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Foundations of applied mathematics I

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Abstract

This paper aims to study the foundations of applied mathematics, using a formalized base theory for applied mathematics: \( \mathsf {ZFCA}_{\sigma }\) (Zermelo–Fraenkel set theory (with Choice) with atoms, where the subscript used refers to a signature specific to the application. Examples are given, illustrating the following five features of applied mathematics: comprehension principles, application conditionals, representation hypotheses, transfer principles and abstract equivalents.

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Notes

  1. Examples include: Frege (1884), Hilbert (1899), Hölder (1901), Russell (1903), Einstein (1921), Carnap (1928), Ramsey (1929), McKinsey et al. (1953), Tarski (1959), Suppes (1960), Putnam (1967), Putnam (1971), Krantz et al. (1971), Sneed (1971), Stegmüller (1976), Field (1980), Burgess (1984), Lewis (1991), Burgess and Rosen (1997), Shapiro (1997), Resnik (1997), Steiner (1998), Ketland (1998), Colyvan (2001), Leng (2010), Ketland (2011), Bueno and Colyvan (2011), Andréka et al. (2012), Pincock (2012), Mycielski (2013), Halvorson (2012), Halvorson (2016), Bueno and French (2018).

  2. For reasons of space, I have set aside discussion of a number of closely-related topics, which cannot be dealt with adequately here. These topics are: indispensability arguments (e.g., Colyvan 2001; Melia 2000; Leng 2010; Colyvan 2019); mathematical explanation in science (e.g., Baker 2005; Mancosu 2018); representation/modelling (e.g., Pincock 2012; Frigg 2012); measurement theory (e.g., Krantz et al. 1971; Roberts 1985; Narens 1985; Suppes 2002; Tal 2015); approximation and idealization (e.g., Batterman 2002; Pincock 2012) and recent so-called “mapping accounts” of applied mathematics (e.g., Pincock 2012; Bueno and Colyvan 2011; Bueno and French 2018). That said, some of these issues are, albeit briefly, touched on at points below. Representation and modelling are illustrated in Example , Sects. 6, , 6.5 and 8, Example  and Example 6. I comment sceptically on a difficulty for the “mapping accounts” in Sect. 7.3 connected to differential equations. An example of mathematical explanation (of quantized atomic orbitals) is mentioned in Sect. 7.4.

  3. There are other measurement devices, like compasses, of course, Hall probes, ...

  4. I took this diagram from Harary (1969, p. 2). I return to this problem several times below.

  5. A (simple, undirected) graph G is a pair (VE) where V is a set, the vertices of G, and E is a set of distinct pairs \(\{x,y\}\) drawn from V, the set of edges of G. An (undirected) multigraph is a kind of graph in which a pair of vertices may be connected by more than one edge. It may be represented as a triple (VEf), where V is the set of vertices, E is a (disjoint from V) set of edges, and f is a connection function, which assigns to each edge in E a pair \(\{x,y\}\) drawn from V.

  6. See Harary (1969: p. 64).

  7. If I understand their meaning right, \(P_K\) is the “assumed structure” that Bueno and Colyvan (2011) refer to and the “empirical set-up” that Bueno and French (2018) refer to.

  8. Though it could be: \(P_K\) is isomorphic to a certain abstract graph, \(G_K\): see Sect.  below.

  9. The approach here is fairly close to that described in Carnap’s works: Carnap (1956), Carnap (1958), Carnap (1966). See also, for example, Suppes (1960), Sneed (1971), Stegmüller (1976), Field (1980), Burgess (1984), Andréka et al. (2012).

  10. This appears in the axiomatizations of relativity given by Andréka et al, who call it the “worldview relation”. See especially: Andréka et al. (2006) and Andréka et al. (2012).

  11. The point of the qualification “basic” is that further impure and mixed objects can then be defined by standard mathematical constructions: pairs, products, etc.

  12. The first principle is routinely used in applications of arithmetic. See Sect. 8 for more on the second principle. I return to the third theorem in Sect. 7: Example 4 and Sects. 7.3 and 7.4.

  13. Zermelo (1908), Russell (1903), Russell (1908), Russell and Whitehead (1912). Quine (1956) discusses Zermelo’s set theory with atoms (individuals, as Quine calls them: Quine suggests individuals be defined as being “self-membered”). Church’s type theories were formulated with individuals at the base (Church 1940). More recent examples include Quine’s set theory with urelements, \(\textsf {NFU}\) (e.g., Jensen 1968; Forster 1995; Holmes 1998; Chihara 1990, pp. 148–149; Barwise 1975), which focuses on Kripke–Platek set theory with urelements, \(\textsf {KPU}\); Barwise and Moss (1996), Lewis (1991), Mendelson (2010, §4.6.5), Jech (2002, p. 250 ff.), Potter (2004) and Menzel (2014).

  14. An approach that seems to be similar is given in Mycielski (2013), where the base theory invoked is called \( \mathsf {ZF}^{+}\) and scientific theories are then treated as taking the form \( \mathsf {ZF}^{+} + \Sigma \), where the scientific laws, etc., are elements of \(\Sigma \).

  15. The approach developed here is influenced by Jon Barwise’s similar many-sorted formulation of \(\textsf {KPU}\) (Kripke-Platek set theory with urelements) in Barwise (1975). I am grateful to Ali Enayat for drawing my attention to Barwise’s approach.

  16. This would have been the preferred formalism of the logical empiricists, and especially Rudolf Carnap: e.g., Carnap (1928), Carnap (1966). See Leitgeb and Carus (2020) for a detailed survey of Carnap’s proposed formalization systems for science. Because of well-known intertranslatability results, these approaches may well be near notational equivalents, modulo some subtleties about proof-theoretic strength.

  17. See Manzano (1996) for an exposition of many-sorted logic and its applications. Enderton (2001) also explains the main ideas. In his Quine (1956), Quine gives some reasons to prefer a single-sorted formalization of theories, although it’s not clear how persuasive these reasons are given the intertranslatability of many-sorted and one-sorted theories.

  18. In principle, we might introduce a richer system, perhaps with subsorts \(\textsf {ord}\), \(\textsf {real}\), \(\textsf {natnum}\), etc., for sets, and with subsorts \(\textsf {point}\), \(\textsf {line}\), \(\textsf {region}\), etc., for atoms.

  19. For example, if \(\textsf {P}\) is a unary predicate on atoms, then \(\textsf {P}(a_i)\) is well-formed, for any atom variable \(a_i\). But \(\textsf {P}(x_i)\) is also well-formed, where \(x_i\) is a global variable.

  20. However, we shall show how to define them in the object language.

  21. I have not said what this constructor \(\Rightarrow \) “is”. However it doesn’t matter. I am using a type-theory notation here: “\(s_1 \Rightarrow s_2\)” means “the ‘type’ of functions from \(s_1\) to \(s_2\)”. Originally, Church (1940), who had two base sorts, o (booleans) and \(\iota \) (individuals), used a bracketing notation (oo), \((o \iota )\), o(oo), \((o \iota ) \iota \), etc. (and read backwards: so \((o \iota )\) is \(\iota \Rightarrow o\)). See Coquand (2018) on type theory and Benzmüller and Andrews (2019) for Church’s type theory, for further explanation.

  22. Thus, by “\(\tau (s) = \dots \)”, we mean “the sort of symbol s is ...”. The functional notation “\(\tau (-)\)” belongs to the meta-language, not the object language.

  23. Isabelle is a higher-order/type-theoretic theorem-proving system initially designed by Laurence Paulson. See Wenzel et al. (2020) for the user manual for Isabelle.

  24. Our structures interpret variables. So, if v is a variable, then \(v^M\) is an element of the domain.

  25. Definition 20 defines the well-formed terms and primitive formulas of \(L(\sigma _{\in })\). Definition 21 completes the definition of the language \(L(\sigma _{\in })\).

  26. In principle, one could weaken the theory by requiring that \(\phi \) may only belong to the sublanguage \(L(\sigma \upharpoonright _{ \mathsf {atom}})\)—the result would be a kind of predicative weakening.

  27. The most important modification is this. Standardly in many-sorted logic, the logical axiom \(\forall v \phi (v) \rightarrow \phi ^v_t\) requires the term t substituted for v to have same sort as that of v (in order-sorted MSL, the sort of t may be a subsort of that of v). In our system, we also have global variables, and in general, an additional logical theorem \(\forall x \phi (x) \rightarrow \phi ^x_t\), where the sort of t may be of any sort subject to \(\phi ^x_t\) being well-formed. This is handled by the additional quasi-logical subsort axioms specified in Definition 24.

  28. The translation was first given in Schmidt (1938), and later in more detail, Wang (1952).

  29. E.g., “X is the ordered pair of \(x_1\) and \(x_2\)”, “\(X_1\) is a binary relation on \(X_2\)”, “X is a real number”, etc.

  30. This form of separation, for relations, is equivalent to the original form: \(\exists X_1 \forall x_3 \big ( x_3 \in X_1 \leftrightarrow \exists x_1 \exists x_2( x_3 = (x_1, x_2) \in X_2 \wedge \phi (x_1, x_2)) \big )\) if we have Pairing.

  31. Analogously, Choice is equivalent to various other statements, using, say, \(\textsf {Z}\) set theory as a background base theory. Similarly, within the Reverse Mathematics programme of proving the equivalence of certain second-order set existence axioms with certain mathematical statements, using weaker subsystems as a base theory. For example, many interesting claims in analysis are equivalent to \(\textsf {ACA}_0\), with the weaker system \(\textsf {RCA}_0\) as base theory (see Simpson 2009 for further details).

  32. This term seems to originate in da Costa and Chuaqui (1988). The slogan often repeated by Patrick Suppes is: “To axiomatize a theory is to define a predicate in terms of notions of set theory. A predicate so defined is called a set-theoretical predicate” (Suppes 1957, Ch. 12 §12.2).

  33. One might wish to bring all the “models” together, as a set. In general, this is not possible, for class/set reasons. However, it is possible to form the set of models which all live inside a suitably large “container set”, called a universe. In Muller (2011), the container set or universe is a von Neumann universe \(V_{\omega + n}\), for some largish n. In fact, to ensure we get the right equivalence for set theory with atoms, it would need to be \(V_{\omega + n}(A)\), where A is the set of atoms.

  34. If \(\sigma \) is purely atomic, this is true for all finitely axiomatized theories in \(L(\sigma )\). In the broader case, it remains true for certain finitely axiomatized theories in \(L(\sigma _{\in })\), but one must be more careful about the set quantifiers appearing in the axioms of \(\Theta \): these quantifiers need to be bounded in some way, so that they range over a set. In such cases, Suppes Equivalence holds too.

  35. One can simplify a bit too much as well. Steven French has published a new book, French (2020), There Are No Such Things As Theories, claiming theories don’t exist.

  36. Such a view was standard amongst the logical empiricists, and that remained so to the 1960s. For example, Carnap (1928), Carnap (1966), Hempel (1952), Przełȩcki (1969) and others. It is sometimes called “the received view”, or “the syntactic view of theories”.

  37. It is sometimes called “the semantic view of theories” or “the model-theoretic view”. The beginnings of this view are usually traced to articles McKinsey et al. (1953), Suppes (1960), a couple more by Patrick Suppes et al, Suppes’s textbook Suppes (1957) (see Ch. 12, §12.2: “Set-Theoretical Predicates and the Axiomatization of Theories”), and then followed by important and influential developments in Sneed (1971), Suppe (1977) and van Fraassen (1980). Joseph Sneed’s work came to influence the “Munich school”: e.g., Stegmüller (1976), Balzer et al. (1987). A valuable and sympathetic summary of the history of the semantic or model-theoretic or structuralist view is Muller (2011).

  38. A discussion arguing that the alleged flaws are misrepresentations is Lutz (2012). Further criticisms of the semantic view have appeared in Halvorson (2012), Halvorson (2013) and Halvorson (2016), as well as Lutz (2014) and Lutz (2017).

  39. I might add that the first objection, the Friedman-Worrall objection, seems to me not quite relevant. Like any other logician, I present, for example, Peano arithmetic \( \mathsf {PA}\) as a theory in \(L(\sigma _{arith})\), where \(\sigma _{arith} = \{0,s,+,\times \}\). \( \mathsf {PA}\) has six axioms \( \mathsf {PA}1, \dots , \mathsf {PA}6\) and an induction scheme \(\text {Ind}(L(\sigma _{arith}))\). Obviously, I may consider \(Mod_{\sigma _{arith}}( \mathsf {PA})\). For example, Kaye (1991) is a book about these models of \( \mathsf {PA}\). However, this excludes the crucial additional point that \( \mathsf {PA}\) is true. That is, \( \mathsf {PA}\) is a formalized theory of number theory—an interpreted theory in an interpreted language, with a fixed, definite intended interpretation: \(( \mathbb {N}, 0, S, +, \times )\). In saying \( \mathsf {PA}\) is true, I mean that each axiom of \( \mathsf {PA}\) is true in \(( \mathbb {N}, 0, S, +, \times )\).

  40. I call this the Truth Bearer Objection. I’ve occasionally presented it in talks since the 1990s.

  41. For van Fraassen, the additional theoretical hypothesis of interest to him, for epistemological reasons, is not the claim of a theory’s being true, but rather a claim of a theory’s “empirical adequacy”. However, I do not wish to get sidetracked into epistemology.

  42. I treat the codomain axiom for \(\textsf {F}\) as implicit, mainly for ease of presentation.

  43. I am using certain kinds of “disquotation” and “compositionality”. See Halbach (2014) or Cieśliński (2017) for a detailed examination of such things.

  44. The reader might note the analogy with disquotation sentences for a truth predicate: \( {\mathsf {T}}( \ulcorner \phi \urcorner ) \leftrightarrow \phi \). This connection is not a coincidence! We could simply define \( {\mathsf {T}}(X)\) to mean: \(\textsf {Sat}(( \mathbb {A}, \overline{\textsf {P}}), X)\). And then: \( \mathsf {ZA}_{\sigma } \vdash {\mathsf {T}}( \ulcorner \phi \urcorner ) \leftrightarrow \phi \), for all \(\phi \in L(\sigma )\).

  45. I.e., \(f(5) = \{1,2\}\), etc. That is, edge labelled 5 connects vertices labelled 1 and 2. E etc.

  46. These axioms (if we treat the free variables \(a_1, \dots , a_{11}\) as constants) are true of \(P_K\). Moreover, they provide a categorical description of the structured system \(P_K\) we are interested in. I call the first four axioms “structural axioms”: any (undirected) multigraph satisfies these. Axiom \((\theta _5)\) is the vertex axiom: the vertices—i.e., land-areas—are \(a_1, \dots , a_4\). Axioms \((\theta _6)\) is the edge axioms: the edges—i.e., bridges—are \(a_5, \dots , a_{11}\). Axiom \((\theta _7)\) states that these are all distinct. Axiom \((\theta _8)\) is the “connection axiom” stating how the bridges and land-areas are connected in this specific multigraph.

  47. This leaves open the possibility that \(\Theta \) may be categorical, but not provably so. Sentences of the form “\(\Theta \) is categorical” (including when \(\Theta \) is “second-order”: i.e., has set quantifiers) are hardly elementary, or even arithmetic sentences, so this must be a real possibility.

  48. \(B_{ \mathbb {R}^d}\) and \(C_{ \mathbb {R}^d}\) are betweenness and congruence relations defined on \( \mathbb {R}^d\). See Tarski (1959, p. 21), for their definitions (as applied to an arbitrary real-closed field F. Here that field is \( \mathbb {R}\)). I have changed notation a little bit.

  49. The subscript on the (model-theoretic) satisfaction predicate \(\textsf {Sat}_2\) is there to indicate that \(\textsf {Sat}_2(M, \ldots )\) means “M is a full model of ...”.

  50. One can repeat this trick for any finitely axiomatized categorical second-order theory whose categoricity is provable in set theory. For example, \(\textsf {PA}_2\) and \(\textsf {RCF}_2\).

  51. A claim which ultimately has empirical content. The surface area A of any sphere with radius R metres around some point p (imagine a gigantic spherical “balloon”, millions of kilometres in radius, centred on the Sun) should satisfy \(A = 4 \pi R^2\) (in \(\text {metres}^2\)). If \(A < 4\pi R^2\) or \(A > 4\pi R^2\), then the (spatial) geometry is non-Euclidean.

  52. This was a topic of enormous interest to, and huge stimulus to, the newly forming group of logical empiricists, Moritz Schlick, Rudolf Carnap and Hans Reichenbach. See Schlick (1918), Carnap (1922), Reichenbach (1924).

  53. The function F might be a temperature function, or an electric field, or a wavefunction, or ...; the set X might be the set of impacts between a billiard ball and the cushion, or the set of voters in a region, or ....

  54. Here, we have some otherwise unspecified function \(F : \mathbb {R}\rightarrow \mathbb {R}\), and \(k \in \mathbb {R}\) is some fixed parameter.

  55. The dots “\(\dots \)” in the antecedent stand for a a couple of implicit structural conditions “F is a function on the reals (\(F : \mathbb {R}\rightarrow \mathbb {R}\)) and F is differentiable”.

  56. Here I simplify a fair amount and use some standard physics notation. The “|x|” means the length of the segment from the point x to some implicit origin O, centred in the proton. Normally, this equation is written in terms of the co-ordinate representation of \(\Psi \) in polar co-ordinates—say \(\psi (r, \theta , \phi )\)—with \(r=0\) being the proton’s location, and then |x| is replaced by r, but I wish to stress that \(\Psi \) really is a (mixed) function on space, even though its co-ordinate representation is a (pure) function \( \mathbb {R}^3 \rightarrow \mathbb {C}\).

  57. See Pincock (2007), Pincock (2012), Bueno and Colyvan (2011) and Bueno and French (2018). The Königsberg Problem is raised and discussed in some detail in both Pincock (2007) and Pincock (2012).

  58. In both cases, I also defined the “assumed structure”. For geometry, the assumed structure is \(( \mathbb {A}, \overline{\textsf {B}}, \overline{\textsf {C}})\). For the Königsberg Problem, the assumed structure is \(P_K\), the Königsberg multigraph. I should mention also that the Königberg problem can be solved without invoking any mapping or isomorphism. One may simply reason directly about \(P_K\) and infer from its properties that each vertex has odd degree, and thus that it lacks a Euler cycle, without a detour involving an abstract copy (i.e., \(G_K\)) of \(P_K\). In fact, the problem can be solved without mathematics at all, but that brings in considerations of conservativeness.

  59. A “structural property” of \(A = (D, R_1, \dots )\) is one that doesn’t depend on aspects of the intrinsic nature of the elements of \(\text {dom}(A)\), but only on the properties and relations \(R_i\) explicitly specified in the structure A. See Korbmacher and Schiemer (2018) for a discussion of such properties and an analysis of them in terms of definability in infinitary languages.

  60. Here the representation is an isomorphism. But it is clear that sometimes weaker morphisms, homomorphisms and embeddings, are in play in applied mathematics.

  61. In this case, structures A and B are isomorphic. What if there is merely a homomorphism \(f :A \rightarrow B\)? Or an embedding, say? Then the class of transferable properties is smaller. The pattern for groups is analysed in Lyndon (1959). For example if \(f: G_1 \rightarrow G_2\) is a surjective homomorphism of groups, then \(G_2\) is Abelian if \(G_1\) is; and \(G_2\) is cyclic if \(G_1\) is; etc.

  62. Although \(\phi _K\) was true, it is, apparently, no longer true; some of the bridges of Königsberg at the time of Euler have been knocked down.

  63. We know, course, from Einstein, that T is not true. The flat spatial geometry \(\textsf {Geom}^{d=3}_2(\textsf {B},\textsf {C})\) is in some sense “approximately true” for smallish regions of space around a point.

  64. One may, of course, take this empirical observation and read it backwards: as empirical evidence for \(d=3\) in our underlying geometric system \(\textsf {Geom}^{d}_2(\textsf {B},\textsf {C})\). This is precisely what one hears in a physics lecture. “Why is \(d=3\)? How do we know? Why isn’t \(d=4\) or \(d=25\)?”, followed by the three finger explanation. Of course, this argument may break down if there are teeny extra dimensions, too small for us to notice them as a new direction into which to “point”. The idea that (spatial) \(d=4\) in fact was introduced by Kaluza (1921) and extended by Klein (1926), suggesting the extra dimension is curled up in a tiny circle with subatomic radius. In such “Kaluza–Klein theories”, by imposing Einstein’s equations on the \(4+1\)-metric, one obtains Maxwell’s equations as a consequence! The idea that \(d=25\) arises in the quantum theory of (bosonic) strings. The quantum theory of a bosonic string in N dimensions only works if \(N = 25 + 1\).

  65. One could extend the sort system S, if one likes, to make it more “user-friendly”. E.g., in geometry, sorts \(\textsf {point}\) and \(\textsf {region}\) treated as atoms; in pure mathematics, sorts for \( \mathbb {N}\), \( \mathbb {Z}\), \( \mathbb {Q}\), \( \mathbb {R}\), and \( \mathbb {C}\), treated as atoms. Or one might include sorts for special kinds of sets, relations, functions, or structures.

  66. I am grateful to two referees who provided very helpful comments and suggestions. I am grateful to Ali Enayat and Christopher Menzel for suggestions. This work was supported by a grant from the National Science Centre (NCN) in Kraków (2018/29/B/HS1/01832).

  67. Usually what I shall call primitive terms and formulas are called atomic terms and atomic formulas. However, “atomic” has a different meaning in this paper.

References

  • Andréka, H., Madarász, J. X., & Németi, I. (2006). Logical axiomatizations of space-time. Samples from the literature. In A. Prékopa, & E. Molnár (Eds.), Non-Euclidean geometries: János Bolyai memorial Volume (Volume 581 of Mathematics and Its Applications) (pp. 155–185). Berlin: Springer.

  • Andréka, H., Madarász, J. X., Németi, I., & Székely, G. (2012). A logic road from special relativity to general relativity. Synthese, 186(3, Logic Meets Physics), 633–649.

  • Baker, A. (2005). Are there genuine mathematical explanations of physical phenomena? Mind, 114, 223–38.

    Article  Google Scholar 

  • Balzer, W., Moulines, C. U., & Sneed, J. D. (1987). An architectonic for science: The structuralist approach. Dordrecht: Reidel.

    Book  Google Scholar 

  • Barwise, J. (1975) Admissible sets and structures: An approach to definability theory (perspectives in mathematical logic (Vol. 7) Berlin: Springer.

  • Barwise, J., & Moss, L. (1996). Vicious circles: On the mathematics of non-Wellfounded phenomena (CSLI Lecture Notes: Number 60). Stanford: CSLI.

    Google Scholar 

  • Batterman, R. (2002). The devil in the details: Asymptotic reasoning in explanation, reduction, and emergence. New York: Oxford University Press.

    Google Scholar 

  • Bays, T. (2005). Review: Set theory and its philosophy: A critical introduction (Michael Potter). Notre Dame Philosophical Reviews. https://ndpr.nd.edu/news/set-theory-and-its-philosophy-a-critical-introduction/.

  • Benzmüller, C. & Andrews, P. (2019). Church’s type theory. The Stanford Encyclopedia of Philosophy. https://plato.stanford.edu/entries/type-theory-church/.

  • Bueno, O., & Colyvan, M. (2011). An inferential conception of the application of mathematics. Noûs, 45(2), 345–374.

    Article  Google Scholar 

  • Bueno, O., & French, S. (2018). Applying mathematics: Immersion, inference, interpretation. Oxford: Oxford University Press.

    Book  Google Scholar 

  • Burgess, J. (1984). Synthetic mechanics. Journal of Philosophical Logic, 13, 319–95.

    Article  Google Scholar 

  • Burgess, J., & Rosen, G. (1997). A subject with no object. Oxford: Clarendon Press.

    Google Scholar 

  • Carnap, R. (1922) . Der Raum: Ein Beitrag zur Wissenschaftslehre. Berlin: Reuther & Reichard. English translation Carnap 2019, Rudolf Carnap: Early writings A.W. Carus, M. Friedman, W. Kienzler, A. Richardson, and S. Schlotter (Eds.) in The collected works of Rudolf Carnap, 1 (pp. 21–208). New York: Oxford University Press.

  • Carnap, R. (1928) . Der Logische Aufbau der Welt. Leipzig: Felix Meiner Verlag. English translation by Rolf A. George, 1967. The Logical Structure of the World. Pseudoproblems in Philosophy. University of California Press.

  • Carnap, R. (1956). The methodological character of theoretical concepts. In H. Feigl & M. Scriven (Eds.), Minnesota studies in the philosophy of science i (pp. 38–76). Minneapolis: University of Minnesota Press.

    Google Scholar 

  • Carnap, R. (1958). Beobachtungssprache und theoretische Sprache. Dialectica, 12(3–4), 236–248. Translation: “Observational Language and Theoretical Language” in J. Hintikka (ed.) 1975, Rudolf Carnap. Logical Empiricist, Dordrecht: Reidel (pp. 75–85).

  • Carnap, R. (1966). The philosophical foundations of physics. New York: Basic Books. Edited by Martin Gardner. Reprinted as An Introduction to the Philosophy of Science, New York: Dover Books, 1995.

  • Chihara, C. (1990). Constructibility and mathematical existence. Oxford: Oxford University Press.

    Google Scholar 

  • Church, A. (1940). A formulation of the simple theory of types. Journal of Symbolic Logic, 5, 56–68.

    Article  Google Scholar 

  • Cieśliński, C. (2017). The epistemic lighteness of truth: Deflationism and its logic. Cambridge: Cambridge University Press.

    Book  Google Scholar 

  • Colyvan, M. (2001). The indispensability of mathematics. Oxford: Oxford University Press.

    Book  Google Scholar 

  • Colyvan, M. (2019). Indispensability arguments in the philosophy of mathematics. Encyclopedia of Mathematics. First published Mon Dec 21, 1998; substantive revision Thu Feb 28, 2019. https://plato.stanford.edu/entries/mathphil-indis/.

  • Coquand, T. (2018) . Type theory. Stanford Encyclopedia of Philosophy. https://plato.stanford.edu/entries/type-theory/.

  • da Costa, N., & Chuaqui, R. (1988). On Suppes’ set-theoretical predicates. Erkenntnis, 29, 95–112.

    Article  Google Scholar 

  • Einstein, A. (1921). Geometry and experience. Address to The Prussian Academy of Sciences (27 January 1921). Translated by G. B. Jeffrey & W. Perret in A. Einstein, 1922: Sidelights on Relativity. London: Methuen.

  • Enderton, H. (2001). A mathematical introduction to logic, revised second edition. London: Harcourt/Academic Press.

    Google Scholar 

  • Euler, L. (1736). Solutio problematis ad geometriam situs pertinentis. Comment Acad Sci U Petrop, 8, 128–140.

  • Feynman, R. (1992). 1965: The character of the physical law (new edition). London: Penguin.

    Google Scholar 

  • Field, H. (2016). 1980: Science without numbers (2nd ed.). Princeton: Princeton University Press/Oxford University Press.

    Book  Google Scholar 

  • Forster, T. (1995). Set theory with a universal set (Oxford logic guides 31. 2nd ed). Oxford: Oxford University Press.

  • Frege, G. (1884) . Die Grundlagen der Arithmetic: Eine logisch-mathematische Untersuchung über den Begriff der Zahl. Breslau: Verlage Wilhelm Koebner. Translation by J.L. Austin as The Foundations of Arithmetic. Oxford: Blackwell, 1950.

  • French, S. (2020). There are no such things as theories. Oxford: Oxford University Press.

    Book  Google Scholar 

  • Friedman, M. (1982). Review: The scientific image (by Bas C. Van Fraassen). The Journal of Philosophy, 79(5), 274–283.

  • Frigg, R. (2012). Models in science. Stanford Encyclopedia of Philosophy. First published Mon Feb 27, 2006; substantive revision Mon Jun 25, 2012. https://plato.stanford.edu/entries/models-science/.

  • Halbach, V. (2014) . Axiomatic theories of truth. Cambridge: Cambridge University Press. First edition 2011. Revised edition 2014.

  • Halvorson, H. (2012). What scientific theories could not be. Philosophy of Science, 79(2), 183–206.

    Article  Google Scholar 

  • Halvorson, H. (2013). The semantic view, if plausible, is syntactic. Philosophy of Science, 80(3), 475–478.

    Article  Google Scholar 

  • Halvorson, H. (2016). Scientific theories. In P. Humphreys (Ed.), Oxford handbook of the philosophy of science. Oxford: Oxford University Press.

    Google Scholar 

  • Harary, F. (1969). Graph theory. Reading, MA: Addison-Wesley.

    Book  Google Scholar 

  • Hempel, C. (Ed.). (1952). Fundamentals of concept formation in empirical science. Chicago: University of Chicago Press.

    Google Scholar 

  • Hilbert, D. (1899) . Grundlagen der Geometrie. Leibzig: Verlag Von B.G. Teubner. Translated (by E.J.Townsend) as The foundations of geometry. Chicago: Open Court.

  • Hölder, O. (1901). Die Axiome der Quantität und die Lehre vom Mass”. Berichten der mathematisch-physischen Classe der Königlisch Sächsischen Gesellschaft der Wissenschaften zu Leipzig, Mathematisch-Physikaliche Classe, 53, 1–63. Translations of this work by Joel Michell and Catherine Ernst appear in O. Hölder, 1996: “The Axioms of Quantity and the Theory of Measurement”. Journal of Mathematical Psychology 40(3): 235–252 and O. Hölder, 1997: “The Axioms of Quantity and the Theory of Measurement”. Journal of Mathematical Psychology 41(4): 345–356.

  • Holmes, M. (1998) . Elementary set theory with a universal set. Louvain-la-Neuve: Université Catholique de Louvain, Département de Philosophie: Cahiers du Centre de Logique (Vol. 10).

  • Jech, T. (2002). Set theory: The third millennium edition. New York: Springer. 1st edition published in 1978 and the second edition in 1997. This is the 3rd revised and enlarged edition.

  • Jensen, R. (1968). On the consistency of a slight (?) modification of Quine’s ‘new foundations’. Synthese, 19, 250–263.

    Article  Google Scholar 

  • Kaluza, T. (1921). Zum Unitätsproblem in der Physik. Sitzungsber Preuss Akad Wiss Berlin (Math Phys) (pp. 966–972).

  • Kaye, R. (1991). Models of peano arithmetic (Oxford logic guides 15). Oxford: Oxford University Press..

  • Ketland, J. (1998). The mathematicization of nature. Ph.D Thesis: London School of Economics.

  • Ketland, J. (2011). Nominalistic adequacy. Proceedings of the Aristotelian Society, 111, 201–217.

    Article  Google Scholar 

  • Klein, O. (1926). Quantentheorie und fünfdimensionale Relativitätstheorie. Zeitschrift für Physik A, 27(12), 895–906.

    Article  Google Scholar 

  • Korbmacher, J., & Schiemer, G. (2018). What are structural properties? Philosophia Mathematica, 26(3), 295–323.

    Article  Google Scholar 

  • Krantz, D., Luce, R., Suppes, P., & Tversky, A. (1971). Foundations of measurement (Vol. 1). London: Academic Press.

  • Leitgeb, H. & Carus, A. (2020). Rudolf Carnap. Stanford Encyclopedia of Philosophy. First published Mon Feb 24, 2020. https://plato.stanford.edu/entries/carnap/.

  • Leng, M. (2010). Mathematics and reality. Oxford: Oxford University Press.

    Book  Google Scholar 

  • Lewis, D. (1991). Parts of classes. Oxford: Oxford University Press.

    Google Scholar 

  • Lutz, S. (2012). On a straw man in the philosophy of science—A defense of the received view. Hopos: The Journal of the International Society for the History of Philosophy of Science, 2(1), 77–120.

  • Lutz, S. (2014). What’s right with a syntactic approach to theories and models? Erkenntnis, 79(8), 1475–1492.

    Article  Google Scholar 

  • Lutz, S. (2017). What was the syntax-semantics debate in the philosophy of science sbout? Philosophy and Phenomenological Research, 95(2), 319–352.

    Article  Google Scholar 

  • Lyndon, R. (1959). Properties preserved under homomorphism. Pacific Journal of Mathematics, 9(1), 143–154.

    Article  Google Scholar 

  • Mancosu, P. (2018). Explanation in mathematics. Stanford Encyclopedia of Philosophy. https://plato.stanford.edu/entries/mathematics-explanation/.

  • Manzano, M. (1996). Extensions of first-order logic., Cambridge tracts in theoretical computer science Cambridge: Cambridge University Press.

    Google Scholar 

  • McKinsey, J., Sugar, A., & Suppes, P. (1953). Axiomatic foundation of classical particle mechanics. Journal of Rational Mechanics and Analysis, 2, 253–272.

    Google Scholar 

  • Melia, J. (2000). Weaseling away the indispensability argument. Mind, 109, 455–480.

    Article  Google Scholar 

  • Mendelson, E. (2010) . An introduction to mathematical logic (Taylor & Francis Group). Boca Raton, FL.: Chapman & Hall/CRC. Fifth edition (first edition published in 1964).

  • Menzel, C. (2014). Wide sets, ZFCU, and the iterative conception. Journal of Philosophy, 111(2), 57–83.

    Article  Google Scholar 

  • Muller, F. (2011). Reflections on the revolution at stanford. Synthese, 183(1), 87–114. Special issue: The Classical Model of Science II: The Axiomatic Method, the Order of Concepts and the Hierarchy of Sciences, edited by Arianne Betti, Willem de Jong and Marije Martijn.

  • Mycielski, J. (2013). On the formalization of theories. Journal of Automated Reasoning, 50, 211–216.

    Article  Google Scholar 

  • Narens, L. (1985). Abstract measurement theory. Cambridge, MA: MIT Press.

    Google Scholar 

  • Pincock, C. (2007). A role for mathematics in the physical sciences. Noûs, 41(2), 253–275.

    Article  Google Scholar 

  • Pincock, C. (2012). Mathematics and scientific representation. Oxford: Oxford University Press.

    Book  Google Scholar 

  • Potter, M. (2004). Set theory and its philosophy: A critical introduction. Oxford: Oxford University Press.

    Book  Google Scholar 

  • Przełȩcki, M. (1969). The logic of empirical theories. London: Routledge & Kegan Paul.

    Google Scholar 

  • Putnam, H. (1967). The thesis that mathematics is logic. In R. Schoenman (Ed.), Bertrand Russell, philosopher of the century. London: Allen and Unwin. Page references to reprint in H. Putnam, 1979: Mathematics, Matter and Method: Philosophical Papers, Volume 1. Cambridge: Cambridge University Press.

  • Putnam, H. (1971). Philosophy of logic. New York: Harper and Row. Reprinted in H. Putnam, 1979: Mathematics, Matter and Method: Philosophical Papers, Volume 1. Cambridge: Cambridge University Press.

  • Quine, W. (1956). Unification of universes in set theory. The Journal of Symbolic Logic, 21(3), 267–279.

    Article  Google Scholar 

  • Quine, W. (1986a). Philosophy of logic. Cambridge, MA: Harvard University Press. Second, revised edition of W. Quine, 1970: Philosophy of Logic. Cambridge, MA: Harvard University Press.

  • Quine, W. (1986b) . Reply to Charles Parsons. In E. Hahn, & P. Schilpp (Eds.), The philosophy of W.V. Quine (Volume 18: Library of living philosophers). Chicago: Open Court.

  • Ramsey, F. (1929). Theories. In R. Braithwaite (Ed.), Foundations of mathematics. Amsterdam: Routledge and Kegan Paul. 1931.

    Google Scholar 

  • Reichenbach, H. (1924). Axiomatik der relativistischen Raum-Zeit-Lehre. Braunschweig: Friedr. Vieweg & Sohn. Engl. transl. The axiomatization of the theory of relativity, by M. Reichenbach with an introduction by W.C. Salmon. Berkeley-Los Angeles: University of California Press.

  • Resnik, M. (1997). Mathematics as a science of patterns. Oxford: Oxford University Press.

    Google Scholar 

  • Roberts, F. (1985). Measurement theory. Cambridge: Cambridge University Press.

    Google Scholar 

  • Russell, B. (1903). The principles of mathematics. Cambridge: Cambridge University Press. Paperback edition: Routledge, London, 1992.

  • Russell, B. (1908). Mathematical logic as based on the theory of types. American Journal of Mathematics, 30(3), 222–262.

    Article  Google Scholar 

  • Russell, B. & Whitehead, A. (1912). Principia Mathematica: Vols. I–III. Cambridge: Cambridge University Press.

  • Schlick, M. (1918). Allgemeine Erkenntnislehre. Berlin: Springer. 2nd edition 1925. Translated with introduction by Blumberg, A & Feigl, H., as General theory of knowledge: Library of Exact Philosophy (Open Court, 1985).

  • Schmidt, F. (1938). Über deductive Theorien mit mehreren Sorten von Grunddingen. Mathematische Annalen, 115, 485–506.

    Article  Google Scholar 

  • Shapiro, S. (1997). Philosophy of mathematics: Structure and ontology. Oxford: Oxford University Press.

    Google Scholar 

  • Simpson, S. (2009). Subsystems of second-order arithmetic. Cambridge: Cambridge University Press. First edition, Berlin: Springer (1998).

  • Sneed, J. (1971). The logical structure of mathematical physics. Dordrecht: Reidel.

    Book  Google Scholar 

  • Stegmüller, W. (1976). The structure and dynamics of theories. Berlin: Springer.

    Book  Google Scholar 

  • Steiner, M. (1998). The applicability of mathematics as a philosophical problem. Cambridge, MA: Harvard University Press.

    Book  Google Scholar 

  • Suppe, F. (Ed.). (1977). The structure of scientific theories. Chicago: University of Chicago Press.

    Google Scholar 

  • Suppes, P. (1957). Introduction to logic. New York: Van Nostrand Reinhold.

    Google Scholar 

  • Suppes, P. (1960). A comparison of the meaning and use of models in mathematics and the empirical sciences. Synthese, 12, 287–301.

    Article  Google Scholar 

  • Suppes, P. (1992). Axiomatic methods in science. In M. Carvallo (Ed.), Nature, cognition and system II (pp. 205–232). Dordrecht: Kluwer.

    Chapter  Google Scholar 

  • Suppes, P. (2002). Representation and invariance of scientific structures (CSLI lecture notes: 130). London: CSLI Publications.

    Google Scholar 

  • Tal, E. (2015). Measurement in science. Stanford Encyclopedia of Phlosophy. First published Mon Jun 15, 2015. https://plato.stanford.edu/entries/measurement-science/.

  • Tarski, A. (1959). What is elementary geometry? In L. Henkin, P. Suppes, & A. Tarski (Eds.), The axiomatic Method. Amsterdam: North Holland. Reprinted in Hintikka, J. ed., 1968: Philosophy of Mathematics. Oxford: Oxford University Press.

  • van Fraassen, B. (1980). The scientific image. Oxford: Clarendon Press.

    Book  Google Scholar 

  • Wang, H. (1952). The logic of many-sorted theories. The Journal of Symbolic Logic, 17(2), 105–116.

    Article  Google Scholar 

  • Wenzel, M., et al. (2020). The Isabelle/Isar reference manual. The manual is available online. https://isabelle.in.tum.de/dist/Isabelle2020/doc/isar-ref.pdf.

  • Worrall, J. (1984). An unreal image (Review of van Fraassen (1980)). British Journal for the Philosophy of Science, 35, 65–80.

    Article  Google Scholar 

  • Zermelo, E. (1908). Untersuchungen über die Grundlagen der Mengenlehre I. Mathematische Annalen 65. English translation by S. Bauer-Mengelberg, “Investigations in the Foundations of Set Theory I”, in van Heijenoort (ed.) 1967.

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Appendix A: Four-sorted system

Appendix A: Four-sorted system

1.1 A.1 Well-formed terms and primitive formulas

Definition 20

(Well-formed) Let \(\sigma \) be an application signature over S. We recursively define the well-formed terms and well-formed primitive formulas:

figure bh

Note: the additional clause in (3) implies that if \(x_1, \dots , x_n\) are global variables, then \(\textsf {P}(x_1, \dots , x_n)\) is always a well-formed primitive formula.

1.2 A.2 Definition of the language \(L(\sigma _{\in })\)

Definition 21

(Definition of \(L(\sigma _{\in })\)) In stages: (1) Variables: Each sort has its own stock of variables (\(\mathbf{b }_i, a_i, X_i, x_i\)). (2) Identity & Membership Predicates: There is a global identity predicate \(=\) and a global membership predicate \(\in \), each with the type declaration: \(\textsf {global} \Rightarrow \textsf {global} \Rightarrow \mathsf {bool}\). (3) Terms: primitive terms are defined to be variables and constants; terms are defined to be primitive terms, along with, for terms \(t_1, \dots , t_n\) and n-ary function symbol \(\textsf {F}\), the function terms \(\textsf {F}(t_1, \dots , t_n)\) constructed inductively, matching the sort restrictions given above.Footnote 67 (4) Primitive Formulas: The primitive formulas are identity and membership primitive formulas, of the form \(t_1 = t_2\) and \(t_1 \in t_2\), along with, for terms \(t_1, \dots , t_n\), formulas \({\mathsf {P}}(t_1, \dots , t_n)\), constructed according to the sort restrictions. (5) Connectives & Quantifiers: We introduce the usual logical connectives \(\lnot , \wedge , \dots \), and quantifiers \(\forall \) and \(\exists \) which can bind any variable. (6) Formulas: Formulas of \(L(\sigma _{\in })\) are primitive formulas, along with formulas obtained recursively by applying connectives and quantifiers, by the obvious modification of the usual definition of “formulas” for a 1-sorted first-order language. (7) Sublanguage \(L(\sigma )\): \(L(\sigma )\) is the corresponding sublanguage over \(\sigma \). (8) Sublanguage \(L(\sigma \upharpoonright _{ \mathsf {atom}})\): \(L(\sigma \upharpoonright _{ \mathsf {atom}})\) is the 1-sorted language over subsignature \(\sigma \upharpoonright _{ \mathsf {atom}}\), in which all variables are atom variables, \(a_i\).

1.3 A.3 Quasi-logical axioms

Definition 22

(Boolean axioms) The following are called boolean axioms:

$$\begin{aligned} \top \ne \bot&\forall \mathbf{b } (\mathbf{b } = \top \vee \mathbf{b } = \bot ). \end{aligned}$$
(96)

Definition 23

(Sort axioms) The following five formulas are called the sort axioms:

figure bi

Definition 24

(Subsort axioms) The following schemes are called “Subsort Axioms”:

figure bj

where it is required that \(\phi (a), \phi (\mathbf{b })\) and \(\phi (X)\) be well-formed.

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Ketland, J. Foundations of applied mathematics I. Synthese 199, 4151–4193 (2021). https://doi.org/10.1007/s11229-020-02973-w

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