Skip to main content
SpringerLink
Log in

Standard Formalization

  • Original Paper
  • Published:
Axiomathes Aims and scope Submit manuscript

Abstract

A standard formalization of a scientific theory is a system of axioms for that theory in a first-order language (possibly many-sorted; possibly with the membership primitive \(\in\)). Suppes (in: Carvallo M (ed) Nature, cognition and system II. Kluwer, Dordrecht, 1992) expressed skepticism about whether there is a “simple or elegant method” for presenting mathematicized scientific theories in such a standard formalization, because they “assume a great deal of mathematics as part of their substructure”. The major difficulties amount to these. First, as the theories of interest are mathematicized, one must specify the underlying applied mathematics base theory, which the physical axioms live on top of. Second, such theories are typically geometric, concerning quantities or trajectories in space/time: so, one must specify the underlying physical geometry. Third, the differential equations involved generally refer to coordinate representations of these physical quantities with respect to some implicit coordinate chart, not to the original quantities. These issues may be resolved. Once this is done, constructing standard formalizations is not so difficult—at least for the theories where the mathematics has been worked out rigorously. Here we give what may be claimed to be a simple and elegant means of doing that. This is for mathematicized scientific theories comprising differential equations for \(\mathbb{R}\)-valued quantities Q (that is, scalar fields), defined on n (“spatial” or “temporal”) dimensions, taken to be isomorphic to the usual Euclidean space \(\mathbb{R}^n\). For illustration, I give standard (in a sense, “text-book”) formalizations: for the simple harmonic oscillator equation in one-dimension and for the Laplace equation in two dimensions.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

Notes

  1. There is a vast literature on differential equations in mathematical physics. Classics are Sommerfeld (1964), Courant and Hilbert (1924) and Jeffreys and Jeffreys (1966), and Arnold (1989). There are many standard undergraduate level texts, which I would recommend: e.g, Arfken and Weber (2005) for physicists and Olver (2014) for mathematicians. There exist very good advanced undergraduate/graduate level texts which develop the framework of differential geometry: e.g., Schutz (1980); and the quite advanced Frankel (2011). Additionally, there are “differential equation solvers” for software packages (e.g., Matlab, Mathematica and R). I use one of these below in Sect. 7.2 to compare an analytic solution of Laplace’s equation to a numerically integrated one. The interested reader who knows R programming might consult the book Soetaert et al. (2012).

  2. Nice explanations of differential geometry may be found in, say, Schutz (1980), Wald (1984), Frankel (2011), or Robin and Salamon (2012).

  3. Epistemologically, perhaps not so trivial, I need to add: as, in their own ways, Hume, Popper, Hempel and Goodman have all pointed out.

  4. Here I use type/sort notation: \((\forall x: \mathbb{R})\). It’s equivalent to \((\forall x \in \mathbb{R})\).

  5. On the usual reduction of reals to sets, each real lives inside \(V_{\omega +1}\) (which is, more or less, \(\mathcal{P}( \mathbb{N})\)), where the \(V_{\alpha }\)’s are “the von Neumann levels” in the hierarchy of pure (well-founded) sets. \(V_0\) is defined to be \(\varnothing\); and, for any ordinal \(\alpha\), \(V_{\alpha + 1}\) is defined to be \(P(V_{\alpha })\) (i.e., the power set of \(V_{\alpha }\)); and, if \(\lambda\) is a limit ordinal (i.e., \(\lambda\) is not equal to \(\alpha + 1\), for any ordinal \(\alpha\)), then \(V_{\lambda }\) is defined to be the union of all the \(V_{\alpha }\)’s with \(\alpha < \lambda\). This is the definition of the von Neumann hierarchy. It is analogous to defining the factorial function by, \(F(0) = 1\) and \(F(n+1) = (n+1)F(n)\), except that it uses ordinals instead of numbers and it has a special “infinity clause”. So, since \(\omega\) is the smallest infinite ordinal, \(V_{\omega }\) is the smallest infinite level: it is the set of all well-founded finite pure sets. From the definition, \(V_{\omega +1}\) is \(P(V_{\omega })\). There are several ways to encode \(\mathbb{N}\) into \(V_{\omega }\) (including a bijective way, called the Ackermann encoding; computer scientists call it the BIT predicate). And there are ways to encode each real number as a subset \(X \subseteq \mathbb{N}\), and conversely too, and indeed bijectively. So, one can think of \(V_{\omega +1}\) as (encoding) the set of reals. Each function \(F: \mathbb{R}\rightarrow \mathbb{R}\) lives inside \(V_{\omega +2}\); and thus a class of such functions lives inside \(V_{\omega + 3}\). Somewhat loosely speaking, the arena of almost all theoretical physics—the function spaces, Lie groups & algebras, topological spaces, manifolds, distribution spaces, bundles, etc.—is \(V_{\omega + n}\), for some smallish n. This material is explained in a set theory textbook, such as Jech (2002), Potter (2004), or Enderton (1977).

  6. A sketch of the construction of the formula \(\Phi _{\text{C}^2( \mathbb{R})}(F)\) is given below, in Sect. 5, Example 3.

  7. In data science, to “smooth” a dataset (usually a time series), one interpolates a smooth curve between the datapoints, usually not lying on those points, but rather averages. For example, the usual visualizations for the global temperature time series are smoothed using a moving average. A “low pass filter” is also a special kind of smoothing function (which eliminates high frequencies). Likewise, image processing filters.

  8. A real-valued Bessel function (on the reals) is a function \(F: \mathbb{R}\rightarrow \mathbb{R}\) satisfying:

    $$\begin{aligned} x^{2}{\frac{d^{2}F}{dx^{2}}}+x{\frac{dF}{dx}}+\left( x^{2}-\alpha ^{2}\right) F=0 \end{aligned}$$
    (19)

    for some constant \(\alpha\). Such functions turn up in many applications involving waves.

  9. Feynman’s puzzle here corresponds to what Hartry Field calls “Heavy Duty Platonism”. See Field (1989): 186–200. The rejection of Heavy Duty Platonism plays a central role also in Field’s main anti-nominalist arguments in Field (1980). See also Knowles (2015) for a recent discussion of this. Knowles defines it as “the view that physical magnitudes, such as mass and temperature, are cases of physical objects being related to numbers”.

  10. I allude here to the Quine–Putnam Indispensability Argument (see Colyvan (2019)). Quine’s views on this are scattered across several decades (e.g., Quine (1948, 1986)). Putnam’s canonical formulations are given in Putnam (1971, 1975).

  11. For details, see, for example, Bennett (1995), Theorem 1, p. 72.

  12. I must stress that the claim that the points of space do indeed carry such relations is an approximation. Yet it holds to an extremely good approximation.

  13. The reason for this is that when \(\mathsf{PA}_2\) is treated as a two-sorted first-order theory, one may use The Completeness Theorem—Henkin’s Completeness Theorem (Henkin 1950)—to transfer between model-theoretic and proof-theoretic results.

  14. See, e.g., Halbach (2014) or Cieśliński (2017).

  15. I have seen it remarked that a formalized theory, in particular for scientific theories, must always be stated directly in machine code. This is incorrect. The requirement is only that it uses predicates and function symbols which are definable. Axioms “higher-up”, so to speak, are stated using these defined predicates. \(\textsf{CT}1\)-\(\textsf{CT}4\) above are examples. But there are many others, well-known in mathematical logic.

  16. One example is the following: Georg Kreisel conjectured that if there is some fixed k such that, for all \(n \in \mathbb{N}\), \(\mathsf{PA}\) proves \(\phi (S^n(0))\) in at most k steps, then \(\mathsf{PA}\) proves \(\forall x \phi (x)\). Does Kreisel’s Conjecture (KC) hold for Peano arithmetic? It turns out to be implementation dependent. Parikh (1973) showed that KC holds for an implementation called \(\mathsf{PA}^{*}\), where the function symbols \(+\), \(\cdot\) are replaced by appropriate predicates, say, A(x, y, z) and M(x, y, z), with suitable axioms stating existence and uniqueness. The proof of this is quite difficult. But, cleverly choosing \(\phi (x)\), it is easy to give a formulation \(\mathsf{PA}^{\phi }\) of \(\mathsf{PA}\), in the same language \(L(0,S,+, \cdot )\) and with the same theorems as standard \(\mathsf{PA}\), such that KC does not hold. See Cavagnetto (2009), Proposition 1.2 (we choose \(\phi (x)\) to be any formula witnessing \(\omega\)-incompleteness).

  17. The theories of Russell and Zermelo included atoms (Zermelo (1908), Russell (1903), Russell (1908)) as did Church’s type theory (Church 1940). Recent examples are: Mendelson (2010) (§4.6.5); Jech (2002) (p. 250 ff.); Potter (2004). A detailed exposition of many-sorted logic appears in Manzano (1996) and recent applications to questions concerning theory formalization in Halvorson (2019).

  18. So \(v_i = v_j\) and \(v_i \in v_j\) are well-formed primitive formulas irrespective of the sorts of the variables \(v_i, v_j\).

  19. See, e.g., Jech (2002) or Enderton (1977) for a detailed exposition of these axioms.

  20. So, what is called “\(\mathsf{ZCA}_{\sigma }\)” below is, strictly speaking, a definitional extension of the given axioms.

  21. An interesting discussion of the mathematical needs of scientifically applicable mathematics is Feferman (1992), in which Feferman discusses a predicative system W (descended from that set out by Hermann Weyl in Weyl (1918)) which is proof-theoretically conservative over \(\mathsf{PA}\). However, this all seems besides the central point, which derives from Quine and Putnam: physical quantities, fields, and so on, are themselves mixed functions and impure structures. It seems irrelevant then whether the set theory one uses, say, is conservative over \(\mathsf{PA}\) or not. Even if it is, these physical, mixed quantities exist (and their values, etc.) and that contradicts nominalism. The realism of Quine and Putnam in question concerns these physical, mixed quantities.

  22. See Hawking and Ellis (1973), Ch. 7, especially pp. 249–251; see Wald (1984), Theorem 10.2.2, p. 264.

  23. Note that, using numerical integration, one may also model geometry on a discrete grid. An example of this is briefly explained in Sect. 7.2, where we look at the analytic solution for Laplace equation in 2 dimensions (with Dirichlet boundary conditions on a unit square) and compare that with the numerical solution obtained using a software partial differential equation solver.

  24. To obtain \(L_2(\sigma )\), we add second-order variables \(X_i^{(j)}\) (i.e., the \(i\hbox{th}\) variable for j-place relations), and atomic formulas \(X_i^{(j)}(p_1, \dots , p_j)\) (and equations \(X_i^{(j)} = X_k^{(j)}\)), and corresponding second-order quantifiers. In a clear sense, one can interpret \(L_2(\sigma )\) into the full set-theoretic extended language \(L(\sigma _{\in })\). For the set-theoretic translation \((X_i^{(j)}(p_1, \dots , p_j))^{\circ }\) of an atomic formula \(X_i^{(j)}(p_1, \dots , p_j)\) is given as: \((p_1, \dots , p_j) \in X_i^{(j)}\), using pairing.

  25. I conflate the function symbol \(\textsf{B}\) and the set constant \(\textsf{B}\) naming its extension; likewise \(\equiv\). But this conflation is harmless, since context always disambiguates.

  26. This is analogous to the fact that one can prove a Representation Theorem for second-order Peano arithmetic \(\mathsf{PA}_2\) inside set theory: any (full) model \(M \models \mathsf{PA}_2\) is isomorphic to \((\omega , \varnothing , (.)^{+}, +_{\omega }, \cdot _{\omega })\).

  27. See also Väänänen and Wang (2015), Button and Walsh (2018) (pp. 223–250), Ketland (2021).

  28. For us, it is important that \(\textsf{EG}^{(n)}\) be expressed as a single axiom, and one can do this because the Continuity Axiom is a single second-order sentence quantifying over sets of points.

  29. Strictly speaking, this is a bit sloppy as I have not defined the product sort \(\mathbb{R}^n\). In our setting, with the given basic sorts, the correct sort of \(\varphi\) is \(\mathbb{P} \Rightarrow \mathbb{S}\); but we add a new axiom (called a codomain axiom in Ketland (2021)): \((\forall p \in \mathbb{P})(\varphi (p) \in \mathbb{R}^n)\).

  30. One may then regard the symbol “\(\varphi\)” as a skolem constant for the existential quantifier in “\((\exists \varphi : \mathbb{P} \rightarrow \mathbb{R}^n) \ (P_{\textsf{geom}} \overset{\varphi }{\cong }\mathbb{E}^{n})\)”.

  31. Strictly speaking, again, the sort declaration given for \(Q_i\) is a bit sloppy as I have not defined the relevant sort \(\mathbb{V}\) (i.e., elements of some vector space). In our setting, with the given basic sorts, the correct sort of \(Q_i\) is \(\mathbb{P} \Rightarrow \mathbb{S}\); but, for each \(Q_i\), we add a new codomain axiom: \((\forall p \in \mathbb{P})(Q_i(p) \in \mathbb{V})\), where \(X \in \mathbb{V}\) is the \(L_{\in }\)-formula defining \(\mathbb{V}\) (e.g., \(\mathbb{R}\) or \(\mathbb{C}\), ...).

  32. Henceforth I shall suppress reference to explicit definitions such as \(\textsf{Dfn}(Q^{\varphi })\): one might consider them absorbed into a definitional extension of \(\mathsf{ZCA}_{\sigma }\).

  33. It seems fairly clear that much the same can be done for General Relativity, where the geometric axiom will be: \((\mathbb{P}, \texttt{chart}, \texttt{metric}) \text{ is a relativistic spacetime}\).

  34. As I noted above, I have also deliberately ignored singularities and distributions. There are ways to handle these, but they introduce orthogonal and, in a sense, “higher-level” complexities not relevant to the current level of analysis: i.e., geometric structure and point fields on the geometry satisfying some differential equation.

  35. This is taught to first-year undergraduates, as how to solve the differential equation \((\partial _{xx} + k^2) F = 0\), though not quite in the way we describe it here!

  36. E.g., for the quantum harmonic oscillator, these additional parameters are m, \(\hbar\), \(\omega\), E.

  37. The first four are: \(H_0(x) = 1\); \(H_1(x) = 2x\); \(H_2(x) = 4x^2-2\); \(H_3(x) = 8x^3-12x\); ...

  38. These methods are described in any textbook on differential equations. See, for example, Arfken and Weber (2005), Ch. 9, §9.4 (p. 554), for explanation of separation of variables.

  39. Though it is clear that this is so, it is certainly not easily so—I do not volunteer to specify every detail and lemma. One could, in principle, formalize all this in a theorem-proving assistant like Isabelle, Lean and so on. But it would be something like a year-long MSc research project for a mathematics or computer science student.

  40. In this case, the Axiom of Choice isn’t needed. As I have tried to stress above, such metatheoretic conclusions will be largely invariant with respect the choice of foundational system, so long as it is not too weak. For example, I am reasonably certain this could be formalized inside the HOL/Isabelle theorem-proving assistant, or within Mizar, Coq or Lean.

  41. The numerical solution was worked out using the package ReacTran (for the programming language \(\texttt{R}\)), based on modifying code provided in Soetaert et al. (2012) (Ch. 9, “Solving Partial Differential Equations in R”, p. 167).

References

  • Andréka H, Madarász JX, Németi I, Székely G (2012) A logic road from special relativity to general relativity. Synthese 186(3):633–649

    Article  Google Scholar 

  • Arfken G (1970) Mathematical methods for physicists, 2nd edn. Academic Press, New York

    Google Scholar 

  • Arfken G, Weber H (2005) Mathematical methods for physicists. Elsevier, Amsterdam

    Google Scholar 

  • Arnold V (1989) Mathematical methods of classical mechanics. New York: Springer. Second edition. This is the English translation by K. Vogtmann and A. Weinstein of the second Russian edition. (First Russian edition 1974.)

  • Barwise J (1975) Admissible sets and structures: an approach to definability theory (Perspectives in Mathematical Logic, Volume 7). Berlin, Springer

  • Bennett M (1995) Affine and projective geometry. Wiley, New York

    Book  Google Scholar 

  • Borsuk K, Szmielew W (1960) Foundations of Geometry: Euclidean, Bolyai-Lobachevskian Geometry and Projective Geometry. Amsterdam: North-Holland Publishing Company. Revised English translation by Erwin Marquit. Reprinted by Dover Books

  • Burgess J (1984) Synthetic mechanics. J Philos Log 13:379–95

    Article  Google Scholar 

  • Burgess J (1988) Sets and point-sets: five grades of set-theoretic involvement in geometry. PSA: Proceedings of the Biennial Meeting of the Philosophy of Science Association 1988(2):456–463

    Google Scholar 

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

    Google Scholar 

  • Button T, Walsh S (2018) Philosophy and model theory. Oxford University Press, Oxford

    Book  Google Scholar 

  • 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 (1939) Foundations of logic and mathematics (foundations of the unity of science). University of Chicago Press, Chicago

    Google Scholar 

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

    Google Scholar 

  • 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

  • Cavagnetto S (2009) The lengths of proofs: Kreisel’s conjecture and Gödel’s speed-up theorem. J Math Sci 158:689–707

    Article  Google Scholar 

  • Choquet-Bruhat Y, Geroch R (1969) Global aspects of the Cauchy problem in general relativity. Commun Math Phys 14:329–335

    Article  Google Scholar 

  • Church A (1940) A formulation of the simple theory of types. J Symb Log 5:56–68

    Article  Google Scholar 

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

    Book  Google Scholar 

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

  • Courant R, Hilbert D (1924) 1953/1962: methods of mathematical physics (Vols 1 and 2), 1st edn. Interscience, New York (in German)

    Google Scholar 

  • Enderton H (1977) Elements of set theory. Academic Press, New York

    Google Scholar 

  • Feferman S (1992) Why a little bit goes a long way: logical foundations of scientifically applicable mathematics. In: Proceedings of the 1992 philosophy of science association 2: 442–455

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

    Google Scholar 

  • Feynman R (1970) Feynman lectures on physics: Vols I–III. Reading: Addison-Wesley. Edited by R.B. Leighton & M. Sands. Page references to 2006 edition

  • Field H (1980) Science without numbers, 2nd edn. Princeton University Press, Princeton

    Google Scholar 

  • Field H (1989) Realism, mathematics and modality. Blackwell, Oxford

    Google Scholar 

  • Frankel T (2011) The geometry of physics: an introduction, 3rd edn. Cambridge University Press, Cambridge

    Book  Google Scholar 

  • Hájek P, Pudlák P (2017) Metamathematics of first-order arithmetic. Cambridge University Press, Cambridge

    Book  Google Scholar 

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

  • Halvorson H (2019) The logic in the philosophy of science. Cambridge University Press, Cambridge

    Book  Google Scholar 

  • Hawking S, Ellis G (1973) The large scale structure of space-time. Cambridge University Press, Cambridge

    Book  Google Scholar 

  • Henkin L (1950) Completeness in the theory of types. J Symb Logic 15(2):81–91

    Article  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 (1950)

  • Hintikka J (ed) (1968) Philosophy of mathematics. Oxford University Press, Oxford

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

  • Jeffreys H, Jeffreys B (1966) Methods of mathematical physics. Cambridge University Press, Cambridge. Corrected third edition. (First edition, 1946)

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

  • Ketland J (2021) Foundations of applied mathematics I. Synthese Forthcoming

  • Knowles R (2015) Heavy duty platonism. Erkenntnis 80:1255–1270

    Article  Google Scholar 

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

    Book  Google Scholar 

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

    Google Scholar 

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

  • Olver PJ (2014) Introduction to partial differential equations. Springer. Undergraduate texts in mathematics

  • Pambuccian V (2011) The axiomatics of ordered geometry I. Ordered incidence spaces. Expositiones Mathematicae 29:24–66

    Article  Google Scholar 

  • Parikh R (1973) Some results on the length of proofs. Trans Am Math Soc 177:29–36

    Article  Google Scholar 

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

    Book  Google Scholar 

  • Putnam H (1971) Philosophy of logic. Reprinted in Putnam (1979). Harper and Row, New York

    Google Scholar 

  • Putnam H (1975) What is mathematical truth? In: Putnam H (ed) Mathematics, matter and method. Cambridge University Press, Cambridge

    Google Scholar 

  • Putnam H (1979) Mathematics, matter and method: philosophical papers, vol 1. Cambridge University Press, Cambridge

    Book  Google Scholar 

  • Quine W (1948) On what there is. The review of metaphysics 2: 21–38. In Quine (1953) and Quine (1980)

  • Quine W (1953) From a logical point of view: nine-logico-philosophical essays. Harvard University Press, Cambridge

    Google Scholar 

  • Quine W (1980) From a logical point of view: nine-logico-philosophical essays. Harvard University Press, Cambridge. Second, revised edition of Quine (1953)

  • Quine W (1986) Reply to Charles Parsons. In: Hahn E, Schilpp P (eds) The philosophy of W.V. Quine (Volume 18: Library of Living Philosophers). Chicago: Open Court

  • Robin J, Salamon D (2012) Introduction to differential geometry. ETH, Zurich

    Google Scholar 

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

  • Russell B (1908) Mathematical logic as based on the theory of types. Am J Math 30(3):222–262

    Article  Google Scholar 

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

    Google Scholar 

  • Sbierski J (2016) On the existence of a maximal Cauchy development for the Einstein equations: a dezornification. Ann Henri Poincaré 17:301–329

    Article  Google Scholar 

  • Schutz B (1980) Geometrical methods of mathematical physics. Cambridge University Press, Cambridge

    Book  Google Scholar 

  • Simpson S (2009) Subsystems of second-order arithmetic. Cambridge University Press, Cambridge, p 1998

    Book  Google Scholar 

  • Soetaert K, Cash J, Mazzia F (2012) Solving differential equations in R. Springer, New York

    Book  Google Scholar 

  • Sommerfeld A (1964) Partial differential equations in physics (lectures on theoretical physics: vol VI). Academic Press, New York. First edition, 1949, translated by Straus, Ernst G., New York: Academic Press

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

    Chapter  Google Scholar 

  • Suppes P (2002) Representation and invariance of scientific structures (CSLI Lecture Notes: 130). CSLI Publications

  • Tarski A (1959) What is elementary geometry? In: Henkin L, Suppes P, Tarski A (eds) The axiomatic method. North Holland. Reprinted in Hintikka (1968), Amsterdam

    Google Scholar 

  • Tarski A, Givant S (1999) Tarski’s system of geometry. Bull Symbolic Logic 5:175–214

    Article  Google Scholar 

  • Väänänen J, Wang T (2015) Internal categoricity in arithmetic and set theory. Notre Dame J Formal Logic 56(1):121–134

    Article  Google Scholar 

  • Wald R (1984) General relativity. University of Chicago Press, Chicago

    Book  Google Scholar 

  • Weyl H (1918) Das Kontinuum: Kritische Untersuchungen über die Grundlagen der Analysis. Verlag von Veit & comp, Leipzig

    Book  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

Download references

Funding

Funding was supported by Narodowe Centrum Nauki (Grant Nos. 2018/29/B/HS1/01832, 2020/39/B/HS1/02020).

Author information

Authors and Affiliations

  1. Institution Faculty of Philosophy, University of Warsaw, 3 Krakowskie Przedmiescie, 00-927, Warsaw, Poland

    Jeffrey Ketland

Authors
  1. Jeffrey Ketland
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Jeffrey Ketland.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Ketland, J. Standard Formalization. Axiomathes 32 (Suppl 3), 711–748 (2022). https://doi.org/10.1007/s10516-022-09625-3

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10516-022-09625-3

Keywords

Search

Navigation