Abstract
This paper pursues two goals. Its first goal is to clear up the “identity problem” faced by the structuralist interpretation of mathematics. Its second goal, through the consideration of examples coming in particular from the theory of permutations, is to examine cases of misunderstandings in mathematics fit to cast some light on mathematical understanding in general. The common thread shared by these two goals is the notion of setting. The study of a mathematical object almost always goes together with the choice of a particular setting, and the understanding of the workings of mathematical settings is an essential component of mathematical knowledge. It is claimed that the recognition of mathematical settings, as features distinct from both mathematical structures and the systems which instantiate those structures, allows one to classify most of understandable misunderstandings in mathematics, and also to solve the identity problem.
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Notes
Although other structuralist philosophies of mathematics could be mentioned, the identity problem pertains foremost to Shapiro’s positions and was discussed in the recent past mainly by Shapiro, which is why Shapiro’s structuralism only will be examined in this paper.
Shapiro (1997, pp. 73–74).
Shapiro (1997, pp. 82–83 and p. 89).
The signature of a structure is the signature of the formal language used to lay down the axioms specifying the interrelationships which define that structure.
Shapiro (2006b, p. 115).
This conception thus entails that a mathematical object has no independent existence. For a discussion of the difficulties raised by this “dependence claim,” see Linnebo (2008).
Keränen (2001).
See Shapiro (1997, pp. 111–113 and pp. 130–132).
Leitgeb and Ladyman (2008, p. 393). Privileged examples, according to Leitgeb and Ladyman, are unlabelled graphs with two distinct but isomorphic connected components, such as:
Shapiro (2006b, pp. 134–138).
Quite generally, the collection of all the objects of a certain system S instantiating a certain structure S can always be well-ordered. The system \(S'\) obtained from S when S is endowed with such a well-ordering \(\prec \), is still an instance of S, but the language enriched with an extra relation symbol for \(\prec \) is now sufficient to distinguish any two objects of \(S'\).
This point is made by Bermúdez (2007, pp. 113–114).
Keränen (2006, pp. 147–148).
Shapiro (2006a, p. 171).
Keränen (2006, p. 149).
Shapiro (2008a, p. 295).
Roberts (2003, p. 321) (the example is due to Irene Heim).
Shapiro (2012, p. 24).
Roberts (2003, pp. 306–307).
Roberts (2003, p. 289).
The rank of an item in a sequence (or in a row) simply is its ordinal position in that sequence (or in that row), when the latter is scanned from left to right.
The distinction, overlooked by Dummy, between numerals as numbers and numerals as floating ranks, is somewhat reminiscent of the distinction that Peirce establishes between symbols and indices, among the three classes of signs that he mentions (icons, indices and symbols). If a numeral as a number corresponds to a symbol (in Peirce’s sense), a floating rank indicates a contextual position, by virtue of the place of its first occurrence in the first row, thus “by virtue of a real connection” with the object that it stands for, and on that account can be compared to an index, as Peirce says a yardstick is, in Peirce (1998, p. 14). On that score, Dummy’s confused interpretation could be described in Peircean terms as the confusion of a symbol with an index.
A note about the choice of the terms “item” and “situation” is in order. Both have been chosen for their neutrality. A mathematical item is not an “element,” since the things that labels label are not necessarily considered as members of some set. Besides, it is not an “object” either (in the sense of the place in a structure), because it cannot be identified independently of its label and to that extent differs from an object whose individuality comes from its place in the structure, independently of any labelling. On the other hand, a mathematical situation simply is the target of some mathematical investigation: The term is used informally, as John Baez uses it in the quote. A mathematical situation is not the same thing as a mathematical structure, because a mathematical investigation may concern several structures without the latter being considered as making up a single structure or a single structure of structures.
Mathematical objects are sometimes renamed, in particular for abbreviatory purposes, but re-labellings are very seldom in mathematics and would consist anyway in introducing new labels, not in exchanging existing labels.
However, a label may sometimes be like a proper name (the main case being that of the numerals ‘1’, ‘2’, ...).
Whitehead (1911, p. 117 and p. 228).
See Magnus et al. (1976, pp. 4–8).
Coxeter (1989, p. 270).
Baez (2009).
Knobloch (2001, p. 145).
A classical result of the theory of determinants is, for instance, that the exchange of two rows or two columns of a matrix leaves the determinant of the matrix unchanged. On all this, see Knobloch (2001, especially pp. 149–154).
See Dochtermann (2009, Example 6.7, pp. 505–506).
Shapiro (2008a, p. 304).
(Leitgeb and Ladyman 2008, pp. 390–391, in particular fn. 4).
The same move seems to be made by Michael Resnik, when he mentions a triangle ABC as a context where the points A, B and C are distinguished—and constitutively so. A natural question to be asked to Resnik is: What is the difference between the triangle ABC and the triangle BCA? Resnik suggests that the labelling of the distinguished positions of a pattern is really part of this pattern [see Resnik (1997, p. 211)].
Shapiro (2006a, pp. 168–169).
Shapiro points out that the ordering can be carried out, in some cases, by a numbering and that such a numbering can always be changed, for instance by adding n to each number. Yet he very meaningfully describes the renumbering \(x\mapsto x + n\) as an “automorphism” of \(\langle \mathbf{S}, <\rangle \).
Keränen (2006, p. 153).
Despite the parallel that may be suggested between Peircean symbols and indices on the one hand, numbers and ranks on the other, the trichotomy of icons, indices and symbols proposed by Peirce has nothing to do with the trichotomy of systems, settings and structures.
An important example is Whitehead’s Theorem, which says that any continuous map \(f : X\rightarrow Y\) between two CW complexes X and Y which induces isomorphisms between the homotopy groups of X and Y, is an homotopy equivalence. This theorem does not imply that X and Y are homotopy equivalent as soon as their respective homotopy groups are isomorphic: It is essential that the isomorphisms between the homotopy groups be, not abstract isomorphisms, but isomorphisms induced by f.
Shapiro (2008a, pp. 297–301) and Shapiro (2012, p. 4). Also, Richard Pettigrew calls “dedicated free variable” any singular term introduced through existential elimination or universal generalization [see Pettigrew (2008, pp. 316–317)]. Such a free variable is said to be contextually dedicated to the extent that the mathematical context determines the stipulation by which the variable is introduced as an (ultimately discharged) assumption for conditional proof.
Shapiro (2008a, p. 297).
Shapiro (2008a, p. 298).
It is not a structure coordinatized by its own automorphisms either, as described by Rizza about extensive domains [see Rizza (2010, in particular pp. 185–187)]. The problem with the latter option is that all the automorphisms of an extensive domain can be thus defined only if the ability to refer individually to each element of the domain is presupposed (see p. 187), which amounts to begging the discriminability question.
Given an L-structure M and a language \(\hbox {L}'\) whose signature extends that of L, an L’-expansion of M is an \(\hbox {L}'\)-structure \(M'\) which, viewed as an L-structure, coincides with M. A typical example of expansion of an L-structure M is the \(\hbox {L}'\)-structure \(M'\), with \(\hbox {L'} = \hbox {L} \cup \{c_a\}_{a\in |M|}\), obtained from M by interpreting each \(c_a\) by a. As one can see, the whole operation presupposes that all the elements of the domain |M| of M can be referred to in the metalanguage, and the expansion of M simply consists in expliciting these expressive resources in a new object-language \(\hbox {L}'\).
See for instance Lengnink and Schlimm (2010). It would be worthwhile to examine how the mathematics education literature discusses misunderstandings similar to those of concern in this paper, and in particular whether it brings up a notion similar to that of setting.
What is said here of concrete drawings does not concern geometric diagrams and, more generally, mathematical diagrams. Rather, the latter certainly provide important cases of settings, although studying the connections between mathematical settings and mathematical diagrams would deserve a study in its own right, which goes beyond the limits of this paper.
What about undefinable real numbers, or the least non-recursive ordinal \(\omega _1^{{\textsc {ck}}}\)? Do not some mathematical objects elude any notational system, and thus any mathematical setting? No, because the proof of existence of such objects is in fact a result about notational systems themselves. This kind of result refers to second-order mathematical settings: mathematical settings for settings taken as mathematical objects in their own right. This does not detract from the existence of mathematical settings for all (at least most) mathematical objects. And it stresses in a new way what has been said about torsors, namely that certain mathematical settings have elicited a mathematical treatment per se.
This point could be the starting point for a reconsideration of Benacerraf’s dilemma. One indication of the link between the identity problem and Benacerraf’s dilemma is the common stress put on singular terms by Benacerraf’s platonist and by Shapiro [see Benacerraf (1973, p. 664, p. 666, pp. 668–669), and Shapiro (2008a, p. 289, p. 297, p. 303, p. 307, respectively)]. Getting back to Benacerraf’s “Mathematical truth,” one can express the mistake of the epistemic horn as the wrong thesis that mathematical settings do not present anything, and the mistake of the semantic horn—the mistake of ante rem structuralism—as the wrong thesis that mathematical structures present “themselves.”
Of course, other descriptions of the nth cyclic group are sometimes needed, based on settings which are not presentations by generators and relations. But, precisely, the mathematical investigation of a given structure naturally involves shifting from one setting to another.
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Acknowledgements
I wish to thank Denis-Charles Cisinski, Étienne Fieux, Gerhard Heinzmann and Marco Panza, as well as anonymous referees, for very valuable suggestions and comments.
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Halimi, B. Settings and misunderstandings in mathematics. Synthese 196, 4623–4656 (2019). https://doi.org/10.1007/s11229-017-1671-x
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DOI: https://doi.org/10.1007/s11229-017-1671-x