(Previous title, now sub-title)
Theorems About Triangles, and Implications for Biological Evolution and AI
The Median Stretch, Side Stretch, Triangle Sum, and Triangle Area Theorems
Old and new proofs.
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This file is http://www.cs.bham.ac.uk/research/projects/cogaff/misc/triangle-theorem.html
also available as http://tinyurl.com/CogMisc/triangle-theorem.html
A messy PDF version will be automatically generated from time to time:
A partial index of discussion notes in this directory is in
See also this discussion of "Toddler Theorems":
This document illustrates some points made in a draft, incomplete, discussion of transitions
in information-processing, in biological evolution, development, learning, etc. here.
That document and this one are both parts of the Meta-Morphogenesis project, partly
inspired by Turing's 1952 paper on morphogenesis.
I suggest below that James Gibson's theory of perception of affordances, is very closely
related to mathematical perception of structure, possibilities for change, and constraints
on changes (structural invariants). Gibson's ideas are summarised, criticised and extended here:
A core aspect is perceiving what is possible -- i.e. acquiring information about structures
and processes that do not exist but could have existed, and might exist in future -- and
grasping some of the constraints on those possibilities.
(This is not to be confused with discovering probabilities: possibilities
are obviously more basic. The differences between learning about constraints
on what's possible and learning probabilities seem to have been ignored by
most researchers studying probabilistic learning mechanisms, e.g. Bayesian
The examples I'll present look very simple but have hidden depths, as a result of which
there is, as far as I know, nothing in AI that is even close to modelling those animal
competences, and nothing in neuroscience that I know of that addresses the problem of
explaining how such competences could be implemented in brains. (I am not claiming that
computer-based machines cannot model them, as Roger Penrose does, only that the current
ways of thinking in AI, Computer science, Neuroscience, Cognitive Science, Philosophy
of mind and Philosophy of mathematics, need to be extended. I'll be happy to be informed
of working models, or even outline designs, implementing such extensions.)
For reasons that will become clearer below this could be dubbed the problem of accounting
for "mathematical qualia", or "contents of mathematical/geometric consciousness" --
their evolution, their cognitive functions, and the mechanisms that implement them.
I have some ideas about the layers of meta-cognitive, and meta-meta-cognitive
mechanisms that are involved in these processes, which I think are related to
Annette Karmiloff-Smith's ideas about "Representational Redescription", (1992)
but I shall not expand on those ideas here: the purpose of this document is
to present the problem.
For more on this see the Meta-Morphogenesis project:
An online English version of Euclid's Elements is here:
That was, arguably, the most important, and most influential, book ever written, ignoring
highly influential books with mythical or false contents. Unfortunately, this seems to have
dropped out of modern education with very sad results.
The discoveries organised and presented in Euclid's Elements were madeI suspect that important subsets of those competences evolved independently in several
using products of biological evolution that humans share with several other
species of animals that can perceive, understand, reason about, construct,
and make use of, structures and processes in the environment-- competences
that are also present in pre-verbal humans, e.g. toddlers.
Human toddlers, and some other animals, seem to be able to make such
discoveries, but they lack the meta-cognitive competences that enable
older humans to inspect and reason about those competences, and the
discoveries they give rise to.
Research on "tool-use" in young children and other animals usually has misguided motivations and should be replaced by research on "matter-manipulation" including use of matter to manipulate matter. But that's a topic for another occasion.
Very often these spatial reasoning competences are confused with very different
competences, such as abilities to learn empirical generalisations from experience,
and to reason probabilistically. In contrast, this discussion is concerned with abilities
to discover what is possible, and constraints on possibilities, i.e. necessities. (These
abilities were also noticed by Immanuel Kant, who, I suspect, would have been actively
attempting to use Artificial Intelligence modelling techniques to do philosophy, had
he been alive now.)
In young humans the mathematical competences discussed here normally become evident
in the context of formal education, and as a result it is sometimes suggested, mistakenly,
that social processes not only play a role in communicating the competences, or the results
of using them, but also determine which forms of reasoning are valid -- a muddle I'll ignore
here, apart from commenting that early forms of these competences seem to be evident in
pre-school children and other animals, though experimental tests are often inconclusive:
we need a deep theory more than we need empirical data.
The capabilities illustrated here are, to the best of my knowledge, not yet replicated
in any AI system, though some machines (e.g. some graphics engines used in computer
games), may appear to have superficially similar capabilities if their limitations
(discussed below) are not exposed. I am not claiming that computers cannot do these
things, merely that novel forms of representation and reasoning, and possibly new
information-processing architectures, will be required -- developing a claim I first
made in Sloman(1971), though I did not then expect it would take so long to replicate
these animal capabilities. That is partly because I did not then understand the full
implications of the claims, especially the connection with some of J.J. Gibson's
ideas about the functions of perception in animals discussed below, and the distinction
between online intelligence and offline intelligence also discussed later, which challenges
some claims made recently about "embodied cognition" and "enactivism", claims that I
regard as deeply confused, because they focus on only a subset of competences associated
with being embodied and inhabiting space and time.
The ideas presented here overlap somewhat with ideas of Jean Mandler on early
conceptual development in children and her use of the notion of an "image schema"
representation, though she seems not to have noticed the need to account for
competences shared with other animals. Studying humans, and trying to model or
replicate their competences, while ignoring other species, and the precocial-altricial
spectrum in animal development, can lead to serious misconceptions.
(I am grateful to Frank Guerin for reminding me of Mandler's work, accessible at
Another colleague recently drew my attention to this paper:
Roger N. Shepard,
The Step to Rationality: The Efficacy of Thought Experiments in Science, Ethics, and Free Will,
In Cognitive Science, Vol 32, 2008,
It's one thing to notice the importance of these concepts and modes of reasoning.
Finding a good characterisation and developing a good explanatory model are very
different, more difficult, tasks.
[Note added: 3 Jan 2013]
This document is also closely related to my 1962 DPhil Thesis attempting to explain and
defend Immanuel Kant's claim (1781) that mathematical knowledge includes propositions that
are necessarily true (i.e. it's impossible for them to be false) but are not provable using
only definitions and logic -- i.e. they are not analytic: they are synthetic necessary truths.
The thesis is available online in the form of scanned in PDF files, kindly provided by the
university of Oxford library:
Aaron Sloman, Knowing and Understanding: Relations between meaning and truth,[End Note]
meaning and necessary truth, meaning and synthetic necessary truth
The most directly relevant section is Chapter 7 "Kinds of Necessary Truth".
It is available in faint but readable format in this file
Also in the Oxford library here.
Very many people have learnt (memorised) the triangle sum theorem, which states that the
interior angles of any triangle (in a plane) add up to half a rotation, i.e. 180 degrees,
or a straight line, even if they have never seen or understood a proof of theorem.
Many who have been shown a proof cannot remember or reconstruct it. I'll introduce a
wonderful proof due to Mary Pardoe later on. For now, notice that the theorem is not at
all obvious if you merely look at an arbitrary triangle, such as Figure T:
Please stare at that for a while and decide what you can learn about triangles from
it. Later you may find that you missed some interesting things that you are capable
I don't know whether the similarity between this exercise and some of the exercises
described by Susan Blackmore in her little book Zen and the Art of Consciousness,
discussed here, is spurious or reflects some deep connection.
Some ways of thinking about triangles and what can be done with them, including
ways of proving the triangle sum theorem, will be presented later. Before that, I'll
introduce some simpler theorems concerning ways of deforming a triangle, and
considering whether and how the enclosed area must change when the triangle is
Note that that's "must change", not "will change", nor "will change with a high
probability". These mathematical discoveries are about what must be the case.
Sometimes researchers who don't understand this regard mathematical knowledge
as a limiting case of empirical knowledge, with a probability of 1.0. Mathematical
necessity has nothing to do with probabilities, but everything to do with constraints
on possibilities, as I hope will be illustrated below.
Why focus on the human ability to notice and prove some invariant property of triangles?
Because it draws attention to abilities to perceive and understand things that are
closely related to what James Gibson called "affordances" in the environment, namely:
animals can obtain information about possibilities for action and constraints on action
that allow actions to be selected and controlled. An example might be detecting that a
gap in a wall is too narrow to walk through normally, but not too narrow if you rotate
your torso through a right angle and then walk (or sidle) sideways through the gap.
You can notice the possibility, and think about it, without making use of it. Use of
offline intelligence is neither a matter of performing actions at the time or in the
immediate future, nor making predictions. The ability to discover such a possibility is
not always tied to be able to make use of it in the near future.
Video 6 here illustrates an 19 month old toddler's grasp of affordances related to a
broom, railings and walls: http://tinyurl.com/BhamCog/movies/vid
The video which shows the child manipulating a broom, includes a variety of actions in
which the child seems to understand the constraints on motion of the broom and performs
appropriate actions, including moving it so as to escape the restrictions on motion that
exist when the broom handle is between upright rails, moving the broom backwards away from
a skirting board in order to be able to rotate it so that it can be pushed down the
corridor, and changing the orientation of the vertical plane containing the broom so that
by the time it reaches the doorway on the right at the end of the corridor the broom is
ready to be pushed through the doorway.
NB: I am not claiming that the child understands what he is doing, or proves theorems.
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Using the ordering information (when available), you can infer that although forward
motion through the gap is impossible, sideways motion through it is possible. It is
important that your understanding is not limited to exactly this spatial configuration
(this precise gap width, this precise starting location, this precise colour of shoe,
this kind of floor material on which you are standing), since you can abstract away
from those details to form a generic understanding of a class of situations in which
a problem can arise and can be solved. The key features of the situation are
relational, e.g. the gap is narrower than one of your dimensions but greater than
another of your dimensions. It does not require absolute measurements. If you learn
this abstraction as a child confronted with a particular size gap you can still use
what you have learnt as an adult confronted with a larger gap, that the child could
have gone through by walking forward.
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Observation of actions of many different animals, for instance nest building birds,
squirrels, hunting mammals, orangutans moving through foliage, and young pre-verbal
children, indicates to the educated observer (especially observers with experience of
the problems of designing intelligent robots) many animal capabilities apparently
based on abilities to perceive, understand and use affordances, some of them more
complex than any discussed by Gibson, including "epistemic affordances" concerned
with possible ways of gaining information, illustrated here. For example, If you are in
a corridor outside a room with an open door, and you move in a straight line towards
the centre of the doorway, you will see more of the room, and will therefore have access
to more information, an epistemic affordance.
(This illustrates the connection between theorems in Euclidean geometry and visual
affordances that are usable by humans, other animals, and future robots.)
I suspect, but will not argue here, that the human ability to make mathematical
discoveries thousands of years ago, that were eventually gathered into a system and
published as Euclid's Elements, depended on the same capability to discover
affordances, enhanced by additional meta-cognitive abilities to think about the
discoveries, communicate them to others, argue about them, and point out and
(These social processes are important but sometimes misconstrued, e.g. byThe information-processing (thinking) required in offline intelligence is sometimes too
conventionalist philosophers of mathematics. They will not be discussed here.)
The roles of external representational media in discovering re-usable generalisations
is different from their meta-cognitive role in reasoning about the status of those
generalisations, e.g. proving that they are theorems. That difference is illustrated
but not explained or modelled in this document.
In this document I have chosen some very simple, somewhat artificial, cases, simply to
illustrate some of the properties of offline thinking competences, in particular, how
they differ from the ability of modern computer simulation engines that can be given
an initial configuration from which they compute in great detail, with great
precision what will happen thereafter. That simulation ability is very different from
the ability to think about a collection of possible trajectories, features they have
in common and ways in which they differ, a requirement for the ability to create
multi-stage plans. (I am not sure Kenneth Craik understood this difference when he
proposed that intelligent animals could use internal models to predict consequences
of possible actions, in (Craik, 1943).)
There are attempts to give machines this more general ability to learn about and use
affordances, by allowing them to learn and use probability distributions, but I shall
try to explain below why that is a very different capability, which lacks the
richness and power of the abilities discussed here, though it is sometimes useful. In
particular, the probability-based mechanisms lack the ability required to do
mathematics and make mathematical discoveries of the kinds illustrated below and in
other documents, including "toddler theorems" of the sorts pre-verbal children seem
able to discover and use, though there are many individual differences between
children: not all can discover the same theorems, nor do they make discoveries in the
One of several motivations for this work is to draw attention to some of what needs
to be explained about biological evolution, for the capabilities discussed here all
depend on evolved competences -- though some may also be implemented in future
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The small successes of the embodied/enactivist approaches (which are, at best, barely
adequate to explain competences of some insects) have diverted attention from the huge
and important gaps in our understanding of animal cognition and the implications for
understanding human cognition and producing robots with human-like intelligence.
Although the online intelligence displayed by the BigDog robot made by Boston Dynamics
[REF] is very impressive, it remains insect-like, though perhaps not all insects are
restricted to "online" intelligence, which involves reacting to the environment under
the control of sensorimotor feedback loops, in contrast with "offline" intelligence,
which involves being able to consider, reason about and make use of possibilities (not
to be confused with probabilities) some of which are used and some avoided. Related
points were made two decades ago by David Kirsh, (though he mistakenly suggests
that tying shoelaces is a non-cerebral competence, possibly because it can become one
through training, as can many other competences initially based on reasoning about
I'll now return to mathematical reasoning about triangles, hoping that readers will
see the connection between that and the ability to use offline intelligence to reason
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When you look at a diagram like Figure T, above, you are able to think of it as
representing a whole class of triangles, and you can also think about processes that
change some aspect of the triangle, such as its shape, size, orientation, or area, in
a way that is not restricted to that particular triangle.
How you think about and reason about possible changes in a spatial configuration is a
deep question, relevant to understanding human and animal cognition -- e.g. perception
of and reasoning about spatial affordances, and also relevant to the task of designing
future intelligent machines. Several examples will be presented and discussed below.
As far as I know, there is no current AI or robotic system that can perform these
although many can do something superficially similar, but much less powerful, namely
answer questions about, or make predictions about, a very specific process, starting
from precisely specified initial conditions. That is not the same as having the
ability to reason about an infinite variety of cases. Machines can now do that using
equivalent algebraic problems, but they don't understand the equivalence between the
algebraic and the geometric problems, discovered by Descartes.
The perception of possible changes in the environment, and constraints on such changes,
is an important biological competence, identified by James Gibson as perception of
"affordances". However, I think he noticed and understood only a small subset of types
of affordance. His ideas are presented and generalised in a presentation on his ideas
(and Marr's ideas) mentioned above.
I shall present several examples of your ability to perceive and reason about possibilities
for change, and constraints on those possibilities inherent in a spatial configuration,
extending the discussion in my 1996 "Actual Possibilities" paper.
In particular, we need to discuss your ability to:
Several proofs of simple theorems will be presented below, making use of your ability
to perceive and reason about possible changes in spatial configurations. I'll start
with some deceptively simple examples relating to the area enclosed by a triangle.
A developmental neuroscience researcher whose work seems to be closely related to
this is Annette Karmiloff-Smith, whose ideas about "Representational Redescription"
in her 1992 book "Beyond Modularity" are discussed here.
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A median of a triangle is a straight line between the midpoint of one side of the
triangle to the opposite vertex (corner). The dashed arrows in Figure M (a) and (b)
lie on medians of the triangles composed of solid lines. The dashed arrows in
triangles (a) and (b) have both been extended beyond the median, which terminates
at the vertex. The dotted lines indicate the new locations that would be produced for
the sides of the triangle if the vertex were moved out, as shown.
Consider what happens if we draw a median in a triangle, namely a line from the
midpoint of one side through the opposite vertex, and then move that vertex along the
extension of the median, as shown in figure M(a). You should find it very obvious
that moving the vertex in one direction along the median increases the area of the
triangle, and movement in the other direction decreases the area. Why?
We can formulate the "Median stretch theorem" (MST) in two parts:
IF a vertex of a triangle is moved along a median away from the opposite side,
THEN the area of the triangle increases.
IF a vertex of a triangle is moved along a median towards the opposite side,
THEN the area of the triangle decreases.
As figure M(b) shows, it makes no difference if the vertex is not perpendicularly
above the opposite side: the diagrammatic proof displays an invariant that is not
sensitive to alteration of the initial shape of the triangle, e.g. changing the slant
of the median, and changing the initial position of the vertex in relation to the
opposite side makes no difference to the truth of theorem. Why?
A problem to think about:
How can you be sure that there is no counter-example to the theorem, e.g. that
stretching or rotating the triangle, or making it a different colour, or painting it
on a different material, or transporting it to Mars, will not make any difference to
the truth of (MST-out) or (MST-in)?
As far as I know the median stretch theorem has never been stated previously, though
I suspect it has been used many times as an "obvious" truth in many contexts, both
mathematical and non-mathematical.
If you know of any statement or discussion of the theorem, please let me know.
Note added 24 Feb 2013:
Readers may find it obvious that the median stretch theorem is a special case of a more
general stretch theorem that can be formulated by relaxing one of the constraints on the
lines in the diagram. Figuring out the generalisation is left as an exercise for the
reader. (Feel free to email me about this.) Compare (Lakatos, 1976).
Added 13 Feb 2013
Julian Bradfield pointed out, in conversation, that one way to think about the truth of
MST-OUT is to notice that the change of vertex adds two triangles to the original
triangle. Likewise, in support of MST-IN, moving the vertex inwards subtracts two
triangles from the original area.
(Below I suggest decomposing the proof into two applications of the Side-Stretch-Theorem
(SST), which can also be thought of as involving the addition or subtraction of a triangle.)
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The concept of "area" used here may seem intuitive and obvious, but
generalising it to figures with arbitrary boundaries is far from obvious and
requires the use of sophisticated mathematical reasoning about limits of
For example, how can you compare the areas of an ellipse and a circle,
neither of which completely encloses the other? What are we asking when we
ask whether the blue circle or the red ellipse has a larger area in Figure A, below?
It is obvious that the black square contains less space than the blue circle, and
also contains less space than the red ellipse, simply because all the space in the
square is also in side the circle and inside the ellipse. But what does it mean to
ask whether one object contains more space than another if each cannot fit inside
Some teachers try to get young children to think about this sort of question by
cutting out figures and weighing them. But that assumes that the concept of
weight is understood. In any case we are not asking whether the portion of paper
(or screen!) included in the circle weighs more than the portion included in the
ellipse. There is a correlation between area and weight (why?) but it is not a
The standard mathematical way of defining the area of a region includes
imagining ways of dividing up non-rectangular regions into combinations of
regions bounded by straight lines (e.g. thin triangles, or small squares),
using the sum of many small areas as an approximation to the large area. The
smaller the squares the better the approximation, in normal cases.
(Why? -- Another area theorem).
For our purposes in considering the triangles in Figure M and Figure S, most of
those difficulties can be ignored, since we can, for now, use just the trivial
fact that if one region totally encloses another then it has a larger area than
the region it encloses, leaving open the question of how to define "area", or
what it means to say that area A1 is larger than area A2, when neither encloses
the other. Our theorems about stretching (MST above and SST below) only require
consideration of area comparisons when one area completely encloses another.
It is very easy for experimental researchers studying animals or young children to
ask whether they do or do not understand areas (or volumes, or lengths of curved
lines), and devise tests to check for understanding, without the researchers
themselves having anything like a full understanding of these concepts that
troubled many great mathematicians for centuries. (I have checked this
by talking to some of the researchers, who had not realised that the
resources for thinking about areas and volumes in very young children
might support only a partial ordering of areas.
These problems are usually made explicit only to students doing a degree in
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If you think about why the MST must always be true, you may notice that the problem
can be broken down into two parts, because the original triangle has two triangular
parts, one on each side of the median; and moving the vertex along the median always
either increases the area of each part or decreases the area of each part. If the
area of each part of the triangle is increased by a movement of the vertex, then the
area of the whole triangle must be increased. Likewise for a decrease in the area of
(See discussion above on "What does 'area' mean?")
If you think about your reasoning about the change in area of each of the two
sub-triangles, you may notice another theorem, which could be called "The side
stretch theorem" illustrated in figure S.
We can formulate the "Side stretch theorem" (SST) in two parts:
(SST-out)Comparing Figure S, with Figure M (a) or Figure M (b) should make it clear that when
IF a vertex of a triangle is moved along an extended side away
from the interior of the side (as in Figure S)
THEN the area of the triangle increases.
IF a vertex of a triangle is moved along a side towards the interior
of that side,
THEN the area of the triangle decreases.
(Draw your own figure for this case.)
Moreover, when the shared vertex in Figure M (a) or (b) moves along the median, both
of the smaller triangles either increase decrease in area, simultaneously, from which
it follows that their combined area must increase when the vertex moves along the
median away from the opposite side and decrease when the vertex moves along the
median towards the opposite side.
For now, I'll leave open the question whether the Side Stretch Theorem (SST-in/out)
be derived from something more basic and obvious. Instead, let's consider what we
mean by "area" before returning to shape changes and their consequences.
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It is also the case that during continuous motion in the same direction a "virtual
discontinuity" can occur. If the vertex starts beyond the original position and moves
back towards the other end of the line, then the area will be continuously
decreasing. But for an observer that has stored information about the original
position, or for that matter any other position on the line, there will be a
discontinuous change from an area greater than the original area to an area less than
the original area -- with "instantaneous equality" separating the two phases of
motion. This discontinuity is not intrinsic to the motion, but involves an external
relationship to a previous state. There are many cases where understanding
mathematical relationships or understanding affordances involves being able to detect
such relational discontinuities (phase changes of a sort).
The relationship between direction of motion of the vertex and whether the area
increases or decreases can be seen to be an invariant relationship. But it is not
clear what information-processing mechanisms make it possible to discover that
invariance, or necessity. Notice that this is utterly different from the kind of
discovery currently made by collecting large numbers of observations and then seeking
statistical relationships in the data generated, which is how much robot learning is
now done. The kind of learning described here, when done by a human does not require
large amounts of data, nor use of statistics. There are no probabilities involved,
only invariant relationships: if the perpendicular distance increases the area must
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In Figure Para, below, two new dotted triangles have been added, a red one and a blue
one, both with vertices on the dashed line, parallel to the base of the original triangle,
and both sharing a side (the base) with the original triangle.
The figure shows that moving the top vertex of the original triangle along a line
parallel to the opposite side will definitely not produce a triangle that encloses
the original, because, whichever way the vertex is moved on that line (the dashed
line in Figure Para) the change produces a triangle with two new sides, one partly
inside the old triangle and the other outside the old triangle. So the new triangle
cannot enclose the old one, or be enclosed by it.
Proving the theorem that moving a vertex of a triangle in a direction parallel to the
opposite side does not alter the area is left as an exercise for the reader, though I
shall return to it below.
There is a standard proof used to establish a formula for the area of a triangle,
which requires consideration of different configurations, as we'll see below.
(The need for case analysis is a common feature of mathematical proof Lakatos 1976).
Exercise for the reader:Try to formulate a theorem about what happens to the sides of a triangle if a vertex moves along a line that goes through the vertex but does not go through the triangle, like the dashed line in Figure Para, above.
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Note:The concept of an infinitely large set being used here is subtle and complex and (as Immanuel Kant noted) raises deep questions about how it is possible to grasp such a concept. For the purposes of this discussion it will suffice to note that if we are considering a range of cases and have a means of producing a new case different from previously considered cases, then that supports an unbounded collection of cases. For example, in Figure S, where a vertex of a triangle is moved along an extension of a side of the triangle, between any two positions of the vertex there is at least one additional possible position, and however far along the extended side the vertex has been moved there are always further locations to which it could be moved. So anyone who is squeamish about referring to infinite sets can, for our purposes, refer to unbounded sets. Below I'll discuss some implications for meta-cognition in biological information processing.
In some cases the new configurations thought about include additional geometrical
features, specifying constraints on the new triangle, for example the constraint that
a vertex is on a median, or extension of a median, of the original triangle, or on a
particular line parallel to one of the sides. Such constraints, involving lines or
circles or other shapes can be used to limit the possible variants of the original
shape, while still leaving infinitely many different cases to be considered.
However, the infinity of possibilities is reduced to a small number of cases by
making use of common features, or invariants, among the infinity of cases.
For instance the common feature may be a vertex lying on a particular line, such as a
median of the original triangle (as in the Median Stretch Theorem (MST) above,
or an extended side of the original triangle (as in the Side Stretch Theorem (SST
above)). Then we can divide the infinity of cases of change of length to two
subsets: a change that increases the length and a change that decreases the length,
as was done for each of the theorems. Each subset has an invariant that can be
inspected by a perceiver or thinker with suitable meta-cognitive capabilities,
discussed further below.
There are more complex cases, as we'll see when considering vertices lying along a
perpendicular to one of the sides of the triangle, required for proving a theorem
about how to calculate the area of a triangle. The complication is that there are
infinitely many perpendiculars to a given line, whereas there is only one extension
to the line, as shown in Figure M and Figure S. However, in all figures required for
discoveries of the sorts we are discussion, there is an additional infinity of cases
because of possible variations in the original triangle considered, before effects of
motion of the vertex are studied.
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NOTE: for now we can ignore the difference between a set being dense andThis ability to notice that some perceived structure or process is continuous, and
being continuous -- a difference that mathematicians did not fully understand until
the 19th Century. I shall go on referring loosely to 'continuity' to cover both cases.
NOTE: The transitions in biological information processing required for organisms
to have this sort of meta-cognitive competence have largely gone unnoticed. But I
suspect they form a very important feature of animal intelligence that later
provided part of the basis for further transitions, including development of
meta-meta-meta... competences required for human intelligence.
These meta-cognitive abilities are superficially related to, but very different from,
abilities using statistical pattern recognition techniques to cluster sets of
measurements on the basis of co-occurrences. Examples of non-statistical competences
include being able to notice that certain differences between cases are irrelevant to
some relationship of interest, or being able to notice a way of partitioning a
continuous set of cases into two or more non-overlapping sub-sets, possibly with
partially indeterminate (fuzzy) boundaries between them. In contrast, many of the
statistical techniques require use of large numbers of precise measures in order to
detect some pattern in the collection of measures (e.g. an average, or the amount of
deviation from the average, or the existence of clusters).
For example, you should find it obvious that the arguments used above based on Figure
M and Figure S to prove the Median Stretch and Side Stretch theorems (MST and SST) do
not depend on the sizes or shapes of the original triangles. So the argument covers
infinitely many different triangular shapes. The features that change if a vertex is
moved away from the opposite side along a median or along a side will always change
in the same direction, namely, increasing the area.
Noticing an invariant topological or geometrical relationship by abstracting away
from details of one particular case is very different from searching for correlations
in a large number of particular cases represented in precise detail. For example
computation of averages and various other statistics requires availability of
many particular, precise, measurements, whereas the discovery process demonstrated
above does not require even one precisely measured case. The messy and blurred Figure
S-b will do just as well to support the reasoning used in connection with the more
precise Figure S, though even that has lines that are not infinitely thin.
Most of these points were made, though less clearly in (Sloman, 1971) which also
emphasised the fact that for mathematical reasoning the use of external diagrams is
sometimes essential because the complexities of some reasoning are too great for a
mental diagram. (These points were generalised in Sloman 1978 Chapter 6). Every
mathematician who reasons with the help of a blackboard or sheet of paper knows this,
and understands the difference between using something in the environment to reason
with and using physical apparatus to do empirical research, though it took some time
for many philosophers of mind to notice that minds are extended. (The point was also
made in relation to reference to the past in P.F. Strawson's 1959 book, Individuals,
An essay in descriptive metaphysics.)
NOTE ADDED 12 Sep 2012: DIAGRAMS CAN BE SLOPPY
In many cases a mathematician constructing a proof will draw a diagram withoutBACK TO CONTENTS
bothering to ensure that the lines are perfectly straight, or perfectly
circular, etc., or that they are infinitely thin (difficult with line drawing
devices available to us). That's because what is being studied is not the
particular physical line or lines drawn on paper or sand, etc. The lines drawn
are merely representations of perfect Euclidean lines whose properties are
actually very different, and very difficult to represent accurately on a
blackboard or on paper. E.g. drawing an infinitely thin line has been a problem.
In fact, the lines don't need to be drawn physically at all: they can be imagined
and reasoned about, though in some cases a physical drawing can help with
both memory and reasoning.
All this seems to b closely related to the ability of animals to perceive affordances
of various kinds, as discussed in http://tinyurl.com/BhamCog/talks/#gibson .
In particular, the kind of mathematical reasoning about infinite ranges of
possibilities and implications of constraints, seems to be closely related to the
ability of young children and other animals to discover possibilities for change in
their environments, and abilities to reason about invariants in subsets of
possibilities that can be relied on when planning actions in the environment.
This leads to the notion of a "toddler theorem" discussed in
I suspect that the reasoning using schematic diagrams illustrated above, and also
illustrated below in Pardoe's proof of the Triangle Sum Theorem (TST), shares features
with animal reasoning about affordances, in which conclusions are reliably drawn about
invariants that are preserved in a process, or about impossibilities in some cases --
e.g. it is impossible to completely enclose a bounded area using only two straight lines.
Why is it impossible?
There have been attempts to simulate mathematical reasoning using diagrams by giving
machines the ability to construct and run simulations of physical processes. But that
misses the point: a computer running a simulation in order to derive a conclusion can
handle only the specific values (angles, lengths, speeds, for example) that occur in
the initial and predicted end states when the simulation runs. Moreover, the
simulation mechanisms have to be carefully crafted to be accurate. In contrast, as
pointed out above, a human reasoning about a geometrical theorem does not require
precision in the diagrams and the conclusion drawn is typically not restricted to the
particular lengths, angles, areas, etc. but can be understood to apply to infinitely
many different configurations satisfying the initial conditions of the proof.
(See the recent discussion between Mary Leng and Mateja Jamnik, in The Reasoner.)
This seems to require something very different from the ability to run a simulation:
it requires the ability to manipulate an abstract representation and to interpret
the results of the manipulation in the light of the representational function of the
representations manipulated. In other words mathematical thinking using diagrams and
imagined transformations of geometrical structures, as illustrated above, inherently
requires meta-cognitive abilities to notice and reason about features of a process in
which semantically interpreted structures are manipulated. The noticing and reasoning
need not itself be noticed or reasoned about, although that may develop later (as
seems to happen, in different degrees, in humans).
I suspect that many animals, and also pre-verbal human children have simplified
versions of that ability, but do not know that they have it. They cannot inspect
their reasoning, evaluate it, communicate it to others, wonder whether they have
covered all cases, etc. There seems to be a kind of meta-cognitive development that
occurs in humans, perhaps partly as a result of learning to communicate and to think
using an external language. It may be that some highly intelligent non-human
reasoners have something closely related. But we shall need more detailed
specifications of the reasoning processes and the mechanisms required, before we
can check that conjecture.
(Annette Karmiloff-Smith's ideas about "Representational Redescription", in
"Beyond Modularity" are also relevant.)
We also need more detailed specifications in order to build robots with these
"pre-historic" mathematical reasoning capabilities -- which, as far as I can tell, no
AI systems have at present. Unlike Roger Penrose, who seems to me to have noticed
similar features of mathematical reasoning, I don't think there is any obvious reason
why computer based systems cannot have similar capabilities. However it may turn out
that there is something about animal abilities to perceive, or imagine, processes of
continuous change at the same time as noticing logically expressible constraints or
invariants of those processes that requires information processing mechanisms that
have so far not been understood. Alternatively, it may simply be that no high calibre
AI programmers have attempted to implement competences of the sorts required to
invent and understand Pardoe's proof, or many of the traditional proofs used in
These ideas suggest a host of possible investigations of ways in which human
capabilities change, along with the reasoning competences of intelligent animals such
as squirrels, elephants, apes, cetaceans, octopuses, and others.
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There is a standard way (or small set of standard ways) of proving the theorem
Triangle Sum Theorem (TST): The interior angles of a triangle add up to a
straight line, or half a rotation (180 degrees).
These standard methods all make use of some version of Euclid's parallel
(Axiom 5 in Euclid's elements) which can be formulated in several equivalent ways, e.g.
Two straight lines L1 and L2 are parallel if and only if they are co-planar and
have no point in common, no matter how far they are extended.
Given a straight line L in a plane, and a point P in the plane not on L, there
is exactly one line through P that is in the plane and parallel to L.
All of this presupposes the concept of "straightness" of a line. For now I'll take
that concept for granted, without attempting to define it, though we can note that if
a line is straight it is also symmetric about itself (it coincides with its
reflection) and also it can be slid along itself without any gaps appearing. If it
were possible to view a straight line from one end it would appear as a point.
The "standard" ways of proving the TST make use of properties of angles formed
when a straight line joins or crosses a pair of parallel lines:
COR: Corresponding angles are equal:BACK TO CONTENTS
If two lines L1, L2 are parallel and a third line L3 is drawn from any point P1
on L1 to a point P2 on L2 and continued beyond P2,
then the angle that L1 makes with the line L3 at point P1, and the angle L2
makes with the line L3 at point P2 (where the angles are on the same side of
both lines) are equal.
ALT: Alternate angles are equal:
If two lines L1, L2 are parallel and a third line L3 is drawn from any point P1
on L1 to a point P2 on L2,
then the angle L1 makes with the line L3 at point P1, and the angle L2 makes
with the line L3 at point P2 (on the opposite sides of both lines) are equal.
For more on transversals and relations between the angles they create, see
That page teaches concepts with some interactive illustrations, but presents no proofs.
The Euclidean proofs of COR and ALT are presented here:
Warning: I have found some online proofs of theorems in Euclidean geometry with bugs
apparently due to carelessness, so it is important to check every such proof found
online. The fact that individual thinkers can check such a proof is in part of what
needs to be explained.
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Her proof just involves rotating a single directed line segment (or arrow, or pencil,
or ...) through each of the angles in turn at the corners of the triangle, which must
result in its ending up in its initial location pointing in the opposite direction,
without ever crossing over itself.
So the total rotation angle is equivalent to a straight line, or half rotation, i.e.
180 degrees, using the convention that a full rotation is 360 degrees.
The proof is illustrated below in Figure Ang2.
In order to understand the proof, think of the blue arrow, labelled "1", as starting
on line AC, pointing from A to C, and then being rotated first around point A, then
point B, then point C until it ends up on the original line but pointing in the
direction of the dark grey arrow, labelled "4".
So, understanding the proof involves considering what happens if
A "time-lapse" presentation of the proof may be clearer, as shown in Figure Ang3:
It may be best to think of the proof not as a static diagram but as a process, with
stages represented from left to right in Figure Ang3. In the first stage, the pale
blue arrow starts on the bottom side of the triangle, pointing to the right then is
rotated through each of the internal angles A, B, C, always rotated in the same
direction (counter-clockwise in this case), so that it lies on each of the other
sides in succession, until it is finally rotated through the third angle, c, after
which it lies on the original side of the triangle, but obviously pointing in the
opposite direction. Some people may prefer to rotate something like a pencil rather
than imagining a rotation depicted by snapshots.
In this triangle the sides are not very different in length, which conceals a problem
that can arise if the first side the arrow is on is very short and the other two are
longer. If the length of the arrow is fixed by the length of the first side, you
would need to imagine either that the arrow stretches or shrinks as it rotates, or
that it slides along a line after reaching it so as to be able to rotate around the
next vertex. Alternatively you can imagine that the depicted arrow is part of a much
longer invisible arrow, so that, as the invisible arrow rotates from one side to
another, it always extends beyond both ends of the new side, and can then rotate
around the next vertex. I leave it to the reader to think about these alternatives
and what difference they make to the proof, and to the cognitive competences required
to construct and understand the proof.
For an arrow to be rotated in a plane and end up lying in its original position it
must have been rotated through some number of half-rotations. (Each half rotation
brings it back to the original orientation, but pointing in alternate directions.)
Since (1) the arrow at no point crossed over its original orientation, and (2)
up pointing in the opposite direction to its original orientation, the total rotation was
through a half circle -- which is clear if you actually perform the rotations using a
physical object, such as a pencil.
And since that rotation was made up of combined rotations through angles A, B, and C,
those three angles must add up to a half circle, i.e. 180 degrees.
A crucial feature of our ability to think about the diagram and the process, is that
we (presumably including you, the reader) can see that the key features of the
process could have been replicated, no matter what the size or orientation of the
triangle, no matter what the lengths of the sides or the sizes of the angles, no
matter which side the arrow starts on, no matter which way it is pointing initially,
and no matter in which order the rotations are performed, e.g. A then B then C, or C
reversed, then B reversed, then A reversed.
This proof of the triangle sum theorem, using a rotating moving arrow, works for all
possible triangles on a plane -- as do the standard Euclidean proofs using parallel
This proof is unlike standard proofs in Euclidean geometry since it involves
consideration of continuous processes, and therefore involves time and temporal
ordering, whereas Euclidean geometry does not explicitly mention time or processes --
though there are some theorems about the locus of point or line satisfying certain
constraints, which can be interpreted either as specifying properties of processes
extended in time, or as properties of static trajectories, e.g. properties of lines
http://tinyurl.com/CogMisc/p-geometry.html presents a more detailed, but still
incomplete, discussion, of the geometrical prerequisites for some of the above
reasoning. It introduces the idea of P-geometry, which is intended to be Euclidean
geometry without the Axiom of Parallels (Euclid's Axiom 5), but with time and motion
added, including translation and rotation of rigid line-segments. When I get time, I
should add the side-stretch theorem to it (SST above).
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The presentations produced no responses -- either critical or approving, except thatAaron Sloman, 2008, Kantian Philosophy of Mathematics and Young Robots, in Intelligent Computer Mathematics, Eds. Autexier, S., Campbell, J., Rubio, J., Sorge, V., Suzuki, M., and Wiedijk, F., LLNCS no 5144, pp. 558-573, Springer, http://www.cs.bham.ac.uk/research/projects/cosy/papers#tr0802 Aaron Sloman, 2010, If Learning Maths Requires a Teacher, Where did the First Teachers Come From? In Proceedings Symposium on Mathematical Practice and Cognition, AISB 2010 Convention, De Montfort University, Leicester http://www.cs.bham.ac.uk/research/projects/cogaff/10.html#1001 And in talks on mathematical cognition and philosophy of mathematics here: http://www.cs.bham.ac.uk/research/projects/cogaff/talks/
I tried searching for online proofs to see if anyone else had discovered this proof
or used it, but nothing turned up. The proof using rotation is so simple and so
effective that both Mary Pardoe and I feel sure it must have been discovered
NOTE ADDED 6 Oct 2012:
I have very recently discovered that as a result of the discussion I stirred up in
2010 on the MKM-IG email list, Andrea Asperti mentioned the proof (and the email
discussion) in this paper, discussing related issues:
Andrea Asperti, Proof, Message and Certificate, in AISC/MKM/Calculemus, 2012, pp. 17--31, Online: http://www.cs.unibo.it/~asperti/PAPERS/proofs.pdf http://dx.doi.org/10.1007/978-3-642-31374-5_2NOTE:
NB: The programs that present such proofs do not themselves understand the proofs.
They can be powerful "cognitive prosthetics" for humans learning mathematics, but the
programs do not know what they have done, or why they have done it, and do not understand
the invariants involved -- e.g. essentially the same proof could have started with a
triangle with different angles, or a triangle of a different size.
I'll now return to the consideration of areas of triangles and how the area of a
triangle is altered by moving one vertex, extending the ideas used in discussing the
Median Stretch Theorem and Side Stretch Theorem, above.
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It should be obvious from the figure that it presents a proof that the exterior anti-clockwise
angles of a triangle sum to a circle (360 degrees) as do the exterior clockwise angles,
not shown in the figure.
Added 19 Mar 2013: This was named "The total turtle trip theorem" by Seymour Papert,
in his Mindstorms: Children, Computers, and Powerful Ideas (1978), though it was well
known long before then. (It can be generalised to smooth simple closed curves. See also
The exterior anti-clockwise angles are those obtained by extending each side in turn in
one direction then rotating the extension to line up with the next side. So, for example,
in Figure Ang4, the internal angles are a, b and c and the exterior anti-clockwise
angles A, B and C are got by extending the first side to location 1 then rotating
the extension through angle A to the next side, then extending that side to location
2 and rotating the extension through angle B to the second side, and so on.
Because results of all those rotations bring the rotated arrows back to the original
orientation, indicated at 1 in the figure, and the rotated arrow does not pass through
its original direction, the total external anti-clockwise rotation must be a full circle
(i.e. 360 degrees). An exercise left to the reader is to show that that's true not only
for triangles but for all polygons, and, by symmetry, must also be true for the sum of the
clockwise external angles. So:
Theorem External: A + B + C = 360
But each of the internal angles is the complement of the adjacent internal angle,
because they sum to a straight line. So we have these three truths:
Theorem: A + a = 180 therefore a = 180 - A
Theorem: B + b = 180 therefore b = 180 - A
Theorem: C + c = 180 therefore c = 180 - A
So, the sum of the internal angles is
a + b + c = (180 - A) + (180 - B) + (180 - C)
= 180 + (180 + 180) - (A + B + C)
= 180 + 360 - (A + B + C)
Then substituting from Theorem External:
= 180 + 360 - 360
So, we have another proof of the standard Triangle Sum Theorem:
Theorem Internal: a + b + c = 180
I tried searching for that proof using google and did not find a previous occurrence of
it, though there seem to be many web sites that mention both the triangle sum theorem for
interior angles and the theorem about exterior angles always summing to 360.
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However there are some theorems in Euclidean geometry whose proof requires use of
more than one diagram, because the theorem has a kind of generality that covers
structurally different cases. An example of such a theorem is a proof that the area
of a triangle is half the area of a rectangle with the same base length and the same
height: Area = 0.5 x Base x Height. The reason for requiring more than one diagram
(unless there is a proof I have not encountered) will be explained below.
A non-diagrammatic algebraic proof may be possible using the Cartesian-coordinate
based representation of geometry, but that is not what this discussion is about.
It is highly regrettable that our educational system produces many people who have
simply memorised the Area formula, without ever discovering a proof or being shown
one, or even being told that there is a proof, though some may have done experiments
weighing triangular and rectangular cards. I shall try to explain how this formula
could be proved, though I'll expand the usual proof to help bring out differences
between this theorem and previous theorems, explaining why this theorem requires
different cases to be dealt with differently.
Consider a theorem related to Figure P-a below, which is subtly different from
Figure M (a),
Figure P-a includes a straight line drawn between a vertex of the triangle and the
opposite side, extended beyond the vertex as indicated by the dashed arrow. In
figure M the line used was a median, joining the mid-point of a side to the opposite
vertex. Here the line is not a median but is perpendicular to the opposite side.
(In some cases the median and the perpendicular are the same line. Which cases?)
You should find it obvious that if the top vertex of the triangle with solid
sides shown in Figure P-a, above, is moved further away from the opposite side
(the base), along a line perpendicular to the opposite side (the dashed arrow), then
the area enclosed by the triangle must increase. This could be called the
"Perpendicular Stretch Theorem" (PST), in contrast with the "Median Stretch Theorem"
(MST), which used a line drawn from the middle of the base.
In this figure it is obvious that moving the vertex up the perpendicular will produce
a new triangle that encloses the original one. Figure P-a shows why it is obvious,
though the Side Stretch Theorem shown in figure S, above, could used to prove this,
by dividing the figure into two parts, just as it was used to prove the Median
stretch theorem. (As with MST, there is a corresponding theorem about the area
decreasing if the vertex moves in the opposite direction on the perpendicular.)
But there is a problem, which you may have noticed, a problem that did not arise for
the median stretch theorem. The problem is that whereas any median from the midpoint
of one side to the opposite vertex will go through the interior of the triangle, the
perpendicular from a side to the opposite vertex may not go through the interior of
the triangle, a problem portended by part (b) of Figure M.
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In this case, moving the top vertex upwards will not produce a new triangle enclosing
in the old one, because one of the sides of the triangle will move so as to cross the
triangle, as illustrated in Figure P-b. So now the proof that the area increases cannot
be based on containment: the new triangle produced by moving the vertex upward does
not include the old triangle, as in the previous configuration. Is there a way of
reasoning about this new configuration so as to demonstrate an invariant relation
between direction of motion of the vertex and whether the area of the triangle
increases or decreases?
Some readers may notice a way of modifying the proof to deal with figure P-b, thereby
extending the proof that moving the vertex further from the line in which the
opposite side lies, always increases the area. It is an extension insofar as it
covers more cases. Of course, the original proof covered an infinite set of cases,
but that infinite set can be extended.
A clue as to how to proceed can come from considering how to prove that moving the
vertex of a triangle parallel to the opposite side, as illustrated in Figure P-c, below,
cannot change the area.
TO BE CONTINUED
I shall later extend this discussion by showing how to relate the area of a triangle
to the area of a rectangle enclosing it. It will turn out that the triangle must
always have half the area of the rectangle, if the rectangle has one side equal in
length to a side of the triangle and the other side equal in length to the
perpendicular of the triangle. Proving this requires dealing with figures P-a and P-b
The proof using a rectangle requires introducing a new discontinuity into the
configuration: dividing up regions of the plane so that they can be compared, added,
Some readers will be tempted to prove the result by using a standard formula for the
area of a triangle. In that case they first need to prove that the formula covers all
cases, including the sort of triangle shown in Figure P-b.
For anyone interested, here's a hint. Consider Figure TriRect, below. Try to prove
that every triangle can be given an enclosing rectangle, such that every vertex of
the triangle is on a side of the rectangle and two of the vertices are on one side of
the rectangle, and at least two of the vertices of the triangle lie on vertices of the
Can you prove something about the area of a triangle by considering such enclosing
Many mathematical proofs are concerned with cases that differ in ways that require
different proofs, though sometimes there is a way of re-formulating the proof so that
the same reasoning applies to all the structurally distinct cases. A fascinating
series of examples from the history of mathematics is presented in (Lakatos, 1976)
A hard problem for human and animal psychology, and studies of evolution of
cognition, is to explain how humans (and presumably some other animals capable
of intelligent reasoning about their affordances), are able to perform these
feats. How do their brains, or their minds (the virtual information-processing
machines running on their brains), become aware that the special case being
perceived shares structure and consequences of that structure, with infinitely
many other configurations, the majority of which have never before been seen or
For an explanation of the notion of a virtual machine made of other concurrently
active virtual machines, some of which also interact with the environment, see
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Even longer before a robot mathematician spontaneously re-invents Pardoe's proof?
(Or the proofs in Nelsen's book.)
For some speculations about evolution of mathematical competences see
If there are differences I suspect they may depend on some of the following:
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PremissesFor someone who does not find this obvious, the proof can first be transformed into
(P1) All Humans are Mortal (or (All x)H(x) -> M(x))
(P2) All Greeks are Humans (or (All x)G(x) -> H(x))
(C) All Greeks are Mortal (or (All x)G(x) -> M(x))
This can be thought of as a process, but the steps are distinct and there are not
meaningful intermediate stages, e.g in which the antecedent "H(x)" and the
implication arrow "->" are gradually removed from the original implication, and the
word "Socrates" gradually replaces the variable "x". Nevertheless the proof can be
expressed diagrammatically using Euler Circles as in Figure Syll (often confused
with Venn Diagrams, which could also be used).
In (Sloman, 1971) I suggested that both types of proof could be regarded as involving
operations on representations that are guaranteed to "preserve denotation". This is
an oversimplification, but perhaps an extension of that idea can be made to work.
In the Pardoe proof, "preserving denotation" would have to imply that a process
starting with the initial configuration in Figure Ang3, and keeping the triangle
unchanged throughout, could go through the stages in the successive configurations
depicted, without anything in the state of affairs being depicted changing to
accommodate the depiction, apart from the changes in position and orientation of the
arrow, as depicted. This implies, for example, that there are no damaging operations
on the material of which the structures are composed. (I suspect there is a better
way to express all this.)
Cathy Legg has presented some of the ideas of C.S. Peirce on diagrammatic reasoning
in (Legg 2011) It is not clear to me whether Peirce's ideas can be usefully applied to
the kinds of reasoning discussed here, which are concerned with geometrical reasoning
as a biological phenomenon with roots in pre-human cognition, and properties that
I suspect could be replicated in robots, but have not yet been, in part because the
phenomena have not yet been understood.
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.
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,
including Mary Pardoe's proof.
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?
School of Computer Science
The University of Birmingham