http://www.cs.bham.ac.uk/research/projects/cogaff/misc/torus.html

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by Olaf Klinke
German Cancer Research Center (DKFZ)
http://www.dkfz.de/en/f100/mitarbeiter/o_klinke.html
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Date: 7 Jul 2014 (Re-formatted: 11 May 2015)

[Olaf] > as you remarked towards the end of your document, discovering some of > the theorems about the torus requires the ability of self-inspection, > or, as I would rather call it, the ability to perform a > "Gedankenexperiment".[Aaron]

I have no quarrel with that, as it also fits many of the other examples I have discussed, including reasoning about areas of triangles, reasoning about adding up angles of a triangle.

But there are different sorts of "Gedankenexperimenten" (?) and different things that become transparent during such experiments, only a subset of which lead to mathematical discoveries, as opposed to new novels, painting of imaginary situations, wishful thinking, idle speculation, etc. So I am trying to understand what accounts for that difference.

A subset of cases uses logical and algebraic reasoning. Later I want to challenge two assumptions (a) that those are the only kinds of reasoning that are mathematically valid -- e.g. demonstrate necessary connections, or impossibilities -- and (b) that it's obvious why those methods justify their conclusions, i.e. demonstrate that if the premisses (or instances of the premisses) are true then the conclusions (or corresponding instances of the conclusions) must also be true, in any possible environment.

[Olaf] > In this particular case, the non-mathematical human would go: What if > the depicted torus was some object I could touch and move, and what if > I grabbed this line and moved it over there? Thus what is necessary > for the discovery of such theorems is at least a good mental model of > the three-dimensional physical world around us. The step from a couple > of mental attempts to an actual theorem is still somewhat obscure for > me, though.

[Aaron]

I agree. The only difference between us is that I would not describe Euclid and
Euclid's predecessors as 'non-mathematical' humans. There were superb
mathematicians long before Descartes, Frege, etc. A formal logical, or algebraic
proof is just another powerful form of "Gedankenexperiment". We also have more
advanced meta-mathematical theorems and theories about what they do. But I
suspect there are similar meta-mathematical theories waiting to be developed
regarding what the ancient mathematicians did -- and what I did as a child
learning Euclidean geometry as part of my school education before the days when
such things were abandoned by mathematical educators, for bad reasons!

I suspect young children have those abilities and use them, but that their
discoveries go unnoticed by parents and nursery-school teachers. I call them
'toddler theorems' and have started collecting examples here:

http://www.cs.bham.ac.uk/research/projects/cogaff/misc/toddler-theorems.html

[Olaf] > If you gave a computer the possibility to simulate a torus and move > about curves on it, obeying the constraints of no cutting or joining, > when would it conclude that Y1 is not deformable into R1?

[Aaron]

At the moment (I claim) we have no method of implementing the kind of discovery
mechanism that a human uses. We can give a graphical engine very specific
specifications of 2-D or 3-D configurations, and specifications of a type of
transformation, and ask the machine to read off features of the results of
performing those transformations on those configurations. But nothing in AI or
robotics that I know of is capable of discovering from a few experiments that
the precise details are irrelevant and the constraints discovered in one case
apply to an infinite set of cases.

I suspect that will require both new representations of spatial structures and processes and new metacognitive abilities to inspect and reason about operations on those representations subject to various constraints.

[Olaf] > The mathematical approach to problems as the torus is, as I call it, > the victory of calculi over intuition.[Aaron]

I believe that that is the "standard" view among many logicians, philosophers of mathematics, and mathematicians. But not all agree. E.g. I heard Mandelbrot give a talk about 20 years ago in which he complained bitterly that mathematical education was now seriously incomplete because of the removal of geometry.

[Olaf] > The cleverness in Descartes' coordinate system (or algebraic topology) > is to de-couple the mathematical representation of space from man's > intuition about space, making it possible to manipulate more complex > collections of facts and hypotheses as before.

[Aaron]

I do not wish to disparage the importance of Descartes' achievement. It has been
pointed out that without the arithmetisation of geometry Newton could not have
developed his mechanics, in part because the differential and integral calculus
requires mappings from geometry to arithmetic.

[Olaf] > Subsequently, mathematicians developed an intuition about the calculi > themselves (as Achim already suggested, mathematicians are just adept > pattern matchers), pushing the boundaries of complexity even further.

[Aaron]

I agree that that's an important part of mathematics (and also computer science,
and software engineering, including the processes that lead to invention of new
more powerful, or application specific programming languages). But something
similar happened earlier, somehow produced by biological evolution, that we
don't yet understand. I even have an open mind as to whether it requires a new
type of (non-Turing) computer.

Could chemical computation, with its deep mixture of discrete and continuous transformations, provide some clues? It seems to be essential for living things, including the processes that build brains.

Maintained by
Aaron Sloman

School of Computer Science

The University of Birmingham