School of Computer Science THE UNIVERSITY OF BIRMINGHAM CN-CR Ghost Machine

From Molecules to Mathematicians:
How could evolution produce mathematicians
     from a cloud of cosmic dust?

What might Turing have done if he had lived longer?
How might it have helped us understand biological evolution,
and how evolution produced mathematicians?

Notes for a presentation to the Midlands Logic Seminar
http://midlandslogic.org.uk/2013_2014/Term1/Week5.html
Room 310, School of Mathematics
University of Birmingham
Friday 1st Nov 2013 -- 5pm

(DRAFT: Liable to change. Please save links, not copies.)

Aaron Sloman
http://www.cs.bham.ac.uk/~axs
School of Computer Science, University of Birmingham.

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Installed: 26 Oct 2013
Last updated:
28 Oct 2013; ... 5 Nov 2013; 7 Nov 2013; ... 21 Nov 2013; 18 Dec 2013

This document is
http://www.cs.bham.ac.uk/research/projects/cogaff/misc/mathsem.html
It is part of the Meta-Morphogenesis project.

A partial index of discussion notes is in
http://www.cs.bham.ac.uk/research/projects/cogaff/misc/AREADME.html

Further reading related to this project is below (to be extended).
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Overview (Revised Dec 2013)

All human mathematicians are (in part) products of biological evolution. Why should
members of any animal species have mathematical interests and capabilities? Because
the world in which organisms develop, consume, compete, flourish and reproduce is
rich in mathematical domains, and mathematical mastery of those domains brings
biological benefits.

But that's not enough: the mechanisms of natural selection must have rich enough
powers to generate mathematical mechanisms before mathematical mechanisms can be
selected. So there may be universes, or planets, where the powers of natural
selection are much more limited than they were on this planet 4.5 billion years ago.
Those limited powers might not have supported evolution of the same range of
mathematical opportunities, mechanisms, and powers as emerged on Earth. What made
that possible? Answer: the original mechanisms and new ones produced by evolution?
What original mechanisms? How were new ones produced?

At all stages there are mathematical structures involved in the evolutionary
progress, and evolution "blindly" discovers and uses them.

In later stages of evolution, individual organisms develop abilities to discover and use
mathematical structures and processes, though without realising what they are doing.
At that stage both evolution and some of its products including non-human species
that are still flourishing, are "blind" mathematicians.

Later still, humans began to think explicitly about these processes and to discuss
their properties. The earliest "non-blind" mathematicians used new biological
mechanisms to notice and reflect on their mathematical reasoning.

Those developments led, eventually, to the collaborative production of Euclid's
Elements. That was definitely a collection of mathematical discoveries even though
neither the formalisms nor the proof methods used were of the type that many (though
not all) mathematicians now seem to regard as defining mathematics, namely use of
logically formulated axioms and rules of inference, and accepting only proofs that
are logically valid derivations from axioms.

At present there's very little we know about the actual history of evolutionary
developments and the pre-history of human mathematics. I'll try to show how the
meta-morphogenesis (M-M) project sets out a strategy for trying to fill some of the gaps,
with the hope of answering not only evolutionary questions, but also philosophical
questions about the nature of mathematics and how mathematical reasoning and
knowledge differs from other kinds. It also involves metaphysical claims about the
possibility of finding mathematical structures in the world.

Perhaps the project will eventually help us to develop robots and AI systems that are
far more intelligent than the current models are, and at last provide good explanatory
models of human and animal intelligence.

This may also lead to much better mathematical education systems based on deeper
insights into the nature of mathematics and into the nature of biological processes
of learning and discovery.
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NOTE:
Some of these ideas will be presented in a 'Frontiers of Science' talk at the
Association for Science Education (ASE) conference in Birmingham, on 11th Jan 2014.
"Could a baby robot grow up to be a mathematician?"
http://www.cs.bham.ac.uk/research/projects/cogaff/misc/ase-2014-abs.html
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Planes question (Part 1):
Is it possible for three plane surfaces to bound (completely enclose) a finite volume?

Think about this and remember how you think about it. The question will be followed
up below.

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Comments, criticisms, suggestions, about the planes question or anything else here,
to a.sloman[AT]cs.bham.ac.uk

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Introduction: Mathematics and Meta-Morphogenesis

"For mathematics is after all an anthropological phenomenon"
(L. Wittgenstein, Remarks on the Foundations of Mathematics)

I'll argue that Wittgenstein was completely wrong.

Mathematical activities include biological phenomena whose possibility depends on
metaphysical facts but for which natural selection could not have achieved so much,
starting from a lifeless planet.

The metaphysical facts are that there are infinitely many mathematical domains,
including many whose instances are intimately involved in solutions to biological
problems found by natural selection.

A mathematical domain is a (possibly infinite) collection of possible structures and
processes. Many, though not all, mathematical domains can have physical instances,
including physical and chemical structures and processes.

Many mathematical discoveries were made and used (blindly) by evolution and its
products long before there were any humans on the planet. So anthropology, the
study of human diversity, is too limited in scope to cover mathematics.

Many biologically important mathematical domains have instances that include
construction and use of information structures. More examples are in this draft list.
Some of the links with biology are presented in outline below.

Other views of philosophers of mathematics are challenged by the biological
role of mathematics, and the history of mathematical discoveries. One is the
view of mathematics as essentially concerned with discovering logically
derivable consequences of logically formulated sets of axioms. The question
about planes, posed above and mentioned below illustrates this challenge
to logicist, axiom-based theories about the nature of mathematics -- unless
the reader is one of the few who use logic to reason geometrically.

What Alan Turing might have done.

One of Turing's last papers, on Chemical Morphogenesis (1952) -- now his most
cited paper -- investigated ideas about how two chemicals diffusing at different
rates, and interacting when they meet, could produce a wide variety of patterns on
the surface of a developing organism, including dots, groups of dots, stripes,
and blotches (as on a cow).

When I read it in 2011 in preparation for writing a paper about his work I wondered
whether, if he had lived on, he might have combined this late work on mixed discrete
and continuous control with his earlier work on discrete forms of computation, in
Turing Machines.
A Protoplanetary Dust Cloud?
Protoplanetary disk
[NASA artist's impression of a protoplanetary disk, from WikiMedia]

Perhaps he could have made significant progress on a problem that has baffled
biologists, chemists, physicists and others: how a planet-wide ecosystem full of
billions of living things of millions of types [*], including mathematicians, could have
assembled itself from a state with no life, but with some external sources of energy,
including solar and cosmic radiation and asteroid impacts.

      [*] http://www.bbc.co.uk/news/science-environment-14616161
      BBC News, 23 August 2011: Species count put at 8.7 million

I can't do what Turing would have done, but I've given a Turing-inspired name to the
project, namely "Meta-Morphogenesis" (since products of morphogenesis in evolution
change the mechanisms of morphogenesis), and I've begun to identify some of the
types of work to be done, including a major multidisciplinary collaborative investigation
of changes in forms of information-processing.

That investigation complements existing biological studies of changes and variations
in morphology, in behaviours, in habitats, in modes of reproduction, and in DNA.

This investigation includes assembling and organising evidence-based, theory-based
and conjectural examples of such transitions. Then, as evidence accumulates and our
theories grow, we may think of new explanations of how these changes came about, and
their implications for later changes. This task will be easier if guided by ideas
about many intermediate stages in the transitions from molecules to mathematicians,
some of which are conjectured below. A longer list, addressing a wider range of
evolutionary transitions can be found here.

A (messy) overview of the whole M-M project can be found on this web site:
http://tinyurl.com/CogMisc/meta-morphogenesis.html

Why we need the M-M project

Psychologists, neuroscientists, cognitive scientists, AI researchers (including
roboticists) and others, already attempt to find out what evolution produced and how
the products work, by studying existing systems and trying to unravel and model their
mechanisms. I suspect these approaches cannot succeed, though they produce many
indications of progress -- but progress towards partial answers only.

One of the problems of studying current species, especially recently evolved species,
is the invisibility of most of the information processing and the unobviousness of
many of the requirements that drove evolution of the functions and mechanisms. A
related problem is the rich interconnectedness of solutions to various sub-problems.
Attempts by neuroscientists and psychologists to do laboratory experiments probing
separate mechanisms still depend on the whole human to understand instructions and
perform actions. So what appear to be properties of sub-mechanisms may merely be
properties of experimental paradigms.

If we can use the observed existing diversity of competences, fossil records, and
many other clues, to suggest (a) intermediate forms of information processing and how
they changed, and (b) possible mechanisms supporting those forms of processing, then
we may be able to build a very large map that will continually guide further research,
both towards new contents for the map, and also new explanatory mechanisms and
architectures. Merlin Donald (2002) also uses evolutionary considerations to drive
informed speculation about the architecture of a human mind, but his ideas about
information-processing mechanisms do not seem to be based on practical model building
experience.

Aspects of evolution:
Generative mechanisms and selection mechanisms

The explanatory power of natural selection includes one element that is often cited
and one that is rarely cited -- at least in popular presentations. The first is the ability
of some biologically useful changes to be preserved and the numbers of individuals
inheriting some of those changes to grow faster than numbers of individuals without
those changes, in a competitive environment.

The second, more fundamental, pre-requisite for natural selection is the existence of
underlying generative mechanisms of reproduction and development that are capable
of producing novel structures, mechanisms, forms of information-processing, and
behaviours, from which natural selection can choose useful subsets.

Not all physical substrates would be capable of supporting mechanisms with those
properties. A planet formed only from grains of sand held together by gravitational
forces would probably not support such generative mechanisms, and in that case
evolution of the biological diversity found on our planet could not have happened.

There may be different collections of generative physical mechanisms that all support
some sort of evolutionary change. But only a special subset could generate the
opportunities for selection that occurred on this planet -- and they depend crucially
on the properties of chemistry, as contrasted, for example, with the properties of
point masses and rigid bodies studied in Newtonian mechanics. (Some of the reasons
for the importance of chemistry are presented in the work of Tibor Ganti (2003) and
Stuart Kauffman).

If we can understand the requirements for those mechanisms, and how chemical
mechanisms, with their rich mixture of continuous and discrete processes, meet the
requirements for information-processing in living things, and why Turing machines and
logical or arithmetical inference engines do not, we may be better able to understand
how we, and many other life forms, evolved. In particular, that might explain the
evolution of mechanisms supporting mental states and processes, including mechanisms
supporting processes of mathematical discovery.
(See the Further Reading below.)

Of course, this challenges the hypothesis that a Turing machine, or collection of
Turing machines, could simulate everything in the physical universe.

The mixture of physical and chemical structures and processes produced when this
planet formed had the potential to support a particularly rich variety of possible
structures and processes of varying size and complexity.

It also (crucially) had the ability to support more and more diversity as size and
complexity increased, including discontinuous changes in types of information
processing -- e.g. from handling information about current sensory and motor signals
to handling information about things that endure when not sensed or acted on (from a
somatic ontology to an exo-somatic ontology).

It seems that evolution had to be able to produce some major discontinuities in
functionality and design. Those transitions include transitions in mathematical
domains instantiated in products of evolution.

Example:
Continuous change in a plane surface can produce a curved line growing
longer and longer, with a fixed curvature. But when when the "growing"
end meets the other end there is a major discontinuity, partitioning the
whole plane into two disconnected regions. Such discontinuities can occur
in evolution and in development.
(Is Catastrophe Theory relevant? http://en.wikipedia.org/wiki/Catastrophe_theory )
I suspect that many different mathematical domains have been directly involved in
providing new opportunities throughout evolution, with the variety of domains
and options increasing as complexity of organisms increased.

Mathematics can not only generate ever increasing complexity and diversity of
structures and processes, it also provides opportunities to tame complexity and
diversity through the discovery of powerful new abstractions that can be instantiated
in diverse ways, a point made in this document on The Nature of Mathematics, although
I think it over-emphasises the role of logic in mathematics:
http://www.project2061.org/publications/sfaa/online/chap2.htm
American Association for the Advancement of Science, 1990.

I suspect that long before human mathematicians discovered new abstractions to tame
complexity, natural selection did that, many times, as explained below.

Mathematical domains required for evolution to work
(Expanded: 21 Nov 2013)

A domain is a set of related possible structures and processes. Processes involve
transformations between structures. The processes can include addition or removal of
sub-structures (or both) and alterations of properties and relationships. Any actual
situation will typically involve instances of several domains, along with possibilities
for change in accordance with processes in those domains.

For example, a sphere located on a surface is an instance of a domain that includes
possibilities of sliding, rolling along the surface and rotating at a point on the
surface, all of which are processes that can introduce new relationships. If the
sphere has marks on the surface those the processes will also alter relationships
between those marks and parts of the plane, or other things on the plane. If there
are other objects on the plane, then possible motions of the other objects also
constitute domains. Domains can be combined to form larger domains, or subdivided
into smaller domains, as implied in the description above of different possible forms
of motion of the sphere on the plane.

One of the sources of mathematics is the existence of constraints in many domains.
For example among the domain of straight line segments in a plane, one of the
constraints, that may not be obvious to everyone at first, is that no two straight
lines can enclose a bounded region of the plane. Why not? Think about possible
configurations of two straight line segments and how they can separate parts of a
plane from other parts.

Atoms and molecules and their possible interactions provide an enormous variety of
domains of structures and processes, which can be combined in various ways to produce
ever more complex domains. Among the (relatively late) products of biological
evolution are organisms that can construct and think about domains of which perceived
physical configurations are instances. J.J.Gibson's notion of perceiving affordances
includes a small subset of such cases. Different organisms can detect and reason
about different sets of affordances -- in different domains. But evolution itself
detects uses domains in developing its abilities to produce varying types of
organisms with varying sorts of capabilities.

Living organisms need many types of parts, made of many types of materials, with many
types of relationships, forming many types of structure, and interacting in many types
of ways to produce many types of process that produce and change many types of state.

Many types of material, relationship, structure, interaction, process, and state have
properties that can be characterised mathematically, and can be broken down into
instances of different mathematical domains: domains concerned with spatial
structures, spatial processes, forces, energy transfer, and also types of information,
forms of representation, forms of information-processing, including especially
types of control -- perhaps the most basic use of information, from which
others (such as referring, stating, asking, intending) can be constructed.
(Sloman 1985), (Sloman 2011)

The more complex structures, processes and capabilities cannot (normally?) be
assembled by gradual accretion of the simplest elements. Rather, achieving a
particular level and type of complexity, with new building blocks and relationships
for assembling novel structures, may be required for some further increases in
biological sophistication to become possible. The evolutionary transition from motion
on four limbs to bipedal walking and running could not have passed through all
possible waist angles if forelimb lengths did not increase enormously. When
appropriate geometry and muscle strengths had developed, perhaps some discontinuous
change in control mechanisms in the brain then enabled upright walking and running.
If so, evolution of locomotion may have passed through many slightly different
mathematical domains and also made some jumps between very different domains.

Similar discontinuities have occurred in the history of computing systems design in the
last 70 years or so. We now have many kinds of virtual machine that can provide platforms
on which more sophisticated types can be built but which could not be built simply by
specifying manipulations of bit patterns in the computer's memory.
[To be added: Why not?]

Note (Added 21 Nov 2013):
A large topic that will be discussed separately later is the difference
between the idea of a set of possible worlds, and a domain. I have many
doubts about the usefulness of the notion of 'this world' let alone
reference to sets of possible alternatives to this world. Moreover it is
clear that there are many people who are able to think about possibilities
and constraints on possibilities without thinking about alternative
possible worlds. Instead they think about alternative configurations of a
portion of the world, and can discover constraints limiting such
alternatives. So I suggest that our normal "modal" concepts of possibility
and necessary derive from an understanding of domains and invariants in
domains, which in turn derives from earlier biological mechanisms for
handling sets of possibilities in some portion of the world.

Even if it is possible to give a coherent analysis of what "this world" is
and what the alternative possible worlds are, that would turn out merely to
be a special, very complicated case, of this more primitive notion of a
domain including many possibilities. For a partial discussion of this,
notion, and its importance for intelligent agents, see Sloman 1996.

The relevance of pictures of impossible objects to understanding human and
animal perception of space is discussed (briefly) in a paper on unsolved
problems concerning animal and robot vision in this section, and the
preceding section on partial orders:
http://www.cs.bham.ac.uk/research/projects/cogaff/misc/vision/#impossible

Conjecture: "Blind" mathematical discovery

Evolution initially "blindly" discovered and made use of many kinds of mathematics,
insofar as many products of natural selection depend on the existence of classes
of structures and processes with mathematical properties that strongly constrain the
structures and processes that their interactions can produce.

A simple example is the need for some rigid materials and some flexible materials.
Rigidity and flexibility are properties of materials that constrain the properties of
processes in which they are involved: both sorts can change their locations relative
to other objects, whereas rigid materials prevent changes of spatial relationships
between parts of an object and flexible materials do not. Flexible materials can have
different sorts of mathematical properties affecting the types of change that can
occur, e.g. elastic, and inelastic changes. An extreme case of flexibility is being
a liquid.

A more complex example is the topological property of hollow 3-D impermeable (or,
more importantly, semi-permeable) membranes: separation of different mixtures of
chemicals inside and outside the membrane, thereby maintaining an internal
combination that would otherwise be diluted and polluted. Mechanisms for forming such
structures and allowing them to reproduce with their contents seem to have been early
products of evolution. Tibor Ganti (2003).

A related example is use of selective, controlled, flow through such an enclosing
membrane to allow useful supplies of materials and energy to come in and waste
products to go out. Kinematic topology?

Another example is use of sensor values to modify mechanisms that allow rates
of flow to increase or decrease: homeostatic mechanisms whose mathematical
properties prevent fatal monotonic or violently fluctuating changes.

Over millions, or billions, of evolutionary steps there were many other mathematical
discoveries, some concerned with properties of structures, e.g. how to make strong,
light skeletons, how to give joints or contact points the right sort of compliance to
support accurate control across a range of task situations, how to configure joints
and bone structures to allow required patterns of movement at the far ends of limbs,
how to produce an increasing variety of external and internal sensors capable of
acquiring information of many kinds, how to use information in increasingly diverse
ways including immediate use control of behaviour and postponed use when the
opportunity arises (e.g. retaining information about location of a food store found
while foraging).

Later, as the information to be acquired, manipulated, derived, stored, combined, and
used became increasingly complex (e.g. information about entities enduring when
unobserved with some changing and some unchanging properties and relationships,
including enduring spatial relationships in some cases), evolution discovered forms
of information acquisition, storage and manipulation that allowed such information to
be used at different times, e.g. in planning manipulative actions on objects, or
planning routes across extended terrain.

In doing all that it implicitly proved that various design solutions were possible,
by creating working instances. In short, the "blind watchmaker" metaphor for natural
selection may be less illuminating than a "blind theorem-prover" metaphor. The
corresponding proofs would be the evolutionary steps from some early state of the
planet to the existence of working instances of new structures or mechanisms.

Alas, the vast majority of such proofs have not been recorded, and reconstructing
them will be very difficult, using much guess-work and conjectural interpretation of
evidence. But if enough linked chains of hypothesised proof-steps can be linked to
form a coherent large network, we one day have the best evidence that can be hoped
for, even if it is not conclusive.

Getting individuals to take on the burden of discovery

Making all those discoveries by natural selection is abominably slow, but evidently
evolution found ways of speeding up its operation, by making mathematical discoveries
about classes of structures, processes, and information, and producing species whose
individuals could instantiate those abstractions differently in different environments, by
interacting with the environments, thereby shifting part of the discovery process from
evolution to processes of development and learning, which are much faster.

This is related to what has been called "The Baldwin Effect" briefly discussed in
http://en.wikipedia.org/wiki/Baldwin_effect

Some of the discovery processes were collaborative: individuals moving around
responding to their immediate environment and current needs could leave chemical
trails and the combinations of chemical trails of many individuals could change the
environment into an important source of information about where to go. This is one of
many examples of "stigmergy" http://en.wikipedia.org/wiki/Stigmergy

It seems that later on some of those processes were mirrored in or replaced by
changes in brain structures in individuals -- e.g. since stable pheromone trails are
not usable by flying animals or animals in fast flowing water.

Note: Some changes to the environment produced by collective behaviour of individuals
last a short time then are removed by influences such as rain, wind, sandstorms, etc.
Others, such as paths hacked through forests where trees do not reappear quickly,
roads built with enduring materials, tunnels and pyramids can endure long enough to
affect many generations. I suspect there are many examples of environmental changes
produced by a species having a long term effect on the genome, in the same way as
climatic or other externally produced changes can.

An example may be the evolution of abilities to use languages for communication,
after evolution of "internal languages" required for perception, control of actions,
learning, motivation, intention, plan formation,, intelligent collaboration,
imitation, etc. Some speculations about how that might have happened can be found in
http://www.cs.bham.ac.uk/research/projects/cogaff/talks/#glang
(Compare Dawkins on "The Extended Phenotype").

The evolved information storage mechanisms include (at least) two mathematical domains:
(a) the domain of information contents, e.g. facts about terrain, and (b) the domain
of structures and processes in the information medium (chemical trails, neural
connections, stored data-structures, etc.)

It is often assumed, wrongly, that isomorphism between related components of
type (a) -- semantic contents -- and type (b) -- syntactic structure -- is necessary.

Getting the right mathematical relationships between different parts of the
genome controlling different aspects of physical, behavioural, and information
processing capabilities is extremely difficult. Perhaps evolution found ways of
coordinating some of the changes it produced, e.g. relating the internal information
structures for controlling actions, to the degrees of freedom and scale of movement
of the effectors.

This makes evolution a blind theorem prover -- where the theorems are about what
works under what conditions, and the derivations are the evolutionary trails. At that
stage there may be nothing in biology that can distinguish pseudo-theorems that just
happen to be useful in a combination of circumstances that cannot be relied on to
persist, and theorems that cannot possibly have exceptions because of the
mathematical relationships. Discovering a pattern of motion that permits escape from
a predator with particular appearance might be an example of the former. Discovering
a way to modify the second derivative of a homeostatic loop as well as the first
derivative in order to prevent overshooting would be an example of the second.

My observations suggest that human children spend far more time exploring different
mathematical domains in the first few years of life than anyone has noticed. This is
driven by what I've called architecture-based motivation rather than reward-based
motivation, since the individuals have no possible way of knowing what rewards they
will get from their explorations, and with architecturally triggered motives they do
not need any rewards as explained in this paper:
http://www.cs.bham.ac.uk/research/projects/cogaff/architecture-based-motivation.pdf

A video recording of an 11 month old child apparently exploring several different
domains associated with a piano can be viewed as video 4 here. The other videos
on that page illustrate other examples of exploration of structured domains.

It seems that Emre Ugur has demonstrated such processes of active exploration,
followed by re-organisation of information, in a robot, in his PhD Thesis (2010) and
related publications.

Providing mathematical meta-cognition for some species

Later still, evolution may have found ways to produce organisms that, in addition to
having the above abilities, also have an extra (meta-cognitive) layer of processing
that develops relatively late and inspects what has been learnt so far, and then
finds ways of improving it, and perhaps later on, or in parallel, develops mechanisms
that allow individuals making such discoveries to communicate the results to others,
permitting cultural evolution to speed up natural selection. Later still evolution
might extend the mechanisms to allow not only results but improved methods of
discovery to be noticed, thought about, and communicated, and later jointly debugged.
Compare Karmiloff-Smith, 1992. http://tinyurl.com/CogMisc/beyond-modularity.html

I suspect the more intelligent of current biological species depend on many different
examples of earlier transitions in types of metacognition, in different environments,
on different scales, with widely varying mathematical complexity and power. It may
take many decades, or perhaps centuries, of interdisciplinary collaboration to fill
in the gaps and extend our knowledge to the level that will permit us to build
machines with squirrel-like or crow-like, or elephant-like or ape-like or human-like
capabilities.

Collaborative mathematical meta-cognition
They will need the ability to make mathematical discoveries and share and criticise
them in collaborative ways, perhaps in something like the processes by which our
ancestors made discoveries that contributed to Euclid's Elements. I suspect some of
those abilities will fit Immanuel Kant's specification for the ability to discover
synthetic necessary truths (in his Critique of Pure Reason): 
 http://www.marxists.org/reference/subject/ethics/kant/reason/ch01.htm
  "THERE can be no doubt that all our knowledge begins with experience.
  For how should our faculty of knowledge be awakened into action did not
  objects affecting our senses partly of themselves produce
  representations, partly arouse the activity of our understanding to
  compare these representations, and, by combining or separating them,
  work up the raw material of the sensible impressions into that
  knowledge of objects which is entitled experience? In the order of
  time, therefore, we have no knowledge antecedent to experience, and
  with experience all our knowledge begins.

  But though all our knowledge begins with experience, it does not follow
  that it all arises out of experience. For it may well be that even our
  empirical knowledge is made up of what we receive through impressions
  and of what our own faculty of knowledge (sensible impressions serving
  merely as the occasion) supplies from itself. If our faculty of
  knowledge makes any such addition, it may be that we are not in a
  position to distinguish it from the raw material, until with long
  practice of attention we have become skilled in separating it. This,
  then, is a question which at least calls for closer examination, and
  does not allow of any off-hand answer: -- whether there is any
  knowledge that is thus independent of experience and even of all
  impressions of the senses. Such knowledge is entitled a priori, and
  distinguished from the empirical, which has its sources a posteriori,
  that is, in experience. ...
  ...
  In what follows, therefore, we shall understand by a priori knowledge,
  not knowledge independent of this or that experience, but knowledge
  absolutely independent of all experience. Opposed to it is empirical
  knowledge, which is knowledge possible only a posteriori, that is,
  through experience.
  ...
We should soon be able to say more than Kant, e.g. about the meta-competences,
meta-meta-competences, meta-meta-....competences provided by evolution and by
genomes interacting with individual development, that make all this possible.
For an introduction to a distinction between pre-configured and meta-configured
competences (Jackie Chappell and Aaron Sloman, 2007) see:
http://www.cs.bham.ac.uk/research/projects/cosy/papers/#tr0609

To be continued, with more examples of mathematical competences
that normally go unnoticed.

For examples of "toddler theorems" see:
http://www.cs.bham.ac.uk/research/projects/cogaff/misc/toddler-theorems.html
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Planes question (Part 2):
The question was:
Is it possible for three plane surfaces to bound a finite volume?
If you have answered it can you specify how you answered it?
Did you derive the result from a set of axioms?

If you have not found an answer to the question, consider this alternative question:
Is it possible for two straight lines to bound (completely enclose) a finite area?

If not, why not?

If it is possible, then please provide an example (and email me!).

The questions about what can be enclosed by planes and lines are concerned with
spaces in which entities can move and change relationships continuously. Not all
mathematical domains are like Euclidean space.

There are some hybrid domains that are partly like euclidean space and partly
discrete. The next section provides examples that not everyone will fine familiar.
_____________________________________________________________________________________

Example: diagonal motion on a rectangular grid:

Suppose you have a large rectangular grid made of small squares arranged in lines and
columns. The grid can be arbitrarily large.

Case 1:
Suppose you have four blue buttons and four red buttons, in arbitrary locations
on the grid. You are allowed to move the blue buttons but only diagonally (through
the corners of the squares, not horizontally or vertically).

Case 2:
Like case 1, but with five red and five blue buttons.

Question:
Using only diagonal moves of the blue buttons
will it always be possible to get each blue button to a red button, so that each
red button has a different blue button on it?



               XX
     Can the blue buttons in grid (a) be transferred to the locations of the
     red buttons using only diagonal moves. What about grid (b)?

Supplementary question:
Does the answer depend on whether the red and the blue buttons are arranged in
similar ways, e.g. both in a horizontal row, or both in a vertical row, or one
vertical one horizontal, or some other configuration?

Does the answer depend on the fact that five is a prime number and four is not?

Does extending the grid so that larger detours are possible make a difference to
the answer?

If you find both an answer to a question and a proof that it is a correct answer I'll
be interested to learn how you proved it. There isn't one right way, though some proofs
are more elegant and more general than others. Can you characterise the domain of
structures (including processes) to which your proof applies?

An alternative exercise
In how many different ways can you prove that interior angles of a planar triangle
add up to half a rotation (=180 degrees)?
What happens to your proof on the surface of a sphere.
To what mathematical domain of structures is your proof relevant?

To find out more about that problem see
http://www.cs.bham.ac.uk/research/projects/cogaff/misc/triangle-sum.html

Question
What are the information-processing requirements for a future robot to be able to
think about such problems and discover proofs?
Can computer-based machines do this now? Discussed in:
http://www.cs.bham.ac.uk/research/projects/cogaff/misc/triangle-theorem.html
http://www.cs.bham.ac.uk/research/projects/cogaff/misc/triangle-sum.html

See also Mary Pardoe's proof of the triangle sum theorem:
http://www.cs.bham.ac.uk/~axs/fig/triangles-notext.jpg

Can you be sure it will work on any triangle?
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NOTE (Added 5 Nov 2013):
Lurking behind the scenes in this document is my belief that Immanuel Kant's theory
that mathematical discoveries lead to what he described as "synthetic apriori knowledge"
is basically correct, even though he used Euclidean geometry as an example and
Euclidean geometry turned out to have an empirical component that was refuted by
Eddington's observation of stellar shift during a solar eclipse in 1919, which
confirmed Einstein's prediction based in the general theory of relativity, according
to which space is non-euclidean in the presence of physical masses. The argument
(which I encountered among philosophers in Oxford in the late 1950s), that Kant had
been shown to be wrong by Einstein ignored the richness of Euclidean geometry and the
fact that the vast majority of it remained intact. After I learnt about Artificial
Intelligence and started programming, around 1970 I acquired the ambition to show
that Kant was correct, by demonstrating how his claimed processes of non-empirical
discovery, triggered by empirical discoveries, but based on genetically endowed
abilities to find a deeper structure, could be demonstrated in a 'baby' robot
interacting with its environment and making discoveries in something like the way
human children do, including discoveries of the sort that might have contributed to
Euclid's elements, and other forms of mathematics.

For various reasons this challenge turned out to be far more difficult than I
expected, so the ideas presented here are merely a report on work in progress, in a
difficult long term research project.
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Planes question (Part 3):
(Updated: 22 Nov 2013)

The question was:
Is it possible for three plane surfaces to bound a finite volume?

The alternative, easier, question was:
Is it possible for two straight lines to bound (completely enclose) a finite area?

I believe it is possible for non-mathematicians to think about both questions,
especially the second, and reach a conclusion that is essentially understood as
a necessary truth (e.g. two straight lines cannot bound a finite area, though they
can (if they are infinitely long) divide the plane containing them into three or four
disjoint infinite areas.

Likewise by considering alternative possible configurations of flat surfaces, first
one surface, then two, then three, you should be able to convince yourself that three
planes cannot completely enclose a volume though they can divide a 3-D space into
separate volumes in various ways (one way would be as an infinite tube of fixed
cross-section).

Sometimes impossibilities are unobvious. For example the painting by Pieter Breugel
entitled "The Magpie on the Gallows" (1568) includes a 3-D object that is impossible
if all the parts of the gallows are interpreted in a natural way, though the impossibility
may not be noticed at first: http://im-possible.info/english/art/classic/bruegel.html

Breugel was not the only artist to discover so-called "impossible objects" before
Roger Penrose discovered his impossible triangle. The Swedish artist Oscar
Reutersvärd produced a similar but more interesting version in 1934.

If you were provided with a pile of rectangular blocks of various sorts could you
construct a configuration like the one depicted in the picture below?

        badblocks

That example was inspired by the first of Reutersvärd's impossible objects,
apparently discovered while developing a doodle derived from a star, shown on the
commemorative stamp here.

The main point I want to make is that many humans, presumably including the original
developers of Euclidean geometry are able to think about a type of spatial
configuration, specified either verbally or in a picture of how it should look, or a
specification of how to construct one, and then discover that the configuration is
impossible, even if many that are close to it are possible. Adding the last brick
will always produce something different from what is depicted or otherwise specified.

The discovery of such impossibilities seems to make use of deep biological
capabilities required for reasoning about positive and negative affordances in the
physical environment, or more generally reasoning about possible ways in which the
environment might have been different or could be made different from how it is.

Right now I don't think there are any AI reasoning systems, nor any known brain
mechanisms, capable of explaining how such discoveries are made. One thing is clear:
we do not need to start from a set of axioms expressed in a logical formalism and
then derive a formula stating the impossibility. The possibility of constructing that
sort of proof was discovered long after Euclid, and it should be obvious that whatever
formalisation of geometry is used, the discovery that formula F ('such and such is
impossible') can be derived logically from a set of axioms is not the sort of
discovery that I have been talking about and which led to Euclid's elements.

Rather, possibilities and impossibilities (including positive and negative affordances)
can be discovered by inspecting particular configurations in a certain way that I
suspect has never been characterised accurately, though Kant seems to have been the
first to try, and many anti-logicist, anti-formalist mathematicians have tried since Kant.

For more on this see the section on impossible objects in the discussion of gaps in
current theories and models of visual perception: here, and the discussion of
"hidden depths in triangle qualia" here.
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Conclusion (provisional)

Since the precursors of Euclid whose discoveries led to Euclid's Elements, and the
branch of mathematics that we now think of as Euclidean geometry, could not have used
explicit axioms and explicit inference rules in making those discoveries, there must
be a kind of mathematical discovery procedure that is independent of such apparatus.

I believe that is close to what Immanuel Kant was claiming when he argued that
mathematical knowledge could be necessary (non-contingent), non-empirical (though
"awakened" or "triggered" by experience) and synthetic (not true by definition).

The later development of the axiomatic method merely introduced a new branch of
mathematics, and confused some mathematicians (not all) into thinking that was the
only true mathematics. Benoit Mandelbrot fulminated against that view of mathematics
at guest lecture at IBM UK about 20 years ago.

I agree with him. The question about planes bounding a volume was discussed in my
DPhil thesis 1962, as one of many examples supporting Kant.
http://www.cs.bham.ac.uk/research/projects/cogaff/07.html#706

The questions about sliding blue and red buttons around a rectangular grid can easily
be answered if you notice that the grid squares can be divided into two colours, e.g.
white and black on a chess board, and consider what changes diagonal moves of a
button can make.

Although the plane surface is continuous, and motions of buttons are continuous, in
the domain of motions between squares there are discontinuities in relationships that
turn this into a partly discrete domain.

I have presented the questions about what planes and straight lines are capable of
enclosing and the questions about sliding buttons, because I think everyone able to
read this document is also capable of making mathematical discoveries, in something
like the way precursors of Euclid might have done (possibly with a little help in
some cases). If you do that you will have the type of experience that I think Kant
was attempting to characterise in his theory about the nature of mathematical
discovery.

In doing so, you do not need to start from a set of axioms specifying assumptions in
a logical notation, nor a set of logical inference rules against which you check your
reasoning steps. That sort of activity defines a type of mathematical domain that was
not discovered by mathematicians/logicians until the nineteenth century, although
Aristotle, Leibniz and probably others took some preliminary steps.

We shall have a better understanding of the nature of mathematics when we know how to
build artificial mathematicians that can make discoveries in the same sorts of ways
as human mathematicians have been doing for over two thousand years.

So far we don't know how to do that, despite some partly relevant artificial
reasoners, that can go through the motions, though without knowing what they are
doing or why, in most cases.

Note that I am not claiming, and I don't believe Kant was claiming, that humans are
infallible when they discover proofs. It is clear from the work of Imre Lakatos that
mathematicians, even great mathematicians and communities of mathematicians, can
make mistakes of various kinds, e.g. failing to notice special cases. However,
mathematical mistakes of various kinds can be detected and corrected: and that is part
of the normal process of mathematical discovery. I have some examples in a
discussion of ways of thinking about areas of triangles here:
http://www.cs.bham.ac.uk/research/projects/cogaff/misc/triangle-theorem.html
I'll add more on that later.

_______________________________________________________________________________

NOTE: I am not claiming that I have explained how the forms of mathematical reasoning
discussed here are possible. One form of explanation would be a specification for
design of a robot that can be shown to enable a 'baby' robot to make mathematical
discoveries as humans do partly as a result of exploring the environment and partly
as a result of use of very powerful forms of reasoning whose possibility is somehow
encoded in its "genome", but which has to be developed by being applied to the
environment.
Compare John McCarthy on The Well Designed Child.
and Chappell and Sloman (2007) on pre-configured and meta-configured competences:
http://www.cs.bham.ac.uk/research/projects/cosy/papers/#tr0609
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Further Reading
(To be expanded)

TO BE EXPANDED


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Maintained by Aaron Sloman
School of Computer Science
The University of Birmingham