Vincenzo Fiore Marrese

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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
  1. d'Errico, Francesco, Renata García Moreno and Riaan F. Rifkin. "Technological elemental and colorimeric analysis of an engraved ochre fragment from the Middle Stone Age levels of Klasies River Cave I, South Africa." Journal of Archaeological Science 39, issue 4 (April 2012): 942-952. https://doi.org/10.1016/j.jas.2011.10.032
  2. Dutkiewicz, Ewa, Gabriele Russo, Saetbyul Lee and Christian Bentz. "SignBase, a collection of geometric signs on mobile objects in the Paleolithic." Scientific Data 7, article 364 (October 2020). https://doi.org/10.1038
  3. Goren-Inbar, Nama. "Quneitra: a Mousterian site on the Golan Heights." Qedem 31, (1990):III–239.
  4. Greif, Hajo. "Likeness-making and the evolution of cognition." Biology & Philosophy 37, issue 1, article 1 (February 2022). https://doi.org/10.1007/s10539-021-09830-1
  5. Joordens, Josephine, Francesco d'Errico, Frank Wesselingh, Stephen Munro, John Vos, Jakob Wallinga, Christina Ankjærgaard, Tony Reimann, Jan R. Wijbrans, Klaudia F. Kuiper, Herman Mücher, Hélène Coqueugniot, Vincent Prié, Ineke Joosten, Bertil Van Os, Anne Schulp, Michel Panuel, Victoria Haas, Wim Lustenhouwer and Wil Roebroeks. "Homo Erectus at Trinil on Java Used Shells for Tool Production and Engraving." Nature 518, (2015):228–231. https://doi.org/10.1038/nature13962
  6. Kastens, Björn. "Geometrie für Germanien. Walther Lietzmann und seine „Frühgeschichte der Geometrie auf germanischem Boden“." Bonjour Geschichte (December 2011). urn:nbn:de:gbv:46-00102220-10
  7. Lietzmann, Walther. "Geometrie und Prähistorie." Isis 20, no. 3 (January 1934): 436-439.
  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
  9. Shaham, Dana, Anna Belfer-Cohen, Rivka Rabinovich and Naama Goren-Inbar. "A Mousterian Engraved Bone: Principles of Perception in Middle Paleolithic Art." Current Anthropology 60, no. 5 (October 2019):708–716. https://doi.org/10.1086/705677
  10. Struik, Dirk Jan. "Stone Age Mathematics : The earliest comprehension of number and geometry appears to go back farther than the time of the Egyptians and Babylonians." Scientific American 179, no. 6 (December 1948):44-49.
  11. Struik, Dirk Jan. A Concise History of Mathematics. 4th ed. New York: Dover Publications, 1987.
  12. Texier, Pierre-Jean, Guillaume Porraz, John Parkington, Jean-Philippe Rigaud, Cedric Poggenpoel, Christopher Miller, Chantal Tribolo, Caroline Cartwright, Aude Coudenneau, Richard Klein, Teresa Steele and Christine Verna. "A Howiesons Poort Tradition of Engraving Ostrich Eggshell Containers Dated to 60,000 Years Ago at Diepkloof Rock Shelter, South Africa." Proceedings of the National Academy of Sciences of the United States of America 107, no. 14 (March 2010):6180-6185. https://doi.org/10.1073/pnas.0913047107
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