Could the two rings have been cut out of two separate blocks of stone and then assembled?

o How do you know the answer to this?

Curves on the surface of a torus
(Skip during workshop presentation)
In a planar or spherical 2D surface S, if C is a (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 the surface that joins a point in S1 and a point in S2 *must* also cross the curve C.

If S is a toroidal surface, e.g. the surface of a ring, then the above is true for some closed curves in S but not all. (Think of closed curves drawn on the surface of the inflated inner tube of a car wheel.)

Now consider the figure below, containing five curves, B1, B2 both blue, R1 red, and Y1, Y2 yellow. These are all closed curves: they have no free ends.

Closed curves on a torus

You can probably tell that the two yellow closed curves Y1 and Y2 are mutually continuously transformable: each can be smoothly moved in the surface of the torus to occupy the exact location of the other. How do you convince yourself that it is possible? Do you have to physically create a succession of intermediate curves, or is it enough to imagine them? Do you have to imagine all of the intermediate locations? Is the equivalence obvious at a more abstract level? What brain mechanisms could discover such obviousness?

Note added 22 Jul 2021
Suppose there are two non-self-crossing continuous closed curves C1 and C2 on the surface of a torus, we can say they are in the same "equivalence class" if C1 can be continuously deformed into C2. (Is it possible for C1 to be continuously deformable into C2, but not vice versa? How could you know such asymmetry is impossible?) A new question arises out of this: how many equivalence classes of continuous closed curves are there? What sort of brain makes it possible for you to work out the answer simply by thinking about the problem, without having to be trained on thousands, or millions, ... or an infinite supply ... of different sets of curves? Does any current robot have an artificial brain with such powers? Could you design such a robot?

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Ancient mathematics
In humans, mechanisms of spatial cognition enabled ancient mathematicians, centuries before Euclid, to make discoveries regarding possibility, impossibility and necessity in spatial structures and processes, without making use of modern mathematical formalisms in which theorems are derivable using only logical inferences: instead they used spatial forms of reasoning, e.g. reasoning based on real or imagined spatial structures rather than logical/algebraic formalisms.

Some intelligent non-human animals (e.g. squirrels, some nest-building birds, octopuses) also seem to have such abilities. They can make use of spatial possibility, impossibility, and necessity in selecting and performing actions, though they cannot communicate their knowledge, and may not have brain mechanisms required for reflecting on what is known, or describing such knowledge.

Many philosophers of mathematics seem to believe that everything that could be done using ancient spatial reasoning mechanisms can be replicated in modern discrete, logic-based, reasoning systems. It's possible that the ancient forms can be modelled in the new forms, but without being replicated (e.g. using physical mechanisms with similar speeds and energy costs).

During the discussion after my presentation Pat Hayes commented that a great deal of geometry and topology has been expressed in logic-based formal systems using only discrete symbols and discrete operations on symbols. The most obvious and well known example is Hilbert axioms using logic, as the great mathematician David Hilbert did when he "axiomatized" Euclidean geometry Hilbert(1899).

But I am not claiming that discrete, logic-based formalisms cannot be used to replicate (or model) ancient geometric discoveries. I claim only that there are alternative mechanisms using spatial reasoning rather than symbolic manipulations, that were used by ancient mathematicians, engineers, architects and also ordinary folk, long before humans discovered the space of formal systems developed in the 19th and 20th Centuries, including Hilbert's logic-based formalisation of the subset of geometry captured by Euclid's informal axiomatisation.

There are still modern mathematicians who present proofs that are based on spatial reasoning, supplemented with verbal commentaries, such as the wonderful demonstrations by mathematician Cem Tezer in these two video presentations on properties of triangles and related structures:

https://www.youtube.com/watch?v=AJvjtK2mmpU
MATH 373 - Geometry I - Week 1 Lecture 1

https://www.youtube.com/watch?v=1hNR-iCuw7g
MATH 373 - Geometry I - Week 1 Lecture 2

I suspect, and I think Turing thought, that those ancient, and not so ancient, spatial reasoning processes cannot all be replicated/modelled in logic based systems. In effect Immanuel Kant made related claims about some forms of mathematical reasoning in 1781, but my claims in this presentation do not depend on whether purely symbolic, logic-based, reasoning cannot replicate the results of the ancient forms of mathematical reasoning. I claim only that there were ancient (and not so ancient) mathematicians whose geometric reasoning went beyond use of logic and definitions.

Kant vs Hume
Immanuel Kant's characterisation of ancient mathematical knowledge, in his Critique of Pure Reason (1781) drew attention to three features of such cognition, using three distinctions that are ignored in current psychology, neuroscience and neural-net based AI: the non-empirical/empirical, analytic/synthetic, and necessary/contingent distinctions.

David Hume and Immanuel Kant (from Wikimedia)

         

Very crudely, David Hume, depicted above, on the left, claimed that there are only two kinds of knowledge:

1. empirical knowledge that comes through sensory mechanisms, possibly aided by measuring devices of various kinds; which he called knowledge of "matters of fact and real existence";
   
2. what he called "relations of ideas", which we can think of as things that are true by definition, such as "All bachelors are unmarried", and (if I've understood Hume rightly) all mathematical knowledge, for example ancient knowledge of arithmetic and geometry, which Hume's words seemed to suggest was no more informative than the bachelor example.

"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:
[SKIP DURING PRESENTATION!]

• The mathematical truths such as Pythagoras' theorem and others proved by Euclid are not analytic but synthetic, i.e. they are not provable simply using definitions and logic -- for example spatial reasoning is required. and

• They also are not derived from sensory experiences by generalising from examples, in such a way that if the experiences had been different the knowledge acquired would have been different, like the knowledge that eating berries with a certain appearance can make you feel ill. You cannot discover that fact simply by reasoning about berries, whereas mathematical discoveries e.g. about numbers or triangles can be discovered by reasoning, so they are apriori not empirical. (Note that "apriori" does not imply "innate". It merely rules out knowledge being derived from observations of the world that could have yielded different results, as is true of most of our knowledge about the world.)

• Moreover, the world could not have been different in ways that would have made our mathematical discoveries false: they are necessarily true, not contingently true, like propositions that are true but could have been false, such as the proposition that we are in the midst of a Covid-19 pandemic. That could have been false, perhaps if certain human behaviours had been different.

  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.

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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.

People who have been deprived of the traditional type of mathematical education normally fail to discover these distinctions themselves, and may not understand them if they read about them in Kant, or commentators.

Both logic-based and neural-net-based AI systems are incapable of replicating the ancient aspects natural intelligence that allow discoveries based on spatial reasoning. Moreover, the ancient mathematical capabilities described by Kant are usually ignored in currently fashionable theories and models of mathematical cognition in psychology, neuroscience, philosophy and AI, including theories that focus on social aspects of mathematics.

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.

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, insofar as ancient number knowledge was derived from properties of the one-to-one correspondence relation, e.g. the fact that it is necessarily transitive and symmetric -- which young humans seem to be unable to grasp before the age of five or six, as shown by Piaget in 1952.

This refutes theories about innateness of knowledge of cardinality, unlike knowledge of numerosity, which lacks the precision of cardinality. The numerosity of a visible part of the environment can be thought of as roughly the product of the average density (e.g. of leaves visible on a tree, or light-points visible in a portion of the night sky) and the proportion of the visual field occupied. That product gives an approximate but unreliable estimate of the actual number of distinct items visible. It can be applied to scenes where there are too many objects to count, such as leaves visible on a tree or stars/planets visible in a region of the sky at night. The numeracy estimate does not make use of one-to-one correspondence.

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'll raise questions about the nature of those mechanisms linking many different sorts of parts of a developing organism and how they can be accommodated inside a fully occupied egg.

REFERENCES AND LINKS
To be extended

Terry F. Godlove, Jr., (Jun, 2009), Poincare, Kant, and the Scope of Mathematical Intuition, in The Review of Metaphysics, Vol 62, No 4, pp. 779--801, Philosophy Education Society Inc.,
https://www.jstor.org/stable/40387765

David Hilbert, 1899, The Foundations of Geometry,, available at Project Gutenberg, Salt Lake City, http://www.gutenberg.org/ebooks/17384 2005, Translated 1902 by E.J. Townsend, from 1899 German edition,

Marvin L. Minsky, 1995, "Virtual Molecular Reality", in Prospects in Nanotechnology, Eds. Krummenacker and Lewis, Wiley,
https://web.media.mit.edu/~minsky/papers/VirtualMolecularReality.html

Aaron Sloman, 1965, "Necessary", "A Priori" and "Analytic", Analysis, Vol 26, No 1, pp. 12--16.
http://www.cs.bham.ac.uk/research/projects/cogaff/62-80.html#1965-02

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Maintained by Aaron Sloman
School of Computer Science
The University of Birmingham