School of Computer Science

Rotation in a discrete universe.
Help requested from quantum physicists
(DRAFT: Liable to change)

Aaron Sloman
School of Computer Science University of Birmingham,UK
(Philosopher in a Computer Science department)

Installed: 10 Mar 2014
Last updated:
22 Apr 2015: added reference to web page illustrating distortions caused by rotations.
5 Apr 2014; 6 Jul 2014; 19 Dec 2014; 26 Mar 2021 (new format)
This paper is
(A PDF version may be added later.)

A partial index of discussion notes is in

Letter to a quantum theorist

I wrote this document after watching this discussion, and a number of others:

Extract from the web page:
So Zeno asked: Is time continuous, or is it made up of discrete moments? Such age-old questions have important implications for the modern-day field of digital physics, for which the notion of analog (continuous) vs. digital (discrete) plays a critical role when considering a reality made of coded information. Here, members of the distinguished panel weigh in with their unique points of view.

Juergen Schmidhuber, Edward Fredkin, Seth Lloyd, John Hockenberry (Chair)
There's an additional, unnamed person on the stage who does not say anything in
this clip.

I have had a nagging worry for some time about using discrete machines to model continuous processes, despite having once accepted the argument that continuous processes can be modelled as accurately as required by choosing a suitable resolution (Sloman 1978).

My worry surfaced again a few years ago when I read (and understood some of) Turing's 1952 paper on Chemical morphogenesis, which left me wondering what he would have done had he lived on. I proposed the label 'Meta-Morphogenesis' for the project, since one of its postulates (well supported empirically) is that the mechanisms that inform and constrain the processes of natural selection are capable of producing new mechanisms that inform and control the processes.

I.e. Mechanisms of morphogenesis can produce consequences that modify the mechanisms of morphogenesis. Hence "Meta-Morphogenesis".

The project is about far more than evolution, because the details of evolution are deeply inter-twined with many other aspects of life forms, including information-based control even in the simplest life forms. Though the types of information used, the sensing devices used to obtain it, the variety of syntactic and other structural roles.

It seems clear that one of the features of chemistry that has been important in biology is its mixture of continuous and discrete changes. Continuous changes happen during translation, rotations, folding, twisting and other processes. Discrete changes occur when bonds form or are unlocked, e.g. by catalysts. I wonder whether this is why Turing wrote in his Mind 1950 paper "In the nervous system chemical phenomena are at least as important as electrical", a remark that seems to have been ignored by all commentators.

About 40 years ago, while first learning to program, I discovered the problem of rotating a pattern in a discrete rectangular array. For angles other than multiples of 90 degrees, any rotation of an array of numbers (e.g. image intensities) will lose information. Moreover, the amount of distortion varies with distance from the centre of rotation. In an image with two or more objects rotating around different centres (e.g. gears, levers, etc.) there will be different distortions in different parts of the process. The same applies to rotations of 3-D structures modelled discretely.

The distortion is illustrated in a separate document, using both rotation of an image represented by an array of pixel values and rotation of the same image represented by coordinates and equations:

[I understand that the use of a fourier transform can reduce (or remove??) the information loss, but presumably that's no help when modelling multiple moving and rotating objects whose spatial interactions play causal roles.]

I understand there is now some disagreement among quantum theorists as to whether quantum physics implies that space is discrete.

Is there someone better qualified than I am, who can comment on the relevance of the problem of rotating objects for the discrete space view. Is there something I've missed that deflates the problem?

Part of the background to this question is my attempt to understand how humans managed to discover Euclidean geometry and what sort of knowledge we get from Euclidean proofs. I always thought Kant was right, as opposed to Hume, Russell and others, and hoped to show why he was right by designing a baby robot that could make discoveries about space and spatial structures and Euclid's predecessors must have done, without mathematics teachers. (I have also begun to try to understand whether Euclidean geometry without the parallel axiom might suffice for intelligent agents, much of the time.)

But getting computers to reason diagrammatically as humans do has proved very difficult. I suspect this has something to do with perception of affordances (possibilities for change, constraints on possibilities, and consequences of possible changes) in non-human animal precursors to humans, and others -- e.g. nest-building birds.

I've been trying to spell out some of the requirements for such reasoning in connection with theorems about triangles and other continuously deformable objects here:
    Hidden Depths of Triangle Qualia
    Theorems About Triangles, and Implications for Biological Evolution and AI
    The Triangle Sum Theorem
    Old and new proofs concerning the sum of interior angles of a triangle.
    (More on the hidden depths of triangle qualia.)
    Reasoning About Continuous Deformation of Curves on a torus and other things.
    Shirt Mathematics
    Illustrating topological and semi-metrical reasoning  in everyday life.
If anyone has any thoughts on these problems, especially the suggestion that rotation poses a problem for a theory that treats space (or space-time) as discrete, I'd very much like to hear them.

I don't think any of the points about discreteness made in the World Science Festival video address that issue.

A related document:
Construction kits required for biological evolution
(Including evolution of minds and mathematical abilities.)
The scientific/metaphysical explanatory role of construction kits

Aaron Sloman, 1978,
The Computer Revolution in Philosophy, Philosophy, Science and Models of Mind,
Harvester Press (and Humanities Press),

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