(DRAFT: Liable to change)
Installed: 26 May 2014
Last updated: 28 May 2014, 4 Jun 2014; 5 Jun 2014
This web site (with four subsidiary sites listed below) is
A PDF version may be added later.
A partial index of discussion notes is in
This is part of a collection of examples of abilities of humans to reason in a
mathematical way even if they don't know they are doing it.
For more examples, see the 'Toddler Theorems' Web page:
including examples of reasoning about triangles, e.g.
and this discussion of blankets, strings, and other things for pulling objects
The processes involved in detecting and understanding knots are also relevant to
problems in designing perceptual (especially visual) systems. It seems that
"knottedness", like "connectedness" can be computationally hard to detect, and I
suspect that like connectedness it is highly resistant to parallel visual
processing in a fixed number of steps. For an incomplete discussion of
limitations of current AI/Robotics visual mechanisms and theories of vision in
psychology/neuroscience (work in progress) see:
Knotty and not-knotty perceptual problems
This directory contains several sets of images of strings with and without knots.
Each set of images is associated with a problem about possibilities for motion
of a string or rope lying on a flat surface, and curled or looped in various
ways, some with some and without knots. In some cases, it is hard to tell
whether a knot exists without attempting to pull the ends of the string apart.
This is related to an earlier collection of examples (here) (loosely inspired by
some of Piaget's experiments with young children, involving use of blankets,
string, and other things to bring an object within reach. [REF needed]
Four sets of example problems: in pictures
The photographs are not very good, and may later be replaced by clearer images.
In some cases the thicker string (rope) was a little too stiff for the purposes
of these displays, so I may try again with a more flexible rope.
Distinguishing knotted and knot-free (1)
Distinguishing knotted and knot-free (2)
Which configurations are simple modifications of others?
Recognizing a configuration from different viewpoints
Some of these are easy others much more difficult. (Why?)
For each set of images there is a set of questions which you can try to answer
by examining and thinking about possible changes in the string configurations,
or effects of changes of viewpoint. Some of the examples are very simple, others
more challenging. (Answers to the questions will be added later.)
Here are some additional questions:
Can any existing machine vision system perform any of these tasks?
Can any known machine learning mechanism enable a robot vision system to
perform these tasks?
At what ages can young humans perform tasks like these?
What changes in an individual between not being able and being able to perform
E.g. are new forms of learning needed? Are new forms of representation needed?
Are new ontological extensions (adding types of structure, adding types of
Are new forms of reasoning required?
Do new brain mechanisms have to be grown, or activated? If so, will the
developments be genetically programmed, or based partly on the genome and
partly on what has been learnt earlier, as suggested in Chappell & Sloman (2007)?
Can logical/algebraic/symbolic-computational forms of representation suffice to
implement these abilities?
Can individuals have and use such abilities without being aware of the fact?
What sort of knowledge about strings, space, motion, and continuity do you need
in order to understand why the potential to be straightened without a knot is
preserved by moving only parts of a string between its two fixed ends.
Could you prove this using logic?
It is possible in principle to program a machine to use an algorithm that
traverses its percept of a string from one end to the other recording the
crossing points as 'going over' or 'going under' to build a 'characteristic'
abstract description of the configuration. This description will be preserved if
portions of the string are moved around on the surface while the ends of the
string are held in place. [Problems about this sort of algorithm.....]
Mathematicians have studied knots for some time, as indicated in Braungardt's
introduction (REF below). One of the interesting facts about knots, illustrated
by the difficulty of some of the above tasks, is that there are mathematically
specifiable types of knots that can have many different appearances, and which
are very difficult for a human visual system to distinguish. This is one of many
ways in which the biological mechanisms for perceiving and representing types of
physical structure are limited in the complexity that they can handle, despite
being extremely fast and extremely versatile in the domains that they cope with,
unlike current artificial vision systems.
The possibility of mathematical understanding of knots (and many other things)
that extends far beyond the limits of what humans can perceive (or learn to
perceive) is one of several objections to statistics-based models of human
learning and intelligence (e.g. Bayesian brains...), since those can only
provide statistical generalisations, not mathematical proofs, as in Euclidean
geometry, topology, etc. [To be expanded.]
Background: Possibility transducers
(Read this after you have tried the string problems, above.)
Sloman (1996) suggested that an important aspect of human and, more generally,
animal intelligence is the ability to perceive, think about and make use of
possible variants of a perceived or known structure, process or situation.
Moreover we can discover systematic relationships between the set of possible
changes that can be made to some entity (inputs) and the set of possible
outcomes of those changes (outputs). In that sense particular objects, processes
or situations can be thought of as possibility transducers.
The ability to think about possibilities and to recognise properties of possibility
transducers is part of the ability to perceive and make use of what J.J. Gibson
referred to as "affordances" in the environment, a concept that can be extended
far beyond the sorts of cases Gibson considered. (REF) This is also related to
philosophical discussions regarding the nature of causation and causal powers.
(Many philosophers base their analyses on the notion of "sets of possible worlds"
some including this world. I don't believe that any such semantic competence was
available to our ancestors and other animals that make intelligent use of
possibilities and constraints in the environment. Instead we and they need to be
able to think about possible variations in an object, a process or a state of
affairs that is a bounded portion of this universe.)
The examples above present some of the ways of thinking of a flexible string
resting on a flat surface as a transducer of sets of possibilities in which the
string is preserved, its total length does not change, but its shape, the
geometrical and topological relationships of parts of the string and portions
of space that they occupy, change.
Learning about such things seems to be important for some nest-building animals
(most obviously weaver birds) and for young children growing up in certain
environments, e.g. with shoe-laces, slings, bows and arrows, tents with tent
ropes, and others. As far as I know very little is understood about how
biological brains acquire and use such competences, and I don't know of any
work on intelligent robotics addressing this problem, though the problem is
recognized in Cabalar and Santos (2011)
There are now many online videos of weaver birds. Here's a short BBC video:
(To be expanded)
Aaron Sloman (1996)
Principles of Knowledge Representation and Reasoning:
Proc. 5th Int. Conf. (KR `96),
Eds. L.C. Aiello and S.C. Shapiro,
Morgan Kaufmann Publishers, Boston, MA, pp. 627--638,
Pulling an object towards you: blankets, planks and string
Jackie Chappell and Aaron Sloman, (2007)
Natural and artificial meta-configured altricial information-processing systems,
International Journal of Unconventional Computing 2007, pp. 211--239,
Pedro Cabalar and Paulo E. Santos, (2011)
Formalising the Fisherman's Folly puzzle,
Artificial Intelligence, 175, 1, John McCarthy's Legacy issue, pp. 346--377,
NOTE: They propose a computational approach to solving such puzzles, on
the assumption that the puzzle has been expressed in a logical formalism for use
with a theorem prover. However, they do not claim that any current AI system can
convert such visually presented problems into their logical format. Compare Frege's
objections to Hilbert's axiomatisation of geometry. The same objection can be
made to Frege's logicisation of arithmetic!
Juergen Braungardt's introduction to Knot Theory
Marvin Minsky and Seymour Papert,
Perceptrons: An Introduction to Computational Geometry,
The MIT Press, Cambridge MA,
1972 (2nd edition with corrections, first edition 1969)