meta-construction kit" able to extend, modify or combine previously created construction kits as human engineers have recently learnt to do in debugging toolkits. Evolution needs to be able to create new meta-construction kits using natural selection. Natural selection, the great creator/meta-creator, is now spectacularly aided and abetted by its products, especially humans!

4  Construction kits generate possibilities and impossibilities

In some kits, features of components, such as shape, are inherited by constructed objects. E.g. objects composed only of Lego bricks joined in the "standard" way have external surfaces that are divisible into faces parallel to the surfaces of the first brick used. However, if two Lego bricks are joined at a corner only, using only one stud and one socket, it is possible to have continuous relative rotation (because studs and sockets are circular), violating that constraint, as Ron Chrisley pointed out in a conversation. This illustrates the fact that constructed objects can have "emergent" features none of the components have, e.g. a hinge is a non-rigid object that can be made from two rigid objects with aligned holes through which a screw is passed.

So, a construction kit that makes some things possible and others impossible can be extended so as to remove some of the impossibilities, e.g. by adding a hinge to Lego, or adding new parts from which hinges can be assembled.

4.1  Construction kits for making information-users

Biological construction kits for producing information-processing mechanisms evolved at different times. [Sloman 1993] discusses the diversity of uses of information from biological sensors, including sharing of sensor information between different uses, either concurrently or sequentially. Some of the mechanisms use intermediaries, such as sound or light, to gain information about the source or reflector of the sound or light; used in taking decisions, e.g. whether to flee, or used in controlling actions such as grasping or walking past a source of information.

Some mechanisms that use information seem to be direct products of biological evolution, such as mechanisms that control reflex protective blinking. Others are grown in individuals by epigenetic mechanisms influenced by context, as explained in Section 2.3. For example, humans in different cultures start with a generic language construction kit (sometimes misleadingly labelled a "universal grammar") which is extended and modified to produce locally useful language understanding/generating mechanisms. Language-specific mechanisms, such as mechanisms for acquiring, producing, understanding and correcting textual information evolved long after mechanisms able to use visual information for avoiding obstacles or grasping objects, shared between far more types of animal. In some species there may be diversity in the construction kits produced by individual genomes, leading to even greater diversity in adults, if they develop in different physical and cultural environments using epigenetic mechanisms discussed above.

4.2  Different roles for information

Information relating to targets and how to achieve or maintain them is control information: the most basic type of biological information, from which all others are derived. A simple case is a thermostatic control, discussed in [McCarthy 1979]. It has two sorts of information: (a) a target temperature (desire-like information) (b) current temperature (belief-like information). A discrepancy between them causes the thermostat to select between turning a heater on, or off, or doing nothing. This very simple homeostatic mechanism uses information and a source of energy to achieve or maintain a state. There are very many variants on this schema, according to the type of target (e.g. a measured state or some complex relationship) the type of control (on, off, or variable, with single or multiple effectors), and the mechanisms by which control actions are selected, which may be modified by learning, and may use simple actions or complex plans.

As [Gibson 1966] pointed out, acquisition of information often requires cooperation between processes of sensing and acting. Saccades are visual actions that constantly select new information samples from the environment (or the optic cone). Uses of the information vary widely according to context, e.g. controlling grasping, controlling preparation for a jump, controlling avoidance actions, or sampling text to be read. A particular sensor can therefore be shared between many control subsystems [Sloman 1993], and the significance of the sensor state will depend partly on which subsystems are connected to the sensor at the time, and partly on which other mechanisms receive information from the sensor (which may change dynamically - a possible cause of some types of "change blindness").

The study of varieties of use of information in organisms is exploding, and includes many mechanisms on molecular scales as well as many intermediate levels of informed control, including sub-cellular levels (e.g. metabolism), physiological processes of breathing, temperature maintenance, digestion, blood circulation, control of locomotion, feeding and mating of large animals and coordination in communities, such as collaborative foraging in insects and trading systems of humans. Slime moulds include spectacular examples in which modes of acquisition and use of information change dramatically.²⁵

Figure 4: Evolutionary transitions from molecules to intelligent animals
Between the simplest and most sophisticated organisms there are many intermediate forms with very different information processing requirements and capabilities.

The earliest organisms must have acquired and used information about things inside themselves and in their immediate vicinity, e.g. using chemical detectors in an enclosing membrane. Later, evolution extended those capabilities in dramatic ways (crudely indicated in Fig. 4). In the simplest cases, local information is used immediately to select between alternative possible actions, as in a heating control, or trail-following mechanism. Uses of motion in haptic and tactile sensing and use of saccades, changing vergence, and other movements in visual perception, all use the interplay between sensing and doing, in "online intelligence". But there are cases ignored by Gibson and anti-cognitivists, namely organisms that exhibit "offline intelligence", using perceptual information for tasks other than controlling immediate reactions, for example, reasoning about remote future possibilities or attempting to explain something observed, or working out that bending a straight piece of wire will enable a basket of food to be lifted out of a tube as illustrated in Fig. 4 [Weir, Chappell,  Kacelnik . 2002]. Doing that requires use of previously acquired information about the environment including particular information about individual objects and their locations or states, general information about learnt laws or correlations and information about what is and is not possible (Note [13]).

An information-bearing structure (e.g. the impression of a foot or the shape of a rock) can provide very different information to different information-users, or to the same individual at different times, depending on (a) what kinds of sensors they have, (b) what sorts of information-processing (storing, analysing, comparing, combining, synthesizing, retrieving, deriving, using...) mechanisms they have, (c) what sorts of needs or goals they can serve by using various sorts of information (knowingly or not), and (d) what information they already have. So, from the fact that changes in some portion of a brain correlate with changes in some aspect of the environment we cannot conclude much about what information about the environment the brain acquires and uses or how it does that - since typically that will depend on context.

4.3  Motivational mechanisms

Unforeseeable biological benefits of automatically triggered motives include acquisition of new information by sampling properties of the environment. The new information may not be immediately usable, but in combination with information acquired later and genetic tendencies activated later, as indicated in Fig. 3, it may turn out to be important, during hunting, caring for young, or learning a language. A toddler may have no conception of the later potential uses of information gained in play, though the ancestors of that individual may have benefited from the presence of the information gathering reflexes. In humans this seems to be crucial for mathematical development.

During evolution, and also during individual development, the sensor mechanisms, the types of information-processing, and the uses to which various types of information are put, become more diverse and more complex, while the information-processing architectures allow more of the processes to occur in parallel (e.g. competing, collaborating, invoking, extending, recording, controlling, redirecting, enriching, training, abstracting, refuting, or terminating). Without understanding how the architecture grows, which information-processing functions it supports, and how they diversify and interact, we are likely to reach wrong conclusions about biological functions of the parts: e.g. over-simplifying the functions of sensory subsystems, or over-simplifying the variety of concurrent control mechanisms producing behaviours. Moreover, the architectural knowledge about how such a systems works, like information about the architecture of a computer operating system, may not be expressible in sets of equations, or statistical learning mechanisms and relationships. (Ideas about architectures for human information-processing can be found in [Simon 1967,Minsky 1987,Minsky 2006,Laird, Newell,  Rosenbloom . 1987,Sloman 2003,Sun 2006], among many others.)

Construction kits for building information-processing architectures, with multiple sensors and motor subsystems, in complex and varied environments, differ widely in the designs they can produce. Understanding that variety is not helped by disputes about which architecture is best. A more complete discussion would need to survey the design options and relate them to actual choices made by evolution or by individuals interacting with their environments.

5  Mathematics: Some constructions exclude or necessitate others

Features of a physical construction kit - including the shapes and materials of the basic components, ways in which the parts can be assembled into larger wholes, kinds of relationships between parts and the processes that can occur involving them - explain the possibility of entities that can be constructed and the possibility of processes, including processes of construction and behaviours of constructs.

Construction kits can also explain necessity and impossibility. A construction kit with a large initial set of generative powers can be used to build a structure realising some of the kit's possibilities, in which some further possibilities are excluded, namely all extensions that do not include what has so far been constructed. If a Meccano construction has two parts in a substructure that fixes them a certain distance apart, then no extension can include a new part that is wider than that distance in all dimensions and is in the gap. Some extensions to the part-built structure that were previously possible become impossible unless something is undone. That example involves a limit produced by a gap size. There are many more examples of impossibilities that arise from features of the construction kit.

Euclidean geometry includes a construction kit that enables construction of closed planar polygons (triangles, quadrilaterals, pentagons, etc.), with interior angles whose sizes can be summed. If the polygon has three sides, i.e. it is a triangle, then the interior angles must add up to exactly half a rotation. Why? In this case, no physical properties of a structure (e.g. rigidity or impenetrability of materials) are involved, only spatial relationships. Figure 5 provides one way to answer the question, unlike the standard proofs, which use parallel lines. It presents a proof, found by Mary Pardoe, that internal angles of a planar triangle sum to a straight line, or 180 degrees. Most humans are able to look at a physical situation, or a diagram representing a class of physical situations, and reason about constraints on a class of possibilities sharing a common feature. This may have evolved from earlier abilities to reason about changing affordances in the environment [Gibson 1979]. Current AI perceptual and reasoning systems still lack most of these abilities, and neuroscience cannot yet explain what's going on (as opposed to where it's going on?). (See Note [13]).

These illustrate mathematical properties of construction kits (partly analogous to mathematical properties of formal deductive systems and AI problem solving systems). As parts (or instances of parts) of the FCK are combined, structural relations between components of the kit have two opposed sorts of consequences: they make some further structures possible (e.g. constructing a circle that passes through all the vertices of the triangle), and other structures impossible (e.g. relocating the corners of the triangle so that the angles add up to 370 degrees). These possibilities and impossibilities are necessary consequences of previous selection steps. The examples illustrate how a construction kit with mathematical relationships can provide the basis for necessary truths and necessary falsehoods in some constructions (as argued in [Sloman 1962,Chap 7]).²⁶ Being able to think about and reason about alterations in some limited portion of the environment is a very common requirement for intelligent action [Sloman 1996 1]. It seems to be partly shared with other intelligent species, e.g. squirrels, nest-builders, elephants, apes, etc. Since our examples of making things possible or impossible, or changing ranges of possibilities, are examples of causation (mathematical causation), this also provides the basis for a Kantian notion of causation based on mathematical necessity [Kant 1781], so that not all uses of the notion of "cause" are Humean (i.e. based on correlations), even if some are. Compare Section 5.3.²⁷

Neuroscientific theories about information-processing in brains currently seem to me to omit the processes involved in such mathematical discoveries, so AI researchers influenced too much by neuroscience may fail to replicate important brain functions. Progress may require major conceptual advances regarding what the problems are and what sorts of answers are relevant.

We now consider ways in which evolution itself can be understood as discovering mathematical proofs - proofs of possibilities.

5.1  Proof-like features of evolution

Moreover, there is not just one sequence: different evolutionary lineages evolving in parallel can produce different DCKs. According to the "Symbiogenesis" theory, different DCKs produced independently can sometimes merge to support new forms of life combining different evolutionary strands.²⁸ Creation of new DCKs in parallel evolutionary streams with combinable products can hugely reduce part of the search space for complex designs, at the cost of excluding parts of the search space reachable from the FCK. For example, use of DCKs in the human genome may speed up development of language and typical human cognitive competences, while excluding the possibility of "evolving back" to microbe forms that might be the only survivors after a cataclysm.

5.2  Euclid's construction kit

The ability of humans to discover such things must depend on evolved information-processing capabilities of brains that are as yet unknown and not yet replicated in AI reasoning systems. The idea of a space of possibilities generated by a physical construction kit may be easier for most people to understand than the comparison with generative powers of grammars, formal systems, or geometric constructions, though the two are related, since grammars and mathematical systems are abstract construction kits that can be parts of hybrid construction kits.

Concrete construction kits corresponding to grammars can be built out of physical structures: for example a collection of small squares with letters and punctuation marks, and some blanks, can be used to form sequences that correspond to the words in a lexicon. A cursive ("joined up") script requires a more complex physical construction kit. Human sign-languages are far more demanding, since they involve multiple body parts moving concurrently.

[Expand the following:]
Some challenges for construction kits used by evolution, and also challenges for artificial intelligence and philosophy, arise from the need to explain both (a) how natural selection makes use of mathematical properties of construction kits related to geometry and topology, in producing organisms with spatial structures and spatial competences, and also (b) how various subsets of those organisms (e.g. nest-building birds) developed specific topological and geometrical reasoning abilities used in controlling actions or solving problems, and finally (c) how at least one species developed abilities to reflect on the nature of those competences and eventually, through unknown processes of individual development and social interaction, using unknown representational and reasoning mechanisms, managed to produce the rich, deep and highly organised body of knowledge published as Euclid's Elements^[1]. There are important aspects of those mathematical competences that as far as I know have not yet been replicated in Artificial Intelligence or Robotics²⁹. Is it possible that currently understood forms of digital computation are inadequate for the tasks, whereas chemistry-based information-processing systems used in brains are richer, because they combine both discrete and continuous operations, as discussed in Section 2.5? (That's not a rhetorical question: I don't know the answer.)

5.3  Mathematical discoveries based on exploring construction kits

Many mathematical domains (perhaps all of them) can be thought of as sets of possibilities generated by construction kits. Physicists and engineers deal with hybrid concrete and abstract construction kits. The space of possible construction kits is also an example, though as far as I know this domain has not been explored systematically by mathematicians, though many special cases have.

In order to understand biological evolution on this planet we need to understand the sorts of construction kits made possible by the existence of the physical universe, and in particular the variety of construction kits inherent in the physics and chemistry of the materials of which our planet was formed, along with the influences of its environment (e.g. solar radiation, asteroid impacts). An open research question is whether a construction kit capable of producing all the non-living structures on the planet would also suffice for evolution of all the forms of life on this planet, or whether life and evolution have additional requirements, e.g. cosmic radiation?

5.4  Evolution's (blind) mathematical discoveries

Some construction kits and their products have mathematical properties, so life and mathematics are closely connected. More complex relationships arise after evolution of mathematical meta-cognitive mechanisms. On the way to achieving those results, natural selection often works as "a blind theorem-prover". The theorems are mainly about new possible structures, processes, organisms, ecosystems, etc. The proofs that they are possible are implicit in the evolutionary trajectories that lead to occurrences. Proofs are often thought of as abstract entities that can be represented physically in different ways (using different formalisms) for communication, persuasion (including self-persuasion), predicting, explaining and planning. A physical sequence produced unintentionally, e.g. by natural selection, or by growth in a plant, that leads to a new sort of entity is a proof that some construction kit makes that sort of entity possible. The evolutionary or developmental trail, like a geometric construction, answers the question: "how is that sort of thing possible?" So biological evolution can be construed as a "blind theorem prover", despite there being no intention, or explicit recognition, regarding the proof. Proofs of impossibility (or necessity) raise more complex issues, to be discussed elsewhere.

These observations seem to support a new kind of "Biological-evolutionary" foundation for mathematics, that is closely related to Immanuel Kant's philosophy of mathematics in his Critique of Pure Reason (1781), and my attempt to defend his ideas in [Sloman 1962]. This answers questions like "How is it possible for things that make mathematical discoveries to exist?", an example of explaining a possibility (See Note 4). Attempting to go too directly from hypothesized properties of the primordial construction kit to explaining advanced capabilities such as human self-awareness, without specifying all the relevant construction kits, including required temporary scaffolding (Sect. 2.7) will fail, because short-cuts omit essential details of both the problems and the solutions, like mathematical proofs with gaps.

Many of the "mathematical discoveries" (or inventions?) produced (blindly) by evolution, depend on mathematical properties of physical structures or processes or problem types, whether they are specific solutions to particular problems (e.g. use of negative feedback control loops), or new construction-kit components that are usable across a very wide range of different species (e.g. the use of a powerful "genetic code", the use of various kinds of learning from experience, the use of new forms of representation for information, use of new physical morphologies to support sensing, or locomotion, or consumption of nutrients etc.)

These mathematical "discoveries" started happening long before there were any humans doing mathematics (which refutes claims that humans create mathematics). Many of the discoveries were concerned with what is possible, either absolutely or under certain conditions, or for a particular sort of construction-kit. Other discoveries, closer to what are conventionally thought of as mathematical discoveries, are concerned with limitations on what is possible, i.e. necessary truths. Some discoveries are concerned with probabilities derived from statistical learning, but I think the relative importance of statistical learning in biology has been vastly over-rated because of misinterpretations of evidence. (To be discussed elsewhere.) In particular, the discovery that something important is possible does not require collection of statistics: A single instance suffices. And no amount of statistical evidence can show that something is impossible: structural constraints need to be understood. For human evolution, a particularly important subtype of mathematical discovery was unwitting discovery and use of mathematical (e.g. topological) structures in the environment, a discovery process that starts in human children before they are aware of what they are doing, and in some species without any use of language for communication. Examples are discussed in the "Toddler Theorems" document referenced in Note. 24.

6  Varieties of Derived Construction Kit

What sort of kit makes it possible for a child to acquire competence in any one of the thousands of different human languages (spoken or signed) in the first few years of life? Children do not merely learn pre-existing languages: they construct languages that are new for them, constrained by the need to communicate with conspecifics, as shown dramatically by Nicaraguan deaf children who developed a sign language going beyond what their teachers understood [Senghas 2005]. There are also languages that might have developed but have not (yet). Evolution of human spoken language may have gone from purely internal languages needed for perception, intention, etc., through collaborative actions then signed communication, then spoken communication, as argued in [Sloman 2008].

If language acquisition were mainly a matter of learning from expert users, human languages could not have existed, since initially there were no expert users to learn from, and learning could not get started. This argument applies to any competence thought to be based entirely on learning from experts, including mathematical expertise. So data-mining in samples of expert behaviours will never produce AI systems with human competences - only inferior subsets at best, though some narrowly focused machines based on very large data-sets or massive computational power may outperform humans (e.g. IBM's Deep Blue chess machine and WATSON).³⁰

The history of computing since the earliest calculators illustrates changes that can occur when new construction kits are developed. There were not only changes of size, speed and memory capacity: there have also been profound qualitative changes, in new layers of virtual machinery, such as new sorts of mutually interacting causal loops linking virtual machine control states with portions of external environments, as in use of GPS-based navigation. Long before that, evolved virtual machines provided semantic contents referring to non-physical structures and processes, e.g. mathematical problems, rules of games, and mental contents referring to possible future mental contents ("What will I see if...?") including contents of other minds.

I claim, but will not argue here, that some new machines cannot be fully described in the language of the FCK even though they are fully implemented in physical reality. (See Section 2.2 on ontologies.) We now understand many key components and many modes of composition that provide platforms on which human-designed layers of computation can be constructed, including subsystems closely but not rigidly coupled to the environment (e.g. a hand-held video camera).

Several different "basic" abstract construction kits have been proposed as sufficient for the forms of (discrete) computation required by mathematicians: namely Turing machines, Post's production systems, Church's Lambda Calculus, and several more, each capable of generating the others. The Church-Turing thesis claims that each is sufficient for all forms of computation.³¹ There has been an enormous amount of research in computer science, and computer systems engineering, on forms of computation that can be built from such components. One interpretation of the Church-Turing thesis is that these construction kits generate all possible forms of information-processing. But it is not at all obvious that those discrete mechanisms suffice for all biological forms of information-processing. For example, chemistry-based forms of computation include both discrete mechanisms (e.g. forming or releasing chemical bonds) of the sort Schrödinger discussed, and continuous process, e.g. folding and twisting used in reproduction. [Ganti 2003] shows how a chemical construction-kit can support forms of biological information-processing that don't depend only on external energy sources (a feature shared with battery-powered computers), and also supports growth and reproduction using internal mechanisms, which human-made computers cannot do (yet).

There seem to be many different sorts of construction-kit that allow different sorts of information-processing to be supported, including some that we don't yet understand. In particular, the physical/chemical mechanisms that support the construction of both physical structures and information-processing mechanisms in living organisms may have abilities not available in digital computers.³²

6.1  A new type of research project

Most biological processes and associated materials and mechanisms are not well understood, though knowledge is increasing rapidly. As far as I know, very few of the derived construction kits have been identified and studied, and I am not aware of any systematic attempt to identify features of the FCK that suffice to explain the possibility of all known evolved biological DCKs. Researchers in fundamental physics or cosmology do not normally attempt to ensure that their theories explain the many materials and process types that have been explored by natural selection and its products, in addition to known facts about physics and chemistry. Schrödinger () pointed out that a theory of the physical basis of life should explain such phenomena, though he could not have appreciated some of the requirements for sophisticated forms of information-processing, because, at the time he wrote, scientists and engineers had not learnt what we now know. Curiously, although he mentioned the need to explain the occurrence of metamorphosis in organisms, the example he mentioned was the transformation from a tadpole to a frog. He could have mentioned more spectacular examples, such as the transformation from a caterpillar to a butterfly via an intermediate stage as a chemical soup in an outer case, from which the butterfly later emerges.³³

[Penrose 1994] attempted to show how features of quantum physics explain obscure features of human consciousness, especially mathematical consciousness, but he ignored all the intermediate products of biological evolution on which animal mental functions build. Human mathematics, at least the ancient mathematics done before the advent of modern algebra and logic, seems to build on animal abilities, such as abilities to see various types of affordance. The use of diagrams and spatial models by Penrose may be an example. It is unlikely that there are very abstract human mathematical abilities that somehow grow directly out of quantum mechanical aspects of the FCK, without depending on the mostly unknown layers of perceptual, learning, motivational, planning, and reasoning competences produced by billions of years of evolution.

20th century biologists understood some of the achievements of the FCK in meeting physical and chemical requirements of various forms of life, though they used different terminology from mine, e.g. Haldane.³⁴ However, the task can never be finished, since the process of construction of new derived biological construction kits may continue indefinitely, producing new kits with components and modes of composition that allow production of increasingly complex types of structure and behaviour in organisms. That idea is familiar to computer scientists and engineers since thousands of new sorts of computational construction kit (new programming languages, new operating systems, new virtual machines, new development toolkits) have been developed from old ones in the last half century, making possible new kinds of computing system that could not previously be built from the original computing machinery without introducing new intermediate layers, including new virtual machines that are able to detect and record their own operations, a capability that is often essential for debugging and extending computing systems. [Sloman 2013 1] discusses the importance of virtual machines in extending what information-processing systems can do, and the properties they can have.

6.2  Construction-kits for biological information-processing

Later still, construction kits used by evolution produced meta-cognitive mechanisms enabling individuals to notice and reflect on their own discoveries (enabling some of them to notice and remove flaws in their reasoning). In some cases those meta-cognitive capabilities allowed individuals to communicate discoveries to others, discuss them, and organise them into complex highly structured bodies of shared knowledge, such as Euclid's Elements (Note 1). I don't think anyone knows how long all of this took, what the detailed evolutionary changes were, and how the required mechanisms of perception, motivation, intention formation, reasoning and planning evolved. Explaining how that could happen, and what it tells us about the nature of mathematics and biological/evolutionary foundations for mathematical knowledge is a long term goal of the Meta-Morphogenesis project. That includes seeking unnoticed overlaps between the human competences discovered by meta-cognitive mechanisms, and similar competences in animals that lack the metacognition, like young humans making and using mathematical discoveries, on which they are unable to reflect because the required architecture has not yet developed, and similar discoveries in other intelligent species. This could stimulate new research in robotics aimed at replicating the developmental processes.

Most of these naturally occurring mathematical abilities have not yet been replicated in Artificial Intelligence systems or robots, unlike logical, arithmetical, and algebraic competences that are relatively new to humans and (paradoxically?) easier to replicate on computers. Examples of topological reasoning about equivalence classes of closed curves not yet modelled in computers (as far as I know) are referenced in Note 32. Even the ability to reason about alternative ways of putting a shirt on a child (Note 18) is still lacking. It is not clear whether the difficulty of replicating such mathematical reasoning processes is due to the need for a kind of construction-kit that digital computers (e.g. Turing machines) cannot support, or due to our lack of imagination in using computers to replicate some of the products of biological evolution - or both! Perhaps there are important forms of representation or types of information-processing architecture still waiting to be discovered by AI researchers. Alternatively the gaps may be connected with properties of chemistry-based information-processing mechanisms combining discrete and continuous interactions, or other physical properties that cannot be replicated exactly (or even approximately) in familiar forms of computation. (This topic requires more detailed mathematical analysis.)

6.3  Representational blind spots of many scientists

Biological mechanisms include: digestive mechanisms, mechanisms for transporting chemicals, mechanisms for detecting and repairing damage or infection, mechanisms for storing re-usable information about an extended structured environment, mechanisms for creating, storing and using complex percepts, thoughts, questions, values, preferences, desires, intentions and plans, including plans for cooperative behaviours, and mechanisms that transform themselves into new mechanisms with new structures and functions.

Forms of mathematics used by physicists are not necessarily useful for studying such biological mechanisms. Logic, grammars and map-like representations are sometimes more appropriate, though I think little is actually known about the variety of forms of representation (i.e. encodings of information) used in human and animal minds and brains. We may need entirely new forms of mathematics for biology, and therefore for specifying what physicists need to explain.

Many physicists, engineers and mathematicians who move into neuroscience assume that states and processes in brains need to be expressed as collections of numerical measures and their derivatives plus equations linking them, a form of representation that is well supported by widely used tools such as Matlab, but is not necessarily best suited for the majority of mental contents, and probably not even well suited for chemical processes where structures form and interact with multiple changing geometrical and topological relationships - one of the reasons for the invention of symbolic chemical notations (now being extended in computer models of changing, interacting molecular structures).

6.4  Representing rewards, preferences, values

Despite all the sophistication of modern psychology and neuroscience, I believe they currently lack the conceptual resources required to describe either functions of brains in dealing with these matters, including forms of development and learning required, or the mechanisms implementing those functions. In particular, we lack deep explanatory theories about mechanisms that led to: mathematical discoveries over thousands of years, including mechanisms producing conjectures, proofs, counter-examples, proof-revisions, new scientific theories, new works of art and new styles of art. In part that's because models considered so far lack both sufficiently rich forms of information-processing (computation), and sufficiently deep methodologies for identifying what needs to be explained. There are other unexplained phenomena concerned with artistic creation and enjoyment, but that will not be pursued here.

7  Computational/Information-processing construction-kits

7.1  Infinite, or potentially infinite, generative power

Research in fundamental physics is a search for the construction kit that has the generative power to accommodate all the possible forms of matter, structure, process, causation, that exist in our universe. However, physicists generally seek only to ensure that their construction kits are capable of accounting for phenomena observed in the physical sciences. Normally they do not assemble features of living matter, or processes of evolution, development, or learning, found in living organisms and try to ensure that their fundamental theories can account for those features also. There are notable exceptions, such as Schrödinger and others, but most physicists who discuss physics and life (in my experience) ignore most of the details of life, including the variety of forms it can take, the variety of environments coped with, the different ways in which individual organisms cope and change, the ways in which products of evolution become more complex and more diverse over time, and the many kinds of information-processing and control in individuals, in colonies (e.g. ant colonies), societies, and ecosystems.

If cosmologists and other theoretical physicists attempted to take note of a wide range of biological phenomena (including the phenomena discussed here in connection with the Meta-Morphogenesis project) I suspect that they would find considerable explanatory gaps between current physical theories and the diversity of phenomena of life - not because there is something about life that goes beyond what science can explain, but because we do not yet have a sufficiently rich theory of the constitution of the universe (or the Fundamental Construct Kit). In part that could be a consequence of the forms of mathematics known to physicists. (The challenge of [Anderson 1972] is also relevant: see Section 10, below.)

If that is true it may take many years of research to find out what's missing from current physics. Collecting phenomena that need to be explained, and trying as hard as possible to construct detailed explanations of those phenomena is one way to make progress: it may help us to pin-point gaps in our theories and stimulate development of new, more powerful, theories, in something like the profound ways in which our understanding of possible forms of computation has been extended by unending attempts to put computation to new uses. Collecting examples of such challenges helps us assemble tests to be passed by future proposed theories: collections of possibilities that a deep physical theory needs to be able to explain.

Perhaps the most tendentious proposal here is that an expanded physical theory, instead of being expressed mainly in terms of equations relating measures, will need a formalism better suited to specification of a construction kit, perhaps sharing features of grammars, programming languages, partial orderings, topological relationships, architectural specifications, and the structural descriptions in chemistry - all of which will need to make use of appropriate kinds of mathematics for drawing out implications of the theories, including explanations of possibilities, both observed and unobserved, such as possible future forms of intelligence. Theories of utility measures may need to be replaced, or enhanced with new theories of how benefits, evaluations, comparisons and preferences, can be expressed [Sloman 1969]. We must also avoid assuming optimality. Evolution produces designs as diverse as microbes, cockroaches, elephants and orchids, none of which is optimal or rational in any simple sense, yet many of them survive and sometimes proliferate, because they are lucky, at least for a while. Likewise human decisions, policies, preferences, cultures, etc.

8  Types and levels of explanation of possibilities

More generally, the question "How is it possible to create X using construction kit Y?" or, simply, "How is X possible?" has several types of answer, including answers at different levels of abstraction, with varying generality. I'll assume that a particular construction kit is referred to either explicitly or implicitly. The following is not intended to be an exhaustive survey of the possible types of answer: merely as a first experimental foray, preparing the ground for future work:

1 Structural conformity: The first type of answer, structural conformity (grammaticality) merely identifies the parts and relationships between parts that are supported by the kit, showing that X (e.g. a crane of the sort in question) could be composed of such parts arranged in such relationships. An architect's drawings for a building, specifying materials, components, and their spatial and functional relations would provide such an explanation of how a proposed building is possible, including, perhaps, answering questions about how the construction would make the building resistant to very high winds, or to earthquakes up to a specified strength. This can be compared with showing that a sentence is acceptable in a language with a well-defined grammar, by showing how the sentence would be parsed (analysed) in accordance with the grammar of that language. A parse tree (or graph) also shows how the sentence can be built up piecemeal from words and other grammatical units, by assembling various sub-structures and, using them to build larger structures. Compare using a chemical diagram to show how a collection of atoms can make up a particular molecule, e.g. the ring structure of C₆H₆ (Benzene).

Some structures are specified in terms of piece-wise relations, where the whole structure cannot possibly exist, because the relations cannot hold simultaneously, e.g. X is above Y, Y is above Z, Z is above X. It is possible to depict such objects, e.g. in pictures of impossible objects by Reutersvard, Escher, Penrose, and others.³⁵ Some logicians and computer scientists have attempted to design languages in which specifications of impossible entities are necessarily syntactically ill-formed. This leads to impoverished languages with restricted practical uses, e.g. strongly typed programming languages. For some purposes less restricted languages, needing greater care in use, are preferable, including human languages [Sloman 1971].

2 Process possibility: The second type of answer demonstrates constructability by describing a sequence of spatial trajectories by which such a collection of parts could be assembled. This may include processes of assembly of temporary scaffolding (Sect. 2.7) to hold parts in place before the connections have been made that make them self-supporting or before the final supporting structures have been built (as often happens in large engineering projects, such as bridge construction). Many different possible trajectories can lead to the same result. Describing (or demonstrating) any such trajectory explains both how that construction process is possible, and how the end result is possible.

In some cases a complex object has type 1 possibility although not type 2. For example, from a construction kit containing several rings it is possible to assemble a pile of three rings, but not possible to assemble a chain of three rings even though each of the parts of the chain is exactly like the parts of the pile.

3 Process abstraction: Some possibilities are described at a level of abstraction that ignores detailed routes through space, and covers many possible alternatives. For example, instead of specifying precise trajectories for parts as they are assembled, an explanation can specify the initial and final state of each trajectory, where each state-pair may be shared by a vast, or even infinite collection, of different possible trajectories producing the same end state, e.g. in a continuous space.

In some cases the possible trajectories for a moved component are all continuously deformable into one another (i.e. they are topologically equivalent): for example the many spatial routes by which a cup could be moved from a location where it rests on a table to a location where it rests on a saucer on the table, without leaving the volume of space above the table. Those trajectories form a continuum of possibilities that is too rich to be captured by a parametrised equation for a line, with a number of variables. If trajectories include passing through holes, or leaving and entering the room via different doors or windows then the different possible trajectories will not all be continuously deformable into one another: there are different equivalence classes of trajectories sharing common start and end states, for example, the different ways of threading a shoe lace with the same end result.

The ability to abstract away from detailed differences between trajectories sharing start and end points, thereby implicitly recognizing invariant features of an infinite collection of possibilities, is an important aspect of animal intelligence that I don't think has been generally understood. Many researchers assume that intelligence involves finding optimal solutions. So they design mechanisms that search using an optimisation process, ignoring the possibility of mechanisms that can find sets of possible solutions (e.g. routes) initially considered as a class of equivalent options, leaving questions about optimal assembly to be settled later, if needed. These remarks are closely related to the origins of abilities to reason about geometry and topology.³⁶

4 Grouping: Another form of abstraction is related to the difference between 1 and 2. If there is a sub-sequence of assembly processes, whose order makes no difference to the end result, they can be grouped to form an unordered "composite" move, containing an unordered set of moves. If N components are moved from initial to final states in a sequence of N moves, and it makes no difference in what order they are moved, merely specifying the set of N possibilities without regard for order collapses N factorial sets of possible sequences into one composite move. If N is 15, that will collapse 1307674368000 different sequences into one.

Sometimes a subset of moves can be made in parallel. E.g. someone with two hands can move two or more objects at a time, in transferring a collection of items from one place to another. Parallelism is particularly important in many biological processes where different processes occurring in parallel constrain one another so as to ensure that instead of all the possible states that could occur by moving or assembling components separately, only those end states occur that are consistent with parallel constructions. In more complex cases the end state may depend on the relative speeds of sub-processes and also continuously changing spatial relationships. This is important in epigenesis, since all forms of development from a single cell to a multi-celled structure depend on many mutually constraining processes occurring in parallel.

For some construction kits certain constructs made of a collection of sub-assemblies may require different sub-assemblies to be constructed in parallel, if completing some too soon may make the required final configuration unachievable. For example, rings being completed before being joined could prevent formation of a chain.

5 Iterative or recursive abstraction: Some process types involve unspecified numbers of parts or steps, although each instance of the type has a definite number, for example a process of moving chairs by repeatedly carrying a chair to the next room until there are no chairs left to be carried, or building a tower from a collection of bricks, where the number of bricks can be varied. A specification that abstracts from the number can use a notion like "repeat until", or a recursive specification: a very old idea in mathematics, such as Euclid's algorithm for finding the highest common factor of two numbers. Production of such a generic specification can demonstrate a large variety of possibilities inherent in a construction-kit in an extremely powerful and economical way. Many new forms of abstraction of this type have been discovered by computer scientists developing programming languages, for operating not only on numbers but many other structures, e.g. trees and graphs.

Evolution may also have "discovered" many cases, long before humans existed, by taking advantage of mathematical structures inherent in the construction-kits available and the trajectories by which parts can be assembled into larger wholes. This may be one of the ways in which evolution produced powerful new genomes, and re-usable genome components that allowed many different biological assembly processes to result from a single discovery, or a few discoveries, at a high enough level of abstraction.

Some related abstractions may have resulted from parametrisation: processes by which details are removed from specifications in genomes and left to be provided by the context of development of individual organisms, including the physical or social environment. (See Section 2.3 on epigenesis.)

6 Self-assembly: If, unlike construction of a toy Meccano crane or a sentence or a sorting process, the process to be explained is a self-assembly process, like many biological processes, then the explanation of how the assembly is possible will not merely have to specify trajectories through space by which the parts become assembled, but also
- What causes each of the movements (e.g. what manipulators are required)
- Where the energy required comes from (an internal store, or external supply?)
- Whether the process involves pre-specified information about required steps or required end states, and if so what mechanisms can use that information to control the assembly process.
- How that prior information structure (e.g. specification of a goal state to be achieved, or plan specifying actions to be taken) came to exist, e.g. whether it was in the genome as a result of previous evolutionary transitions, or whether it was constructed by some planning or problem-solving mechanism in an individual, or whether it was provided by a communication from an external source.
- How these abilities can be acquired or improved by learning or reasoning processes, or random variation (if they can).

7 Use of explicit intentions and plans: None of the explanation-types above presupposes that the possibility being explained has ever been represented explicitly by the machines or organisms involved. Explaining the possibility of some structure or process that results from intentions or plans would require specifying pre-existing information about the end state and in some cases also intermediate states, namely information that existed before the process began - information that can be used to control the process (e.g. intentions, instructions, or sub-goals, and preferences that help with selections between options). It seems that some of the reproductive mechanisms that depend on parental care make use of mechanisms that generate intentions and possibly also plans in carers, for instance intentions to bring food to an infant, intentions to build nests, intentions to carry an infant to a new nest, intention to migrate to another continent when temperature drops, and many more. Use of intentions that can be carried out in multiple ways selected according to circumstances rather than automatically triggered reflexes could cover a far wider variety of cases, but would require provision of greater intelligence in individuals.

Sometimes an explanation of possibility prior to construction is important for engineering projects where something new is proposed and critics believe that the object in question could not exist, or could not be brought into existence using available known materials and techniques. The designer might answer sceptical critics by combining answers of any of the above types, depending on the reasons for the scepticism.

Concluding comment on explanations of possibilities:
Those are all examples of components of explanations of assembly processes, including self-assembly. In biological reproduction, growth, repair, development, and learning there are far more subdivisions to be considered, some of them already studied piecemeal in a variety of disciplines. In the case of human development, and to a lesser extent development in other species, there are many additional sub-cases involving construction kits both for creating information structures and creating information-processing mechanisms of many kinds, including perception, learning, motive formation, motive comparison, intention formation, plan construction, plan execution, language use, and many more. A subset of cases, with further references can be found in [Sloman 2006].

The different answers to "How is it possible to construct this type of object" may be correct as far as they go, though some provide more detail than others. More subtle cases of explanations of possibility include differences between reproduction via egg-laying and reproduction via parturition, especially when followed by caring for young. The latter allows a parent's influence to continue during development, as does teaching of younger individuals by older ones. This also allows development of cultures suited to different environments.

To conclude this rather messy section: the investigation of different types of generality in modes of explanation for possibilities supported by a construction kit is also relevant to modes of specification of new designs based on the kit. Finding economical forms of abstraction may have many benefits, including reducing search spaces when trying to find a new design and also providing a generic design that covers a broad range of applications tailored to detailed requirements. Of particular relevance in a biological context is the need for designs that can be adjusted over time, e.g. during growth of an organism, or shared across species with slightly different physical features or environments. Many of the points made here are also related to changes in types of computer programming language and software design specification languages. Evolution may have beaten us to important ideas. That these levels of abstraction are possible is a metaphysical feature of the universe, implied by the generality of the FCK.

9  Alan Turing's Construction kits

Another type of construction kit with related properties is Conway's Game of Life,³⁷ a construction kit that creates changing patterns in 2D regular arrays. Stephen Wolfram has written a great deal about the diversity of constructions that can be explored using such cellular automata. Neither a Turing machine nor a Conway game has any external sensors: once started they run according to their stored rules and the current (changing) state of the tape or grid-cells. In principle either of them could be attached to external sensors that could produce changes to the tape of a turing machine or the states of some of the cells in the Life array. However any such extension would significantly alter the powers of the machine, and theorems about what such a machine could or could not do would change.

Modern computers use a variant of the Turing machine idea where each computer has a finite memory but with the advantage of much more direct access between the central computer mechanism and the locations in the memory (a von Neumann architecture). Increasingly, computers have also been provided with a variety of external interfaces connected to sensors or motors so that while running they can acquire information (e.g. from keyboards, buttons, joy-sticks, mice, electronic piano keyboards, network connections, and many more) and can also send signals to external devices. Theorems about disconnected Turing machines may not apply to machines with rich two-way interfaces to an external environment.

Turing machines and Game of Life machines can be described as "self-propelling" because once set up they can be left to run according to the general instructions they have and the initial configuration on the tape or in the array. But they are not really self-propelling: they have to be implemented in physical machines with an external power supply. In contrast, [Ganti 2003] shows how the use of chemistry as a construction kit provides "self-propulsion" for living things, though every now and again the chemicals need to be replenished. A battery driven computer is a bit like that, but someone else has to make the battery.

Living things make and maintain themselves, at least after being given a kick-start by their parent or parents. They do need constant, or at least frequent, external inputs, but, for the simplest organisms, those are only chemicals in the environment, and energy either from chemicals or heat-energy via radiation, conduction or convection. John McCarthy pointed out in a conversation that some animals also use externally supplied mechanical energy, e.g. rising air currents used by birds that soar. Unlike pollen-grains, spores, etc. propagated by wind or water, the birds use internal information-processing mechanisms to control how the wind energy is used, as does a human piloting a glider.

9.1  Beyond Turing machines: chemistry

One of the important differences between types of construction kit mentioned above is the difference between kits supporting only discrete changes (e.g. to a first approximation Lego and Meccano (ignoring variable length strings and variable angle joints) and kits supporting continuous variation, e.g. plasticine and mud (ignoring, for now, the discreteness at the molecular level).

One of the implications of such differences is how they affect abilities to search for solutions to problems. If only big changes in design are possible the precise change needed to solve a problem may be inaccessible (as I am sure many who have played with construction kits will have noticed - when a partial construction produces a gap whose width does not exactly match the width of any available pieces). On the other hand if the kit allows arbitrarily small changes it will, in principle, permit exhaustive searches in some sub-spaces. The exhaustiveness comes at the cost of a very much larger (infinite, or potentially infinite!) search-space. That feature could be useless, unless the space of requirements has a structure that allows approximate solutions to be useful. In that case a mixture of big jumps to get close to a good solution, followed by small jumps to home in on a (locally) optimal solution can be very fruitful: a technique that has been used by Artificial Intelligence researchers, called "simulated annealing".³⁸

A recently published book [Wagner 2014] claims that the structure of the search space generated by the molecules making up the genome increases the chance of useful, approximate, solutions to important problems to be found with relatively little searching (compared with other search spaces), after which small random changes allow improvements to be found. I have not yet read the book but it seems to illustrate the importance for evolution of the types of construction-kit available.³⁹ I have not yet had time to check whether the book discusses uses of abstraction and the evolution of mathematical and meta-mathematical competences discussed here. Nevertheless, it seems to be an (unwitting) contribution to the Meta-Morphogenesis project. Recent work by Jeremy England at MIT⁴⁰ may turn out also to be relevant. ..

9.2  Using properties of a construction-kit to explain possibilities

Likewise, a physical construction kit can be used to demonstrate that some complex physical objects can occur at the end of a construction process. In some cases there are objects that are describable but cannot occur in a construction using that kit: e.g. an object whose outer boundary is a surface that is everywhere curved, cannot be produced in a construction based on Lego bricks or a Meccano set, though one could occur in a construction based on plasticene, or soap-film.

9.3  Bounded and unbounded construction kits

Analysis of chemistry-based construction kits for information-processing systems would range over a far larger class of possible systems than Turing machines (or digital computers), because of the mixture of discrete and continuous changes possible when molecules interact, e.g. moving together, moving apart, folding, twisting, but also locking and unlocking - using catalysts [Kauffman 1995]. I don't know whether anyone has a deep theory of the scope and limits of chemistry-based information-processing.

Recent discoveries indicate that some biological mechanisms use quantum-mechanical features of the FCK that we do not yet fully understand, providing forms of information-processing that are very different from what current computers do. E.g. a presentation by Seth Lloyd, summarises quantum phenomena used in deep sea photosynthesis, avian navigation, and odour classification.⁴¹ This may turn out to be the tip of an iceberg of quantum-based information-processing mechanisms.

There are some unsolved, very hard, partly ill-defined, problems about the variety of functions of biological vision: e.g. simultaneously interpreting a very large, varied and changing collection of visual fragments, perceived from constantly varying viewpoints, e.g. as you walk through a garden with many unfamiliar flowers, shrubs, bushes, etc. moving irregularly in a changing breeze. Could some combination of quantum entanglement and non-local interaction play a role in rapidly and simultaneously processing a large collection of mutual constraints between multiple visual fragments? The ideas are not yet ready for publication, but work in progress is recorded here: http://www.cs.bham.ac.uk/research/projects/cogaff/misc/quantum-evolution.html.

Some related questions about perception of videos of fairly complex moving plant structures are raised here: http://www.cs.bham.ac.uk/research/projects/cogaff/misc/vision/plants/.

10  Conclusion: Construction kits for Meta-Morphogenesis

The idea of a construction kit is offered as a new unifying concept for philosophy of mathematics, philosophy of science, philosophy of biology, philosophy of mind and metaphysics. The aim is to explain how it is possible for minds to exist in a material world and to be produced by natural selection and its products. Related questions arise about the nature of mathematics and its role in life. The ideas are still at an early stage of development and there are probably many more distinctions to be made, and a need for a more formal, mathematical presentation of properties of and relationships between construction kits, including the ways in which new derived construction kits can be related to their predecessors and their successors. The many new types of computer-based virtual machinery produced by human engineers since around 1950 provide examples of non-reductive supervenience (as explained in [Sloman 2013 1]). They are also useful as relatively simple examples to be compared with far more complex products of evolution.

In [Esfeld, Lazarovici, Lam,  Hubert , 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 1996 1,Sloman 2013 1]).

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?⁴²

End Note

"In working on the ACE I am more interested in the possibility of producing models of the actions of the brain than in the practical applications to computing."
http://www.rossashby.info/letters/turing.html

It would be very interesting to know whether he had ever considered the question whether digital computers might be incapable of accurately modelling brains making deep use of chemical processes. He also wrote in [Turing 1950]
"In the nervous system chemical phenomena are at least as important as electrical."
But he did not elaborate on the implications of that claim.⁴³

11  Note on Barry Cooper

I also owe much to the highly intelligent squirrels and magpies in our garden, who have humbled me.

References

Footnotes:

¹ http://www.gutenberg.org/ebooks/21076

²I have argued elsewhere that the concept of "consciousness" is problematic in part because the adjectival forms are more basic than the noun, and the adjectival forms have types of context sensitivity that can lead to truth-values that depend on context in complex ways. Some of the issues are summarised in http://www.cs.bham.ac.uk/research/projects/cogaff/misc/family-resemblance-vs-polymorphism.html

³http://www.cs.bham.ac.uk/research/projects/cogaff/crp/#chap2

⁴ http://www.cs.bham.ac.uk/research/projects/cogaff/misc/explaining-possibility.html

⁵Expanded in http://www.cs.bham.ac.uk/research/projects/cogaff/misc/meta-morphogenesis.html

⁶https://en.wikipedia.org/wiki/Genetic_programming

⁷Assembly mechanisms are part of the organism, as illustrated in a video of grass growing itself from seed https://www.youtube.com/watch?v=JbiQtfr6AYk. In mammals with a placenta, more of the assembly process is shared between mother and offspring.

⁸Implications for evolution of vision and language are discussed in http://www.cs.bham.ac.uk/research/projects/cogaff/talks/#talk111

⁹http://www.cs.bham.ac.uk/research/projects/cogaff/misc/autism.html

¹⁰Examples include: https://en.wikipedia.org/wiki/Parse_tree
https://en.wikipedia.org/wiki/Structural_formula
https://en.wikipedia.org/wiki/Flowchart
https://en.wikipedia.org/wiki/Euclidean_geometry
https://en.wikipedia.org/wiki/Entity-relationship_model
https://en.wikipedia.org/wiki/Programming_language

¹¹http://www.cs.bham.ac.uk/research/projects/cogaff/00-02.html#71

¹²Often misleadingly labelled "non-linear". http://en.wikipedia.org/wiki/Control_theory http://en.wikipedia.org/wiki/Nonlinear_control

¹³E.g. http://www.cs.bham.ac.uk/research/projects/cogaff/misc/impossible.html

¹⁴ https://en.wikipedia.org/wiki/Meccano, https://en.wikipedia.org/wiki/Tinkertoy and https://en.wikipedia.org/wiki/Lego

¹⁵E.g. see James Ashenhurst's tutorial: http://www.masterorganicchemistry.com/2011/11/10/dont-be-futyl-learn-the-butyls/

¹⁶Partly inspired by memories of a talk by Lionel Penrose in Oxford around 1960 about devices he called droguli - singular drogulus. Such naturally occurring multi-stable physical structures seem to me to render redundant the apparatus proposed in [DeaconDeacon 2011] to explain how life apparently goes against the second law of thermodynamics. See https://en.wikipedia.org/wiki/Incomplete_Nature

¹⁷https://en.wikipedia.org/wiki/Paper_doll

¹⁸Such as putting a shirt on a child: http://www.cs.bham.ac.uk/research/projects/cogaff/misc/shirt.html I think Piaget noticed some of the requirements.

¹⁹http://www.cs.bham.ac.uk/research/projects/cogaff/misc/toddler-theorems.html#primes

²⁰[TrehubTrehub 1991] proposed an architecture for vision that allows snapshots from visual saccades to be integrated in a multi-layer fixation-independent visual memory.

²¹http://en.wikipedia.org/wiki/Two-streams_hypothesis

²²The BICA society aims to bring together researchers on biologically inspired cognitive architectures. Some examples are here: http://bicasociety.org/cogarch/

²³Our SimAgent toolkit is an example http://www.cs.bham.ac.uk/research/projects/poplog/packages/simagent.html [Sloman 1996 2].

²⁴As discussed in connection with "toddler theorems" in http://www.cs.bham.ac.uk/research/projects/cogaff/misc/toddler-theorems.html Contributions from observant parents and child-minders are welcome. I think deeper insights come from extended individual developmental trajectories than from statistical snapshots of many individuals.

²⁵ http://www.theguardian.com/cities/2014/feb/18/slime-mould-rail-road-transport-routes

²⁶Such relationships between possibilities provide a deeper, more natural, basis for understanding modality (necessity, possibility, impossibility) than so called "possible world semantics". I doubt that most normal humans who can think about possibilities and impossibilities base that on thinking about truth in the whole world, past, present and future, and in the set of alternative worlds.

²⁷For more on Kantian vs Humean causation see the presentations on different sorts of causal reasoning in humans and other animals, by Chappell and Sloman at the Workshop on Natural and Artificial Cognition (WONAC, Oxford, 2007): http://www.cs.bham.ac.uk/research/projects/cogaff/talks/wonac Varieties of causation that do not involve mathematical necessity, only probabilities (Hume?) or propensities (Popper) will not be discussed here.

²⁸http://en.wikipedia.org/wiki/Symbiogenesis

²⁹Some of them listed in http://www.cs.bham.ac.uk/research/projects/cogaff/misc/mathstuff.html

³⁰This comparison needs further discussion. See http://www.popsci.com/science/article/2012-12/fyi-which-computer-smarter-watson-or-deep-blue

³¹For more on this see: http://en.wikipedia.org/wiki/Church-Turing_thesis

³²Examples of human mathematical reasoning in geometry and topology that have, until now, resisted replication on computers are presented in http://www.cs.bham.ac.uk/research/projects/cogaff/misc/torus.html and http://www.cs.bham.ac.uk/research/projects/cogaff/misc/triangle-sum.html

³³http://en.wikipedia.org/wiki/Pupa

http://en.wikipedia.org/wiki/Holometabolism

³⁴http://en.wikipedia.org/wiki/J.\_B.\_S.\_Haldane

³⁵http://www.cs.bham.ac.uk/research/projects/cogaff/misc/impossible.html

³⁶Illustrated in these discussion notes:

http://www.cs.bham.ac.uk/research/projects/cogaff/misc/changing-affordances.html

http://www.cs.bham.ac.uk/research/projects/cogaff/misc/triangle-theorem.html

http://www.cs.bham.ac.uk/research/projects/cogaff/misc/torus.html

³⁷http://en.wikipedia.org/wiki/Conway.27s.Game.of.Life

³⁸One of many online explanations is http://www.theprojectspot.com/tutorial-post/simulated-annealing-algorithm-for-beginners/6

³⁹An interview with the author is online at
https://www.youtube.com/watch?v=wyQgCMZdv6E

⁴⁰ https://www.quantamagazine.org/20140122-a-new-physics-theory-of-life/

⁴¹https://www.youtube.com/watch?v=wcXSpXyZVuY

⁴²Here's a cartoon teasing particle physicists:
http://www.smbc-comics.com/?id=3554

⁴³I think it will turn out that the ideas about "making possible" used here are closely related to Alastair Wilson's ideas about grounding as "metaphysical causation". [Wilson 2015].