2nd June 2021 -- 15:00-18:00 CEST time 14:00-17:00 UK (summer) time (Liable to change)
The material below (including text, images, diagrams, videos) was intended to be used for a live Zoom presentation with shared screen. I was unable to use screen-sharing owing to an unannounced change in Zoom. The actual talk was recorded in a "Talking head" video, which has now been added to the Tubingen Youtube web site here: https://www.youtube.com/watch?v=Bbh8E3Pk0R0
The subsequent discussion was also recorded but there are no plans to add it to
that web site. I have made it available here:
(The recording of the discussion may have to be removed, as I have not yet obtained permission to make it public.)
I was not able to show any of the material below during the talk because the zoom update had made it impossible for me to use screen-sharing mode.
I discovered the remedy too late: a few days after the presentation.
In the presentation much is made of the proceesses of assembly of an animal (vertebrate or invertebrate) by chemical processes inside an egg or cocoon, with no opportunity for learning complex behaviours in the enclosed space. Here is a sequence of images indicating the growing complexity of the foetus inside a chicken's egg. Most of the internal complexity cannot be shown in a few photographs (bones, nerves, blood-vessels, sinews, muscles, etc.)
Some of the stages are listed in a separate document showing
Photographs of chick embryo stages
Later, I'll add more links to related work in biochemistry, embryology,
immunology, developmental biology, varieties of forms and mechanisms of
reasoning, including for example, work by Mike Levin and colleagues:
Some of the ideas presented in the video and this web page were included in a
"COGS" presentation at the University of Sussex on 16th Feb 2021, which made use
of a subset of this web page.
Installed: 2 Jun 2021; Updated 6 Jun 2021
This document is
This page is still under development.
Suggestions for improvement (or relevant links) are welcome.
The talk will present examples of types of spatial intelligence, used in detecting and employing varieties of spatial possibility, necessity, and impossibility, that cannot be explained by currently known mechanisms._______________________________________________________________________________________
Evidence from newly hatched animals suggests that mechanisms using still unknown chemistry-based forms of computation can provide information that is not restricted to regularity detection, but is concerned with possibility spaces and their restrictions. Ancient human spatial intelligence is partly based on multi-generational discovery of what is possible, necessarily the case, or impossible, in complex and changing environments, using mechanisms of spatial cognition, centuries before Euclid, that enabled discoveries regarding possibility, impossibility and necessity in spatial structures and processes, long before modern mathematical, symbolic, logic-based, or algebraic formalisms were available.
Immanuel Kant characterised such mathematical cognition in terms of three distinctions largely ignored in contemporary psychology, neuroscience, and AI research: non-empirical/empirical, analytic/synthetic, and necessary/contingent. He argued that ancient geometric cognition was not based simply on empirical generalization, nor on logical deduction from arbitrary definitions. The truths discovered were non-empirical, synthetic, and non-contingent. I'll explain below in more detail how Kant's ideas differed from David Hume's.
Neither formal logic-based characterizations of mathematics (used in automated theorem provers), nor postulated neural networks collecting statistical evidence to derive probabilities, can model or explain such ancient mathematical discoveries. E.g. necessity and impossibility are not extremes on a probability scale.
Unexplained facts about spatial competences of newly hatched animals, before neural networks can be trained in the environment, may be related to mechanisms underlying ancient spatial intelligence in humans and other animals.
Chemical mechanisms inside eggs, available before hatching, somehow co-existing with the developing embryo, apparently suffice to produce the hatchling's spatial intelligence. Such mechanisms may be partly analogous to types of "virtual machinery" only recently developed in sophisticated forms that provide services across the internet (like zoom meetings) that "float persistently" above the constantly changing, particular physical mechanisms at work, without occupying additional space.
While chemical mechanisms in early stages of reproduction are well-studied, little is known about the enormously complex types of machinery required for later stages, e.g., of chick production, including creation of control mechanisms required for actions soon after hatching. I suggest that development of the foetus uses many stages of control by increasingly sophisticated virtual machines controlling and coordinating chemical mechanisms as they create new chemical mechanisms and new layers of virtual machinery.
Different sub-types must have evolved at different times, and the later, more complex virtual machines may have to be assembled by earlier virtual machines, during foetus development, whereas earliest stages of reproduction simply use molecular mechanisms controlling formation and release of chemical bonds linking relatively simple chemical structures.
I suspect Alan Turing's work on chemistry-based morphogenesis (published 1952) was a side effect of deeper, more general, thinking about uses of chemistry-based spatial reasoning in intelligent organisms. But he died without publishing anything to support that suspicion, though he did assert in 1936 that machines can use mathematical ingenuity, but not mathematical intuition, without explaining the difference (on which Kant might have agreed). We may never know how far Turing's thinking had progressed by the time he died.
It will also highlight a serious gap in mathematical education since the middle
of the 20th Century. This disastrous educational change was initially inspired
by the Bourbaki project in France, which emphasised mathematics as a formal,
symbolic, activity, disparaging centuries of mathematics based on spatial
reasoning. Wikipedia gives a useful overview:
I'll include some familiar and unfamiliar examples of spatial mathematical knowledge, and use them to illustrate differences between David Hume's and Immanuel Kant's views on types of knowledge (or types of truth). I am sure Hume must have been familiar with Euclidean geometry, but did not reflect on what he knew, as Kant did.
I'll give an example of very familiar facts about eggs and chickens that I failed to reflect on for many years. Questions about those facts undermine common beliefs about the nature of mathematical knowledge, in surprising ways.
The account of Euclidean geometry in this talk is somewhat short and shallow. There are tutorials online that provide more detail. Below is a sample collection of examples and online tutorials.
Presentation on Euclidean geometry by Zsuzsanna Dancso at MSRI.
Old and new proofs concerning the sum of interior angles of a triangle.
-- Mary Pardoe's proof (about 1970) unwittingly rediscovering an old proof:
Whatever the size and shape of the triangle it is obvious (how) that, so long as it is on a plane surface, three rotations of the blue arrow will combine the sizes of the interior angles of the triangle, necessarily adding up half of a full rotation, i.e. 180 degrees.
Types of spatial (proto-mathematical) understanding (in very young children) of what is possible or impossible/necessary in "everyday" spatial structures and processes.
Is it possible to shut a drawer without squashing any fingers?
(First reported to me by Manfred Kerber.)
Can the child infer something about shutting doors?
Does everything that's impossible have to be learnt from experience?
Pre-verbal Toddler Topologist
I suggest that the ancient mathematical discoveries based on spatial reasoning use brain mechanisms that nobody understands yet, that are closely related to the forms of spatial reasoning found in other intelligent species, e.g. birds that assemble complex nests, such as crows and weaver-birds.
The required spatial reasoning abilities evolved before the development of (human) language-based reasoning, because they seems to exist in many non-human intelligent species without human languages.
Moreover, some examples of intelligent (but presumably unconscious) spatial reasoning are evident in pre-verbal human children (as illustrated in this video of a child with a pencil:
There are more examples on the "Toddler theorems" page.
Compare squirrels and humans with cats and dogs trying to get through a gap while
holding something horizontally in their mouth.
Cat and gate video.
Learning about epistemic affordances
Getting information about the world from the world, and making the directly available information change.
Things you probably know, but did not always know:
- You can get more information about the contents of a room from outside an open doorway
(a) if you move closer to the doorway,
(b) if you keep your distance but move sideways.
Why do those procedures work? How do they differ?
- Why do perceived aspect-ratios of visible objects change as you change your viewpoint?
A circle becomes an ellipse, with changing ratio of lengths of major/minor axes.
Rectangles become parallelograms
- In order to shut a door, why do you sometimes need to push it, sometimes to pull it?
- Why do you need a handle to pull the door shut, but not to push it shut?
- Why do you see different parts of an object as you move round it?
- Why can you use your experience moving round a house to predict your experiences when you move round it in the opposite direction? (Example due to Immanuel Kant).
- When can you can avoid bumping into the left doorpost while going through a doorway by aiming further to the right -- and what problem does that raise?
- How you could use the lid of one coffee tin to open the lid of another which you cannot prise out using your fingers? (Also mentioned above)
How did some species evolve abilities to work out that something is impossible in more cases?
What brain mechanisms enable them to detect impossibility?
Is everything learnt from failure? That could be very inefficient for a tree climbing species where most failures are not survived? E,g. a heavy animal, like an orangutan.
-- Linking and unlinking rings made of solid impenetrable material?
What brain mechanisms allow you to judge that some describable spatial process is impossible? Statistics-based learning can merely show something is improbable, not impossible (or necessarily true).
So ancient mathematical discoveries must have used something more powerful than statistics.
Reasoning about a Torus.
What sorts of closed curves can be drawn on the surface of a torus?
2000 years unsolved: Why is doubling cubes and squaring circles impossible using only straight edge and compasses?
How many ways can circles overlap? - Numberphile
-- Many more examples are available online using Alexander Bogomolny's "live" online demos:
"Morley's Miracle" (trisectors of angles of a triangle form an equilateral triangle"
What you can do with a 5x5 Square Grid and 5 Circles
Complete index to the examples
-- A miscellaneous collection of examples of spatial reasoning involving possibility impossibility and necessity
Ancient mathematicians discovered (and proved) that if squares are constructed
on each side of a right angled planar triangle (i.e. a triangle one of whose
angles is a right angle -- a 90 degree angle) then the area of the square on the
hypotenuse (the side opposite the right angle) must equal
the sum of the areas on the other two sides.
(See https://en.wikipedia.org/wiki/Pythagorean_theorem.) This theorem applies only to triangles in a plane, e.g. not on the surface of a sphere nor any other curved surface.
So in this image (from Wikipedia) the area of Square C is the sum of the areas of the squares A and B:
This result has been proved in hundreds of different ways. Proofs of the theorem had been discovered several hundred years before Pythagoras was born!
Here's a link to a "dynamic" proof, on Wikipedia, which would not have been
valid according to Euclid because it involves moving triangles:
(Readers who are familiar with Euclidean geometry should be able to modify this proof so that it does not involve any motion.)
Core abilities that evolved to support effective interaction with increasingly complex and varied spatial environments in many species, including humans, were later extended by meta-cognitive mechanisms supporting further discovery based on reflection on what has been learnt and communication with other individuals about what is being learnt. In humans this includes use of a "meta-configured" genome, that uses delayed gene-expression for powerful meta-competences, to deal more effectively with environmental influences that change across generations, including cultural and linguistic influences in the case of humans.
(The ideas are summarised in a short video here
One outcome of use of a meta-configured genome is production of a large number
of very different human languages (including signed, vocal, and written
languages), as a result of the same gene expression mechanisms at work in
Also "meta-configured" genome
In contrast, what is now called a "deep learning" mechanism, which provides only learning in a single individual in a single generation, is more accurately described as "multi-level shallow learning".
Competences based on a meta-configured genome, including ancient mathematical abilities, cannot be explained by mechanisms currently known to biologists, neuroscientists, psychologists, physicists, philosophers or AI researchers -- unless I've missed something. I'll provide evidence that they depend on still unknown chemistry-based forms of computation.
Learning produced by a meta-configured genome can include processes that operate during production of a new organism completely enclosed in an eggshell, before the new individual can learn by interacting with its eventual environment.
I shall try to show, at least in outline, how deep discoveries that go beyond detection of regularities at multiple levels, are concerned with possibility spaces, and restrictions within those spaces, i.e. they are concerned with multi-generational discovery of what is possible, necessarily the case, or impossible. within a complex and changing environment.
Examples are ancient mathematical discoveries in geometry, topology, and arithmetic. Ancient number concepts and knowledge of arithmetic were initially based not on abilities to recite number names but on discovering and making practical use of mathematical properties of one-to-one correspondences that can be combined in various ways to solve practical problems (e.g. catching enough fish to feed one's family?), as proposed in this IJCAI 2016 workshop paper: https://www.cs.bham.ac.uk/research/projects/cogaff/16-25.html#1602
Discoveries in logic and set theory in the last two centuries merely extended the range and variety of human mathematical knowledge rather than revealing the foundations of previous knowledge as many now believe.
Necessity and impossibility are not points on a scale of probabilities, so mechanisms (e.g. real or simulated "neural nets") that merely derive probabilities from statistical data are incapable of explaining (or replicating) important types of biologically evolved spatial intelligence (not only in humans) that are central to ancient mathematical discoveries made long before the development of symbolic, logic-based forms of reasoning.
(Most logic-based, symbolic, reasoning makes use of mathematical discoveries in the last two centuries, that were not available to ancient mathematicians, or to other spatially intelligent animals, such as elephants, crows, squirrels, and pre-verbal human toddlers.)
Although many ancient mathematical results (e.g. those summarised in Euclid's Elements) are still taught, very few learners now seem to have personal experience of making such discoveries. So a vast amount of research is now being done by highly intelligent people with a serious gap in their education, which makes them blind to the inadequacies of both statistics-based neural nets and logic-based formal reasoning mechanisms.
They also cannot understand Immanuel Kant's claims (e.g. in his 1781 Critique of Pure Reason) about mathematical knowledge, which is probably why so few researchers seem to know what he claimed (summarised below).
As a result of these educational gaps, forms of spatial cognition required for the ancient mathematical discoveries are ignored in most current theories in neuroscience, psychology, philosophy of mind, philosophy of mathematics, and also in all current AI models that I know of. So current robots cannot use the ancient forms of spatial reasoning about impossibility/necessity that underly spatial intelligence in young humans and many other intelligent animals, including squirrels, elephants, apes, pre-verbal human toddlers, and many nest-building birds, e.g. crows and weaver birds.
There are some AI geometric reasoners that are partly inspired by Kant's ideas or ancient geometers, but use logical and algebraic reasoning mechanisms that were not available to ancient mathematicians or other spatially intelligent animals. E.g. this system is far superior to most humans at proving theorems in Euclidean geometry:
Shang-Ching Chou, Xiao-Shan Gao, Jing-Zhong Zhang, Machine Proofs In Geometry: Automated Production of Readable Proofs for Geometry Theorems, World Scientific, Singapore, 1994, http://www.mmrc.iss.ac.cn/~xgao/paper/book-area.pdf
I first attempted to defend Kant in my 1962 DPhil thesis (now online), and in several published papers, including recent papers, without realising that most modern readers would have no personal experience of what I was discussing.
I shall therefore briefly present some examples of ancient spatial reasoning that would have been familiar to most well-educated researchers in maths, science, engineering, or architecture a century ago, but not most current university graduates (in my experience). I'll also include some non-standard examples so that participants can test their spatial reasoning abilities.
The fact that people have certain cognitive abilities does not imply that they know they have them. For example, even very young children, can somehow understand that it is impossible to separate two linked rings made of solid impenetrable material simply by moving them around in space, without breaking open either of the rings. When stage magicians use specially prepared rings to give the appearance of doing the impossible, even quite young children can tell that something is wrong, without having been taught topology, but they may not be able to state clearly what they assume is not possible.
Another example: different sorts of surface have different topological properties. For example, everyone reading this will find it obvious (after a little thought) that in a planar or spherical 2D surface S, if C is a simple (non-self crossing) closed curve, then C divides the surface S into two non-overlapping portions, S1 and S2, and *every* continuous line L in S that joins a point in region S1 and a point in region S2 *must* also intersect the curve C. Moving everything to a high altitude, or another planet, or changing the colour of the surface cannot make any difference to that necessity.
But there are non-planar, non-spherical surfaces on which exceptions are possible. If S is a toroidal surface, e.g. the surface of a ring, or car inner tube, then the above is true for some closed curves in S but not all. (Think of different sorts of non-self-crossing closed curves drawn on the surface of a doughnut/torus.)
Evolved mechanisms of spatial cognition enabled ancient human mathematicians, centuries before Euclid, to make discoveries regarding possibility, impossibility and necessity in spatial structures and processes, without making use of modern mathematical symbolic, logic-based, or algebraic formalisms. Ancient reasoners and young children can contemplate real or imagined spatial structures, without using algebraic specifications.
How brains achieve such spatial reasoning is not yet known. I suspect Alan Turing's work on chemistry-based morphogenesis (published 1952) was a side effect of deeper, more general thinking about uses of spatial reasoning in intelligent organisms (implicitly extending Kant's theories, about which he seems to have been ignorant). But he died without publishing anything to support that claim, though he did assert in 1936 that computers can use mathematical ingenuity, but not mathematical intuition, without explaining the difference. (Perhaps clues will turn up in his un-published, hand-written notes?)
Very crudely, David Hume, depicted above, on the left, claimed that there are
only two kinds of knowledge:
"True by definition" applies to all truths that can be proved using only logic and definitions.
An example is "No bachelor uncle is an only child", which can easily be proved from the definitions of "bachelor", "uncle" and "only child", using only logical reasoning.
Hume famously claimed that if someone claims to know something that is neither of type 1 (empirical) nor of type 2 (mere relations between ideas, or definitional truths) we should "Commit it then to the flames: for it can contain nothing but sophistry and illusion", which would have included much theological writing. and much philosophical writing by metaphysicians.
[I apologise to Hume and Hume scholars: this presentation over-simplifies Hume's position in order to contrast it with Kant's claims, below.]
Immanuel Kant's response (1781)
In response to Hume, Immanuel Kant, depicted above, on the right, claimed, in his Critique of Pure Reason, that there are some important kinds of knowledge that don't fit into either of Hume's two categories ("Hume's fork"), for they are not mere matters of definition, nor derivable from definitions by using logic.
Kant pointed out that instead of Hume's single distinction between two categories of knowledge we need to take account of three different distinctions:
the analytic/synthetic distinction,
the empirical/non-empirical (empirical/apriori) distinction, and
the necessary/contingent distinction.
(For a more detailed explanation of the three distinctions see Sloman 1965).
Using Kant's distinctions, we can locate ancient mathematical discoveries in relation to three different contrasts:
Another example: I am now in Birmingham in England. In principle I could now have been somewhere else at this time, e.g. in Berlin, in Germany. So that is a contingent truth.
If something is a necessary truth, then there are no possible circumstances in
which it could be false.
There are also necessary falsehoods. E.g. 3 + 5 = 9 is false and could not have been true in any circumstances (without changing the meaning of what is being said. So it is necessarily false and its negation is necessarily true.
In short: Kant replaced Hume's single division of types of knowledge into two categories, with a much richer analysis making use of three different divisions, producing six categories. Not all combinations are possible, however. E.g. something cannot be both apriori and necessarily false.
In his argument against Hume, Kant drew attention to kinds of mathematical knowledge that do not fit into either of Hume's two categories: since we can discover by means of special kinds of non-logical, non-empirical reasoning (that he thought were deeply mysterious, since he was unable to explain how the reasoning mechanisms worked), that "5+3=8" is a necessary truth, but not a mere matter of definition, nor derivable from definitions using only logic. It actually refers to an invariant effect of combining two relatively simple cases if one to one correspondence to form a more complex one to one correspondence. (Details are left as an exercise for the reader. Compare Chapter 8 of Sloman(1978).
(We can now understand, partly thanks to the work of Frege and Russell, among others, how the necessity of mathematical truths about the natural numbers is a consequence of the fact that the relationship of one-to-one correspondence between sets is necessarily transitive and symmetric. They aimed to show that such features were provable using only logic, whereas I think Kant would have claimed, correctly, that such discoveries were originally based on a form of spatial reasoning rather than 19th-20thh century logical reasoning. But the above is a more complex feature involving parallel rather than sequential combinations. Sloman(2016))
Kant thought such mathematical discoveries in arithmetic, and discoveries in Euclidean geometry were synthetic, not analytic and also could not possibly be false, so they are necessary truths, and because they are not based on or subject to refutation by observations of how things are in the world, such knowledge is non-empirical, i.e. a priori.
For a more careful and detailed, but fairly brief explanation of Kant's three
distinctions. apriori/empirical, analytic/synthetic and necessary/contingent,
http://www.cs.bham.ac.uk/research/projects/cogaff/misc/kant-maths.html or Sloman(1965).
Example: slice a vertex off a convex polyhedron
For readers who have not personally had the kind of geometric discovery experience described by Kant here is an example, that, as far as I know does not occur in standard geometry text books but is closely related to examples in Shephard (1968). Try to answer this question without reading any geometry textbooks:
If there is a solid convex polyhedron, and exactly one vertex is sliced off with a single planar cut, e.g. using a very thin planar saw, how will the numbers of vertices, edges and faces (V, E and F) of the new resulting polyhedron be related to the original three numbers?
Note that this question can be given a definite answer only if the polyhedron is convex: i.e. any straight line joining two points on the surface of the polyhedron lies entirely inside the polyhedron, or on its surface.
What mechanisms in your brain could enable you to answer the question (repeated below in blue)?:
Solid, opaque, convex, polyhedron with partly visible faces, edges and vertices.
Using a planar cut, remove exactly one vertex, e.g. one of the two ringed vertices.
How will the numbers of vertices, edges and faces of
the remaining polyhedron differ from the original numbers?
I leave the problem as an exercise. It is possible to work out the answer by using normal spatial reasoning abilities, though the answer is not directly derivable from Euclid's axioms, which said nothing about slicing 3D objects! (However, there may be an equivalent question that mentions only Euclidean points, lines, and surfaces.)
Hint (read this paragraph only if stuck):
If you try to answer the question, your reasoning can use the fact that if V is the vertex that is sliced off, then there will be some number (N) of edges that meet at V. So the cut must go through each of those edges, producing a new polyhedron. Some of the original edges will have been shortened and some of the original planes have had a vertex removed, namely the vertex V. However a straight cut removing exactly one vertex of a planar polygon will change the number of vertices and edges of that polygon. How?
This example goes beyond Euclidean geometry, which does not include the concepts of viewpoint or visibility, or slicing portions off objects! I have found that some people who have never studied geometry formally, and are unfamiliar with Euclid's specification of geometry, are able to think about the problem and work out the answer, intuitively understanding (as an ancient, pre-Euclid mathematician might have) what the answer must be. However, there is a complication mentioned in the note below, which I did not notice until I stumbled across it about two years after I first used this example in discussions. (Some readers may be able to think of the complication without looking at the note!)
I have just discovered today (1 Jun 2021) that there is an entertaining
semi-serious online video by Margaret Gruss about slicing polyhedra on a kitchen
Implications for "trainable neural net" models of cognition
The fact that mathematical discoveries are concerned with what is possible and with impossibilities and necessary connections, implies that they cannot be achieved by mechanisms (e.g. statistics-based neural nets) that are capable only of using statistical evidence to discover probabilities. A neural net collecting statistical data cannot discover that something is impossible, or necessarily true. It therefore cannot replicate, or provide an explanation for, the discoveries made thousands of years ago by ancient mathematicians using spatial reasoning.
Immanuel Kant understood that necessity and impossibility had nothing to do with probability: necessity and impossibility are not extremes on a probability scale.
So mechanisms whose mode of operation is to collect statistics and derive probabilities (based on numerical ratios) cannot model or explain or replicate ancient human abilities to discover geometrical impossibility or necessity. I suggest that very young (pre-verbal) children and some intelligent non-human animals are capable of detecting and making use of spatial impossibilities and necessities in choosing actions to achieve spatial goals. Slightly older children seem to be aware that merely changing the spatial location or orientation of a pair of interlocked solid metal rings will not allow the rings to be moved apart.
Current AI also includes powerful theorem-proving and problem-solving machines that use modern logic and arithmetic. However those logical techniques were discovered only relatively recently, and there is no evidence that such logical machinery was available in the brains of great ancient mathematicians at the time of Euclid, and earlier.
I conclude that neither of the two dominant current strands in AI, namely statistics based neural nets and logic based theorem provers, can explain the forms of reasoning used by ancient mathematicians, or by intelligent animals solving spatial reasoning problems in their environments, of sorts illustrated above.
For example, a bird approaching a part-built nest holding a twig horizontally in its beak and confronted by two branches forming an upright "V" on the near side of the nest may understand that it is impossible to move the twig to the nest unless either the orientation of the twig is altered (e.g. so that one end of the twig goes through the V first) or a new route to the nest is chosen, e.g. going round the side of one of the branches of the V. Some animals (including crows, rooks and other corvids, and various apes) apparently have that kind of intelligence but many do not, like the poor cat with corn cob in this video (which has a happy ending): https://www.youtube.com/watch?v=rd_b52ifNxo
However, discovering empirically that our physical space is not Euclidean no
more refutes Kant's commendation of discoveries by Euclid and others than
discovering that the surface of a sphere or a teapot is non-Euclidean proves
Kant (or Euclid) wrong. If different types, or regions, of space with different
properties exist, there can still be a particular type that necessarily has the
properties discussed by Euclid (and Kant), in consequence of having more
(For more details see my DPhil thesis Sloman1962, which included a defence of Kant against Hempel).
Did Einstein refute Kant?
It is widely, but erroneously, believed that Immanuel Kant's philosophy of mathematics in his Critique of Pure Reason (1781) was disproved by Einstein's theory of general relativity (confirmed by Eddington's observations of the solar eclipse in 1919, establishing that physical space is non-Euclidean).
This no more refutes Kant's position (as I understand it) than demonstrating that the surface of a sphere is inconsistent with Euclidean geometry. E.g. the closest thing to a straight line on a sphere is a geodesic, and it is easy to make a triangle on a sphere bounded by three geodesics where all the angles of the triangle are right angles, adding up to 270 degrees not 180 degrees as in Euclidean geometry. The surface of a teapot with spout and handle deviates even more from a Euclidean surface.
The belief that Kant's view was refuted by the discovery that physical space is non-Euclidean is erroneous if Kant was not making a claim about physical space but about one of the types of space that we can think about, e.g. by imagining some basic features, or abstracting them from perceived objects (e.g. the 2D space on the surface of a spherical object, or an egg shaped object, or a toroidal -- circular tube shaped -- object), and then deriving implications of these basic features, by thinking about the features themselves, i.e. not merely manipulating sentences describing those features. Using defining features of a type of space to derive consequences yields conclusions that do not depend on what exists in the physical universe.
Without much difficulty you should be able, for example, to think of alternatives to a circular tube forming a 3D ring, or toroid, by imagining various deformations of that shape, e.g. twisting it into a figure 8-like shape, or introducing sharp corners and flat surfaces, turning the tube into a square picture-frame like shape, perhaps with with a very thick frame. Some of the mathematically possible deviations from familiar Euclidean space are much harder to think about than others. Compare thinking about 1000-dimensional shapes embedded in a 1001-dimensional space.
A task for brain theorists
We can use sentences in a spoken, written, or internal language to consider new possibilities and then derive consequences of those possibilities. However, we can also use non-linguistic forms of representation to visualise possibilities and then derive consequences.
That's the sort of thing ancient mathematicians did when they first made their discoveries, and similar exercises of spatial imagination play a role in the thinking of mechanical engineers, architects, designers of new furniture or tools, dress-makers, and many others who work on spatial structures, including inventing new, useful, types. How that is possible needs to be explained by a theory of how brain mechanisms perform those tasks. Insofar as such performances include detecting impossibilities and necessary consequences of certain spatial structures and operations, they are neither simply
Non-human spatial intelligence (e.g. Betty the crow)
I suspect other intelligent animals can do something similar to a limited extent, but can't talk about it or reflect on their discoveries using an internal language, as humans can. A striking example was Betty the New Caledonian Crow whose creative problem-solving abilities were made famous in 2002 by a video showing her bending a piece of wire in order to lift a bucket of food out of a vertical glass tube. Details are available here
The researchers did not think it worth mentioning in their reports that Betty solved the problem in several different ways, as shown by videos available on their web site!
Did Einstein refute Kant?
As a mathematics student around 1958 I encountered philosophers claiming that Kant had been proved wrong because Einstein and Eddington had demonstrated that Euclid's results were not true of physical space, I felt that they were mistaken because Kant's claims corresponded to my experience of doing mathematics. So I obtained permission to switch from mathematics to philosophy in order to defend Kant. My 1962 DPhil thesis (now online) defended a slightly modified version of Kant's claim that many important mathematical discoveries are non-empirical, non-contingent, and non-analytic (i.e. not just logical consequences of axioms and definitions), but did not explain how brains or machines could make such discoveries.
In any case, demonstrating that our 3D environment is non-Euclidean no more refutes Kant (or Euclid) than demonstrating that the surface of a ball is non-Euclidean. The main point is that we are able to think about Euclidean spaces and reason about them, including making discoveries about necessary features of those spaces. In doing that we may be inspired by related features of our physical space, but that does not imply that we have to assume that physical space is exactly Euclidean in order to study properties of Euclidean spaces, which ancient mathematicians clearly were able to do long before Euclid produced his axiomatic specification.
A biological hypothesis
I now want to suggest that some of the more complex mechanisms involved in building a brain (e.g. building the brain of a new chick inside the egg) may have to select actions (chemical construction/assembly processes) on the basis of a form of spatial reasoning about distances, available routes, connections to be made, available materials, constructions built so far. Normally such decisions would have to be taken by a designer physically separate from the object being constructed. But there is no chick designer physically separate from the partly constructed chick.
But perhaps the processes occur in virtual machinery developed in brains as part of the process of developing more complex brain mechanisms in the new organism. If physical mechanisms needed for the virtual machinery taking control decisions can be shared with physical mechanisms being constructed as part of the new organism, then all the physical processes can be contained within the egg although they support distinct processes in different virtual machines. I think someone cleverer than I am will need to transform that proposal into something deeper and better worked out: for now it seems to me to be the only route to an explanation of the things we know result from construction processes in the egg -- including production of a highly competent chick whose competences cannot be based on learning in the physical environment.
Summary: Hume vs Kant
Kant's characterisation of ancient mathematical cognition, in his Critique of Pure Reason (1781) drew attention to three features of such cognition, using three distinctions that are ignored in most current psychology, neuroscience and neural-net based AI: namely: non-empirical/empirical, analytic/synthetic, and necessary/contingent, as summarised above.
Both logic-based and neural-net-based AI systems are incapable of replicating the ancient aspects of natural intelligence that allow discoveries of non-empirical, synthetic, necessary truths. Current psychology cannot explain them. Most psychologists don't recognise the need.
Theories at odds with Kant's insights include both 'formal', logic-based, characterisations of mathematics, used in modern automated theorem provers, that reason by manipulating discrete symbolic structures, and also neural theories that attempt to explain or model mathematical discovery processes in terms of neural networks that collect statistical evidence that is used to derive probabilities. Necessity and impossibility are not extremes on probability scales.
Kant made related claims about arithmetical knowledge, which can be defended by showing how natural number concepts depend on properties of the one-to-one correspondence relation: it is necessarily transitive and symmetric. Humans can understand this on the basis of spatial reasoning, though Piaget showed that young children seem unable to do that until they are 5 or 6 years old, which implies that they do not understand number concepts until then, despite claims made by psychologists on the basis of misleading evidence.
As Kant realised, ancient knowledge of geometry, (including mathematical discoveries made centuries before Euclid), is neither simply composed of results of empirical generalisation from experience of special cases (i.e. empirical knowledge of probabilities), nor mere logical consequences of definitions. I.e. they are non-empirical (a priori), and synthetic (not derivable from definitions using only logic, and the truths they identify are non-contingent, despite some claimed counter-examples. It is less obvious that he was also right about arithmetical knowledge,
This refutes theories about innateness of knowledge of cardinality, unlike knowledge of numerosity, which lacks the precision of cardinality.
My talk will raise questions about mechanisms available for explaining spatial intelligence in humans and other animals, based on hitherto unexplained facts about spatial competences of newly hatched animals, such as chicks, ducklings, turtles and crocodiles, whose abilities cannot be explained by neural networks trained after hatching. They must be explained by chemical mechanisms inside their eggs, available before hatching.
I don't believe anyone now understands how those chemical processes are controlled, or how the controlling mechanisms can be accommodated inside a fully occupied egg. I suggest that that challenge can be met using the concept of "virtual machinery", only recently developed in current sophisticated forms providing services across the internet that "float persistently" above the constantly changing particular physical mechanisms at work, but without occupying additional space. An example is the type of zoom virtual machine now used to support many "online" meetings. The creation of a new zoom meeting, and its extension as each new participant joins does not require creation of new physical links connecting all the participants.
A lot is known about chemical mechanisms that make possible the early stages of the reproductive processes, and researchers in several disciplines are constantly extending what is known. In contrast, I suspect very little is known about the enormously complex types of virtual machinery required for later stages of (e.g.) chick production, including the creation of control mechanisms for actions required soon after hatching. I suspect that the processes of development of the foetus require many stages of control by increasingly sophisticated virtual machines controlling and coordinating chemical mechanisms as they create new chemical mechanism.
Different sub-machines must have evolved at different times, and the later, more complex virtual machines may have to be assembled by earlier virtual machines, during individual development, although the earliest control machines are simply molecular mechanisms controlling formation and release of chemical bonds linking relatively simple chemical structures.
It seems clear that mechanisms of biological evolution "discovered" and used the powers of virtual machinery long before human engineers did. What are the implications for current theories of fundamental physics?
I suspect that Alan Turing's 1952 paper 'The chemical basis of morphogenesis', which discusses chemistry-based pattern formation on the surfaces of plants and animals, was a side effect of his research on the much deeper publication on the role of chemical mechanisms in control processes inside eggs, and the recently evolved thought processes used by human mathematicians. Turing unwittingly rediscovered Kant's problems and began to develop an answer. Perhaps we'll never know how far he had got by the time he died.
These ideas are still lacking in clarity, precision and detailed evidence. Filling the gaps will be a very difficult long term project, requiring contributions from several different disciplines, including new ideas about the specific types of control machinery required at different stages of reproduction of both physical and virtual machines in an animal as complex as a baby chicken.
The presentation will add a lot more detail including (speculative) discussion of the problems of accounting for what needs to happen inside an egg (e.g. of chicken, duck, turtle, alligator, etc) to produce highly competent hatchlings that have forms of spatial intelligence that could not have been acquired after hatching and before they were needed.
I'll present some "first-draft" ideas about how the processes inside the egg might include use of virtual machinery that coexists with, and is implemented in, physical machinery and can be used for increasingly complex control tasks during the hatching process, without requiring special additional physical material for the task (for which there is no space inside the egg).
Instead, what changes are the routes taken by different sorts of signals: which are not material objects but patterns of activity, such as transmission of bit-patterns, whose effects can be to alter patterns stored in various parts of the system, e.g. your bank's computer files. Storing such patterns now serves purposes previously served by storing physical objects, e.g. coins, or paper documents.
(This sort of topic should be, but is not yet, part of the standard philosophy syllabus for students studying metaphysics.)
The same network of worldwide facilities can simultaneously host multiple zoom meetings starting and ending at different times, growing and shrinking in size, without changes in network wiring. Those processes occur in parallel with many other constantly changing activities (e.g. email messaging, grocery ordering, salaries and rents being paid, and network health-monitoring signals being sent to various control-subsystems managing parts of the network), all sharing major parts of the same networks, or sharing sub-nets in some cases.
Of course, all this activity implemented in virtual machinery can also have physical effects, through the mediation of terminals used by humans or signals to robots or humans in large warehouses combining physical items into new packages to be physically delivered to purchasers later. Other examples include patterns of activity that cause seats to be reserved on trains or planes, or rooms in hotels, enabling various human activities following use of a booking system -- perhaps including arrival of a pre-booked taxi for the journey to the airport.
I am suggesting that biological evolution discovered the powers of sophisticated chemistry-based virtual machinery long before we discovered (less sophisticated) varieties implemented using human designed electronic machinery.
A key point is that for a system controlling assembly of very complex physical systems, with multiple developments going on in parallel, there is a need for the controlling mechanisms to acquire and use information about what is and is not spatially possible -- e.g. one sequence of actions may produce a desirable result that is impossible if the actions are performed in a different order. (E.g. putting an object into a container, and delivering the container to a particular destination.)
-- the mechanisms that originally evolved for use in forms of spatial intelligence required in the relatively high level control mechanisms controlling assembly of multiple complex body parts in a foetus, taking decisions on the basis of which sequences can and which cannot achieve required results,
were later copied and used in brains of complex animals
-- for taking decisions about actions in a spatial environment, on the basis of which spatial movements of body parts and other objects can, and which cannot achieve desired goals of the whole organism, e.g. getting to some food, or climbing up a tree to a place out of reach of a predator.
Readers should easily think of many more examples, relevant to different species at different stages of development in different environments.
One consequence of all this is that it pays organisms that live a long time to
have forms of motivation that are not restricted to satisfying immediate
biological needs, but instead generate actions that obtain information, that
may turn out to be useful later, as proposed in
Another consequence is that if the original controlling mechanisms were implemented in chemical machinery used for assembling organisms, including building brains, then it may be that even after the evolution of brains for various high level processes of information acquisition and use, the brain mechanisms used for spatial reasoning might continue to be based on variants of the chemical mechanisms evolved for spatial control before brains existed.
The implications for current theories which assume that neural nets are the basis of human intelligence should be obvious, which is just as well since neural nets are incapable of discovering and reasoning about necessity and impossibility.
Whether my answers are correct (or more likely wrong or seriously incomplete) the biological facts presented in the talk will undermine current widely accepted theories in neuroscience, psychology, philosophy, and AI, with many practical implications, e.g. for mathematical education?
The talk will be followed by extended discussion (after a short break) during which participants may wish to discuss the implications -- e.g. for philosophy, psychology, neuroscience, biology, AI, and education.
These ideas are still lacking in clarity, precision and detailed evidence. Filling the gaps will be a very difficult long term project requiring contributions from several different disciplines, including new ideas about the specific types of control machinery required at different stages of reproduction of both physical and virtual machines in an animal as complex as a baby chicken.
Perhaps at the end of the discussion we can consider setting up some organisational machinery to encourage, support, and coordinate results of further (multidisciplinary) research on these topics, especially attempts to understand developmental processes in a variety of organisms, including humans, whose virtual machinery continues to develop for longer than any other species, with far more use made of trans-generational cooperation to extend the powers already being used.
I suspect these or similar ideas are part of the unstated motivation for Alan Turing's last published paper (in 1952) "The chemical basis of morphogenesis".
This is an expanded (hopefully improved) version of a zoom talk presented at
Sussex University on 16th Feb 2021, available here:
Search down for 16th Feb.
Note on the sliced polyhedron example
People to whom I present the sliced polyhedron example above, normally do not notice that the specification of a planar slice as removing a single vertex leaves open the possibility of the slice going through one or more of the remaining vertices without removing an additional vertex. Thinking about possible ways in which that can occur, and how they change the answer to the question, is left as an exercise for readers. I first thought of the sliced polyhedron example to illustrate geometric reasoning abilities that are not purely logical, nor simply examples of empirical generalisation, during a discussion at a workshop in 2018. It wasn't until two years later (following a discussion with Manfred Kerber) that I first noticed the additional possibilities: yet another illustration of the fact that brain mechanisms that enable new mathematical discoveries, concerning what is necessarily the case or impossible, are not infallible. (Compare the long and chequered history of Euler's theorem, discussed in Lakatos(1976).)
Philip Ball (2015),
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a commentary on Turing (1952) 'The chemical basis of morphogenesis',
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Detailed information about contents (not on publisher's site):
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Meta-Morphogenesis and Toddler Theorems: Case Studies.
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Open Access Government
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