Abstract
The aim of the paper is to argue for the cognitive unity of the mathematical results ascribed by ancient authors to Thales. These results are late ascriptions and so it is difficult to say anything certain about them on philological grounds. I will seek characteristic features of the cognitive unity of the mathematical results ascribed to Thales by comparing them with Galilean physics. This might seem at a first sight a rather unusual move. Nevertheless, I suggest viewing the process of turning geometry into an axiomatic-deductive science as a process of idealization in mathematics that is parallel to the process of idealization in physics. In Kvasz (Acta Phys Slovaca 62:519–614, 2012) I offered an epistemological reconstruction of the process of idealization in physics during the scientific revolution of the seventeenth century. In the present paper I try to employ these epistemological insights in the process of idealization in physics and propose a reconstruction of the cognitive unity of the mathematical results ascribed to Thales, who can, on the basis of these ascriptions, be seen as one of the initiators of idealization in mathematics.
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Notes
One of the recent books on ancient mathematics, The Shaping of Deduction in Greek Mathematics by Netz (2004, p. 272), contains the frequently-quoted sentence, “Pythagoras the Mathematician Perished in AD 1962”. The year 1962 is the year of the publication of the original German edition of Burkert’s monograph, Lore and Science in Ancient Pythagoreanism. It seems that there is a consensus among classical philologists that the early period of ancient mathematics, represented by Thales and Pythagoras, cannot be reliably reconstructed.
More precisely, a series of three articles dedicated to Thales, Pythagoras and Euclid, forming a mathematical counterpart to the three papers describing the process of formation of early modern physics: Galilean physics in light of Husserlian phenomenology (Kvasz 2002); The mathematization of nature and Cartesian physics (Kvasz 2003); and The Mathematization of Nature and Newtonian Physics (Kvasz 2005).
The metaphor of crystallization can make the impression of a foreign element in the history of science. The aim of this paper is to propose a new approach to the reconstruction of ancient mathematical texts, so that metaphorical language may seem to be too vague. However, I consider crystallization as an example of phase transitions and suggest that we see the birth of mathematics not as a causal, deterministic process, but rather as a phase transition of the cognitive system. It is an open question whether a sufficiently comprehensive theory of such phase transitions can be developed, but I believe that the metaphor of crystallization offers at least a context in which we can understand the character of the problem.
I consider the relation of the two methods (the historical and the teleological) as analogous to the relation of micro- and macro-description in statistical physics. This is just another aspect of the metaphor of the cognitive phase transition.
In the history of science, the view of history as a process directed to our present state is subject to sharp criticism. However, when there is no better alternative, we must be satisfied (at least for the time being) with such a teleological reconstruction; and in a sense, we can see the teleological reconstruction of the emergence of geometry as parallel to the analytic method, where the final state (the solution of the problem) is also considered and then a backwards path sought, one that would connect this final state with the given data.
Husserl (1954) formulated the aim of using an understanding of the process of the constitution of ancient Greek geometry to reactivate the authentic meaning of geometry used in Galilean physics. The present paper can be seen as a proposal to use the link envisioned by Husserl in the opposite direction. Naturally I am fully aware that, according to Husserl, Galilean science lost contact with the life-world and that Husserl turned to the constitution of geometry in order to recover this contact. However, I do not share Husserl’s negative evaluation of Galilean science. I believe Galileo had an authentic grasp of the meaning of geometry and an authentic contact with the life-world, so I see no need for any reactivation of the meaning of geometry for understanding Galileo. Nevertheless, I believe that by pointing to the connection between Galilean science and the emergence of geometry, Husserl pointed to a structural connection that can be used to understand the process of the constitution of ancient Greek geometry. The idea is that if there really is a connection between Galilean science and the origins of geometry (as postulated by Husserl) then our understanding of Galilean science can be used to shed light on the early stages of the development of Greek geometry, for which we almost entirely lack textual evidence.
See the papers mentioned in footnote 3.
The deficiencies of Galilean physics are usually not mentioned in the literature. The general trend in the history of science is to present Galilean physics in a positive light. Interestingly enough, this is in sharp contrast to the case of Cartesian physics, where the general trend is to be over-critical and to question whether Descartes managed any outstanding scientific achievements except the discovery of the law of refraction. In this paper I will also characterize, as precisely as possible, the deficiencies of the Galilean system. I am led in this by the belief that we can learn from Galileo not only by analyzing his achievements but also by studying his failures.
By a cognitive style I mean features characterizing the work of a particular mathematician, such as the way of argumentation (e.g., willingness to use analogies or their total rejection); strictness of argumentation (e.g., care for an explicit formulation of all implicit assumptions or a lack of such care); degree of universality of assertions (e.g., a systematic effort to search for general principles or satisfaction with the analysis of particular cases); kind of unity of the subject-matter (e.g., attempts to use a principle of classification or a habit of jumping from one case to the next); methodological unity (e.g., a willingness to follow a proclaimed and clearly formulated method or satisfaction with ad hoc tricks); and manner of presentation (e.g., a willingness to explain the motivations of every step leading to a particular discovery or efforts at hiding any motivation). At later stages in the development of the language of mathematics these features of a cognitive style can be more precisely defined as the logical power, expressive power, methodical power, integrative power, explanatory power, and metaphorical power of the language of mathematics (see Kvasz 2008, p. 16). However, for characterizing the emergence of the language of mathematics I prefer to use the less strict concept of a cognitive style.
Deficiencies that we can clearly recognize thanks to our epistemological reconstruction of Galilean physics, but of which the authors, ascribing these deficient mathematical results to Thales, could not have been aware.
I do not see these deficiencies negatively as errors. I view them rather as characteristic features of the language used by Galileo in the development of his physics. Thus, I consider the notion of deficiency as an epistemological category that has a similar role in the description of idealizations as the logical and expressive boundaries of language have in the description of re-codings (see Kvasz 2008, p. 16).
In this respect they differ, for instance, from the law of conservation of energy, which applies to all physical systems, i.e., as much as to a falling body as a pendulum.
In comparison with Aristotelian physics, Galileo made an important step forward, because in Galilean physics uniform motion is not restricted to the heavens. Nevertheless, this step was not great enough. Inertial motion was not yet recognized by Galileo as rectilinear and the principle of inertia was not recognized as a universal principle, valid for all motions.
Galilean physics has a number of further achievements and deficiencies. My goal is not to offer an exhaustive list of them but to focus on those that can be used in the analysis of Thales’ mathematics.
See e.g. (Schramm 1994, p. 572).
For an alternative reading of T2 and T6 see (Kraft 1971, pp. 86–91).
Of course, the proof made by Thales could not be a deductive derivation from axioms, as we understand proofs today. However, if we look at the sentences that the tradition attributes to Thales, it is not difficult to discover that they all can be made evident by some manipulation (in reality or in mind) of the respective geometrical objects. Should we fold a circle over its diameter, the two halves coincide. Similarly, if an isosceles triangle is turned along the axis of the angle opposite its base, it will coincide with the original and so it has equal angles opposite its equal sides. A proof made by Thales could thus have the form of the immediate evidence of the veracity of the proposition on the basis of symmetry discovered by means of an appropriate manipulation (folding, rotating, or shifting) of the particular object (in reality or in mind).
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Acknowledgements
I would like to thank Vojtěch Kolman, Zdeněk Kratochvíl, Anton Markoš and Michael Pockley for helpful comments and Matyáš Havrda and Vladislav Suvák for help with the Greek terminology. I am grateful for the support of the Formal Epistemology—the Future Synthesis project within the framework of the Praemium Academiae program at the Institute of Philosophy of the Czech Academy of Sciences.
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Kvasz, L. Cognitive Unity of Thales’ Mathematics. Found Sci 25, 737–753 (2020). https://doi.org/10.1007/s10699-019-09622-7
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DOI: https://doi.org/10.1007/s10699-019-09622-7