Vincenzo Fiore Marrese

Art and mathematics

Abstract

The intersections between art and mathematics could be very old, though challenging to frame. Prehistoric artefacts, like any object in general, face questions related to the ontology of art. Likewise, without written documentation, addressing prehistoric artefacts in mathematics is difficult. Taking care of these conditions, we can evaluate some hypotheses.

In his "A Concise History of Mathematics", Dirk Jan Struik states that palaeolithic paintings show a "remarkable understanding of form", which, in mathematical terms, means a "two-dimensional mapping of objects in space" [note 1].

Likewise, Struik states a connection between some patterns and mathematics, i.e. geometry [note 2]. He called these patterns "geometrical patterns" [note 3].

The adjective "geometrical" or "geometric" open up some questions about an ontology of shape. I'll momentarily pass on.

According to scholars, one can find "geometric signs" in Paleolithic artefacts [note 4].

The artefact known as DUB1006-fL is an engraved shell that could trace back to 0.54 (± 10) and 0.43 (± 0.05) million years [note 5]. So, one can say a maximum age of roughly 500.000 years. It comes from Trinil and is part of the Dubois collection in the museum Naturalis (Leiden, The Netherlands) [note 6]. It shows a geometric pattern that reminds the shape of the Latin alphabet letters 'Z', 'M', or a "mirrored" 'N', and "parallel lines" [note 7].

The artefact labelled KRM 13 is a possible engraved stone with a maximum age of 100.000 years and a minimum age of 85.00 years [note 8]. It comes from the Klasies River Caves [note 9]. It shows a geometric pattern of "sub-parallel lines" [note 10].

The artefact labelled DRS N7 - Frank 2447 is an engraved eggshell dating back approximately 60.000 years [note 11]. It comes from the Diepkloof Rock Shelter and is part of a 270 fragments group [note 12]. It shows a geometric pattern based on a "hatched band motif" [note 13].

The artefact known as BQ85-11732 is an engraved stone dating back to the Middle Paleolithic from a site with traces dating back 53.900 ± 5.900 years [note 14]. So, one can say a maximum age of roughly 50.000 years. It comes from Quneitra [note 15] and is part of the collection of the Israel Museum [note 16]. It shows a geometric pattern that reminds "concentric arcs" and "parallel lines" [note 17].

Struik seems focused on Neolithic artefacts [note 18], although he assumes the same for Paleolithic artefacts [note 19]. He writes that geometrical patterns on Neolithic pottery are "expressions of interest in mathematics" [note 20]. If we assume that, he stresses, the origin of mathematics is related to some notions of "numerical and geometrical relations" [note 21].

One of Struik's references is "Geometrie und Prähistorie" by Walther Lietzmann [note 22]. According to Björn Kastens, Lietzmann used to include racial "and National Socialist ideology in his texts", opposing what he called "Northern geometry" to Greek geometry [note 23]. However, Kastens writes that in "Geometrie und Prähistorie", published by the University of Chicago, there is no trace of this ideology [note 24]. Nevertheless, he assumes there are opportunistic reasons towards the American readers behind this choice [note 25].

It was a matter to criticise the "traditional opinion" [note 26] about the origin of mathematics. On one side, there was the idea that mathematics comes from practical needs, and Lietzmann questioned it [note 27]. Then there was the idea that Egyptians and Babylonians invented mathematics, and Struik doubted it [note 28]. Both critics used as pieces of evidence the prehistoric artefacts with geometric patterns [note 29].

Lietzmann talks about "künstlerischem Ornament" and "prähistorische Ornamentik". For the latter, he states as source Max Dehn [note 30].

Bibliography

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8. Marshack, Alexander. "A Middle Paleolithic symbolic composition from the Golan Heights: the earliest known depictive image." Current Anthropology 37, no. 2 (April 1996):357–365. https://doi.org/10.1086/204499
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Notes

1. Struik, A Concise History of Mathematics, 9. back to the text
2. Struik, 11-14. back to the text
3. Struik, 13. back to the text
4. See, e.g. Ewa Dutkiewicz et. al., "SignBase, a collection of geometric signs on mobile objects in the Paleolithic," 1. back to the text
5. Josephine Joordens et. al., "Homo Erectus at Trinil on Java Used Shells for Tool Production and Engraving," 228. back to the text
6. Josephine Joordens et. al., 228. back to the text
7. Josephine Joordens et. al., 229. back to the text
8. Francesco d'Errico et. al., "Technological elemental and colorimeric analysis of an engraved ochre fragment from the Middle Stone Age levels of Klasies River Cave I, South Africa," 942, 946. back to the text
9. Francesco d'Errico et. al., 943. back to the text
10. Francesco d'Errico et. al., 948. back to the text
11. Pierre-Jean Texier et. al., "A Howiesons Poort Tradition of Engraving Ostrich Eggshell Containers Dated to 60,000 Years Ago at Diepkloof Rock Shelter, South Africa," 6180-6181. back to the text
12. Pierre-Jean Texier et. al., 6180. back to the text
13. Pierre-Jean Texier et. al., 6181. back to the text
14. Dana Shaham et. al., "A Mousterian Engraved Bone: Principles of Perception in Middle Paleolithic Art.," 708. back to the text
15. Dana Shaham et. al., 708; see also Nama Goren-Inbar, "Quneitra: a Mousterian site on the Golan Heights," III–239. back to the text
16. “Rainbow“ symbol incised on plaque, https://www.imj.org.il/en/collections/330096-0, retrieved on the 10th of April 2023 back to the text
17. Dana Shaham et. al., "A Mousterian Engraved Bone: Principles of Perception in Middle Paleolithic Art," 709; see also Alexander Marshack, "A Middle Paleolithic symbolic composition from the Golan Heights: the earliest known depictive image," 335. back to the text
18. Struik, A Concise History of Mathematics, 7. back to the text
19. Struik, "Stone Age Mathematics", 44. back to the text
20. Struik, 44. back to the text
21. Struik, 44. back to the text
22. Struik, A Concise History of Mathematics, 7. back to the text
23. Kastens, "Geometrie für Germanien," 1. back to the text
24. Kastens, 4. back to the text
25. Kastens, 5. back to the text
26. "die traditionelle Aufassung" in Kastens, 2; see also "It has long been customary to attribute the invention of this science to the Egyptians and Babylonians" in Struik, "Stone Age Mathematics", 44. back to the text
27. Kastens, 2. back to the text
28. Struik, 44. back to the text
29. Lietzmann, "Geometrie und Prähistorie"; Struik, "Stone Age Mathematics". back to the text
30. Lietzmann, "Geometrie und Prähistorie", 436. back to the text