A video recording of the presentation and discussion can be accessed
on the Sussex seminar site, after the abstract dated 16 Feb:
(The web page wrongly gives the year as 2020)
The meeting organiser, Simon Bowes, has kindly added the contents of the chat
panel and zoom's attempt to transcribe what I was saying into textual form (with
a low level of accuracy, unfortunately):
video of talk with chat and transcript.
At the top left of the video display there is a menu containing four options: which can be selected by a mouse click, as follows:
Details: Gives information about the meeting (date, time, people present, etc.)
Discussion: Gives the contents of the "Chat" file, i.e. items typed in by participants.
Captions: Is a stream of text derived from my spoken words. Unfortunately, the transcription is not very accurate.
Hide: hides all the contents of the column.
NOTE ON A SERIOUS GAP IN MANY MATHEMATICAL EDUCATION PROGRAMMES
(Added: 2 Apr 2021)
During presentations and discussions of the ideas in this talk I have discovered that a high proportion of highly intelligent well educated researchers have had no personal experience of discovering and using constructions and proofs in Geometry: previously a standard part of mathematics education. E.g. when I was at school in Southern Africa in the early 1950s I was taught geometry and it was my favourite subject. I discovered later that mathematical education had been changed in many countries so as to exclude spatial/diagrammatic reasoning from mathematical teaching (apparently Bulgaria is an exception, or was, until recently).
This disastrous educational change seems to have been in part inspired by the Bourbaki project in France:
It was partly because of this 20th Century educational disaster that I felt I had to include in my presentation a fairly detailed account of the difference between Hume's and Kant's views on types of knowledge (or types of truth) since Kant's thinking was in part inspired by his knowledge of geometry.
The account of Euclidean geometry in the talk is somewhat short and shallow.
There are some tutorials online that provide more detail. A small sample is
below. I hope to provide a better list later.
Presentation on Euclidean geometry by Zsuzsanna Dancso at MSRI.
2000 years unsolved: Why is doubling cubes and squaring circles impossible?
How many ways can circles overlap? - Numberphile
Old and new proofs concerning the sum of interior angles of a triangle.
A fairly random collection of examples of spatial reasoning.
Discussion of "toddler theorems": types of spatial understanding of what is possible or impossible/necessary in spatial structures and processes.
5x5 Square Grid and 5 Circles
Many more examples are avaiable online using Alexander Bogomolny's "live" online demos:
Reasoning about a Torus.
-- Examples (a random subset):
This talk continues that theme, by invoking biology, including evolution and chemical aspects of biogenesis, in defence of an updated Kantian theory.
This is loosely related to studies of Symbiogenesis, investigating how evolution can sometimes produce new forms of life by combining different evolutionary strands. See https://en.wikipedia.org/wiki/Symbiogenesis
I recently realised that thinking about how an egg produces a chick (or a duckling, or a young alligator, etc....) raises deep questions (including some difficult unobvious questions) about how the powers of biochemical processes are related to spatial reasoning, visual capabilities, and action control processes. The questions generally seem to have gone unnoticed. I did not notice them until I was preparing an invited talk for the APA conference in January 2021: https://www.apaonline.org/page/2021E_program
What happens when an egg produces a chick?
There is a lot of information available about how the physical structure inside the egg changes between the time the hen lays the egg and the time a fully formed young chick emerges. But I have found no analysis of the ways in which the control processes involved (inside the egg) in creating the chick need to change as what has already been created becomes more complex, including developing millions of new molecules of many different sorts during formation and growth of new body parts and mechanisms for transporting required chemicals, disposing of waste matter, controlling parallel assembly of related structures (e.g. bones and other items attached to them). Moreover, what emerges from an egg typically has complex behavioural competences that could not have been acquired by any of the known forms of learning by interacting with the environment. An example from the BBC Springwatch programme showing newly hatched avocets is available here.
I doubt that a combination of all the current skills and knowledge of human physicists, chemical engineers, biochemists, physiologists, neuroscientists, AI researchers, ... (add all other relevant fields) would suffice to explain in detail how egg chemistry produces a new chicken (or other animal created in an egg), even if everything is already known about how the earliest new cells are created). [Is that already known?]
Here's a video showing some of what goes on inside a chicken egg
There is no human-designed automated assembly system that can transform the relatively amorphous contents of an egg into a chick, with the standard competences of a chick.
How does the assembly work? Showing a video gives very little information about all the sub-problems involved, and how they change between the initial processes that come from the presence of a DNA molecule in a single cell, through all the processes of cell division and differentiation, along with integration and functional coordination so that all the many parts work together, performing a large variety of different functions, when the chick has been constructed.
The short video showing avocets (above) indicates that the chemical processes inside an egg can produce extremely complex spatial percepts of changing configurations.
The process continues after hatching: the chick feeds on seeds and other matter and transforms a significant subset of their contents into materials used for growth of all its body parts, until there's an adult chicken with many different competences -- male or female, with different body parts and complex behaviours of many kinds, including mating.
FOR MUCH OF THE PROCESS OF DEVELOPMENT THERE'S NO BRAIN TO CONTROL THINGS: CONTROL OF GROWTH AND MANY INTERNAL PROCESSES, INCLUDING GROWTH, DIGESTION OF FOOD, REPAIR OF DAMAGE, MAINTENANCE OF BODY PARTS IS DONE MAINLY BY CHEMICAL MECHANISMS, INCLUDING MECHANISMS THAT CREATE THE BRAIN AND ITS CONNECTIONS.
So, chemical mechanisms transform the apparently simple contents of an egg into something extraordinarily complicated and also highly competent, in its own way, e.g. a chick that can peck for food, swallow and digest food, follow a hen, drink water, and much more in later life.
That "intelligent" use of chemistry continues during the life of the organism insofar as a vast and complex collection of mechanisms disassemble, transport, and reassemble in new ways, chemicals acquired through eating, drinking and breathing, then deal with waste products, tissue damage, and invaders of various kinds. In the case of female animals there can be additional processes of creation of a new organism, either via an external egg in which the process completes itself, or by internal assembly and nurturing, as in mammals.
These developmental, maintenance, and reproductive processes vary enormously across different forms of life, sizes of organisms, and different habitats, including food sources, prey, predators and many other occupants of the environment.
How can the reproductive processes be explained?
[To be reformulated:]
The competences required in an egg to enable it to produce a chicken cannot be results of learning or training because there is no opportunity for training or practice within the egg to produce all the competences used, although it may be the case that after the new organism has reached a certain level of complexity the subsequent assembly processes require some of the mechanisms produced in the egg to be trained, to enable them to produce more complex structures.
But there is no space in the egg for separate assembly mechanisms: when the egg cracks open only the chick emerges.
Perhaps the assembly mechanisms, insofar as they go beyond mechanisms required in the final product, somehow exist as virtual machinery, distributed across portions of the developing embryo, rather than separate machines like those used in a factory to assemble new machines.
In contrast: an AI-based robot that has not been pre-engineered to do a particular task, would typically require a lot of training involving trial and error before it can walk, pick things up, traverse rough terrain, and follow a hen. But the assembly mechanisms in eggs of many species, just seem to work in the same space as the new developing individual.
Where does the chick's competence come from?
The answer must be: the mechanisms that create the chick inside the egg. Those are all chemical mechanisms, and during most of the creation process they operate without a functioning brain: though they build a brain among other things: eyes, legs, claws, beak, skin, feathers, wings, digestive system, nervous system, lungs, heart, blood vessels, muscles, bones, tendons, ...
Not included in the video:
Similar points can be made about the amazing transformations in both physical form and behavioural competences found in examples of metamorphosis during development of insects, e.g. transformation of a caterpillar into a butterfly inside a pupa as illustrated here (Monarch Butterfly Metamorphosis): https://www.youtube.com/watch?v=ocWgSgMGxOc. Unlike the chick, it builds its own "eggshell", within which it creates a new organism!
INFORMATION BASED CONTROL
The processes that control the assembly of a chicken or any other animal must constantly make use of information, including information about the current state of affairs, information about what needs to be done, information about how to do it, and how to modify actions as the structures manipulated change.
These are control processes. The control is information-based insofar as deciding what to do next makes use of information both about the current state of affairs, including what has already been assembled and what resources are available, and information about what has to be achieved. This may or may not include detailed specifications of how it is to be achieved.
In principle, the mechanisms controlling assembly may be fully distributed, like the mechanisms that allow water molecules poured into a stationary bucket to fill a lower portion of the bucket bounded above by an approximately flat horizontal surface and the inner surfaces of the bucket. If the bucket is rotating fast enough about a central vertical axis, the water will end up spinning with the bucket but differently distributed inside the bucket, with the upper surface no longer flat. Some of the spinning water molecules will move up the inner surface of the bucket. As a result the upper surface of the water will no longer be flat. Some parts will be higher than the surface in a non-spinning bucket, some lower.
So water in a bucket has some self-organising capabilities, but nothing like the intricate self-organising mechanisms that transform chemical substances inside an egg into a chicken whose body is highly differentiated, with complex behavioural abilities available immediately after hatching.
NOTE ON INFORMATION
I shall later argue that neural network based mechanisms that are now extremely popular are incapable of replicating the capabilities of ancient mathematicians, and suggest that chemistry-based mechanisms can.
But in order to explain what I mean I'll have to give a summary of a disagreement between David Hume and Immanuel Kant, with deep implications for the nature of mathematical discovery, especially mathematical discovery based on spatial reasoning. These mathematical discoveries produce new information about what is and is not possible in spatial structures and processes. However, they are closely related to common features of spatial cognition that are not restricted to mathematicians, or even humans. Examples include spatial cognition in nest-building birds, tree-climbing primates, and animals that hunt their prey down and then open them up in complex feeding activities.
All of these activities require making use of information about spatial structures and processes, including abilities to use perceptual information to make choices between possible processes.
I do not use "information" here in the sense of Claude Shannon: his was a syntactic concept that confused hordes of scientists and other thinkers, though Shannon was not confused. Instead I use "information" in the much older, non technical sense that refers to semantic content.
That concept was known by Jane Austen, who repeatedly used it in her novels,
because the sharing, concealing and use of information, but not information in
Shannon's sense, plays such an important role in human relationships. So I call
it "Austen-information", as explained here:
Meanwhile, I want to talk about the nature of ancient mathematical knowledge of geometry and topology, as opposed arithmetic, algebra, and mathematical logic.
I'll describe a disagreement between David Hume and Immanuel Kant, then return to biology and chemistry, with a dose of metaphysics thrown in.
I shall try to show how 'Kantian' features of spatial intelligence are connected with more general questions about biological cognition, especially uses of information about space, spatial structures and spatial processes, during development from a newly fertilised egg to an independent animal. (Plant development is also relevant, but will not be discussed in this talk.)
Spatial reasoning not in Euclid's Elements
There are often many different ways to prove the same thing in mathematics. For example there are several hundred different proofs of Pythagoras' theorem, discussed briefly below. A much simpler theorem, is the triangle sum theorem, which states that the internal angles of a planar triangle add up to half a rotation: 180 degrees.
Many people have learnt at school that that is a fact about geometry, though my informal investigations suggest that most current academics do not know how to prove it. There is a standard proof (used by Euclid) that makes use of parallel lines, and which is taught in the schools that do teach proofs.
The proof below of the triangle sum theorem discovered by Mary Pardoe, a young mathematics teacher, around 1971 would not be valid according to Euclid's Elements, because line segments cannot be rotated in Euclid's version of geometry.
The proof uses a triangle of a particular shape, size and orientation, but it is obvious (why?) that exactly the same demonstration could be given starting from any other triangle. Try imagining various triangles of different shapes and sizes, and convince yourself that the rotating line proof works for all of them, even a triangle with a very short horizontal base containing two vertices with the third vertex a long way off to one side.
is still involved in discussions of mathematical education, as shown
It turned out that Mary was not the first person to have discovered this proof. See the report in this document: https://www.cs.bham.ac.uk/research/projects/cogaff/misc/triangle-sum.html
Some mathematicians don't regard that as a proof. Discussing that debate would take us too far off topic. Does it convince you, the reader?
Use of a rotating line segment is not legal in Euclidean geometry, but it is arguable that that restriction was simply an arbitrary stipulation. There are various additional constructions that were not included by Euclid, such as the Neusis construction, which makes it possible to trisect an arbitrary angle, which can't always be done using Euclid's permitted constructions. For details see: https://www.cs.bham.ac.uk/research/projects/cogaff/misc/trisect.html (Pdf version also available.)
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 several hundred 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.)
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.
So 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 distinctions.
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 was 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.
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,
https://www.cs.bham.ac.uk/research/projects/cogaff/misc/kant-maths.html or Sloman(1965).
Pre-verbal Toddler Mathematicians
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:
For more examples see this (unfinished) discussion of "Toddler theorems".
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. Moreover, 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!)
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 derivable from definitional relations between ideas nor "based on experimental reasoning concerning matter of fact and existence" (as Hume expressed it) -- which is why Kant described the discoveries as synthetic and a priori (non-empirical).
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.
Erwin Schrödinger's thoughts on life and quantum physics
In What is Life Schrödinger, offered an explanation of the role of quantum mechanisms in enabling complex linear molecules to be copied reliably across many generations by making use of catalytic mechanisms to form and release bonds easily that would otherwise require high energy mechanisms. He showed how such quantum mechanisms could underly accurate reproduction across generations.
I don't know whether he realised that the same features are relevant to the uses of chemistry in building a new organism by continually deriving and using specifications for new more complex parts to be assembled within the organism. These processes cannot be explained by Newtonian mechanisms that allows only gravitational attraction and elastic repulsion forces. Even adding electromagnetic mechanisms discovered and analysed by Faraday and Maxwell in the 19th Century could not not explain how strong bonds are formed and turned off very rapidly using low energy catalytic mechanisms.
Only the development of quantum physics in the 20th century provided such explanations, and that was why Schrödinger was able to use quantum theory as part of his explanation of reliable reproduction.
But the same theory is also relevant to processes of constructing new more complex molecules from old fragments by using bonds, and then by altering the structures by temporarily releasing bonds and adding new bonded materials as needed for building new molecules.
Repeated use of such mechanisms can produce new more complex structures during development of an organism. That would include production of a new brain by assembling increasingly complex molecular structures in the right way.
Requirements for this to work
New chemical manipulation mechanisms produce discrete changes in structures using catalytic processes: forming and releasing bonds.
[Fast and using little energy]
But as the structures in the new organism grow more complex and diverse, the assembly mechanisms need to become more complex and more sophisticated.
Can the mechanisms controlling formation of new structures become more complex as the parts of the new animal that need to be extended become more complex?
So there are two parallel streams of development - chick building processes, and construction of chick-building mechanisms.
But the chick-building mechanisms are not new physical mechanisms grown inside the egg alongside the chick that is being grown. If that were the case we might need yet another layer of building machinery to build the chick-building mechanisms. I suggest that instead of being new separate physical machines the chick building mechanisms are controlling mechanisms in virtual machinery, somehow implemented within the same physical space as the newly growing chick.
But as the building processes get more complex the construction control machinery may also require to be expanded, perhaps using using yet more new building mechanisms providing new kinds of sophistication.
If this is correct there are parallel streams of part construction and constructor construction (and possibly also constructor construction construction!).
I suggest that evolution (as in so many other respects) discovered the powers of virtual machinery before we did, and many of the controllers required for part construction could use virtual machinery.
Chick construction processes don't have time to mess around using lots of trial and error learning to find out what to do: instead the mechanisms are evolved over time, in the history of the species, and possibly some related species, and then re-used and extended.
That would apply also to chick brain construction processes.
At later stages of evolution the mechanisms used for internal construction were re-deployed for controlling actions in the environment -- outside the egg, after hatching.
In that case much competence produced by evolutionary processes concerned with spatial manipulation and reasoning may be available at birth.
If that's true of chicks, then why not other organisms, including humans?
Is this a possible origin for Kantian spatial reasoning mechanisms?
There's a huge amount of additional detail to be filled in and a lot of research required into the biochemistry needed.
A huge amount of potentially relevant research being done for other reasons is already going on e.g. industrial biochemistry, immunology, etc. etc.
Perhaps the results can be combined and shown to support new kinds of biological virtual machinery with powerful spatial reasoning capabilities: Justifying Kant's claims.
Watch this space over the next few years.
END FOR THIS PRESENTATION
The remainder of this document will be pruned and reorganised later.
It can be ignored until this note is removed.
After learning about AI around 1969/70, I hoped to use AI to build a computer model of a mathematical reasoner making the same discoveries about space as the ancient geometers had. But this has proved very difficult. I now think that is because brains do things that cannot be done on digital computers, because they use forms of reasoning that include both discrete and continuous changes, whereas digital computers are capable only of discrete changes. One of the consequences of using discrete representations is that thinking about rotations has problems.
If thinking about a rotating ball requires locating the ball in a discrete space, then there will be problems about smooth rotation. That's because during even small rotations points on the ball further from the centre of rotation will move through different numbers of spatial locations and if the motion has to using stepping through a discrete grid, maintaining the shape of the ball during rotation would be impossible. We are able to think about and reason about continuous transformations that would be impossible in a discrete machine. It's clear that brains can reason about and visualise continuous changes, though it is not clear how they do this. Perhaps such thinking makes use of sub-neural chemistry, which combines discrete chemical bonds that can change discretely and continuous motion through space. This thought may have been the motivation for Turing's work on continuously diffusing chemicals that interact in a plane Turing(1952). I suspect that paper was intended only as a preliminary investigation, to be followed up by richer and deeper investigations of 3D chemical processes, including, perhaps, sub-neural brain processes.
Whatever the mechanisms are, it is clear that humans can think about, and visualise continuous motions of various kinds, and it is not clear that digital computers can make the same discoveries about continuous spaces. (Is that why Turing claimed, in his thesis that computers were capable of mathematical ingenuity, but not mathematical intuition? -- a fact I first learnt from Francesco Beccuti in 2018).
Alternative popular accounts of mathematical knowledge are mistaken
Theories of mathematical knowledge put forward by psychologists, neuroscientists, logicians, AI theorists, and recent philosophers of mathematics fail to account for some of the facts noticed by Kant concerning the complexities of mathematical discovery and their relationships with spatial consciousness (in humans and other animals).
There are many researchers in psychology and neuroscience who try to investigate development of number competences, and think they have found evidence that number concepts are innate. That's because they don't understand that a full grasp of natural numbers (1, 2, 3, etc.) depends on understanding that the relationship of one-to-one correspondence is necessarily transitive , a type of understanding that Piaget showed many years ago (1952) does not develop until age 5 or 6 in humans.
Spatial and temporal one-to-one correspondence are two special cases of this discovery. Spatial consciousness formed the basis of ancient human mathematical consciousness in topology and geometry centuries before Euclid, and even longer before the development of logic-based formal foundations for (some) mathematics.
Readers who have never had personal experience of discovering geometric
constructions and proofs may find this online tutorial by Zsuzsanna Dancso
Without such personal experience it is impossible to understand what Immanuel Kant was talking about in his discussions on the nature of mathematical knowledge in his Critique of Pure Reason Kant(1781)
There are many more online geometry tutorials of varying quality. The worst ones merely present geometric facts to be remembered -- like some bad mathematics teaching in schools?
Sub-organism precursors of mathematical competences
I suggest that long before such spatial consciousness evolved in ancient humans and other animals, partly analogous capabilities, namely increasingly sophisticated abilities to detect and make use of spatial structures and relationships, must have evolved for use in ancient mechanisms assembling complex molecular structures during processes of biological reproduction, such as the processes of assembling a chick, or a crocodile, inside an egg.
Evolution confronts spatial complexity
As assembled structures become more complex, so must the mechanisms extending and combining those structures.
So if a developing organism in an egg acquires more interconnected interacting parts, then the mechanisms controlling further development will become more complex. In an egg these developments start before there is a brain. There must be mechanisms in the egg that are capable of building and manipulating continuously changing structures. Could those mechanisms also be responsible for some ancient forms of reasoning about continuous space?
The increasing complexity of such reasoning may include:
(i) increased size of the chemical structures involved in the construction processes,
(ii) increased complexity of the information required for selecting and executing next steps in the construction and
(iii) increased complexity of the mechanisms required to acquire and use the information.
I suspect there is already a lot more known about such mechanisms than I am aware of, though its full significance for reproductive processes may not yet be known! The existence of newly hatched chicks, crocodiles and other self-assembled organisms is evidence for the existence of the mechanisms, but not the details of their operation, nor my claim that as the assembly progresses, new assembly mechanisms must be constructed to deal with greater complexity of the assembly tasks, including use of increasingly complex information-based control mechanisms.
In cases where the assembly continues without any interaction with details of the environment that the organism will later inhabit, the competences produced in the new chic, or other hatchling, cannot be derived from trial and error learning mechanisms, such as neural nets used to gain information about the environment by interacting with it.
These ideas lead to a (still under-developed) biology-based interpretation of Immanuel Kant's theory of mathematical consciousness, referring to evolutionary and developmental precursors of spatial consciousness in vertebrates, and some other spatially intelligent species, such as octopuses Godfrey-Smith (2017).
The relationship between what is explicitly in the genome, i.e. directly encoded in DNA, and the "long range" influence of the genome in an adult organism is complex and indirect. This is perhaps clearest in the case of human linguistic competences: only the human genome, among known species, has the ability to produce adult individuals communicating and understanding (giving and receiving information) in a rich human language.
But the languages used for communication by adult humans vary enormously, in syntactic structures used, semantic contents expressed, communicative functions, and the modes of production (e.g. spoken languages, signed languages, written languages, typed languages (using keys on a computer keyboard), and the skin-contact language developed by Helen Keller's teacher Anne Sullivan for communication with the deaf and blind child. (See https://en.wikipedia.org/wiki/Helen_Keller).
Such diversity in the relationships between what is in the genome and the long
term effects of genome contents raises serious challenges for simple
theories of what genes are, what they can do, and how they are related to their
products. Compare the criticisms in
(Note: Unpublished research by Francesca Bellazzi at Reading University is also relevant.)
The Meta-Configured genome (MCG) theory, initially developed with biologist
Jackie Chappell, and more recently with Peter Tino, attempts to show how this
lack of specificity in the genome can illustrate both the great power of
products of biological evolution and the diversity of mechanisms and
sub-processes used in gene-expression. An introduction to the MCG idea
(including a short video) can be found here:
Gene expression in a Meta-Configured Genome
Cascaded, staggered, developmental trajectories, in complex organisms, in which later processes of gene expression (down arrows more to the right) use "parameters" acquired from results of earlier processes (down arrows more to the left) in increasingly complex ways.
In human versions of these mechanisms, spatial-cognitive abilities concerned with perception and action are later extended by additional self-reflective mechanisms, used in diagnosing flaws in behaviours, repairing them, or teaching them to others. The development of linguistic competences (whether spoken, signed, written or (nowadays) typed language) illustrates complexities of meta-configured genomes.
Some non-human primates share aspects of these competences -- partly illustrated by their ability to engage in various kinds of play with humans and among themselves (e.g. between parents and offspring) found in orangutans and gorillas, for example. Compare the complex relations between humans and some varieties of dogs (e.g. sheep-dogs).
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).)
A KEY IDEA:
As more components, especially more complex components, are assembled, the number of opportunities for further assembly actions obviously increases. But most of the available, i.e. physically/chemically possible, options will not be appropriate for the particular organism at that stage of development.
So the later, more complex, assembly processes require abilities both to detect possibilities and constraints (impossibilities) determining what can be done next and to decide which options should be selected at that time in various parts of the developing organism. The choices should be determined by information-based reasoning process, since use of trial and error would explode the time required for assembly (because of the combinatorics) and could lead to too many fatal options: bad choices with no return route, which would form an increasing proportion of the options as the developing organism becomes more complex.
So, not only the components of the new organism, but also the components of the construction mechanisms building the organism, must be developed chemically by pre-hatching, pre-natal, mechanisms, including the information-processing abilities required for controlling increasingly complex construction processes.
From eggs to minds
In a subset of organisms, evolution seems to have found ways of making the sorts of spatial reasoning competences required for development of the embryo also available to the completed organism, as shown by spatial intelligence in squirrels, nest-building birds, and in a modified form in foals that can run with the herd within hours of birth, and many others.
In a small subset of those species, most obviously humans, the processes of learning to detect and use spatial impossibility and necessity continue after birth, in some cases still somehow using sub-cellular chemical control mechanisms for spatial reasoning that were previously used to control assembly of parts of the organism itself. Building a weaver bird's nest may be more or less complex than building a weaver bird, but it would be useful for the nest building processes to make use of the powerful bird-building mechanisms, if such abilities can be made available in the fully formed bird, as illustrated in this video showing the abilities of weaver birds: https://www.youtube.com/watch?v=qbWM1QAVGzs.
In humans, additional reflective mechanisms continue to be built after birth, so that (as Piaget discovered) they can make proto-mathematical discoveries, such as the necessary transitivity of one-to-one correspondence in the fifth or sixth year. This is a pre-requisite for a full understanding of numbers and their uses in counting sets.
There are many gaps still to be filled in this theory, including explaining in detail how the chemical bootstrapping mechanisms originally provided by DNA extend themselves at later stages in a developing organism so as to include useful abilities to detect impossibility and necessity, required for effective control of assembly of more complex physiological structures and mechanisms -- a process children playing with construction kits such as Meccano, Lego, Tinker-toys, Fischer-technic, etc. develop spontaneously and unwittingly and use in controlling more complex assembly processes. Other intelligent species seem to have similar abilities, though only humans seem to be able to go on to reflect on, discuss, and explicitly teach such competences. But the core mechanisms are needed in the initial assembly of all organisms that grow themselves starting from fertilised egg-cells and creating long chains of increasingly complex processes of assembly, interleaved with processes that create new assembly capabilities.
The requirements are different in organisms like trees that do not move about as a whole, though their life cycles involve many motions of parts.
The genetically based mechanisms that develop spatial reasoning competences during reproduction are important because understanding of (e.g. spatial) impossibility (and necessity) cannot be learnt empirically, e.g. because no amount of failure proves impossibility. This also points to serious limitations of artificial neural nets whose learning is based entirely on collection of statistics and derivation of probabilities.
More than mere failure, or success, is required to explain reasons for failure or success. It also requires insight into the structures of problems: the type of insight without which the development of many types and branches of mathematics, and their application in practical activities, e.g. making clothes, building shelters, building machines to help with construction processes, would have been impossible. The recently fashionable theory that mathematical competences depend on uses of symbolic, logic-based, reasoning cannot account for the much older forms of mathematical discovery based on spatial reasoning abilities, some of which seem to be partially shared with other intelligent species.
So we need alternatives to both logic-based symbolic reasoning mechanisms and statistics-based probabilistic reasoning to explain spatial mathematical intelligence, or to replicate it in future machines.
Alan Turing on chemical morphogenesis
There is fragmentary evidence that Alan Turing was thinking about a project of this sort when he wrote his 1952 paper on chemistry-based morphogenesis, explaining formation of surface patterns on organisms. I suspect his unstated long-term intention was much deeper and more important than explaining how visible patterns form. There is a sentence about the importance of chemistry in brains in his Mind 1950 paper that seems to be relevant. Also relevant is the distinction he made in his thesis between mathematical ingenuity and mathematical intuition, claiming that unlike human mathematicians (Turing-equivalent) computers are capable of mathematical ingenuity but not mathematical intuition. He did not explain why not.
The label "Meta-Morphogenesis" was introduced to refer to that conjectured more
ambitious project in Sloman(2013).
Continued development of
the project since then is reported in a growing collection of online documents referenced in
https://www.cs.bham.ac.uk/research/projects/cogaff/misc/meta-morphogenesis.html, which include a theory of evolved construction-kits, including construction-kits created during processes of development of individual organisms in fertilized eggs, or seeds.
There seems to be little or no recognition of these processes and their implications in current philosophy of mind, psychology, neuroscience and AI. So theories developed in those fields are incapable of producing adequate explanations of a variety of phenomena, including spatial learning and reasoning in many species, ancient processes of mathematical discovery in geometry and topology, long before Euclid, and important aspects of human consciousness, including forms of proto-consciousness involved in multiple layers of increasingly complex information-based control mechanisms during development from fertilised eggs. Insofar as the key processes crucially involve both discrete and continuous change they cannot be fully replicated on digital computers, though they can be implemented in chemical processes for reasons pointed out in Schrödinger's 1944 Book, though he apparently did not notice their importance beyond explaining the possibility of reliable biological reproduction.
Penrose and Hameroff
These ideas are related to, but different from, claims made and developed by Roger Penrose since 1989 (in The Emperor's New Mind) and in subsequent work partly in collaboration with Stuart Hameroff, illustrated in a recent joint presentation online here:
Consciousness and the physics of the brain May 12, 2020
They don't seem to have noticed the types of multi-stage chemistry-based intelligence that are required during construction of a new complex organism (including construction of brains) starting from a fertilised egg-cell, Perhaps the microtubules occurring during development of a foetus are relevant long before the microtubules in brains, emphasised by Hameroff. (See https://science.sciencemag.org/content/357/6354/882.1)
NOTE FOR PHILOSOPHY TEACHERS
I suggest that, in view of what we now know about life, and the rate at which such knowledge is being extended, teaching philosophy of mind and philosophy of mathematics without teaching any evolutionary and developmental biology is educationally misguided. Likewise failing to teach philosophy students about varieties of types of virtual machinery that have been discovered or invented since mid-twentieth century.
For more information about the rich variety of types of virtual machines see this messy document (which I hope to rewrite): https://www.cs.bham.ac.uk/research/projects/cogaff/misc/vm-functionalism.html
DAY 1: Appearance of embryonic tissue.
DAY 2: Tissue development very visible. Appearance of blood vessels.
DAY 3: Heart beats. Blood vessels very visible.
DAY 4: Eye pigmented.
DAY 5: Appearance of elbows and knees.
DAY 6: Appearance of beak. Voluntary movements begin.
DAY 7: Comb growth begins. Egg tooth begins to appear.
DAY 8: Feather tracts seen. Upper and lower beak equal in length.
DAY 9: Embryo starts to look bird-like. Mouth opening occurs.
DAY 10: Egg tooth prominent. Toe nails visible.
DAY 11: Cob serrated. Tail feathers apparent.
DAY 12: Toes fully formed. First few visible feathers.
DAY 13: Appearance of scales. Body covered lightly with feathers.
DAY 14: Embryo turns head towards large end of egg.
DAY 15: Gut is drawn into abdominal cavity.
DAY 16: Feathers cover complete body. Albumen nearly gone.
DAY 17: Amniotic fluid decreases. Head is between legs.
DAY 18: Growth of embryo nearly complete. Yolk sac remains outside of embryo. Head is under right wing.
DAY 19: Yolk sac draws into body cavity. Amniotic fluid gone. Embryo occupies most of space within egg (not in the air cell).
DAY 20: Yolk sac drawn completely into body. Embryo becomes a chick (breathing air with its lungs). Internal and external pipping occurs.
Chickens mate -- chicks follow:
Baby Crocs Hone Hunting Skills -- National Geographic:
"they are born already knowing what to do"
Ducklings first feed after hatching -- First Swimming baby ducks
(Incubated -- no mother.)
Erkurt M. (2018) Emergence of form in embryogenesis. in Interface, 15: 20180454. Journal of the Royal Soc. https://dx.doi.org/10.1098/rsif.2018.0454
Godfrey-Smith, P. (2007). Innateness and Genetic Information. In P. Carruthers,
S. Laurence, & S. Stich (Eds.),
The Innate Mind Volume 3: Foundations and the Future
(pp. 55-105). OUP.
Carl G. Hempel (1945), Geometry and Empirical Science, in American Mathematical Monthly, 52, 1945, also in Readings in Philosophical Analysis eds. H. Feigl and W. Sellars, New York: Appleton-Century-Crofts, 1949, https://www.ditext.com/hempel/geo.html
Kant, Immanuel (1781). Critique of pure reason, (Translated (1929) by Norman Kemp Smith), London: Macmillan. Retrieved from https://archive.org/details/immanuelkantscri032379mbp/page/n10/mode/2up
J. Schmidhuber, 2014, Deep Learning in Neural Networks: An Overview, Technical Report IDSIA-03-14, IDSIA, https://arxiv.org/abs/1404.7828
A. Sloman, (1962), Knowing and Understanding: Relations between meaning and truth, meaning and necessary truth, meaning and synthetic necessary truth. DPhil Thesis, Oxford University, https://www.cs.bham.ac.uk/research/projects/cogaff/62-80.html#1962
A. Sloman, (1965) `Necessary', `A Priori' and `Analytic', in Analysis 26, pp. 12--16, https://www.cs.bham.ac.uk/research/projects/cogaff/62-80.html#1965-02
A. Sloman, (2013). Virtual machinery and evolution of mind (part 3) Meta-morphogenesis: Evolution of information-processing machinery. In S. B. Cooper & J. van Leeuwen (Eds.), Alan Turing - His Work and Impact (p. 849-856), Amsterdam: Elsevier. https://www.cs.bham.ac.uk/research/projects/cogaff/11.html#1106d
A. Sloman, (2013--2020),
Jane Austen's concept of information (Not Claude Shannon's).
Online technical report, University of Birmingham,
A. M. Turing, 1952, The Chemical Basis Of Morphogenesis,
Phil. Trans. R. Soc. London B 237, 237, pp. 37--72. Also reprinted in S. B. Cooper & J. van Leeuwen (Eds.) (2013), Alan Turing - His Work and Impact (p. 849-856), Amsterdam: Elsevier.
Alastair Wilson, 2017,
(He also has a book published later.)
Hens and chicks
MURGI Hen Harvesting Eggs to Chicks new "BORN" Roosters and Hens Small Birds
Ducklings around the lake. (4.24 starting to paddle)
Newly Hatched Ducklings  -- All waiting for last eggs to hatch.
'Peak hype': why the driverless car revolution has stalled
-- into pond after 1:31
-- climbing out at 2:30
Baby Alligators Hunt
Croc Babies Hunt Quickly
Croc camera captures incredible footage
(Spy in the Wild - BBC) https://www.youtube.com/watch?v=9jRSgZVhWvw Hen and baby chicks go to 1:54 to see chicks feeding. https://www.youtube.com/watch?v=L9-EA1ZnGwQ Hen and baby chicks. https://www.youtube.com/watch?v=QPqcSKhtxKk Ducklings around the lake
I thank Mike Ferguson for pointing out errors and infelicities in this document, in June 2021.