Abstract
Representations, in particular diagrammatic representations, allegedly contribute to new insights in mathematics. Here I explore the phenomenon of a “free ride” and to what extent it occurs in mathematics. A free ride, according to Shimojima (Artif Intell Rev 15: 5–27, 2001), is the property of some representations that whenever certain pieces of information have been represented then a new piece of consequential information can be read off for free. I will take Shimojima’s (informal) framework as a tool to analyse the occurrence and properties of them. I consider a number of different examples from mathematical practice that illustrate a variety of uses of free rides in mathematics. Analysing these examples I find that mathematical free rides are sometimes based on syntactic and semantic properties of diagrams.
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Notes
The quote is from (Ishiguro 1972, p. 35).
When I mention Shimojima’s examples, I refer to the examples given in the before-mentioned contributions (Shimojima 1996, 2001). In later articles Shimojima elaborates further on his framework, analysing, for example, so-called ‘observational advantages’ of Euler diagrams, see Staplelton et al. (2017). In the book, (Shimojima 2015) the notion of a free ride is further qualified. It is one of four potential cognitive advantages of diagrams which are based on semantic constraints on the representations.
Note that in one sense this study is more restricted than Shimojima’s since I consider only examples from one field: mathematics. In another sense, it is wider since I will allow for free rides that might not be classified as such in Shimojima’s framework.
Adding the term ‘sentential’ makes it possible, if needed, to distinguish between purely linguistic representations and sentential representations. The following is a linguistic presentation of a circle: a circle is a geometric, plane figure consisting of a closed line defined by the fact that all its points have the same distance to a given point whereas the expression \(B_r=\{(x,y)| (x-c_1)^2 +(y-c_2)^2=r^2\}\) is sentential.
I do not exclude linear diagrams, for example, a linear arrow diagram such as a short exact sequence or a single line segment in geometry. Since they are composed of ‘diagrammatic’ parts such as arrows or straight lines, I will classify them as diagrams.
The permutation, \(\pi \), appears in the following expression (\(B_i\) denotes \(m\times n\) random matrices, i.e., matrices with random variables as entries, \(Tr_n\) denotes trace, and \({\mathbb {E}}\) is expectation): \({\mathbb {E}}\circ Tr_n[B_1^*B_{\pi (1)}\cdot \ldots B^*_pB_{\pi (p)}]=m^{e({\hat{\pi }})}\cdot n^{o({\hat{\pi }})}\). Drawing diagrams as in Fig. 3 and 4 together with other diagrams representing equivalence classes of the set \(\{1, 2, \ldots p\}\) made it clear that the numbers \(e({\hat{\pi }})\) and \(o({\hat{\pi }})\) depend on whether the permutation is crossing or not. See (Carter 2010).
The proof of this proposition in the published article is not a diagrammatic proof, but a reformulation of it.
The condition \(\frac{w(z)- w(z')}{z-z'}=\frac{w(z)-w(z'')}{z-z''}\) expressed in polar coordinates becomes \( \frac{R'\cdot e^{i \psi '}}{r'\cdot e^{i\phi '}}=\frac{R''\cdot e^{i \psi ''}}{r''\cdot e^{i\phi ''}} \). Manipulating terms of this identity leads to \((\psi ''-\psi ') = (\phi ''-\phi ')\).
The fact that information from different representations is combined is a remarkable and fruitful practice often used in mathematics. In this case, for example, properties of complex functions (and so complex numbers) and figures from plane geometry are combined. See Carter (2012) for a further elaboration of this point.
A co-exact property of a representation is a property that will not be affected by continuous deformations of the representation. An often used example of such a property is the crossing of the circles in Euclid’s proposition I.1.
The construction can be used to form, for example, the rational numbers from the integers by adding multiplicative inverses to all numbers except zero.
Rule 3 on the morphisms of S ensures that composed morphisms can be turned into the same form.
I refer to topological properties in this example, since the representation exploits that certain locations are chosen and that the arrows between them introduce a kind of orientation.
See Goodman (1969) for further requirements posed on notational systems.
A \(C^*\)-algebra is an algebra over the complex numbers on which is defined a norm, \(||\cdot ||\), and a \(^*\)-operation that fulfil the following: For a, b elements of the algebra, i) \(a^{**}=a\), ii) \((ab)^*=b^*a^*\), iii) \(||a^*a||= ||a||^2\), iv) \(||a^*||=||a||\). The definitions of both \(K_0\) and \(K_1\) involve several advanced concepts such as unital and projective elements of the algebra, split exact sequences and forming equivalence classes—too long to formulate in a footnote.
Vertices and edges form so-called generators. The range and source of edges are used to formulate relations on these generators so that they form a group. A construction referred to as ‘the analytic completion’ of this group yields a particular \(C^*\)-algebra.
\({\mathbb {Z}}V\) is the free abelian group generated by V. Elements of this group have the form \(x=\sum _{i=1}^{\infty } z_i\cdot v_i\), where \(v_i\in V\) and \(z_i\in {\mathbb {Z}}\), only a finite number is non-zero.
A linear map T fulfils that \(T(ax+by)=aT(x) +bT(y)\) for a, b scalars.
Shimojima (2015) distinguishes between the information that can directly be inferred from the given semantic convention and ‘meaning derivation’. The reading of a linear map from a directed graph presumably belongs to the second category. Note also that Shimojima (2015) no longer refers to nomic constraints as underlying free rides and over specificity. Shimojima instead refers to the more general notion of ‘spatial constraints’ of the diagrammatic representation.
In the semiotics of Peirce, one might say these representations are iconic metaphors, or icons depending on symbolic elements (see 3.363 in the collected works of Peirce).
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Earlier versions of this paper were presented on a number of occasions. I am grateful for the input that I have received in each case. I thank in particular Emiliano Ippoliti for inviting me to participate in the workshop ‘Explanatory and Heuristic Power of Mathematics’ held at the Sapienza University of Rome, June 2019, and the participants for their insightful comments. I also thank the two anonymous referees provided by Synthese for their constructive criticism of the earlier version of the article.
This article belongs to the topical collection "Explanatory and Heuristic Power of Mathematics", edited by Sorin Bangu, Emiliano Ippoliti, and Marianna Antonutti.
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Carter, J. “Free rides” in Mathematics. Synthese 199, 10475–10498 (2021). https://doi.org/10.1007/s11229-021-03255-9
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DOI: https://doi.org/10.1007/s11229-021-03255-9