Nature Of Mathematics Textbook Pdf

Abstract

How does mathematics apply to something non-mathematical? We distinguish between a general application problem and a special application problem. A critical examination of the answer that structural mapping accounts offer to the former problem leads us to identify a lacuna in these accounts: they have to presuppose that target systems are structured and yet leave this presupposition unexplained. We propose to fill this gap with an account that attributes structures to targets through structure generating descriptions. These descriptions are physical descriptions and so there is no such thing as a solely mathematical account of a target system.

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Notes

1. 1.

   For the purpose of this paper it is not necessary to take a stance on whether models or theories are the units of scientific representation, nor does it matter whether models should be identified with mathematical structures (for a discussion of these issues see our 2017a,b). Our point of departure here is the hopefully uncontroversial observation that whatever does the representing often involves mathematics.

2. 2.

   We note that our choice of example is inline with much of the recent discussion on the applicability of mathematics where examples from biology have occupied centre stage; cf. Baker's cicadas (2005), van Fraassen's deer counting (2008, Chap. 11), and Lyon and Colyvan's bees (2008).

3. 3.

   Generating a strictly speaking false claim does not imply that the application is completely unsuccessful. Falsity comes in degrees, and so does being an unsuccessful application. There is clearly a sense in which some applications of mathematics generate claims about systems that are only approximately true (e.g. Newton's use of calculus to describe planetary motions), and yet one would not want to dismiss them as a unsuccessful tout court. However, quantifying degrees of success (or lack thereof) is tantamount to giving degrees of approximate truth, which is a problem we cannot tackle in this paper. For a review see Oddie's (2016). We would like to thank an anonymous referee for drawing our attention to this issue.

4. 4.

   Pincock (2012, Chap. 7) provides numerous more realistic examples of failed applications.

5. 5.

   Explaining the application of mathematics by appeal to structures looks most natural in the context of structuralist philosophy of mathematics such as Shaprio's (1997) or Resnik's (1997). However, the mapping account does not presuppose this approach to mathematics and could be adopted by proponents of other accounts. We briefly return to this issue in Sect. 6.

6. 6.

   The standard notion of a structure is introduced, for instance, in Hodges' (1997) and Enderton's (1972/2001). We note, however, that in the context of logic a structure is sometimes also taken to include a language and an interpretation function, which are absent from the notion of a structure as used in the current context.

7. 7.

   For a discussion of these see, for instance, Suppes' (1960/1969), van Fraassen's (1980, 2008), Da Costa and French's (2003), French and Ladyman's (1999), Bueno and French's (2011), Bartels' (2006), Mundy's (1986), Pero and Suárez's (2016).

8. 8.

   For a discussion of these approaches see Portides' (2017).

9. 9.

   It's worth noting here that Bueno and Colyvan call their account an "inferential account" of the application of mathematics rather than a "mapping account". Whilst it's true that their account is much richer than the simple version of the mapping account defined above, it does rely crucially on mappings to and from target systems and mathematical structures, and in this sense is an advanced version of, rather than an alternative to, the mapping account.

10. 10.

    A related point is made by Weisberg (2013, Sect. 3.3) who points out that in cases where mathematical structures are used to represent a physical system, an assignment and intended scope are needed to specify which parts of the structure are mapped to which parts of the target system. Where the structure contains parts that don't correspond to any purported part of the target system (e.g. where irrational numbers satisfy the Lotka–Volterra equations) it should be made explicit that this is not supposed to have physical import.

11. 11.

    For a critical discussion (that pre-dates van Fraassen's (2008) development) see Brading and Landry's (2006).

12. 12.

    A similar position is found in Resnik's (1997, p. 204). Due to the fact that some mathematical structures are not instantiated by any physical system, Shapiro further distinguishes between whether or not structures as universals should be thought of as ante rem universals, in re universals (eliminative structuralism) or in modal terms (Shapiro 1997). These distinctions are immaterial to our question. But a word of warning about terminology is in order here. Shapiro often uses the term "exemplified" to refer to the relationship between a universal and a physical system. We prefer "instantiates" given that that "exemplification" is used in a slightly different way in the literature about representation following Goodman and Elgin, and has been incorporated into the literature on scientific representation in our (forthcoming).

13. 13.

    For a detailed discussion of the theorem see Ketland's (2004). However, the basic point is straightforward and can shown as follows. Let \(\langle U,R\rangle \) be a structure such that \(|U|=k\). Let \(T=\{x_1 ,\ldots ,x_k \}\) be a set consisting of the k objects from the target system. Since \(|T|=|U|=k\) there is a bijection \(f{:}\;T\rightarrow U\). Using this bijection, for each \(r_i \in R\), we can construct a set \(r^{\prime }_i \) consisting of n-tuples of objects from T as follows: \(\{\langle x_i ,\ldots ,x_j \rangle :\langle f(x_i ),\ldots ,f(x_j )\rangle \in r_i \}.\) Collecting these relations together gives a structure, \(\langle T,R^{\prime }\rangle \) which is isomorphic to \(\langle U,R\rangle \) by construction.

14. 14.

    For a discussion of the various responses see Ainsworth's (2009).

15. 15.

    What follows is a variation on an example discussed in Frigg's (2006, pp. 57–58).

16. 16.

    Weisberg (2013, Sect. 5.3.1) makes a similar point when he draws a distinction between phenomena and target systems. He claims that the former stand in a one-to-many relationship with the latter, where a model's "intended scope" specifies a (proper) subset (target system) of the total state of the system (phenomena). Having said that, our point here goes beyond his: even once a target system of interest is fixed, there still remains the question concerning its structure (cf. the example of the methane molecule which would seem to count as a target system on any reasonable reading of the term).

17. 17.

    See Bueno and Colyvan's (2011, p. 347) quoted above. As discussed, Bueno and Colyvan allow for revisions of the "assumed structure" downstream. But our concern is prior: where did the original structure come from in the first place?

18. 18.

    Abstraction can be explicated a number of different ways. One way that we find conducive is Cartwright's (1999, Chap. 3), but nothing in what follows depends on what account of abstraction one adopts.

19. 19.

    In our example of the dynamics of a population this allows for embedding mappings between discrete structures like \(S_{T}\) and continuous structures involving \({\mathbb {R}}\), as used, for example, in the Lotka–Volterra equations. These discrete-to-continuous mappings are not restricted to biology; see Maddy's (1995, pp. 254–255) for a nice discussion of Feynman's concerns that time could be discrete despite the fact continuous structures are used to represent it. We are grateful to Mark Colyvan for bringing this example to our attention.

20. 20.

    The classic source is Krantz et al's (1971). Brown (1999, Chap. 4) offers an introduction; Díez (1997a,b) provides an overview of the development of the theory.

21. 21.

    Those who worry that labeling rods with numbers is already an application of mathematics and that \({{\varvec{D}}}_\mathbf{l }\) therefore presupposes what it is supposed to provide can replace the names \({{\varvec{o}}}_{{\varvec{k}}}\) by ordinary proper names and name the rods "Jim", "Mina", "Emily", etc.

22. 22.

    Recall that Bueno and Colyvan describe their account as an "inferential" conception of the application of mathematics, rather than a mapping account, and they take it that their account has the resources to account for mathematics playing an explanatory role. They take it that it is the choice of mapping at the interpretation and immersion stages that allows explanations to be obtained (2011, p. 366).

23. 23.

    It's worth noting here that nothing we say in this paper has any bearing on other arguments for or against nominalism, for example appeals to the scientific practice of representing non-actual states of systems via phase spaces (Lyon and Colyvan 2008), or whether, once a representation theorem has been proven, everything that might be said about the mathematical structure can be rephrased in terms of the empirical structure (see Balaguer's 1998, p. 112 and Pincock's 2007, Sect. III for further discussion).

24. 24.

    Blumenthal attributes this quote to Hilbert in his biography included in Hilbert (1935).

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Acknowledgements

We would like to thank the participants in JN's 'Models and Representation' spring 2016 research seminar at the University of Notre Dame, Otávio Bueno, Mark Colyvan, Susana Lucero, and two anonymous referees for helpful discussions and/or comments on earlier drafts. Thanks also to the audiences in Nikosia and Rostock for helpful feedback. Special thanks goes to Demetris Portides for inviting us to be part of this project.

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Affiliations

1. History and Philosophy of Science Program, University of Notre Dame, Notre Dame, IN, USA

   James Nguyen

2. Centre for Philosophy of Natural and Social Science, London School of Economics and Political Science (LSE), London, UK

   James Nguyen & Roman Frigg

3. Department of Philosophy, Logic and Scientific Method, London School of Economics and Political Science (LSE), London, UK

   Roman Frigg

Corresponding author

Correspondence to Roman Frigg.

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Nguyen, J., Frigg, R. Mathematics is not the only language in the book of nature. Synthese (2017). https://doi.org/10.1007/s11229-017-1526-5

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• Published: 28 August 2017

• DOI : https://doi.org/10.1007/s11229-017-1526-5

Keywords

• Application of mathematics
• Structure
• Mapping account
• Representation
• Isomorphism
• Physical descriptions

Nature Of Mathematics Textbook Pdf

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