Russell and the foundations of qualitative spatial reasoning: the first steps

ABSTRACT

We argue that the ideas of Bertrand Russell, a famous English philosopher and logician, have a bearing on the philosophical foundations of one of the sub–fields of AI, called qualitative spatial reasoning. The research conducted within that field focuses on non–numerical reasoning about regions of space designed to mimic human spatial behaviour and thus avoid the artificiality of the numerical approach. After briefly describing the main characteristics of this field, we analyse Russell's works on geometry. We show that despite major differences in how the subject matter is treated, these publications do have a common core that is related to the non–numerical, qualitative parts of geometry. Therefore, we argue that Russell should be viewed as a forefather of qualitative spatial reasoning on par with Whitehead or Leśniewski. Moreover, we believe that the efforts within qualitative spatial reasoning should be geared more towards the types of geometry he describes.

KEYWORDS:

• Bertrand Russell
• foundations of geometry
• qualitative spatial reasoning
• affine geometry
• projective geometry
• region-based geometry
• non-metrical geometry
• mereology

Acknowledgements

The author wishes to thank Ian Pratt-Hartmann, Bernard Linsky, Sébastien Gandon and the staff at the Bertrand Russell Archives in Hamilton, Canada.

Disclosure statement

No potential conflict of interest was reported by the author(s).

Notes

1 Two sets are in contact, if their boundaries share at least one point.

2 The authors of Randell, Cui, and Cohn (1992) write that the primitives of RCC8 can be viewed as physical objects, regions and other sets of entities, and provide a pictorial representation to guide intuition.

3 Roughly speaking, a regular set is one with no pin-holes or boundary ‘cracks'’.

4 A region closed, if we take it together with its boundary, and is open otherwise.

5 This is by no means the final word in the quest for well-behavedness, see Lando and Scott (2019).

6 A set is connected if there is an uninterrupted path between any two points in that region.

7 The only author that touches on this but with a different goal in mind is S. Gandon in Gandon (2012). We also believe that this approach permeated throughout Russell’s philosophical and even logical works as well, and thus the weight of Russell's geometrical studies in his formative years can be potentially even greater. Such a claim, however, requires a separate study.

8 Cf. Russell (1903, 419–428).

9 This is the term Russell uses to denote non-Euclidean geometry.

10 Russell (1897, 199). See also Russell (1897, 33).

11 See Monge (1839, p. xvi).

12 Let us note that this omission was not even corrected in the French edition of his book, see (Russell 1901).

13 This construction is known as the harmonic conjugate. The sought-after projective invariant, this construction was a landmark in projective geometry and was deemed important by Russell for philosophical reasons as well.

14 Russell writes about the period of the development of geometry in which projective methods were worked out, with Cayley playing a leading role, that ‘almost every important proposition, though misleading in its obvious interpretation, has nevertheless, when rightly interpreted, a wide philosophical bearing' (Russell 1897, 28).

15 The choice of words is also perhaps revealing of Russell's attitude (Russell 1897; 122).

16 Russell (1897, 32). We will not go into technical details here as it would obscure a more general point we are trying to make. See e.g. (von Staudt 1847) for details.

17 These, in turn, are related to compositions of perspectivities, i.e. ways, in which an image can be distorted according to the laws of perspective.

18 Mario Pieri was an Italian geometer, heavily influenced by von Staudt and Giuseppe Peano. Pieri's works were discovered by Russell after the writing the Essay. S. Gandon in his study (Gandon 2012) puts forward that the difference between von Staudt and Pieri in terms of approaching projective geometry had some consequences for Russell's philosophical stance.

19 A set is convex, if for any two points in that set, the straight line connecting these two points is also in the set. Convexity is important in many areas of geometry and computer science alike.

20 With some exceptions but most of these can be dealt with using affine geometry as well.

21 A convex hull of a set is the smallest convex set containing it.

22 This is a contentious matter, however, and not one we dealt with in this article.

Additional information

Funding

The research related to this article was supported by the Polish National Science Centre [grant number 2017/26/D/HS1/00200].

Notes on contributors

Adam Trybus

Adam Trybus is an assistant professor in logic at the Institute of Philosophy in Zielona Góra, Poland. His PhD (School of Computer Science, University of Manchester) was devoted to the logical analysis of fragments of affine geometry. He is currently working on Russell’s foundational ideas in geometry (PI in a project funded by Polish National Science Centre, grant number 2017/26/D/HS1/00200).