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Installed: 12 May 2013 (Moved from Preface to the above triangle-theorem document.)
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Design space and niche space and mappings between them
All the designs, all the niches, all the behaviours, all the relationships between
designs and behaviours and between designs and niches and between behaviours and
niches are instances of mathematical domains, though some of the domains are static
for a while, whereas others keep changing, and there is enormous diversity. Life is
not just one pattern instantiated in all species and all niches.
That last statement about domains changing is imprecise and potentially misleading.
The instances of domains can change while remaining instances, if the domains have
sufficient generality. However, for some of the changes in challenges or requirements
and changes in solutions there are major domain transitions. We may need to develop a
new language to express all this, though some of the work already done in computer
science, concerned with formal specifications of requirements, designs,
implementations and behaviours may prove relevant. Compare Sloman and Vernon
(2007).
Summary of key ideas: The blind theorem-prover
A high level messy summary:
The early achievements were produced by evolution, then later by associative
(statistical) learning, then later by other processes involving meta-cognition,
then collaborative (social processes) then co-ordination through formal educational
and research structures.
But it all rests on domains of structure that exist waiting to be discovered.
(The online mathematical doodles of Vi Hart illustrate many
such domains in a highly entertaining and creative way.)
The instances of domains may be produced in many ways: physical and chemical
processes, evolutionary processes, activities of individual organisms, interactions
between organisms, types of environment in which organisms evolve, learn, perceive
and act, cultural processes, individual intentions.
But the types of which they are instances existed earlier, insofar as the instances
were possible before they actually existed.
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Triangle theorem document
An early version of this document appeared as a preface to another document:
http://www.cs.bham.ac.uk/research/projects/cogaff/misc/triangle-theorem.html
Hidden Depths of Triangle Qualia
(Theorems About Triangles, and Implications for Biological Evolution and AI
The Median Stretch, Side Stretch, Triangle Sum, and Triangle Area Theorems)
The Preface grew too large, so it is now growing even larger, on its own.
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When will the first baby robot grow up to be a mathematician?
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I think that's a step in the right direction, but the step is much too short. A bigger
step in the right direction would describe mathematics as a biological phenomenon.
I shall try to describe a deep, but largely unnoticed, collection of relationships
between biological evolution and various forms of mathematical, pre-mathematical,
meta-mathematical and philosophical reasoning capabilities, geometrical reasoning
being an important special case.
One aspect of this is to enrich the metaphor of evolution as a "blind watchmaker" by
construing evolution as a "blind theorem-prover" whose theorems are all about what is
possible. (Compare Chapter 2 of "The Computer Revolution in Philosophy" (1978))
In this document I shall try to show that there are deep connections between:
I shall try to show that human mathematics has several different sources, whose potential
goes beyond human mathematics:
(a) the structural relationships within a multitude of different domains that impose
requirements and opportunities for biological information processing,(b) the evolution and development of abilities that make it possible for some individuals
to discover, reason about and make use of properties of those domains (in some cases
without being aware that they are doing so),(c) the evolution of meta-cognitive competences that made it possible to reflect on,
discuss, teach, and overtly reason about or argue about properties of those domains,(d) the apparently never-ending possibility of creating, or discovering, new domains by
modifying or combining old domains, or by forming new meta-domains by abstracting from
details of previously known domains.(I'll explain below what a domain is, and give examples.)
This means that any complete philosophy of mathematics, including answers to "What is
mathematics?" "What makes it possible?" must discuss
-- opportunities for, and constraints on, biological evolution,
-- structures in the world that evolution reacts or responds to, and
-- later products of evolution building on older ones, including evolved abilities to
by-pass evolution, through learning, a much faster process.
This seems to be a never-ending process of growth.
The myriad possibilities for biological structures and processes are rooted in very
general physical/chemical features of the universe and particular historical conditions in
various portions of the universe.
Human mathematics is rooted in biological phenomena, and grown in stages, initially
mainly by natural selection, starting with micro-organisms whose information-based control
mechanisms evolved so as to engage with increasingly rich mathematical features of
structures and processes in the environment.
Some of those mechanisms have recently been modelled or replicated in computer-based
machines capable of logical, arithmetical or algebraic reasoning. These competences seem to
have developed relatively recently in humans. Paradoxically, some of the much older forms
of reasoning, apparently shared with some other animals, have so far resisted computer-based
replication, as mentioned above.
Later, as organisms became more complex, and their environments changed, new forms of
information-processing became increasingly useful, to address increasingly complex challenges
and opportunities presented by the physical world and its occupants, non-living and living
(including predators, prey and new sources of plant food), along with new challenges and
opportunities continually presented by results of earlier evolutionary developments that
provided new sensors, new manipulators, new information-processing capabilities, new
problems of learning and control, and new forms of mathematics implicit in the
information-processing strategies. Some of those transitions in information-processing are
listed in this document.
Although some species successfully existed for many millions of years with little change,
because their niches changed little, others acquired new forms of information-processing
driven by evolution, combined with processes of learning, development, and social/cultural
change. The key point is that such changes, both in the challenges and in the responses,
involved alterations or extensions in the mathematical structures of physical, biological,
social, and mental processes in organisms. Those changes enabled them to cope better with,
or cope with a wider variety of, naturally occurring problems.
For the purposes of this discussion, John McCarthy was right to suggest in (1979) that
thermostatic control processes are examples of primitive mental processes, implemented
in physical or chemical processes.
The information-processing requirements became ever more demanding as control problems
became more complex, e.g. going beyond switching something on or off, to increasing or
decreasing something (speed, angle, a gap between claws), then to modifying the rate of
increase or decrease (i.e. acceleration or deceleration), then perhaps moving from scalar
changes to structural changes (e.g. changing relationships between parts of grippers and
parts of objects gripped), possibly requiring more parallel control functions (biting
while chasing, climbing while holding a baby) ... .
Different sorts of factors contributed to increasing complexity of information-processing
mechanisms: including both previous evolutionary changes in the organisms themselves as
well as changes in their physical environments and changes in the information-processing
sophistication of their prey, predators and conspecifics. The resulting changes included
new forms of perception, formation of new long-term re-usable information structures,
performing more complex derivations, solving more complex planning problems, forming more
complex plans, using more sophisticated forms of representation, controlling more complex
processes, etc.,
So, changing mathematical properties of niches or task demands, led to changing
mathematical properties of information processing mechanisms and the behaviours they
produced. Many more examples were investigated in relation to child development by Piaget,
including examples in his last two books, on Possibility and Necessity, closely related to
our topic -- partially reviewed here.
The success of this process depended in part on the possibility of separating the enormous
variety of matters to be dealt with into relatively self-contained domains that could be
mastered separately, a divide-and conquer learning strategy that made major achievements
tractable. (The different "gaits" used by four-legged mammals for locomotion at different
speeds and in different circumstances seem to be an example of this point.)
In a universe without this separability of increasingly complex challenges into
distinct domains, biological evolution as we know it would have been impossible. In part
this is because many of the domains with enormously varied contents had a relatively small
'generative basis' from which the rest could be derived, hugely reducing the amount of
empirical learning from examples required. (I suspect that there is a theorem lurking here
about how the universe needs to be generated from a relatively small collection of
structures and laws relating them in order to be so rich and varied -- a variation of
the "Anthropic Principle".)
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Warning: The English word "domain" has several different uses, for example inThe idea of a "Domain" is very important for the thesis summarised here. I'll sometimes
biology, in mathematics, in politics, and no doubt many others. I use the word fairly
loosely, to refer to something like a collection of related types of entity, where
the entities in the domain usually have internal complexity, and are are related in
some systematic way through their formation, their interactions, and possibly also
their relationships to something outside the domain, e.g. things that learn about or
make use of, or contribute to the existence of members of, the domain. This use of
the word seems to be fairly common among a subset of scientists and philosophers,
though it is likely that different groups use different words or phrases for the same
concept. For example I think "micro-world" or "microworld" is often used in a similar
way (e.g. by Seymour Papert), though that is sometimes restricted to domains of very
small living organisms! Outside particular realms of logic or mathematics, no complex
concept can ever be defined explicitly in an unambiguous way, since there will always
be circular connections with other concepts. The current fashion for attempting to
"ground" concepts (very old in philosophy) is based on a failure to understand this
point. Good teachers understand that learning a concept can be more like learning
to find your way around a new town than memorising a formula.
See also http://edutechwiki.unige.ch/en/Microworld
At some future date a better theory of types of domain, and the cognitive challenges and
opportunities they provide for their inhabitants, their users, and individuals learning
about them, will be needed. For now I'll present a compressed intuitive introduction with
no pretense at precision or completeness.
The idea (or a closely related idea) plays an important role in Annette Karmiloff-Smith's
developmental theory, according to which learners first acquire behavioural competence in
a domain empirically through exploration, and experiment, with or without the help of a
teacher or a supporting community, and then after a while transform that competence into
something more powerful through a process of "Representational Redescription":
See http://tinyurl.com/CogMisc/beyond-modularity.html#microdomains
Several examples of domains related to her ideas are discussed in Butler (2007).
It seems clear that her notion of representational redescription is closely connected with
the examples I have been giving of types of (proto-)mathematical discovery of which most
humans, including toddlers (as described here), are capable, whether they are aware that
they make the discoveries or not.
First approximation: a domain is a class of possible structures or processes that can be
created and manipulated, either physically or "in thought" (by thinkers with appropriate
representational/information-processing capabilities). Physical domains contain
perceivable structures and processes, some or all of which may also be manipulable, for
example, configurations of rocks, or piles of sand, or bricks, or sounds produced either
vocally or by manipulating objects in the environment. There are also abstract domains
such as domains of sets of numbers and arithmetical processes transforming those sets.
NOTE: (28 Jun 2013)
For quick introductions to many domains see Vi Hart's amazing high speed
online math-doodles.
Although I have not noticed her using the term "domain", she finds small, fairly
closed, domains everywhere, including various familiar bits of mathematics from
arithmetic and geometry, but also in physical structures and transformations,
e.g. folding an sheet of paper repeatedly, tying long thin balloons together,
arranging furniture, patterns in "dances" performed with fingers and thumbs, etc.
My point is that these are crystallized sample fragments from a much richer
space of domains whose instances include spatial structures and processes and
also more abstract structures and processes in mathematics and in information
processing mechanisms.
The existence of a domain, and even the causal role of a domain, need not depend on anyone
having noticed or thought about the domain. For example the (vast) domain of possible
chemical structures and processes possible very early in the history of this planet made
it possible for the earliest forms of life to come into existence Ganti (2003), but nobody
had to be around to think about what was happening.
Some domains contain only structures and processes that are products of purely physical
processes, e.g. domains of ocean waves or shadows of clouds on a lifeless planet, or
planetary motions; while other domains have physical structures and processes produced
intentionally or unintentionally by humans or other animals, such as: paths in a forest,
sand-castles, patterns made with elastic bands and pins, sentences, poems, piano sonatas,
or numerical calculations. The sets of structures and processes that are possible in
principle in this universe are enormously varied, and the set of domains of structures and
processes that are theoretically possible is unbounded, even if the set that could be
instantiated physically is bounded in various ways, including space available, durations of
processes, numbers of physically distinct components, etc.
The set of domains that can be thought about includes some whose instances could not exist
physically, for example, a domain containing arbitrarily long, infinitely thin, rigid
linear structures that can transmit forces. For now I'll leave aside the question whether
a mathematical continuum exists in physical space.
Instances of a domain will typically be complex objects made of parts standing in various
relationships, or complex processes in which objects change properties and relationships,
and possibly come into or go out of existence, either through processes of assembly and
disassembly of complex wholes, or through appearance or disappearance of 'basic'
components. Processes in a domain can include addition or removal of parts of an instance,
or alterations of the properties or relationships of some or all of the parts.
Domain synthesis
Often a new domain can be defined by combining domains, for example, the domain of
problems of interpreting 2-D line drawings as depicting 3-D configurations of opaque
polyhedra (as in the work of Huffman, Clowes, Waltz, Martin, and others [Add REFS]), or
the domain of processes of production of verbal descriptions of pictures, or things
depicted, or processes of generating pictures from verbal descriptions, or mathematical
specifications, or the code for drawing programs.
Some domains are totally discrete, such as the domain of configurations of zeros and ones
in a 2-D array. Others are continuous, such as the domain of shapes and processes that can
be created by rearranging a piece of string on a flat surface, without any crossings or
contacts of two parts of the string. Extending the domain to include points where one part
of the string crosses or touches another part introduces discontinuities into the domain.
A domain of blocks of different heights arranged in a row can be thought of as discrete if
only the possible orderings are considered, whereas it includes both continuous and
discrete change if spatial processes in which blocks are moved from one location to
another so as to alter the ordering are included in the domain.
Several more examples of domains are mentioned in the discussions of "Beyond Modularity",
and toddler theorems:
Most normal humans, beyond a certain age, seem to be capable of inventing both new
instances within a previously known domain (if the domain's contents have not yet been
exhaustively listed), and many can also invent new theoretical domains such as might form
backdrops to fantasy adventure stories.
Domains of the sorts referred to here, sometimes referred to as "micro-worlds" have often
been the subject of study in Artificial Intelligence, in projects concerned either with
explicitly programming computers to have competence in a domain (e.g. interpreting line
drawings as depicting 3-D polyhedral scenes) or investigating ways in which a computer (or
robot) can acquire such competence as a result of some sort of training process, or a
process of exploration and experiment driven by the robot rather than a tutor presenting
examples.
Domains may be naturally occurring or artificial -- devised for educational or other
purposes. A fairly complex domain my colleagues and I created as a challenge for robot
vision, manipulation and learning is the "Polyflap" domain, most of which is still far
beyond the competences of current robots. Many domains, some explicitly recognized as such
and some not, are associated with board games, sporting activities, musical activities,
domestic rituals and chores, and, of course branches of mathematics, some discovered
centuries ago (e.g. arithmetic and geometry), others noticed only very recently, e.g.
group theory, boolean algebra, and theory of formal grammars.
My claim is that all organisms are confronted with domains of structures and processes, of
various kinds and various degrees of complexity, implicitly specified by relationships
between aspects of the environment in which they exist, perceive and act and by their own
physical competences and their information-processing competences required for use of
sensory information and for initiation and control of behaviours.
It seems that both Karmiloff-Smith and I have been independently thinking about various
sorts of domain relevant to cognition and cognitive development in humans and other
animals, in which we have both been heavily influenced by the work of Piaget. My interest,
unlike hers, was largely driven by problems in the philosophy of mathematics, going back
to my 1962 DPhil thesis. It is very likely that many people in many different
disciplines have noticed the psychological and pedagogical importance of domains, probably
using different terminology for them, though I am not aware of a previous attempt to use
the idea of a domain as a bridge between biological evolution and philosophy of
mathematics.
The transitions required to meet these challenges include: development of new forms of
representation of information (including chemical, neural, behavioural, encodings, and
creation of information structures in the environment, e.g. pheromone trails, worn tracks,
distinctive nests and hives, etc.), development of new information-processing mechanisms
to make use of these forms of representation, development of new ontologies extending
previous semantic contents, and development of new information-processing architectures,
including virtual machine architectures, capable of combining multiple cooperating
information-processing mechanisms and capabilities.
For example, the mathematical properties of continuous feedback control systems in
homeostatic mechanisms are different from the mathematical properties of persistent
information structures recording spatial layout of important locations in the environment
(nest, food sources, obstacles, paths, etc.). The mathematical properties of linear
grammar-based information transmissions and their semantic contents are different from
both. We can now see the need for differential equations, for graph structures, for
logical formalisms, whereas the problems were invisible to our ancestors, and even to most
contemporary humans, who study no mathematics, logic, or computer science.
Some of the biological mathematical competences may have evolved several times, in
different contexts, sometimes merged later on, sometimes not, including abilities to cope
with and reason about sets, about measures (e.g. of time, length, area, volume, angle,
weight, speed, force, and many more), about rates of change, about cardinality, about
orderings and partial orderings. Information about unbounded processes (indefinitely
getting smaller, thinner, longer, straighter, more curved, etc.) may have come from the
disadvantages of pre-specified bounds or limits in forms of representation (as I think
Kant noticed).
The meta-mathematical modes of thinking and reasoning required to describe and compare
all those are very recent mathematical products, on this planet. But some have very old
pre-cursors. For history of human communication technologies see Dyson (1997).
In principle, not only is it possible for human cultures to discover and engage with
different mathematical domains, or sub-domains, it is also possible for other species to
discover and engage with domains relevant to their forms of life, including for example,
animals that live only in deep water, or shallow water, animals that spend most of their
life in flight, animals that lack a vision system, and so on. So it is true only for a
subset of mathematics that it is an anthropological phenomenon, and even that subset is
beginning to be extended by machines blazing trails that humans cannot follow.
a collection of "engineering solutions" to biological problems aboutNot all the problems are identified by human engineers, or other animals, since many are
structures and processes -- especially problems concerned with
information, and how it can be processed; along with problems that
have not yet been solved and a collection of strategies for generating
new problems.
Although mathematics teachers and philosophers of mathematics attempt to identify a
logically structured order of presentation of concepts, problems, theorems and proofs, the
order of discovery may be completely different from any such sequence, often driven by
processes of playful exploration, or in some cases the order in which the world happens to
present problems to be solved. Educational systems that assume there is a single "right"
order through which all children should acquire complex concepts, theories and competences
are doomed to harm a subset of learners. (This seems to have been one of the consequences
of bad decisions about how to teach mathematics in schools, influenced by logicist or formalist
philosophies of mathematics.)
Knowledge about any mathematical domain can start as unconnected fragments, that are later
combined and organised, either by outstanding individuals, or through collaborative
processes. The information-processing mechanisms required differ. Developments in AI show
how some powerful mathematical competences can be implemented on computers, but there are
some old and familiar competences, used in elementary geometrical reasoning, that so far
have proved very hard to replicate or model on computers, suggesting that the biological
mechanisms used have properties not yet been understood. This topic is discussed in
http://tinyurl.com/CogTalks/#m-m
Many biologists, neuroscientists, philosophers, and AI/Robotics researchers assume that
biological information processing mechanisms are mainly concerned with processes of
recognition or classification used in discovery of regularities in the form of correlations
that can be learnt from examples and in some cases encoded as probabilities because
the correlations have exceptions.
But this misses a different requirement, namely for ways of representing and reasoning
about collections of possibilities, and their limitations, constraints, or invariants.
Evolution seems to have provided many species (including some hunters and some
nest-builders) with powerful ways of dealing with such problems, allowing individuals
confronted with new situations to grasp the possibilities, understand the constraints,
work out positive and negative affordances, and find a novel solution. Sometimes the
solution is understood at a level of abstraction that justifies the description
"discovering toddler theorems", as in http://tinyurl.com/CogMisc/toddler-theorems.html
Newly discovered possibilities need not all be construed as exceptions to regularities.
(Some of those transitions in individual learning and development are described as
processes of 'Representational Redescription' by Karmiloff-Smith.)
Only humans, it seems, have, in addition, developed the metacognitive and meta-semantic
capabilities required for thinking about what they have learnt, noticing what others have
and have not learnt, and helping others to learn. Discussing the problems, criticising and
combining alternative solutions, noticing and mending flaws, all require very sophisticated
biological mechanisms, of types summarised below, not yet replicated in reasoning machines.
After the processes of exploration, discovery and reasoning themselves become subjects of
exploration, discovery, reasoning and education, we have what is recognisable as
mathematics. Before that happens, some individuals may acquire competences and knowledge
that could be described as 'proto-mathematics', as in the examples of
"toddler theorems".
There are many discrete transitions in the development of such knowledge and competences,
often not noticed by psychologists or ethologists untutored in philosophy of mathematics,
investigating understanding of numbers or geometry in children or non-human animals (e.g.
failing to distinguish diverse aspects of number competences, such as understanding of
numerosity, cardinality, and ordinal structures).
Often, areas of mathematics, and the associated human mathematical abilities, turn out to
be capable of generating their own new problems, and, in some cases, solutions, for
example the problem of developing an economical extendable notation for cardinal numbers,
problems of solving simultaneous equations, or problems of propagating length and angle
constraints across geometrical configurations, solved by use of trigonometry.
Which mathematical (and meta-mathematical) competences concepts and knowledge exist at
any time in any group of individuals is a product of biological evolution combined with
individual development, and individual and social learning. That is always a subset of a larger
mathematical space in which new challenges await attention.
Opinions differ regarding the diversity of mathematics. At first sight geometrical,
arithmetical, and logical concepts and knowledge are very different, yet Descartes showed
how geometry could be embedded in arithmetic, and Frege, Russell, and Whitehead showed how
(up to a point) arithmetic could be embedded in logic. Despite such discoveries, the
domains are very different -- as the examples of reasoning about triangles in the triangle
paper should make clear.
Geometry is concerned with the space of geometrical structures and processes in physical
space; (cardinal) arithmetic is concerned with the space of properties of one-one mappings
and how they are transformed by various operations and combinations of operations, and
logic is concerned with a space of syntactic and semantic structures and processes. Other
branches of mathematics, e.g. calculus, deal with different spaces. So the descriptive
role of philosophy of mathematics requires study of those spaces of possible problems,
methods, solutions, mechanisms, forms of representation, modes of reasoning, and their
biological and historical trajectories -- including possible future trajectories. This
philosophical work overlaps with the more theoretical work in mathematics, which is often
highly methodologically aware. (I have been helped to see all this by Ian Hacking.)
Mathematics, much more than politics, is the art of the possible, including the study of
constraints or limitations on possibilities -- e.g. it is not possible for a set of
objects to have a one to one correspondence with two different initial sequences of
numerals, one when counted left to right and one when counted right to left. Discovering
the scopes and limitations of different sets of possibilities typically requires much play
and exploration. Some of the domains explored by children have not yet, as far as I know,
been comprehensively mathematicised, for instance what happens when one plays with
different kinds of stuff (matter) as recommended in Sauvy & Sauvy.
Other domains about which young children and presumably many other animals learn,
such as how visual information travels and can be blocked or unblocked, reflected or
refracted, have been extensively studied, in geometrical optics.
The importance, in mathematics, of collections of possibilities ("domains") has
been eclipsed in philosophical discussions by the apparent centrality of necessity
in mathematical truths. That leads to puzzles as to the source of the necessity, and
several far-fetched answers have been proposed including the suggestion that all
mathematical necessities come from logic, or from human decisions. I think it is best to
frame the questions in terms of constraints on possibilities. In that case we can ask how
the structure of a set of possibilities allows some possibilities and not others. Then,
using the equivalence between 'Necessary(p)' and 'Not(Possible(Not(p))', we get
explanations of mathematical necessities from the impossibilities of various structures
and processes, for example the impossibility of making a polygon with three sides that
does not also have three vertices. A child might discover the existence of prime numbers
by discovering the impossibility of rearranging some collections of blocks into a regular
array, as discussed in http://tinyurl.com/CogMisc/toddler-theorems.html#primes
Mathematical competences are (among other things) solutions to the biological problem
posed in the final chapter of Craik (1947) of enabling animals to reason about possible
actions instead of having to perform them. Some of the kinds of reasoning that lead to
Euclidean geometry are illustrated in the main body of this paper in connection with
reasoning about possible ways of deforming triangles -- a human capability that has proved
hard to implement on computers, for reasons that are not clear.
In order to understand how such mathematical reasoning is possible, we need a very clear
understanding of what the problems are, whereas philosophers mostly try to understand the
solution(s) without understanding the original (biological) problems, and that leads to
narrowly focused, or inaccurately focused, theories about the nature of mathematics, for
example asking how a child reasons using laws of logic or geometry, instead of asking how
a child might discover such a law in particular instances.
Kant, perhaps, was an exception, though limited by the science of his time. This paper is
intended to be a small contribution to understanding the problems in a Kantian framework,
accepting that (contra Hume and Mill) there is something non-empirical about mathematical
discoveries and accepting that what is discovered is neither trivial nor some sort of
stipulation (as Quine seems to suggest).
The topic, discussed inconclusively here, is how it is possible to discover constraints on
possibilities for change in a triangle. I'll try to illustrate some forms of reasoning
that turn up in geometry and can be understood by a child, which seem to be beyond the
scope of current computer-based reasoning systems. Exactly why is not clear. I am not
claiming they are impossible for future robots playing with spatial structures.
It is often assumed that every action performed intentionally must have been selected
because it was expected to provide some reward, possibly satisfaction of curiosity, or a
feeling of pleasure, or an increase in knowledge or competence. But, as Ryle pointed out
in 1949, that's incompatible with an action being selected for it's own sake, e.g. simply
wanting to dance. I have argued elsewhere that some actions serve motives that are
'architecture-based', not 'reward-based'. For more on architecture-based motivation (ABM)
see: http://www.cs.bham.ac.uk/research/projects/cogaff/09.html#907
Emre Ugur's PhD thesis Ugur(2010), referenced there, shows how a robot with ABM
can learn useful facts about its environment, and about itself, e.g. its capabilities and
their limitations. His mechanisms demonstrate a possible way in which playful activity,
done without any ulterior motive, may, as an unintended side effect, produce important
kinds of learning. That may be part of the explanation of how tendencies to play first
evolved.
The side effects can include influencing the growth or strengthening of body parts
concerned with the actions, discovering correlations between motor signals, sensory
information, and other internal states and processes, discovering correlations between
actions produced by the observer, and discovering what kinds of processes in the
environment or changes of objects in the environment can occur, i.e. facts about what is
possible.
Examples of discovery of what's possible can include arranging toys or other objects in
straight lines or in regular arrays, or using body-parts or other devices to record
results of previous experimental actions, or making things move, or distorting things, or
producing motion trails, e.g. in sand or mud. There may in some cases be important
overlaps between what can be learnt in different sorts of activity. Compare using a finger
or stick to draw patterns in sand, in mud, or on a smooth surface. There will be
considerable differences between different sensory or motor contents and relationships,
but more importantly similarities and differences between things that can be learnt about
possible structures in the environment, e.g. thicker or thinner lines, jagged, curved or
straight lines, regularly spaced arrays of objects or drawn structures, e.g. dots, arrows,
straight lines, squares, etc. One of the possibilities is discovering properties of
numbers, e.g. whether they are prime, as discussed in connection with "Toddler Theorems"
here: http://tinyurl.com/CogMisc/toddler-theorems.html
More generally, architecture-based motivation can produce learning processes that develop
exosomatic concepts referring to structures and processes that can occur in the
environment. Initially all that may be discovered is that various forms of process produce
other changes, including changed information-processing. Later the play may create new
experiences or new theories that require new concepts to be developed. For example by
playing with squares or cubes and trying to form regular arrays, a child may discover the
need for a concept of primeness of a number, as illustrated in
http://www.cs.bham.ac.uk/research/projects/cogaff/misc/toddler-theorems.html#primes
What's learnt in different contexts, performing different actions, may be correlations
between actions performed in certain circumstances and consequences. Or it may be new
kinds of possible state or process. Or it may be constraints on possible changes in
certain structures. In some cases that can involve forming a new abstraction that applies
to different situations.
Many such domains are explored by children and other animals, but have never been
explicitly studied by mathematicians. (E.g. ways of putting on a sweater, shirt, vest,
jacket, etc with neck-hole, waist-hole, and arm-holes/sleeves?) But that typically does
not prevent them being mathematically very able.
Many people think that competence in a domain involves learning reliable correlations in
that domain. But there's a deeper kind of learning, about what's *possible* in that
domain. Somehow, evolution developed mechanisms enabling animals to acquire a grasp of
what's possible in certain situations, and some of the limitations on what's possible.
These can be thought of as many domains in which proto-mathematical knowledge/competence
develops. In the past few years I've been exploring lots of little domains that children
can learn to explore and play with, several referred to in the lovely little book by
Sauvy & Sauvy.
For example. there's the domain of shapes that can be made with a rubber band and pins in
pin-board, an outline capital "T" being one example, but not a capital "A". Another domain
is the set of processes and patterns involved in discs placed in a rectangular grid and
moved only diagonally.
Not all domains use the same kind of knowledge and reasoning (e.g. logical knowledge).
Many animals, and young children can acquire familiarity with a domain without being able
to reflect on, or talk about what they have learnt. Humans sometimes can as they get
older, though they may need help from a teacher, e.g. studying spatial structures and
motions.
Which domains get explored, and how far they are explored and what concepts and 'theorems'
are produced is to some extent a collection of historical accidents. But the content
acquired is not all empirical: there are structural constraints in a domain making some
things possible, others impossible.
(E,g, you can turn over a coin once, twice, or any number of times, then pause. Repeating
this produces a sequence of paused states of the coin. If each run turns it over an odd
number of times, then it will never be the case that the same face is up in two successive
pauses. There are also 'mini' theorems about rubber bands and pins, about sliding coins
diagonally on a rectangular grid, about varieties of knots, and many more.)
There are also domains that involve kinds of stuff, e.g. rigid impenetrable stuff, or
uncompressable fluids, or elastic stuff.
Some domains include kinds of information structures and their relationships (e.g.
entailment, contradiction).
The need to deal with some domains comes from evolution and the environment. Others are
just invented 'toy' domains, including some mentioned in this document on toddler theorems:
http://tinyurl.com/CogMisc/toddler-theorems.html
In general what's learnt in a domain can be decomposed and recombined so as to generate
new domains.
Mathematics is the investigation of such domains, which can take many forms, and can be
done alone or in a social group.
But those mathematical activities generate new domains and new tools for reasoning about
them., in what I suspect is a never ending process, although every now and again a complex
collection of domains is found to be be collapsible into special cases of one domain (e.g.
groups). Moreover, some domains generate meta-domains, concerned with aspects of previous
domains.
Which domains have actually been discovered and investigated and what has been learnt
about them is typically in part the history of biology or of a community of individuals,
or a particular individual. However that such domains can in principle be found and
studied (including many not yet discovered) is not a fact of biology or human history.
Neither is it typically a fact about humans that a particular domain includes certain
sorts of variety and certain constraints.
Which primes, or which symmetric geometric shapes, are still waiting to be discovered is
not a fact of human history, though it may enable some human futures.
There's a lot more to be said about the variety of such domains, the cognitive (including
motivational) mechanisms relevant to exploring them, and the ways in which previous
domains can contribute to the construction or discovery of new ones.
From this viewpoint emphasis on "the hard logical must", or on "what follows from what",
just focuses on narrow aspects of the broad variety of mathematical topics that can grow
out of products of evolution. In each case, requests for justifications of claims made are
not to be based on deduction from axioms, but consequences of things discovered by
inspection -- possibly after specifying what's included and excluded from the domain, e.g.
flexible objects.
In principle the investigation of all such mathematical domains could be done by robots.
But in practice there are kinds of human reasoning about what's possible that seem to be
very hard to implement on computers. I have been collecting examples from euclidean
geometry.
It could turn out that these forms of reasoning require development of a new kind of
thinking reasoning machine, that may not be implementable on turing machines. I have an
open mind.
There's lots more to be said about this. I think it pushes in a direction no philosophers
of mathematics that I know of have explored because they have generally not noticed the
connections between human mathematical competences and related competences of other
organisms, especially abilities to survey sets of possibilities and select something useful
in the current situation.
From this point of view, an important task for philosophy of mathematics is to provide an
overview of this collection of domains of knowledge, of requirements for exploring them,
of ways in which the knowledge gained interfaces with other kinds of biological information
processing, e.g. abilities to act, to, communicate, to cooperate, to teach, to learn, to
do science, to do engineering, etc.
And to investigate the information processing capabilities required for the various kinds
of discovery of and application of facts about possibilities and limits of possibilities in
various kinds of domain.
The more detailed work will be shared with other disciplines, biology, neuroscience,
psychology, linguistics, education, engineering, ...
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BACK TO CONTENTS
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A discussion of "Toddler Theorems", with examples:
http://tinyurl.com/CogMisc/toddler-theorems.html
A DRAFT list of types of transitions in biological information-processing:
Varieties of Evolved (Developed, Learnt, ....) Biological Computation, discussed in
http://www.cs.bham.ac.uk/research/projects/cogaff/misc/evolution-info-transitions.html
This is a draft, incomplete, discussion of transitions in information-processing, in
biological evolution, development, learning, etc.
A discussion of the difficulty of using computers to model human geometrical reasoning,
and related kinds of reasoning about affordances apparently done by some other animals can
be found in http://www.cs.bham.ac.uk/research/projects/cogaff/misc/triangle-theorem.html
using examples of very simple proofs of theorems about shape changes that increase or
decrease the area of a triangle, or which produce a new triangle containing or contained
in the old triangle.
Proofs of the Triangle Sum Theorem are compared in
http://www.cs.bham.ac.uk/research/projects/cogaff/misc/triangle-sum.html
For more on the nature of mathematical reasoning, its evolution and its development see:
NOTE:
Craik proposed that biological evolution produced animals with the ability to
work out what the consequences of an action would be without performing the
action, by making use of an abstract model of the situation in which the
action is performed. It is not clear to me that he noticed the difference
between running a detailed model of a specific situation to discover the
specific consequences, which some current AI systems (e.g. game-engines) can do,
and noticing an invariant property of such a process with different starting
configurations as required for understanding why a strategy will work in a
(possibly infinite) class of cases.
I think he came close, but did not quite get there, but I have read only the 1943 book.
Entertaining video tutorial on Pythagoras and irrationality
showing the connection between irrationality of square root of 2, and Pythagoras' theorem.
https://www.youtube.com/watch?v=X1E7I7_r3Cw
"What's up with Pythagoras", by Vi Hart.
See more of her work:
https://www.youtube.com/user/Vihart
Here: http://vihart.com
And here
https://www.khanacademy.org/math/recreational-math/vi-hart
NB
For many of her videos, you may have to pause and replay bits, to take in the details.
Additional influences can be found in the acknowledgements section of the paper on P-Geometry
http://tinyurl.com/CogMisc/p-geometry.html#acknowledge
Offers of help in making progress will be accepted gratefully, especially suggestions
regarding mechanisms that could enable robots to have an intuitive understanding of
space and time that would enable some of them to rediscover Euclidean geometry.
I believe that could turn out to be a deep vindication of Immanuel Kant's
philosophy of mathematics. Some initial thoughts are in my online talks, including
http://www.cs.bham.ac.uk/research/projects/cogaff/talks/#toddler Why (and how) did biological evolution produce mathematicians?
Maintained by
Aaron Sloman
School of Computer Science
The University of Birmingham
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