Preprint updated: July 6, 2016
Natural Vision and Mathematics: Seeing
Impossibilities
Bridging the Gap Workshop IJCAI 2016(Furbach & Schon, 2016)
Aaron Sloman*
School of Computer Science, University of Birmingham, UK
Abstract. The Turing-inspired Meta-Morphogenesis project investigates
forms of biological information processing produced by evolution since
the earliest life forms, especially information processing mechanisms that
made possible the deep mathematical discoveries of Euclid, Archimedes,
and other ancient mathematicians. Such mechanisms must enable per-
ception of possibilities and constraints on possibilities - types of affor-
dance perception not explicitly discussed by Gibson, but suggested by
his ideas. Current AI vision and reasoning systems lack such abilities.
A future AI project might produce a design for "baby" robots that can
"grow up" to become mathematicians able to replicate (and extend) some
of the ancient discoveries, e.g. in the way that Archimedes extended Eu-
clidean geometry to make trisection of an arbitrary angle possible. This
is relevant to many kinds of intelligent organism or machine perceiving
and interacting with structures and processes in the environment. This
would demonstrate the need to extend Dennett's taxonomy of types of
mind to include Euclidean (or Archimedean) minds, and would support
Immanuel Kant's philosophy of mathematics.
Keywords: AI, Kant, Mathematics, Meta-morphogenesis, intuition, Eu-
clid, Geometry,Topology, Kinds-of-minds, Meta-cognition, Meta-meta-
cognition, etc.
Mathematics and computers
It is widely believed that computers will always outperform humans in math-
ematical reasoning. That, however, is based on a narrow conception of math-
ematics that ignores some of the history of mathematics, e.g. achievements of
Euclid and Archimedes, and also ignores kinds of mathematical competence
that are a part of our everyday life, but mostly go unnoticed, e.g. topological
reasoning abilities.1 These are major challenges for AI, especially AI projects to
replicate or model human mathematical competences. I don't think we are ready
to build working systems with these competences, but I'll introduce a research
programme that may, eventually, lead us towards adequate models. It challenges
common assumptions about the nature and evolution of human language and
hints at evolved mechanisms supporting mathematical competences and closely
related perceptual and reasoning competences.
Formal mechanisms based on use of arithmetic, algebra, and logic, dominate
AI models of mathematical reasoning, but the great ancient mathematicians did
not use modern logic and formal systems. Such things are therefore not necessary
for mathematics. they are part of mathematics: a recent addition, but appear not
* This is a snapshot of part of the Turing-inspired Meta-Morphogenesis project.
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Natural Vision and Mathematics
to be needed for older forms of human and animal reasoning (e.g. topological rea-
soning). By studying achievements of ancient mathematicians, pre-verbal human
toddlers, and intelligent non-human animals, especially perception and reasoning
abilities that are not matched by current AI systems, or explained by current
theories of how brains work, we can identify major challenges. Powerful new
internal languages are needed, similar to evolved languages used for perceiving,
thinking about, and reasoning about shapes, structures and spatial processes. If
such internal languages are used by intelligent non-human animals and pre-verbal
toddlers, their evolution must have preceded evolution of human languages for
communication (Sloman, 1978b, 1979, 2015). Structured internal languages (for
storing and using information) must have evolved before external languages, since
there would be nothing to communicate and no use for anything communicated,
without pre-existing internal mechanisms for constructing, manipulating and
using structured percepts, intentions, beliefs, plans, memorised maps, etc.
The simplest organisms (viruses?) may have only passive physical/chemical
reactions, and only trivial decisions and uses of information (apart from genetic
information). Slightly more complex organisms may use information only for
taking Yes/No or More/Less or Start/Stop decisions, or perhaps selections from a
pre-stored collection of possible internal or external actions. (Evolution's menus!)
More complex internal meaning structures are required for cognitive functions
based on information contents that can vary in structure and complexity, like
the Portia spider's ability to study a scene for about 20 minutes and then climb a
branching structure to reach a position above its prey, and then drop down for its
meal (Tarsitano, 2006). This requires an initial process of information collection
and storage in a scene-specific structured form that later allows a pre-computed
branching path to be followed even though the prey is not always visible during
the process, and portions of the scene that are visible keep changing as the
spider moves. Portia is clearly conscious of much of the environment, during and
after plan-construction. As far as I know, nobody understands in detail what the
information processing mechanisms are that enable the spider to take in scene
structures and construct a usable 3-D route plan, though we can analyse the
computational requirements on the basis of half a century of AI experience.
This is one example among many cognitive functions enabling individual
organisms to deal with static structured situations and passively perceived or
actively controlled processes, of varying complexity, including control processes
in which parts of the perceiver change their relationships to one another (e.g.
jaws, claws, legs, etc.) and relationships to other things in the environment (e.g.
food, structures climbed over, or places to shelter).
Abilities to perceive plants in natural environments, such as woodlands or
meadows, and, either immediately or later, make use of them, also requires acqui-
sition, storage and use of information about complex objects of varying struc-
tures, and information about complex processes in which object-parts change
their relationships, and change their visual projections as the perceiver moves.
Acting on perceived structures, e.g. biting or swallowing them, or carrying them
to a part-built nest to be inserted, will normally have to be done differently in
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different contexts, e.g. adding twigs with different sizes and shapes at different
stages in building a nest. How can we make a robot that does this?
Conjecture:
Non-human abilities to create and use information structures of varying complex-
ity are evolutionary precursors of human abilities to use grammars and semantic
rules for languages in which novel sentences are understood in systematic ways
to express different more or less complex percepts, intentions, or plans to solve
practical problems, e.g. using a lexicon, syntactic forms, and compositional
semantics. In particular, a complex new information structure can be assembled
and stored, then later used as an information structure (e.g. plan, hypothesis)
while selecting or controlling actions.
We must not, of course, be deceived by organisms that appear to be intention-
ally creating intended structures but are actually doing something much simpler
that creates the structures as a by-product, like bees huddled together, oozing
wax, vibrating, and thereby creating a hexagonal array of cavities, that look
designed but were not. Bees have no need to count to six to do that. Most nest-
building actions, however, are neither random nor fixed repetitive movements.
They are guided, in part, by missing portions of incomplete structures, where
what's missing and what's added keeps changing. So the builders need internal
languages with generative syntax, structural variability, (context sensitive) com-
positional semantics, and inference mechanisms in order to be able to encode
all the relevant varieties of information needed. Nest building competences in
corvids and weaver birds are examples. Human architects are more complicated!
Abilities to create, perceive, change, manipulate, or use meaning structures
(of varying complexity) enable a perceiver of a novel situation to take in its struc-
ture and reason hypothetically about effects of possible actions - without having
to collect evidence and derive probabilities. The reasoning can be geometric or
topological, without using any statistical evidence: merely the specification of
spatial structures. Reasoning about what is impossible (not merely improbable)
can avoid wasted effort. For example, the "polyflap"domain2 was proposed as
an artificial environment illustrating some challenging cognitive requirements. It
contains arbitrary 2D polygonal shapes each with a single (non-flat) fold forming
a new 3D object. An intelligent agent exploring polyflaps could learn that any
object resting on surfaces where it has a total of two contact points can rotate
in either direction about the line joining the contact points. Noticing this should
allow the agent to work out that achieving stability would require a supporting
surface on which at least one more part of the supported object can rest. In
simple cases all three points are on one horizontal plane: e.g. on a floor. But an
intelligent agent that understands stability should be able to produce stability
with three support points on different, non-co-planar surfaces, e.g. the tops of
three pillars with different heights. Any two of the support points on their own
would allow tilting about the line joining the points. But if the third support
point is not on that line, and a vertical line through the object's centre of gravity
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Natural Vision and Mathematics
goes through the interior of the triangle formed by the support points then the
structure will be stable3.
An intelligent machine should be able to reason in similar ways about novel
configurations. This illustrates a type of perception of affordances in the spirit
of Gibson's theory, but going beyond examples he considered. I don't recall
him mentioning use of geometrical or topological reasoning in deciding what
would be stable. This contradicts a common view that affordances are discovered
through statistical learning. Non-statistical forms of reasoning about affordances
in the environment (possibilities for change and constraints on change) may have
been a major source of the amazing collection of discoveries about topology
and geometry recorded in Euclid's Elements. Such forms of discovery are very
important, but still unexplained.
Intelligent non-human animals (e.g. Portia), and humans, use evolved mecha-
nisms that can build, manipulate and use structured internal information records
whose required complexity can vary and whose information content is derivable
from information about parts, using some form of "compositional semantics",
as is required in human spoken languages, logical languages, and programming
languages. However, the internal languages need not use linear structures, like
sentences. In principle they could be trees, graphs, nets, map-like structures or
types of structure we have not yet thought of. Human sign languages are not
restricted to discrete temporal sequences of simple signs: usually movements of
hands, head and parts of the face go on in parallel. This may be related to use
of non-linear internal languages for encoding perceptual information, including
changing visual information about complex structured scenes and tactile infor-
mation gained by manual exploration of structured objects. In general the 3-D
world of an active 3-D organism is not all usefully linearizable.
The variety of types of animal that can perceive and act intelligently in
relation to novel perceived environmental structures, suggests that many use
"internal languages" in a generalised sense of "language" ("Generalised Lan-
guages" or GLs), with structural variability and context sensitive compositional
semantics, which must have evolved long before human languages were used for
communication (Sloman & Chappell, 2007)4. The use of external, structured,
languages for communication presupposes internal perceptual mechanisms using
(GLs), e.g. for parsing messages and relating them to percepts and intentions.
There are similar requirements for intelligent nest building by birds and for
many forms of complex learning and problem solving by other animals, including
elephants, squirrels, cetaceans, monkeys, apes and pre-verbal human toddlers5.
Is there a circularity?
In the past, philosophers would have argued (scornfully!) that postulating the
need for an internal language IL to be used in understanding an external language
EL, would require yet another internal language for understanding IL, and so
on, leading to an infinite regress. But AI and computer systems engineering
3 I first discovered this "Polyflap stability theorem" while writing this paper.
5
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demonstrate that there need not be an infinite regress. Our understanding has
expanded enormously since the 1960s as AI and software engineering researchers
should know. How brains achieve similar feats is unknown, however.
Mechanisms for communicating information using familiar human languages
would have no point if senders and recipients were not already information
users, otherwise they would have nothing to communicate, and would have no
way to use information communicated to them. Yet there is much resistance
to the idea that rich internal languages used for non-communicative purposes
evolved before communicative languages. That may be partly because many
people do not understand the computational requirements for the competences
displayed by pre-verbal humans and other animals, and partly because they
don't understand complex virtual machines do not require an infinite hierarchy
of internal languages.
These comments about animals able to perceive, manipulate and reason
about varied objects and constructions, apply also to pre-verbal human toddlers
playing with toys and solving problems, including manipulating food, clothing,
and even their parents. (See also Note 5.)
The full repertoire of biological vehicles and mechanisms for information
bearers must include both mechanisms and meta-mechanisms (mechanisms that
construct new mechanisms) produced by natural selection and inherited via
genomes, and also individually discovered/created mechanisms, in humans, and
other altricial species with "meta-configured" competences in the terminology
of (Chappell & Sloman, 2007).
Dennett (e.g. 1995) firmly resists the internal language idea: he argues, on the
contrary, that consciousness followed evolution of language mechanisms used for
external communication, later also used internally for silent self-communication.
But Portia spiders surely did not need ancestors that discussed planned routes
to capture prey before they evolved the ability to reason silently about terrain
and routes and plans.
Creation vs Learning:
Evidence from deaf children in Nicaragua (Senghas, 2005) and subtle clues
in non-deaf children, show that children do not learn languages from existing
users. Rather, they develop mechanisms of increasing power, that enable them
to create languages collaboratively6. Normally children collaborate as a relatively
powerless minority, so the creation process looks like imitative learning.
Many familiar processes of perceiving, learning, intending, reasoning, plan-
ning, plan execution, plan debugging, etc. would be impossible if humans (and
perhaps some other intelligent animals) did not have rich internal languages
and language manipulation mechanisms (GL competences). For more on this see
Note 4. Some philosophers, cognitive scientists and others believe that structure-
based "old fashioned" AI has failed. But the truth is that NO form of AI
has "succeeded", apart froom narrowly focused AI applications. Many of them
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Natural Vision and Mathematics
are very shallow and more restricted than symbolic planners and reasoners7.
Compare (Rescorla, 2015).
We still need to learn much more about the nature of internal GLs, the
mechanisms required, and their functions in various kinds of intelligent animal.
We should not expect them to be much like kinds of human languages or
computer languages we are already familiar with, if various GLs also provide
the internal information media for perceptual contents of intelligent and fast
moving animals like crows, squirrels, hunting mammals, monkeys, and cetaceans.
Taking in information about rapidly changing scenes, needs something different
from Portia's internal language for describing a fixed route. Moreover, languages
for encoding information about changing visual contents will need different sorts
of expressive powers from languages for human conversation about the weather
or the next meal8.
Many people have studied and written about various aspects of non-verbal
communication and reasoning, including, for example, contributors to (Glasgow,
Narayanan, & Chandrasekaran, 1995), and others who have presented papers on
diagrammatic reasoning, or studied the uses of diagrams by young children. But
there are still deep gaps, especially related to mathematical discoveries.
Many of Piaget's books provide examples, some discussed below. He un-
derstood better than most what needed to be explained, but he lacked any
understanding of programming or AI and he therefore sought explanatory models
where they could not be found, e.g. in boolean algebras and group theory.
The importance of Euclid for AI
AI sceptics attack achievements of AI, whereas I am attacking the goals
of researchers who have not noticed the need to explain some very deep, well
known but very poorly understood, human abilities: the abilities that enabled
our ancestors prior to Euclid, without the help of mathematics teachers, to make
the sorts of discoveries that eventually stimulated Euclid, Archimedes and other
ancient mathematicians who made profound non-empirical discoveries, leading
up to what is arguably the single most important book ever written on this
planet: Euclid's Elements9. Thousands of people all around the world are still
putting its discoveries to good use every day even if they have never read it10.
botanical challenge for vision researchers.
9 There seems to be uncertainty about dates and who contributed what. I'll treat
Euclid as a figurehead for a tradition that includes many others, especially
Thales, Pythagoras and Archimedes, perhaps the greatest of them all, and
a mathematical precursor of Leibniz and Newton. (More names are listed
I don't know much about mathematicians on other continents at that time or earlier.
I'll take Euclid to stand for all of them, because of the book that bears his name.
10 Moreover, it does not propagate misleading falsehoods, condone oppression of women
or non-believers, or promote dreadful mind-binding in children.
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As a mathematics graduate student interacting with philosophers around
1958, I formed the impression that the philosopher whose claims about mathe-
matics were closest to what I knew about doing mathematics, especially geom-
etry, was Immanuel Kant . But his claims about our knowledge of Euclidean
geometry seemed to have been contradicted by recent theories of Einstein and
empirical observations by Eddington. Philosophers therefore thought that Kant
had been refuted, ignoring the fact that Euclidean geometry without the parallel
axiom remains a deep and powerful body of geometrical and topological knowl-
edge, and provides a basis for constructing three different types of geometry:
Euclidean, elliptical and hyperbolic, the last two based on alternatives to the
parallel axiom11. We'll see also that it also has an extension that makes trisection
of an arbitrary angle possible, unlike pure Euclidean geometry. These are real
mathematical discoveries about a type of space, not about logic, and not about
observed statistical regularities.
My experience of doing mathematics suggested that Kant was basically right
in his claims against David Hume: many mathematical discoveries provide knowl-
edge that is non-analytic (i.e. synthetic, not proved solely on the basis of logic
and definitions), non-empirical (i.e. possibly triggered by experiences, but not
based on experiences, nor subject to refutation by experiment or observation, if
properly proved), and necessarily true (i.e. incapable of having counter-examples,
not contingent).
This does not imply that human mathematical reasoning is infallible: (Lakatos,
1976) demonstrated that even great mathematicians can make various kinds of
mistakes in exploring something new and important. Once discovered, mistakes
sometimes lead to new knowledge. So a Kantian philosopher of mathematics
need not claim that mathematicians produce only valid reasoning12.
Purely philosophical debates on these issues can be hard to resolve. So when
Max Clowes13 introduced me to AI and programming around 1969 I formed the
intention of showing how a baby robot could grow up to be a mathematician
in a manner consistent with Kant's claims. But that has not yet been achieved.
What sorts of discovery mechanisms would such a robot need?
Around that time, a famous paper by McCarthy and Hayes claimed that
logic would suffice as a form of representation (and therefore also reasoning)
for intelligent robots. The paper discussed the representational requirements for
intelligent machines, and concluded that "... one representation plays a dominant
role and in simpler systems may be the only representation present. This is a
representation by sets of sentences in a suitable formal logical language... with
function symbols, description operator, conditional expressions, sets, etc." They
discussed several kinds of adequacy of forms of representation, including meta-
physical, epistemological and heuristic adequacy (vaguely echoing distinctions
Chomsky had made earlier regarding types of adequacy of linguistic theories).
12 My 1962 DPhil thesis presented Kant's ideas, before I had heard about AI. (Sloman,
1962)
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Despite many changes of detail, a great deal of important AI research has since
been based on the use of logic as a GL, now often enhanced with statistical
mechanisms.
Nevertheless thinking about mathematical discoveries in geometry and topol-
ogy and many aspects of everyday intelligence suggested that McCarthy and
Hayes were wrong about the sufficiency of logic. I tried to show why in (Sloman,
1971) and later papers. Their discussion was more sophisticated than I have indi-
cated here. In particular, they identified different sorts of criteria for evaluating
forms of representation, used for thinking or communicating:
A representation is metaphysically adequate if the world could have
that form without contradicting relevant facts about aspects of reality.
A representation is called epistemologically adequate for a person or
machine if it can be used practically to express the facts that one actually
has about the aspect of the world.
A representation is called heuristically adequate if the reasoning pro-
cesses actually gone through in solving a problem are expressible in the
language.
Ordinary language is obviously adequate to express the facts that
people communicate to each other in ordinary language. It is not adequate
to express what people know about how to recognize a particular face.
They concluded that a form of representation based on logic would be heuris-
tically adequate for intelligent machines observing, reasoning about and acting
in human-like environments. But this does not provide an explanation of what
adequacy of reasoning is. For example, one criterion might be that the reasoning
should be incapable of deriving false conclusions from true premisses.
At that time I was interested in understanding the nature of mathematical
knowledge (as discussed in (Kant, 1781)). I thought it might be possible to
test philosophical theories about mathematical reasoning by demonstrating how
a "baby robot" might begin to make mathematical discoveries (in geometry
and arithmetic) as Euclid and his precursors had. But I did not think logic-
based forms of representation would be heuristically adequate because of the
essential role played by diagrams in the work of mathematicians like Euclid and
Archimedes, even if some modern mathematicians felt such diagrams should be
replaced by formal proofs in axiomatic systems - apparently not noticiing that
that changes the investigation to a different branch of mathematics. The same
can be said about Frege's attempts to embed arithmetic in logic.
(Sloman, 1971) offered alternatives to logical forms of representation, in-
cluding (among others) "analogical" representations that were not based on the
kind of (Fregean) function/argument structure used by logical representations.
Despite explicit disclaimers, the paper is often mis-reported as claiming that
analogical representations are isomorphic with what they represent: which is
clearly not generally true, since a 2-D picture cannot be isomorphic with the 3-
D scene it represents, one of several reasons why AI vision research is so difficult.
A revised, extended, notion of validity of reasoning, was shown to include
changes of pictorial structure that correspond to possible changes in the entities
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or scenes depicted, but this did not explain how to implement a human-like
diagrammatic reasoner in geometry or topology. 45 years later there still seems
to be no AI system that is capable of discovering and understanding deep
diagrammatic proofs of the sorts presented by Euclid, Archimedes and others.
This is associated with inability to act intelligently in a complex and changing
environment that poses novel problems involving spatial structures.
A more subtle challenge comes from the discovery known to Archimedes that
there is a simple and natural way of extending Euclidean geometry by adding
the neusis construction which makes it easy to trisect an arbitrary angle14.
As far as I know no current AI reasoning system could make such a discovery.
It is definitely not connected with statistical learning: that would not provide
insight into mathematical necessity or impossibility. The result was not derived
from axioms: it showed that Euclid's axioms could be extended. Mary Pardoe,
a former student, discovered a related but simpler extension to Euclid, allowing
the triangle sum theorem to be proved without using the parallel axiom15.
As far as I know, nobody in AI has tried to implement abilities to discover
Euclidean geometry, including topological reasoning, and the extensions men-
tioned here, in an AI system or robot with spatial reasoning abilities. I am still
trying to understand why it is so difficult. (But not impossible, I hope.)
Not only competences of adult human mathematicians still await replication
in AI. Many intelligent animals, such as squirrels, nest building birds, elephants
and even octopuses have abilities to discover and use spatial manipulations of ob-
jects in their environment (or their own body parts) and apparently understand
what they are doing. Betty, a New Caledonian crow, made headlines in 2002
when she was observed (in Oxford) making a hook from a straight piece of wire
in order to extract a bucket of food from vertical glass tube (Weir, Chappell, &
Kacelnik, 2002). The online videos demonstrate something not mentioned in the
original published report, namely that Betty was able to make hooks in several
different ways, all of which worked immediately without any visible signs of trial
and error. She clearly understood what was possible, despite not having lived in
an environment containing pieces of wire or any similar material (twigs either
break if bent or tend to straighten when released). It's hard to believe that such
a creature could be using logic, as recommended by McCarthy and Hayes. But
what are the alternatives? Perhaps a better developed theory of GLs will provide
the answer and demonstrate it in a running system.
The McCarthy and Hayes paper partly echoed Frege, who had argued in
1884 that arithmetical knowledge could be completely based on logic, But he
denied that geometry could be (despite Hilbert's axiomatization of Euclidean
geometry). (Whitehead & Russell, 1910-1913) had also attempted to show how
the whole of arithmetic could be derived from logic. though Russell oscillated in
his views about the philosophical significance of what had been demonstrated.
I was unaware of this until I found the Wikipedia article in 2015:
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Natural Vision and Mathematics
Frege was right about geometry: what Hilbert axiomatised was a combination
of logic and arithmetic that demonstrated that arithmetic and algebra contained
a model of Euclidean geometry based on arithmetical analogues of lines, circles,
and operations on them, discovered by Descartes. But doing that did not imply
that the original discoveries were arithmetical discoveries rather than discoveries
about spatial structures, relationships and transformations. (Many mathematical
domains have models in other domains.)
When the ancient geometricians made their discoveries, they were not rea-
soning about relationships between logical symbols in a formal system or about
numbers or equations. This implies that in order to build robots able to repeat
those discoveries it will not suffice merely to give them abilities to derive logical
consequences from axioms expressed in a logical notation, such as predicate
calculus or the extended version discussed by McCarthy and Hayes.
Instead we'll need to understand what humans do when they think about
shapes and the ways they can be constructed, extended, compared, etc. This
requires more than getting machines to answer the same questions in laboratory
experiments, or pass the same tests in mathematical examinations. We need to
develop good theories about what human mathematicians did when they made
the original discoveries, without the help of mathematics teachers, and without
the kind of drill and practice now often found in mathematical classrooms. Those
theories should be sufficiently rich and precise to enable us to produce working
models that demonstrate the explanatory power of the theories.
As far as I know there is still nothing in AI that comes close to enabling robots
to replicate the ancient discoveries in geometry and topology, nor any formalism
that provides the capabilities GLs would need, in order to explain how products
of evolution perceive the environment, solve problems, etc. Many researchers
in AI, psychology and neuroscience, now think the core requirement is a shift
from logical reasoning to statistical/probabilistic reasoning. But that cannot
prove necessary truths. I suspect a deeper advance will come from extending
techniques for reasoning about possibilities, impossibilities, changing topological
relationships and the use of partial orderings (of distance, size, orientation,
curvature, slope, containment, etc.)16.
What about arithmetic?
The arguments against any attempt to redefine geometry in terms of what follows
from Hilbert's axioms can be generalised to argue against Frege's attempt to
redefine arithmetic in terms of what follows from axioms and rules for logical
reasoning. In both cases a previously discovered and partially explored math-
ematical domain was shown to be modelled using logic. But modelling is one
thing: replicating another.
The arithmetical discoveries made by Euclid and others long before the
discovery of modern logic were more like discoveries in geometry than like proofs
in an axiomatic system using only logical inferences. However, arithmetical
knowledge is not concerned only with spatial structures and processes. It involves
affordances.html
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general features of groups or sets of entities, and operations on them. For exam-
ple, acquiring the concept of the number six requires having the ability to relate
different groups of objects in terms of one-to-one correspondences (bijections).
So the basic idea of arithmetic is that two collections of entities may or may
not have a 1-1 relationship. If they do we could call them "equinumeric". The
following groups are equinumeric in that sense (treating different occurrences of
the same character as different items).
[U V W X Y Z] [P P P P P P] [W Y Y G Q P]
If we focus on types of character rather than instances, then the numbers are
different. The first box contains six distinct items, the second box only one type,
and the third box five types. For now, let's focus on instances not types.
The relation of equinumerosity has many practical uses, and one does not
need to know anything about names for numbers, or even to have the concept
of a number as an entity that can be referred to, added to other numbers etc.
in order to make use of equinumerosity. For example, if someone goes fishing to
feed a family and each fish provides a meal for one person, the fisherman could
take the whole family, and as each fish is caught give it to an empty-handed
member of the family, until everyone has a fish. Our intelligent ancestors might
have discovered ways of streamlining that cumbersome process: e.g. instead of
bringing each fish-eater to the river, ask each one to pick up a bowl and place it
on the fisherman's bowl. Then the bowls could be taken instead of the people,
and the fisherman could give each bowl a fish, until there are no more empty
bowls, then carry the laden bowls back.
What sort of brain mechanism would enable the first person who thought
of doing that to realise, merely by thinking about it, that it must produce
the same end result as taking all the people to the river? A non-mathematical
(statistical reasoning) individual would need to be convinced by repetition that
the likelihood of success is high. A mathematical mind would see the necessary
truth. How?
Of course, we also find it obvious that there's no need to take a collection of
bowls or other physical objects to represent individual fish-eaters. We could have
a number of blocks with marks on them, a block with one mark, a block with
two marks, etc., and any one of a number of procedures for matching people to
marks could be used to select a block with the right number of marks to be used
for matching against fish.
Intelligent fishermen could understand that a collection of fish matching the
marks would also match the people. How? Many people now find that obvious
but realising that one-one correspondence is a transitive relation is a major
intellectual achievement, crucial to abilities to use numbers. I doubt that any
neuroscientist knows how brains support this realisation. We also know that it is
not necessary to carry around a material numerosity indicator: we can memorise
a sequence of names and use each name as a label for the numerosity of the
sub-sequence up to that name, as demonstrated in (Sloman, 1978a, Chap8)17. A
human-like intelligent machine would also have to be able to discover such strate-
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gies, and understand why they work. This is totally different from achievements
of systems that do pattern recognition. Or deep statistical learning. Perhaps
studying intermediate competences in other animals will help us understand
what evolution had to do to produce human mathematicians. Understanding
mathematical induction over infinite sets is an even greater challenge. (This is
all much deeper than learning to assign number names.)
Piaget's work showed that five- and six-year old children have trouble under-
standing consequences of transforming 1-1 correlations, e.g. by stretching one of
two matched rows of objects (Piaget, 1952). When they do grasp the transitivity
have they found a way to derive it from some set of logical axioms using explicit
definitions? Or is there another way of grasping that if two collections A and B
are in a 1-1 correspondence and B and C are, then A and C must also be, even
if C is stretched out more in space?
I suspect that for most people this is more like an obvious topological theorem
about patterns of connectivity in a graph rather than something proved by logic.
But why is it obvious to adults and not to 5 year olds? Anyone who thinks it
is merely a probabilistic generalisation that has to be tested in a large number of
cases has not understood the problem, or lacks the relevant mechanisms in nor-
mal human brains. Does any neuroscientist understand what brain mechanisms
support discovery of such mathematical properties, or why they seem not to have
developed before children are five or six years old (if Piaget used good tests).
Much empirical research on number competences grossly over simplifies what
needs to be explained, omitting the role of reasoning about 1-1 correspondences.
It would be possible to use logic to encode the transitivity theorem in a usable
form in the mind of a robot, but it is not clear what would be required to mirror
the developmental processes in a child, or our adult ancestors who first discovered
these properties of 1-1 correspondences. They may have used a more general and
powerful form of relational reasoning of which this theorem is a special case. The
answer is not statistical (e.g. neural-net based) learning. Intelligent human-like
machines would have to discover deep non-statistical structures of the sorts that
Euclid and his precursors discovered.
The machines might not know what they are doing, like young children who
make and use mathematical or grammatical discoveries. But they should have the
meta-cognitive ability to become self-reflective and later make philosophical and
mathematical discoveries. I suspect human mathematical understanding requires
at least four layers of meta-cognition, each adding new capabilities, but will not
defend that here. Perhaps robots with such abilities in a future century will
discover how evolution produced brains with these capabilities (Sloman, 2013).
Close observation of human toddlers shows that before they can talk they
are often able to reason about consequences of spatial processes, including a 17.5
month pre-verbal child apparently testing a sophisticated hypotheses about 3-D
topology, namely: if a pencil can be pushed point-first through a hole in paper
from one side of the sheet then there must be a continuous 3-D trajectory by
which it can be made to go point first through the same hole from the other
Sloman Bridging
13
side of the sheet:18. (I am not claiming that my words accurately describe her
thoughts: but clearly her intention has that sort of complex structure even though
she was incapable of saying any such thing in a spoken language. What sort of
GL was she using? How could we implement that in a baby robot?)
Likewise, one does not need to be a professional mathematician to understand
why when putting a sweater onto a child one should not start by inserting a
hand into a sleeve, even if that is the right sleeve for that arm (Note 1). Records
showing 100% failure in such attempts do not establish impossibility, since they
provide no guarantee that the next experiment will also fail. Understanding
impossibility requires non-statistical reasoning.
Generalising Gibson
James Gibson proposed that the main function of perception is not to provide
information about what occupies various portions of 3-D space surrounding the
perceiver, as most AI researchers and psychologists had previously assumed (e.g.
(Clowes, 1971; Marr, 1982)) but rather to provide information about what the
perceiver can and cannot do in the environment: i.e. information about positive
and negative affordances - types of possibility.
Accordingly, many AI/Robotic researchers now design machines that learn
to perform tasks, like lifting a cup or catching a ball by making many attempts
and inferring probabilities of success of various actions in various circumstances.
But that kind of statistics-based knowledge cannot provide mathematical
understanding of what is impossible, or what the necessary preconditions or con-
sequences of certain spatial configurations and processes are. It cannot provide
the kind of reasoning capabilities that led up to the great discoveries in geometry
(and topology) (e.g. by Euclid and Archimedes) long before the development of
modern logic and the axiomatic method. I suspect these mathematical abilities
evolved out of abilities to perceive a variety of positive and negative affordances,
abilities that are shared with other organisms (e.g. squirrels, crows, elephants,
orangutans) which in humans are supplemented with several layers of metacog-
nition (not all present at birth).
Spelling this out will require a theory of modal semantics that is appro-
priate to relatively simple concepts of possibility, impossibility and necessary
connection, such as a child or intelligent animal may use (and thereby prevent
time-wasting failed attempts).
What sort of modal semantics?
I don't think any of the forms of "possible world" semantics are appropriate to
the reasoning of a child or animal that is in any case incapable of thinking about
the whole of this world let alone sets of alternative possible worlds. Instead,
the learner's modal semantics will have to be based on a grasp of ways in which
properties and relationships in a small portion of the world can change and which
combinations are possible or impossible. E.g. if two solid rings are linked it is
impossible for them to become unlinked through any continuous form of motion
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Natural Vision and Mathematics
or deformation - despite what seems to be happening on a clever magician's
stage. This form of modal semantics, concerned with possible rearrangements
of a portion of the world rather than possible whole worlds was proposed in
(Sloman, 1962). Barbara Vetter seems to share this viewpoint (Vetter, 2013).
Another type of example is in the following figure.
Fig. 1. Possible and impossible configurations of blocks.
(Swedish artist Oscar Reutersvard drew the impossible configuration in 1934)
What sort of visual mechanism is required to tell the difference between the
possible and the impossible configurations. How did such mechanisms evolve?
Which animals have them? How do they develop in humans? Can we easily
give them to robots? How can a robot detect that what it sees depicted is
impossible?19
A child could discover prime numbers by attempting to arrange groups of
blocks into NxM regular arrays. It works for twelve blocks but adding or removing
one makes the task impossible. Has any child ever discovered primeness in that
way? Which robot will be the first to do that? (Pat Hayes once informed me that
a frustrated conference receptionist trying to tidy uncollected name cards made
that discovery without recognizing its significance. She thought her inability to
make a rectangle in some cases was due to stupidity.)
The link to Turing
What might Alan Turing have worked on if he had not died two years after
publishing his 1952 paper on the Chemical basis of morphogenesis? Perhaps
the Meta-Morphogenesis (M-M) project: an attempt to identify significant tran-
sitions in types of information-processing capabilities produced by evolution,
and products of evolution, between the earliest (proto-)life forms and current
organisms, including changes that modify evolutionary mechanisms20.
Conclusion
Natural selection is more a blind mathematician than a blind watchmaker: it
19 Richard Gregory demonstrated that a 3-D structure can be built that looks exactly
like an impossible object, but only from a particular viewpoint, or line of sight.
References
15
discovers and uses "implicit theorems" about possible uses of physics, chem-
istry, topology, geometry, varieties of feedback control, symmetry, parametric
polymorphism, and increasingly powerful cognitive and meta-cognitive mecha-
nisms. Its proofs are implicit in evolutionary and developmental trajectories. So
mathematics is not a human creation, as many believe.
The "blind mathematician" later produced at least one species with meta-
cognitive mechanisms that allow individuals who have previously made "blind"
mathematical discoveries (e.g. what I've called "toddler theorems") to start
noticing, discussing, disputing and building a theory unifying the discoveries.
Later still, meta-meta-(etc?)cognitive mechanisms allowed products of meta-
cognition to be challenged, defended, organised, and communicated, eventually
leading to collaborative advances, and documented discoveries and proofs, e.g.
Euclid's Elements (sadly no longer a standard part of the education of our
brightest learners). Many forms of applied mathematics grew out of the results.
Unfortunately, most of the pre-history is still unknown and may have to be based
on intelligent guesswork and cross-species comparisons. Biologically inspired
future AI research should provide clues as to currently unknown intermediate
forms of biological intelligence. I don't think current AI techniques are anywhere
near this, especially statistics-based AI. Perhaps we'll make more progress by
trying to model less sophisticated reasoners, like Portia Spiders and intelligent
toddlers (with their rich internal languages) than attempting to replicate Euclid
directly.
Acknowledgements:
I am grateful for discussions with Jackie Chappell about animal intelligence,
with Aviv Keren about mathematical cognition, and discussions about life, the
universe, and everything with Birmingham colleagues and Alison Sloman.
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