Abstract
In this paper, I present the case of the discovery of complex numbers by Girolamo Cardano. Cardano acquires the concepts of (specific) complex numbers, complex addition, and complex multiplication. His understanding of these concepts is incomplete. I show that his acquisition of these concepts cannot be explained on the basis of Christopher Peacocke’s Conceptual Role Theory of concept possession. I argue that Strong Conceptual Role Theories that are committed to specifying a set of transitions that is both necessary and sufficient for possession of mathematical concepts will always face counterexamples of the kind illustrated by Cardano. I close by suggesting that we should rely more heavily on resources of Anti-Individualism as a framework for understanding the acquisition and possession of concepts of abstract subject matters.
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Notes
Here and throughout the paper I use contemporary symbolic notation to present the computations performed by Cardano and others. I do so for the sake of convenience. I do not commit to the claim that Cardano’s mathematics should be identified with contemporary mathematics without qualification.
Bombelli comments: “It was a wild thought in the judgment of many; and I too for a long time was of the same opinion. The whole matter seemed to rest on sophistry rather than on truth. Yet I sought so long, until I actually proved this to be the case.” (Burton 1995, p. 328).
The relation between linguistic meaning and mental content is complicated. Often, several concepts are associated with one linguistic form. Often, it depends on context, which concept is expressed by a word. I here merely provide a sketch of the relationship between concepts and linguistic expressions.
Philosophical tradition acknowledges the need to distinguish concepts from conceptions. Discussions of externalism and anti-individalism in the philosophy of mind strongly support making the distinction. I can here only sketch some of the motivations for this distinction. But see Kripke (1980), Putnam (1970, 1975, 1973), Burge (1993, 2012).
My claims about the standard description do not express to a strong position in the ongoing debate between presentism and historicism in the history of mathematics. Historicism is, roughly, the view that one cannot interpret early mathematics in terms of contemporary mathematical problems, aims, and methods. Presentism maintains that one can, and indeed should, do so. (Hodgkin 2005, p. 5ff.).
The actual practice of historians of mathematics betrays a nuanced application of both historicist and presentist methods. In this paper I rely on verdicts by historians of mathematics that specifically investigate Cardano’s case.
My argument relies on the claim that Cardano possessed concepts of complex numbers as explicated a few decades later by Bombelli. Thus I rely on the claim that the correct explication of Cardano’s concepts was available only after Cardano discovered these concepts. I do not assume that Cardano was thinking thoughts with (all) the concept(s) or explications of complex numbers available in contemporary mathematics—e.g. the explication of complex numbers as points on a Riemann surface.
My argument is similarly unaffected by debates about revolutions in mathematics (Gillies 1995). Historians have both claimed (Dauben 1984) and denied (Crowe 1975) that mathematics undergoes proper revolutions or paradigm-changes. This debate is ongoing. Much of the debate concerns the proper notion of a revolution. According to one view, a revolution in mathematics would require abandoning (all of the relevant) earlier mathematics. According to another view, revolutions occur when earlier theories are not abandoned or overthrown, but their significance within mathematics is strongly altered. Neither of the views denies that there is a strong continuity between mathematical theories.
My argument would be affected by this debate if it turned out that there is no continuity in mathematics between Cardano and Bombelli. The argument would be affected if it could be argued that a discontinuity between the two mathematicians altered their concepts. Historians of mathematics specializing on this episode in the history of mathematics, however, instead support the claim that there is a continuity in Cardano and Bombelli’s concepts.
Recent theorizing about conceptual change in the history of science more generally leaves my argument equally unaffected. Thus (Friedman 2002, p. 185ff.) argues that some developments in the history of science lead to changes in theoretical frameworks or paradigms. Theories of conceptual change typically do not claim that a shift in framwork leads to a change in all concepts. Rather, they lead to a change in an important subset of concepts.
I believe that work on externalist theories of meaning by e.g. Kripke (1980), Putnam (1970, 1975, 1973), Burge (1993, 2012) has shown that even during deep changes in scientific theorizing, many concepts typically do not change.
However, all the paper claims is an identity of concepts between Cardano and Bombelli. There is no reason to assume that the truth of Friedman’s general account would affect this claim as against historians of mathematics’ verdict.
Thanks to an anonymous reviewer for prompting these clarifications.
This strategy has been proposed to me by an anonymous reviewer. Guy Longworth independently suggested a similar reinterpretation strategy. I am indebted to both.
I owe this objection to Sheldon Smith. Cf. also (Wilson 2007). I reject a view of mathematics as the mere manipulation of symbols. According to a version of this view, no mathematical thoughts have contents. It is beyond the scope of this paper to address the view at length. There might be independent grounds for thinking that Cardano merely manipulates symbols. This is the challenge I discuss in the main text.
Peacocke does to my knowledge not explicitly state this requirement on mathematical concepts. But both his examples and constraints on possession conditions seem to imply the requirement.
I think the individual should also understand that the individual complex numbers are numbers, hence are of the same ontological type as, say, the integers. Since Bombelli, too, had difficulties accepting complex numbers as numbers, I here only discuss the slightly weaker requirement that the explication requires understanding complex numbers as elements of certain mathematical structures.
For criticism of the notion, cf. (Burge 2012, p. 578).
There are plausibly other instances of this type of case. For example, Saccheri’s development of non-Euclidean geometries—intended as reductio ad absurdum proofs of Euclid’s fifth postulate. (Burton 1995, Chap. 11).
E.g. as a field, as pairs of real numbers, as vectors, matrices, quaternions. (Burton 1995). Another important definition of complex numbers is Euler’s: \(z=r\times e^{i\times \theta }\).
This explication is often called the polar representation of complex numbers. I propose the above term because it highlights the great difference between the approaches leading to the geometrical and the arithmetical explication.
For a full explanation of Anti-Individualism, cf. (Burge 2010, Chap. 3).
“Questions of ’access’ to [the realm of mathematical entities] are on reflection seen to be misconceived.” (Burge 1992, p. 316).
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Acknowledgments
I am indebted to Tyler Burge and Sheldon Smith for extensive help with this paper. Comments by Susan Carey, Guy Longworth, Tony Martin, and two anonymous reviewers have led to improvements. I have benefited from discussions with audiences in Durham and Warwick.
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Buehler, D. Incomplete understanding of complex numbers Girolamo Cardano: a case study in the acquisition of mathematical concepts. Synthese 191, 4231–4252 (2014). https://doi.org/10.1007/s11229-014-0527-x
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DOI: https://doi.org/10.1007/s11229-014-0527-x