A partial index of discussion notes is in http://www.cs.bham.ac.uk/research/projects/cogaff/misc/AREADME.html
Two more papers based on the thesis work were published in 1965 and 1969:"Knowing and Understanding: Relations between meaning and truth, meaning and necessary truth, meaning and synthetic necessary truthhttp://www.cs.bham.ac.uk/research/projects/cogaff/07.html#706This argued (e.g. against Hume) that Immanuel Kant was right in claiming in 1781 that in addition to
there are also truths that are neither empirical nor trivial but provide substantial knowledge, namely truths of mathematics.
- empirical facts that can be refuted in experiments and observations with novel conditions
and- analytic, essentially trivial, truths that depend only on definitions and their logical consequences, and do not extend knowledge
The concepts used here are explained in "'NECESSARY', 'A PRIORI' AND 'ANALYTIC'" (1965)
http://www.cs.bham.ac.uk/research/projects/cogaff/07.html#701
http://www.cs.bham.ac.uk/research/projects/cogaff/07.html#714 Functions and Rogators (1965)
http://www.cs.bham.ac.uk/research/projects/cogaff/07.html#712 Explaining Logical Necessity (1968-9)
Around 1970 Max Clowes introduced me to Artificial Intelligence, especially AI work on Machine vision. That convinced me that a good way to make progress on my problems might be to build a baby robot that could, after some initial learning about the world and what can happen in it, notice the sorts of possibilities and necessities (constraints on possibilities) that characterise mathematical discoveries. My first ever AI conference paper distinguishing "Fregean" from "Analogical" forms of representation was a start on that project, followed up in my 1978 book, especially Chapters 7 and 8.
From about 1973, I was increasingly involved in AI teaching and research and also had research council funding for a project on machine vision, some results of which are summarised in chapter 9 of CRP. Later work (teaching and research) led me in several directions linking AI, Philosophy, language, forms of representation, architectures, relations between affect and cognition, vision, and robotics. Progress on the project of implementing a baby mathematician was very slow, mainly because the various problems (especially about forms of representation) turned out to be much harder than I had anticipated. Moreover, I did not find anyone else interested in the project.
- Interactions between philosophy and AI: The role of intuition and non-logical reasoning in intelligence,
Proc 2nd IJCAI, 1971, London, pp. 209--226,
http://www.cs.bham.ac.uk/research/cogaff/04.html#200407
- CRP: The Computer Revolution in Philosophy: Philosophy, Science and Models of Mind,
Harvester Press (and Humanities Press), 1978, http://www.cs.bham.ac.uk/research/cogaff/crp
In 2008 Mary Leng jolted me back into thinking about mathematics by
inviting me to give a talk in a series on mathematics at Liverpool
University. In that talk and in a collection of subsequent papers and
presentations I tried to collect examples and arguments about how
various aspects of mathematical competence could be seen to arise out of
requirements for interacting with a complex, structured, changeable
environment. I did not find anyone else who shared this interest,
perhaps because the people I met had not spent five years between
the
ages of five and ten playing with meccano?
http://www.cs.bham.ac.uk/research/projects/cosy/photos/crane/
There are also transitions in information-processing capabilities and mechanisms which are much harder to detect, though their consequences may include observable behaviours.
The transitions producing new capabilties and mechanisms are examples of a generalised concept of morphogenesis, originally restricted to transitions producing physical structures and properties.
Among the transitions are changes in the mechanisms for producing
morphogenesis. These are examples of meta-morphogensis (MM). The
examples of information processing competence described here may
occur at various stages during the lives of individuals. The
mechanisms that produce new ways of acquiring or extending
competences are mechanisms of meta-morphogensis, about which little
is known. Piaget identified many of the transitions in children he
observed, and thought that qualitative changes in competence
producing competences were global, occurring in succession, at
different ages, during the development of a child. Karmiloff-Smith,
in Beyond Modularity suggests that transitions between stages
may occur within different domains of competence, and will often be
more a function of the nature of the domain than the age of the
child, though she allowed that there are also some age-related
changes.
See
http://www.cs.bham.ac.uk/research/projects/cogaff/misc/beyond-modularity.html
Transitions occur across species, within a species, within an individual, concurrently in different species, and in some cases in eco-systems or sub-systems involving more than one species.
People who have not designed, tested or debugged working systems may lack the concepts and theories required.
Exploration here does not necessarily refer to geographical exploration. It can
including investigating the space of possible actions on some object or type of
object, e.g. things that can be done with sand, with water, with wooden blocks,
with string, with paper, with diagrams, etc.
[See Sauvy and Sauvy]
A very simple architecture with motives triggered by what is perceived, but with no computation or comparison of rewards, or expected utility.
In particular, the individual may be unaware of what is being done or why it is being done.
The main consequence is that the learner can now work out things that previously had to be learnt empirically, or picked up from teachers, etc. This means that the realm of competence is enormously expanded.
This requires the use of information structures of variable complexity composed of components that can be re-used in novel structures with (context-sensitive) compositional semantics.
I have argued in the past that there are alternative forms of representation that can be used for reasoning, and modelling causal interactions.
[Added 27 Oct 2011]
It is also connected with our discussion of precursors to the use of
language for communication -- in pre-verbal humans, in pre-human
ancestors and in other species. E.g. see
http://www.cs.bham.ac.uk/research/projects/cogaff/talks/#glang
Evolution of minds and languages.
What evolved first and develops first in children: Languages for
communicating, or languages for
thinking (Generalised Languages: GLs)
If we treat language learning as a special case of something more general, found also in pre-verbal children and in other species that can see, think, plan, predict, and control their actions sensibly, that may give us new clues as to the nature of language learning.
An example, going from sensory information in a 2-D discrete retina to assumed continuously moving lines sampled by the retina, or even a 3-D structure (e.g. rotating wire-frame cube) projecting onto the retina, is discussed in http://www.cs.bham.ac.uk/research/projects/cogaff/misc/simplicity-ontology.html
Ontologically non-conservative transitions refute the philosophical theory of concept empiricism (previously refuted by Immanuel Kant), and and also demolish symbol-grounding theory, despite its popularity among researchers in AI and cognitive science.
They also defeat forms of data-mining that look for useful new concepts (or features) that are defined in terms of the pre-existing concepts or features used in presenting the data to be learnt from. (Some work by Stephen Muggleton, using Inductive Logic Programming may be an exception to this, if some of the concepts used to express new abduced hypotheses, are neither included in nor definable in terms of some initial subset of symbols.)
This use of abstraction in mathematics is often confused with use of metaphor, which as normally understood does require the original cases to be retained and constantly referred to.
How to discover relevant possibilities:
Similar observations of other animals can be useful, though for
non-domesticated animals it can be very difficult to find examples
of varied and natural forms of behaviour. TV documentaries available
on Cable Television and the like are a rich source, but it is not
always possible to tell when scenarios are faked.
Some videos that I use to present examples are here:
http://www.cs.bham.ac.uk/research/projects/cogaff/movies/vid
[To be continued.]
Unfortunately the educational experience of many researchers includes neither learning to think like a mathematician nor learning to think like a designer.
E.g. many people who can state Pythagoras' theorem, or the triangle sum theorem have no idea how to prove either, and in some cases don't even know that proofs exist, as opposed to empirical evidence obtained by measuring angles, areas, etc.
[To be continued.]
Below this list is a collection of examples extracted from those papers and presentations, along with some new examples based on things I have read and conversations with friends and colleagues.
PAPERS
This list of examples is a tiny sample. I shall go on extending it.
(Contributions welcome.)
NOTE: The order of the examples presented here is provisional.
Later I'll try to impose a more helpful structure. Some of the examples were inspired by this wonderful little book:J. Sauvy and S. Sauvy, The Child's Discovery of Space: From hopscotch to mazes -- an introduction to intuitive topology, Penguin Education, 1974, Translated from the French by Pam Wells,Provisional list of examples:
(To be extended and re-organised.)
- Problems of alignment when manipulating and stacking objects
- Problems of moving objects in a complex structured environment.
- Learning to think about changes that could happen but are not happening.
- Discovering the triangle sum theorem
- Five year old spatial reasoning: Partial understanding of motions and speeds
- Partly integrated competences in a five year old
- Unanswered deep questions
- More evidence for partial construction of a theory for a domain.
- Sorting or stacking objects by height or size
- From special to general and back again
- Sliding coins diagonally on a grid
- Pulling an object towards you: blankets, planks and string
- The surrogate screwdriver example
- Topological and semi-metrical puzzles
- Learning about numbers
- Learning about one to one mappings (bijections)
- Learning about measures
- The chocolate slab puzzle
- Discovering counter-examples to the chocolate slab theorem
- Another "Lakatosian" counter example to the chocolate theorem
- Learning about epistemic affordances
Getting information about the world from the world- Rearranging wooden blocks
- Playing and exploration can be done in your mind instead of in the world
- Playing can reveal both new possibilities and impossibilities.
(Discovering constraints.)- The drawer shutting theorem
- Carrying things on a tray
- Domains and micro-domains concerned with meccano modelling
- POLYFLAPS - An artificial domain for research in this area
- Window opening theorems
- Imre Lakatos
The importance of Imre Lakatos' writings on Science and mathematics.- Types of dynamical system
At first very young children playing with 'lift out' toys like these find it difficult to insert a cut-out picture into its recess, even if they remember which recess it came from.
E.g. They put the picture down in approximately the right place and if it doesn't go in they may press hard, but not attempt any motion parallel to the picture surface.
After a while they seem to learn that both the recesses and movable objects have boundaries, and that when flat objects are brought together the boundaries may or may not be merged.
At this more advanced stage, a child may place the picture object in roughly the right place and then try sliding and rotating until it falls into the recess.
Still later, the child realises that boundaries can be divided into segments and that segment of the recess boundary may match a segment of the object boundary, and then try to insert the object by first ensuring that matching segments are adjacent and then slightly varying the location and orientation of the piece until it falls into the recess.
Long before they can do this, I suspect they can insert a circular disc into a recess, since there is no problem of alignment. If there are different discs and recesses of different sizes the insertion requires size and location to be perceived and used in controlling the insertion process. When the items are not symmetrical, inserting requires (a) identification of matching portions of the recess and the movable piece, (b) the ability to match locations and orientations of the two boundaries, (c) depending on how tight the fit is, the ability perform slight movements to compensate for imprecision in the placing action, (d) in some cases using a tilted insertion orientation to allow the shape of the recess to guide the inserted piece into the exact location and orientation.
There are similar problems stacking cups, except that in addition to the shape of boundary, the size can be very important, and children may have to learn to order the sizes in order to ensure that all the cups can be stacked. There are probably many intermediate discoveries that can be made and used, some of them red-herrings because they only work by accident in certain conditions, or because they are allow a cup to be stacked but prevent ALL cups being stacked, e.g. placing the smallest cup on or in the largest cup.
This is an example of matter manipulation, a type of competence that subsumes tool-use and many other things that have been studied in children and other animals.
A broom can be thought of as a "tool for shifting dirt on a floor", but in the video is is not being used in that way. Rather the child appears to be moving the broom around for its own sake, rather than for the sake of some other effect.
Such matter-manipulation sometimes has utilitarian functions (e.g. obtaining food, putting on clothes, getting hold of some object that is out of reach) but need not have. With or without serving an explicit goal of the manipulator the processes seem to be a pervasive type of activity in very young children and also some other animals.
Presumably this is because playful, exploratory, manipulation can provide much information about, for example:
Suppose it is formed from a stretched rubber band held in place by pins.
There are many ways the shape, size, orientation and location of the triangle could be transformed, by moving the pins.
Think of some possible changes do-able by moving one, or two or all three pins, and for each change try to work out its consequences.
That is an easy task for a mathematician since much of mathematics is a result of the human (animal?) ability to look at something and think about how it could be changed, and what the consequences would be.
Most humans do it often in everyday life, e.g. when considering rearrangements of furniture.
The ability to do this develops slowly and erratically in children -- and in cultures See also (Piaget & others, 1981, 1983)
Among the many possible ways you could alter the triangle, e.g. moving, or rotating the whole thing there is one that involves moving only one pin, parallel to the opposite side, in either direction, e.g. moving the top pin here, parallel to the opposite side (the "base").
Another possibility involves moving the top pin up or down in either direction perpendicular to the opposite side.
Can you see any interesting difference between those two sets of possible changes to the configuration?
One set of changes will increase or decrease the total area of the interior of the triangle.
The other set of changes will leave the area of the triangle unchanged.
Can you see why that must be so? Here's the explanation:
If you don't recognize what's going on, try writing to me. (I'll add a link to an explanation later.)
The crucial point about such a diagram is that (like all diagrams used in proofs in Euclidean geometry) the relationships perceived in the diagram do not depend on the specific size, shape, colour, location, orientation, etc.
They don't even depend on the diagram being drawn accurately (with perfectly thin, perfectly straight lines). That's because once the proof is understood correctly its scope covers a very large class of abstraction. It's not clear that people not trained in mathematics can easily think that way.
There's an interesting 'bug' in the proof-sketch as shown in the diagram which is related to the need to do proper case analysis. It's a simple example of the sort of phenomenon discussed by Imre Lakatos in Proofs and Refutations, mentioned below. The bug in the 'chocolate' theorem, discussed below, is another example. Identifying the bug is, for now, left as an exercise for the reader, though mathematicians will find it obvious. Max Wertheimer discussed an analogous bug in a proof given by a school teacher regarding the area of a parallelogram, described in his book Productive Thinking. More examples of buggy, but fixable, proofs are given below. [The relationship between this sort of bug and the problems a child has in handling exceptions to grammatical rules in language may be illuminating, as regards information processing architectures and mechanisms required.]
This human ability to reason about necessary consequences of alterations to configurations in the environment may be closely related to Kenneth Craik's hypothesis that some animals can use internal models of the environment to work out consequences of possible actions. (Craik, 1943)
Compare also (Karmiloff-Smith, 1992), and Piaget's work on possibility and necessity, and also Kant's philosophy of mathematics (Kant 1781).
Work that remains to be done includes finding out how a child, or non-human animal, or future robot, could notice that some collection of structures and processes forms a domain that has interesting properties, including invariants that are discoverable by reasoning about the structures and relationships, how the relationships can be discovered and supported by a non-empirical argument, how different domains can be combined to form new domains of expertise, and how all of this can lead to the phenomena of Representational Redescription discussed by K-S.
We also still need to understand how to get robots and other learning machines to go through similar procedures.
Based partly on ideas by Mary Pardoe developed while she was teaching children mathematics.
Consider a slow moving van and a fast moving racing car. They start moving towards each other at the same time.
The racing car on the left moves much faster than the van on the right: Whereabouts will they meet -- more to the left or to the right, or in the middle?
One five year old answered by pointing to a location on the left, somewhere near "b" or "c".
Me: Why?
Child: It's going faster so it will get there sooner.
What produces this answer? Could it be:
Here are some fragments that may have been learnt, but perhaps without all their conditions for applicability fully articulated.
The first premiss is a buggy generalisation: it does not allow for different kinds of "race".
The others have conditions of applicability that need to be checked.
Perhaps the child had not taken in the fact that the problem required the racing car and the van to be travelling for the same length of time, or had not remembered to make use of that information.
Perhaps the child had the information (as could be tested by probing), but lacked the information-processing architecture required to make full and consistent use of it, and to control the derivation of consequences properly?
Is Vygotsky's work relevant?
Some parts of Piaget's theory of "formal operations"?
Compare Karmiloff-Smith on "Representational Redescription",
discussed in
http://www.cs.bham.ac.uk/research/projects/cogaff/misc/beyond-modularity
Could the child's reasoning be evidence for a process of representational
redescription that is still incomplete: i.e. generally useful items of
information that can be recombined in different contexts have been extracted
from the collection of empirically learnt associations. But the conditions for
recombination, and the constraints on applicability of inferences, have not yet
been discovered. In principle, this looks like a type of learning that could be
modelled in terms of construction of a rule-set capable of supporting deductive
inference.
(I think Richard Young's PhD thesis around 1972 was concerned with a process
something like this, but involving ordering of objects by height.)
"Today might be much more hotter than it usually bees"More generally, the phenomena of "U-shaped" language learning provide many clues as to what goes on when information fragments acquired empirically are transformed into a "deductive" system, when the system needs to be capable of handling exceptions -- unlike the systems of topology, geometry, and other kinds of proto-mathematical knowledge.
See the short, tentative, discussion in this PDF presentation:
http://www.cs.bham.ac.uk/research/projects/cogaff/talks/math-order-stacking-sloman.pdf
Does starting from a different configuration change what is
possible?
Can you get from configuration (a) below to configuration
(b) using only diagonal moves?
The next one is harder:
How people work on such problems differs according to prior knowledge and experience.
Sometimes proving that something is impossible can be done by exhaustive search (though understanding the need to ensure that the search is exhaustive is an achievement, as is organising the search so as to ensure exhaustiveness.
A different kind of competence can lead to a much more economical explanation of why the task is impossible. The core characteristic of mathematics, which also frequently motivates its development is productive laziness, which I suspect begins to develop between ages 1 and 3 years.
This is a case where the advance of knowledge involves noticing that
a particular problem is a special case of a general type of problem.
(If a problem is too hard to solve, trying a harder one sometimes
gives new insights.)
If you have not noticed the easy way to solve the above problems consider what difference it would make if the squares were black and white, as on a chess board. Mathematicians can use the notion of "parity" here. E.g. giving squares coordinates, they can be divided into two classes: those whose coordinates sum to an even number and those whose coordinates sum to an odd number. The squares in a horizontal or vertical line will have alternating parity. Squares in a diagonal line will have the same parity. This makes it very easy to check whether a start configuration can be transformed to a target configuration.
Normally such discoveries are made only by adult or bright mathematical learners. My point is that a young child could learn some of the generative facts about the diagonal moving coin domain by playing. Using a two-colour grid will make some things easier to learn. (Why?)
See http://www.cs.bham.ac.uk/research/projects/cogaff/misc/orthogonal-competences.html#blanket
If they cannot find such an object, but they understand what it is
about the screwdriver or spoon that makes it a suitable tool, some
of them will notice the possibility of using the lid of another tin
instead of a screwdriver, to lift the stuck lid.
http://www.cs.bham.ac.uk/research/projects/cogaff/misc/orthogonal-competences.html#lids
Starting from the configuration on the left the aim is to get to the
configuration on the right, without disconnecting the rope from the
two disks at its ends.
(This picture is from the very interesting paper by Cabalar and
Santos, below.)
There are many more puzzles shown and offered for sale at the "MrPuzzle" web site, e.g.
See also http://www.mrpuzzle.com.au/
Dealing with such puzzles requires the ability to think about topology-preserving transformations of physical objects involving flexible inelastic strings, beads, discs, and various rigid objects with holes and slots through which string and other things can pass.
In many cases it is also important to make use of non-topological relationships such as relative size (e.g. a bead is too large to pass through a hole, and a string loop is too short to pass over the far edge of an object).
In such cases, an important kind of discovery is how an alteration that does not transform the topology can transform a metrical relationship. E.g. pulling part of a string from one portion of the puzzle to another portion can increase the size of a loop until some object can pass through it that previously could not.
For each class of puzzle there can be a wide range of possible actions to consider. In particular the learner may need to learn:
There seem to be many different domains/microdomains a learner can explore: including the possible processes associated with a particular puzzle, the possible processes associated with a class of puzzles, and the possibilities created by combining features of different puzzles.NB Looking at the sophisticated logical formalism developed in that paper to enable a computer to reason about such puzzles it seems clear that what their AI system does is very different from what a logically and mathematically naive human might do when looking at the puzzle and thinking about actions that would change relationships, e.g.For more on such puzzles and formal reasoning about them see
Pedro Cabalar and Paulo E. Santos, Formalising the Fisherman's Folly puzzle, AIJ, 175,1,pp 346--377, 2011 http://www.sciencedirect.com/science/article/pii/S0004370210000408
"If I push that disk through the slot, I shall then be able to slide
the ring up over the top of the post, but..."
Such thoughts seem to make intrinsic use of the structure of the
perceived scene in something like the way described in Sloman 1971.
These questions are all related to the question: what sort of understanding of the puzzle (and what form of representation of that understanding) allowed the authors to discover the axioms that characterise it well enough to be used by an AI system? This is also related to the problem of how our ancestors perceived, thought and reasoned about spatial structures and relationships before Euclidean geometry had been codified, and even longer before cartesian coordinates were used to represent geometry arithmetically and algebraically.
It seems very likely that those pre-Euclidean and pre-Logical forms of representation and reasoning are still used, unwittingly, by young children and by other animals with spatial intelligence, e.g. nest-building birds and hunting animals.
The following seems to be a fairly standard (but mostly unnoticed by researchers) way of acquiring cardinality competences, though these components are not learnt in sequence, but interleaved:
It is not always noticed that without the sophisticated apparatus of modern mathematics many measures form only partial orderings.
E.g. at a certain stage areas or volumes may be comparable only if one shape can fit entirely inside another. So a long thin rectangle and a circle whose radius is less than the length and greater than the breadth of the rectangle are not comparable in area, at that stage. (As far as I know this was ignored by Piaget and all the researchers inspired by his work.)
For example, several different competences are required in order to rank the areas A, B, C and D in the following figure.
Someone who can accurately visualise the effect of moving one bounded area while another remains fixed, or who can cut out the area and move it onto another, may discover that area A can fit entirely inside B. So the area of A is less than the area of B.
However, the shape A cannot be contained in C, and C cannot be contained in A. Moreover, C cannot be contained in B, and B cannot be contained in C. This means it is impossible to rank shapes A, B and C in area on that criterion. They form only a partial ordering relative to the containment criterion.
Someone who has (somehow -- this is non-trivial) discovered a way of assigning measures of area to rectangular shapes, and then has discovered that that can be extended to a way of assigning measures to triangles
area = half(base x height)
could realise (how) that any area bounded by straight edges (i.e. any
polygon) can be systematically divided into triangles, so that the area can be
computed by triangulation, followed by adding all the areas of the triangles.
That will enable the three shapes A, B, and C to be given a numerical measure of
area, instead of just a partial ordering of spatial extent defined in terms of
containment.
But a polygon can be divided into triangles in different ways, so the argument assumes that different triangulations of the same total area will produce triangles whose sums are all the same. Is that obviously true? (It may seem to be obvious if you start from the assumption that the measure of area of an arbitrary shape is uniquely defined. But that assumption requires justification. In fact there is a lot of non-trivial mathematics concerned with the investigation of things that seem obvious to non-mathematicians.)
If we attempt to generalise the notion of area to a region not bounded by straight lines, like figure D, then there is no way to convert that region into a set of triangles. Our simple partially ordered notion of relative area defined by containment can still be used. For example, figure A can be re-located to fit entirely inside D, though that may not be obvious to everyone.
But we wish to extend the notion of a measure of area, defining a total ordering, so that it includes shapes with curved boundaries, like D, then a different approach is required. In fact it requires the use of integral calculus and concepts of limits of infinite series, which were invented by geniuses like Newton and Leibniz and not fully clarified until the mathematics of the 19th Century. (Some might say: not even then!).
There are also problems about the justification for talking about cardinality of large collections of objects (like the visible stars on a clear night, or the leaves on a big tree) where we do not have any chance of counting them, e.g. because they exist for a very short time, or because they are in constant motion, or for some other reason.
All this means that when researchers ask whether children or animals have concepts of size or number they often have no idea of the variety of interpretations that their question can have, with different answers being appropriate to the different interpretations. It is probably fair to say that most members of the adult population of any country on this planet lack well-defined concepts of area and volume. (It may be assumed that area and volume can always be defined in terms of the results of weighing, but that typically assumes the notion of uniform density, which in turn assumes notions of weight and volume.)
It is not clear which of these competences (relating to cardinality, mappings and measures) a child can acquire without help. The ontologies required, the invariants, and the applications, all must have been discovered originally piecemeal, perhaps in inconsistent fragments, without help, and then organised into a shared system through some collaborative process, probably over many generations, long before Euclid's time. I don't know if we'll ever find definitive evidence for those aspects of our pre-history. But perhaps we can replicate some of them in future intelligent robots. And if we look carefully, asking the right questions, we may be able to see some of the fragments in child development, though not all fragments will necessarily appear in all children: there are many routes through this maze of ontologies.
http://www.cs.bham.ac.uk/research/projects/cogaff/10.html#1001
If Learning Maths Requires a Teacher, Where did the First Teachers Come From?
You have a slab of chocolate in the form of a 7 by 7 square of pieces divided by grooves, and you want to give 49 friends, each one piece.The puzzle draws attention to a domain of processes of subdivision of a rectangular array into its component elements by a succession of linear slices.
You have a knife that can cut along a groove.
What is the minimum number of groove cuts that will divide the bar into 49 pieces?
RULES FOR CHOCOLATE CHOPPING:
Stacking or overlaying two or more pieces, or abutting two pieces, to divide them both with one cut is not allowed: each cut is applied to exactly one of the pieces of chocolate.
It's an exception because the original argument assumed that every cut divides one piece into two pieces.
With holes, is it a slab or isn't it?
Often a proof in mathematics that seemed valid works for a range of cases, but has counter-examples not thought of when the proof was constructed.
Many such examples connected with the history of Euler's theorem about plane polyhedra were discussed in this famous book.
Imre Lakatos:One of the consequences of our ability to perceive, imagine, or create instances of novel possible configurations is that we can sometimes create new configurations that refute our mathematical conjectures, generalisations or even proofs.
Proofs and refutations: The Logic of Mathematical Discovery
Cambridge University Press, 1976
This is different from the empirical refutation of "All swans are white", which turned up in Australia.
In defining the problem, I had not noticed the need to specify that every cut must go from one boundary point to another: i.e. no cuts may begin or end at a point that is completely surrounded by chocolate.
This example illustrates the relationship between (a) simple everyday activities, and variations that are clearly intelligible to ordinary people with no knowledge of abstruse mathematics, and (b) deep concepts from topology.
Alison Sloman later pointed out that the counter-example might have been ruled out in advance by requiring portions of the slab to be broken rather than cut.
It is important not to inflate Lakatos' argument in {\em Proofs and Refutations} as demonstrating that there is never any real progress in mathematics, or that mathematics is empirical.
On the contrary, every mistake that leads to a revision of a definition, or a statement of a theorem, or a proof adds to our mathematical knowledge: mathematicians can make non-empirical discoveries without being infallible.
Things you probably know, but did not always know:
A circle becomes an ellipse, with changing ratio of lengths of major/minor axes. Rectangles become parallelograms
A child given a set of wooden blocks can do all sorts of experiments -- exploring the space of processes involving the blocks.
Then the child may notice that attempts to rearrange a
configuration into a rectangle always fail:
What kind of
experimentation can that provoke, and what sorts of discoveries can
be made?
How could one be sure that there is NO way of arranging the last collection into a rectangular array, apart from the straight line shown?
Could such a child discover the concept of a prime number?
Could the child discover the fundamental theorem of arithmetic?
(The unique factorization theorem.)
Are some forms of mathematical discovery impossible without a social environment?
Don't assume a teacher with prior knowledge of the theorems has to be involved: someone must have made some of these discoveries without being told them by a teacher.
NOTE
One of the fundamental requirements for mathematical thinking is
being able to organise collections of possibilities and making sure
that you have checked them all.
If you can't do that you don't have a mathematical result, only a guess.
Having discovered those possibilities an animal, or robot, can play with them, e.g. by trying various combinations of possibilities to find out what happens.
We can play in the environment, and we can play in our minds.
Both kinds of experimentation can increase know-how, and support faster problem-solving, using patterns that have been learnt and stored. But we need to account for the differences between learning that is empirical and learning that is more like deductive reasoning, or theorem-proving. (As in "toddler theorems" about opening and shutting drawers and doors, or pulling a piece of string attached to something at the other end.)
He would put both hands on the rim of the open drawer and push: disaster.
Eventually he discovered a different way.
Is the discovery that using the flat of your hand to push a drawer
shut avoids the pain a purely empirical discovery?
Or could the consequence be something that is worked out,
either before or after the action is first performed that way.
Another toddler theorem -- for some toddlers?
What sorts of representational, architectural, and reasoning (information manipulation) capabilities could enable a child to work out
Why it is easier to carry a tray full of cups and saucers using a hand at each side than using only one hand on one side?
For the purposes of research in intelligent robots, we have created
an artificial domain in which humans may have as much to learn as
the robots, and which can start simple, then get increasingly
complex: the domain of polyflaps. See
http://www.cs.bham.ac.uk/research/projects/cogaff/misc/polyflaps
Somatic, exosomatic, meta-semantic....
Maintained by
Aaron Sloman
School of Computer Science
The University of Birmingham