Abstract
In Philosophia Botanica (1751), Carolus Linnaeus (1707–1778) presented a calculation of the number of plant genera that may be distinguished based on his taxonomic concepts. In order to derive that number, he relied upon the organs of fructification, which represent the flower and the fruit, by selecting over 30 elements from them, and then assuming that each could vary by four dimensions. However, while Linnaeus was good in counting stamens and pistils, he and many of his followers who edited or translated Philosophia Botanica were less careful, basing their calculations of the number of possible genera on flawed assumptions, or even introducing basic arithmetic errors. Furthermore, although mathematics was quite advanced in the eighteenth century, the gap between combinatorial and botanical thinking was too deep, preventing Linnaeus to seek a reasonable solution to the problem he raised. The authors demonstrate this by a historical analysis of 15 editions of Philosophia Botanica, plus many references to it, and conclude that the desired number almost always appeared in error during the past 265 years. The German botanist J. G. Gleditsch (1714–1786) was the most successful with respect to Linnaeus’ original intention. Elementary mathematics demonstrates that if Linnaeus’ assumptions were taken seriously, then the possible number of genera would be astronomical. The practice he followed in Genera Plantarum (1754) shows, however, that the fructification dimensions served as a universal set for Linnaeus from which he chose only the relevant ones for describing a particular genus empirically. Based on the corrections and modifications implemented in reworked editions, we suggest an evolutionary network for the historical and modern versions or translations of Philosophia Botanica.
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Notes
Eddy (2010) circumvented the problem by saying that Linnaeus used “three hundred and sixty-five aphoristic sections.”
In the corresponding aphorism of Fundamenta Botanica, Linnaeus lists 6 items for the calyx, 2 for the corolla, 2 for the stamen, 3 for the pistil, 3 for the pericarpium, 2 for the semen and 3 for the receptacle, giving a total of 21. A year later was published the first edition of Genera Plantarum eorumque characteres naturales secundum numerum, figuram, situm, & proportionem omnium fructificationis partium (Leiden 1737) in which the Preface mentions 26 items distributed over the parts as follows: 6 for the calyx, 3 for the corolla, 2 for the stamen, 3 for the pistil, 7 for the pericarp, 2 for the seed and 3 for the receptacle (see Müller-Wille and Reeds 2007).
Linné, Carl von. 1787. Pflanzenphilosophie im Auszuge, nach Linné's Methode; Augsburg: Wolff; Linnaeo, C. 1778. Explicacion de la Filosofia y Fundamentos Botanicos. Madrid: Jardin Botanico (transl. A. Palau y Verdera); Linney, K. 1800. Filosofiya Botaniki. Sanktpeterburg: Akademii Nauk (transl. T. Smelovsky).
Even Linnaeus himself was inconsistent in this regard, because he used dimensions interchangeably with „attributes, modes, mechanical principles, mechanical fundamentals” and—in Fundamenta Botanica—„diversities” (Cain 1994, p. 20).
Nevertheless, in the text below the table the wrong value of 38 remained: „ergo quater triginta octo” (see Fig. 3c).
Genera Plantarum had seven editions published in the lifetime of its author, the first—as said in footnote 1—in Leiden (1737) and the seventh in Frankfurt (1778) (Müller-Wille and Scharf 2009).
Linné: Systema Naturae 1735.
Numbering of elements as in section 86 of PhB, Stockholm 1751.
References
Andrietti, F., & Generali, D. (2002). Storia e storiografia della scienza: il caso della sistematica. Milano: Franco Angeli.
Atran, S. (1987). Origin of the species and genus concepts: An anthropological perspective. Journal of the History of Biology, 20(2), 195–279.
Atran, S. (1990). Cognitive foundations of natural history. Towards an anthropology of science. Cambridge: Cambridge University Press.
Bueno, G. (1998). Los limites de la evolución en el ámbito de la Scala Naturae. In E. Molina, A. Carreras, & J. Puertas (Eds.), Evolucionismo y racionalismo (pp. 49–88). Zaragoza: Institución Fernando el Católico.
Cain, A. J. (1993). Linnaeus’s ordines naturales. Archives of Natural History, 20, 405–415.
Cain, A. J. (1994). Numerus, Figura, Proportio, Situs; Linnaeus’s definitory attributes. Archives of Natural History, 21, 17–36.
Čermáková, L. (2013). Úloha smyslového vnímání při poznávání a popisu přírody v renesanci. (The Role of Sensory Perception in Cognition and Description of Nature in the Renaissance.) MSc thesis, Charles University, Prague.
de Lara Filho, D. (2006). Museu: de espelho do mundo a espaco relaçional. MSc thesis, Univ. Sao Paulo.
Eddy, M. D. (2010). Tools for reordering: commonplacing and the space of words in Linnaeus’s Philosophia Botanica. Intellectual History Review, 20, 227–252.
Ereshefsky, M. (1997). The evolution of the Linnaean hierarchy. Biology and Philosophy, 12, 493–519.
Ereshefsky, M. (2004). The poverty of the Linnaean hierarchy: A philosophical study of biological taxonomy. Cambridge: Cambridge University Press.
Foucault, M. (2002). The order of things: An archaeology of human sciences. London: Routledge. Originally: Les mots et les choses. Paris: Gallimard. 1966.
Freer, S. (2003). Preface to Philosophia Botanica (pp. ix–xiii). Oxford: Oxford Univ. Press.
Géczy, B. (1982). Lamarck és Darwin. Budapest: Gondolat.
Hacker, J. B. (1992). To name a plant – a historical perspective. Queensland Naturalist, 32, 2–13.
Hulth, J. M. (1907). Bibliographia Linnaeana. Matériaux pour servir a une bibliographie linnéenne. Partie I, Livraison I. Uppsala: Kungl. Vetenskaps Societeten.
Kwa, C. (2011). Styles of knowing. A new history of science from ancient times to the present. Pittsburgh, PA: University of Pittsburgh Press. Originally: De ontdekking van het weten: Ein andere geschiedenis van de wetenschap. Amsterdam, 2005.
Larson, J. L. (1971). Reason and experience. The representation of natural order in the work of Carl von Linné. Berkeley: University of California Press.
Lee, J. (1788). An introduction to the science of botany chiefly extracted from the works of Linnaeus, to which are added several new tables and notes and a life of the author. London: Printed for F.C. and J. Rivington, Wilkie and Robinson, J. Walker, White and Co.
Müller-Wille, S. and Scharf, S. (2009). Indexing nature: Carl Linnaeus (1707–1778) and his fact-gathering strategies. Working Papers on The Nature of Evidence: How Well Do ‘Facts’ Travel? No. 36/08.
Müller-Wille, S. (2007). Collection and collation: Theory and practice of Linnaean botany. Studies in History and Philosophy of Biological and Biomedical Sciences, 38, 541–562.
Müller-Wille, S., & Reeds, K. (2007). A translation of Carl Linnaeus’s introduction to Genera plantarum (1737). Studies in History and Philosophy of Biological and Biomedical Sciences, 38, 563–572.
Oittinen, V. (2007). Kant ja Linné. Ajatus, 64, 31–50.
Oittinen, V. (2009). Linné zwischen Wolff und Kant. Zu einigen Kantischen Motiven in Linnés biologischer Klassifikation. In E.-O. Onnasch (Ed.), Kants Philosophie der Natur, ihre Entwicklung im Opus postumum und ihre Wirkung (pp. 51–77). Berlin: Walter de Gruyter.
Pulteney, R. (1805). A general view of the writings of Linnaeus (2nd ed.). London: Mawman.
Roger, J. (1981). Linné et l’ordre de la nature. Arion, 6(5–6), 3–7.
Scopoli, G. A. (1786). Fundamenta Botanica. Vienna.
Soulsby, B. H. (1933). A catalogue of the works of Linnaeus (and publications more immediately relating thereto) preserved in the libraries of the British Museum (Bloomsbury) and the British Museum (Natural History) (South Kensington). London: British Museum.
Stafleu, F. A. (1971). Linnaeus and the Linneans: The spreading of their ideas in systematic botany, 1735–1789. Utrecht: Oosthoek.
Stafleu, F. A. and Cowan, R. S. (1981). Taxonomic literature. A selective guide to botanical publications with dates, commentaries and types. Vol III: Lh–O. Regnum Vegetabile 105.
Stearn, W. T. (1985). Botanical latin. Newton Abbot: David & Charles.
Stemerding, D. (1991). Plants, animals and formulae: Natural history in the light of Latour’s science in action and Foucault’s the order of things. PhD thesis, University of Twente.
Svenson, H. K. (1945). On the descriptive method of Linnaeus. Rhodora, 47, 273–302.
Vasilyeva, L. N., & Stephenson, S. L. (2008). The Linnaean hierarchy and ‘extensional thinking’. The Open Evolution Journal, 2, 55–65.
Weinberger, D. (2007). Everything is miscellaneous: The power of the new digital disorder. New York: Times Books.
Acknowledgments
The authors are grateful to the editor, S. Müller-Wille for his detailed comments which greatly improved the manuscript, and the anonymous referee for the constructive criticism. We thank Veronika Kulin for her help with latin translations, Partícia Dérer for assistance with the Russian language and Enrico Feoli for translating some text in Italian.
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This paper is dedicated to the memory of the father-in-law of the first author, the biologist and educator Gy. Kontra (1925–2007). A decade ago, he surprised J. P. by a gift, an original copy of Philosophia Botanica, its Berlin (1780) edition.
Appendix
Appendix
Here we provide a detailed estimation of \(\mathscr{L}\), based on modern combinatorial thinking and on the detailed list of elements and their dimensions in PhB. First of all, the 31 elements are partitioned into 23 kinds and 8 parts, respectively, as shown in the main text. The number of possibilities is determined for each main part separately, considering whether their elements are parts or kinds. The operation “+” applies to kinds, and “·” to parts. Then, assuming independence between the seven parts, the seven numbers are multiplied to obtain \(\mathscr{L}.\)
I. For part calyx, element 1 (perianthium) has all the four dimensions with the following number of states: numerus 3, figura 4, proportio 3 and situs 3 (Fig. 2).Footnote 8 That is, if these states are freely combined (which is probably not plausible biologically), then we have 3 · 4 · 3 · 3 = 108 possibilities. Element 2 (involucrum) is not described in terms of the four dimensions, while its number may have 6 states (from monophyllum to hexaphyllum, p. 63). The same is true for element 4, the spatha, with three states (p. 64). No dimensions for kinds 3 and 5–7 (amentum, gluma, calyptra and valva) are mentioned at all, so we cannot do any better than counting them only once. In sum, the calyx may have 108 + 6 + 3 + 1 + 1 + 1 + 1 = 121 distinct realizations.
II. For the corolla part, element 8, the petalum is counted with 8 possible states but the other three dimensions are ignored (p. 64). Element 9 (nectarium) is described in terms of figura (with 4 states), proportio (2 states) and locus (2 states). For the corolla, we thus have 8 · 4 · 2 · 2 = 128 possibilities.
III. For the stamen, element 10, the filamentum has all the four dimensions: numerus (ca. 15 states, as we know from Systema naturae), figura (8 states), proportio (4 states) and situs (4 states). Element 11, anthera, has numerus (4 states), figura (5 states) and situs (4 states). Element 12, pollen, has only one dimension, figura, with 7 states. Altogether, for stamen the number of distinct morphologies would be 15 · 8 · 4 · 4 · 4 · 5 · 4 · 7 = 1,075,200.
IV. For the last main part of the flower, the pistillum, element 13 (germen) is not detailed at all and the reader is referred to the pericarp; element 14, the stylus has numerus (we assume 4 states according to Systema naturae), figura (5 states) and situs (4 states). Element 15, the stigma has numerus (5 states) and figura (17 states). The pistillum may therefore give rise to 4 · 5 · 4 · 5 · 17 = 6800 different realizations.
V. The first fruit part, the pericarpium includes eight elements (here kinds) (16: capsula, 17: siliqua, 18: legumen, 19: conceptaculum, 20: drupa, 21: pomum, 22: bacca, 23: strobilus), and only capsula is detailed for numerus (7 states, for example, unicapsularis and multicapsularis), figura (7 states) and situs (3 states). Therefore, this part contributes to the total number of combinations by 7 + 7 · 7 · 3 = 154.
VI. For the semen, we have elements 24–26, the semen, the nux and the propago. Only the semen sensu stricto is detailed in terms of numerus (4 states), figura (5 states) and situs (4 states). We thus have 4 · 5 · 4 + 1 + 1 = 82 versions.
VII. The receptaculum part, containing elements 27 (receptaculum proprium), 28 (receptaculum commune), 29 (umbella), 30 (cyma) and 31 (spadix) is almost completely neglected in the discussion of PhB, only the figura of receptaculum is mentioned with 3 possible states (p. 71). Therefore, it is counted as 1 + 1 + 1 + 1 + 3 = 7.
If all the main parts, elements (kinds plus smaller parts), dimensions and their states may combine freely (which is impossible biologically), then these seven sums obtained above must be multiplied, giving the most conservative mathematical upper bound as \(\mathscr{L}\) = 121 · 128 · 1,075,200 · 6800 · 154 · 82 · 7 = 1019 (ten quintillion) which is astronomical given that the highest estimates for the number of extant species are between 107 and 108. For comparison, note that the total mass of the hydrosphere of our Earth is 1.4 · 1018 (1.4 quintillion) metric tons. Consequently, if we take Linnaeus’ suggestion seriously, the possible number of genera is far beyond reality and, even if we forget about the impossible combinations, the outcome still remains in the realm of incomprehensibly large numbers.
Even though in some instances the number of states and the status of the dimension we used here could be debated, the mathematically correct estimation of the potential number would probably be of similar magnitude as the above result.
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Podani, J., Szilágyi, A. Bad math in Linnaeus’ Philosophia Botanica . HPLS 38, 10 (2016). https://doi.org/10.1007/s40656-016-0110-5
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DOI: https://doi.org/10.1007/s40656-016-0110-5