(*Notations, non-standard concepts, and definitions used commonly in these investigations are detailed in this post.*)

What essentially differentiate our approach of finitary agnosticism to mathematical reasoning in this investigation from both the classical Hilbertian, and the intuitionistic Brouwerian, perspectives is how we address two significant dogmas of the classical first order logic FOL.

**Hilbertian Theism**

In a 1925 address^{[1]} Hilbert had shown that the axiomatisation of classical Aristotlean predicate logic proposed by him as a formal first-order -predicate calculus^{[2]} in which he used a primitive choice-function^{[3]} symbol, `‘, for defining the quantifiers `‘ and `‘ would adequately express—and yield, under a suitable interpretation—Aristotle’s logic of predicates if the -function was interpreted to yield Aristotlean particularisation^{[4]}.

Classical approaches to mathematics—essentially following Hilbert—can be labelled `theistic’ in that they implicitly *assume*—without providing adequate objective criteria—^{[5]} both that:

The standard first order logic FOL is consistent;

and that:

The standard interpretation of FOL is finitarily sound (which implies Aristotle’s particularisation).

The significance of the label `theistic’ is that conventional wisdom tacitly believes that Aristotle’s particularisation remains valid—without qualification—even over infinite domains; a belief that is not unequivocally self-evident, but must be appealed to as an article of faith.

Although intended to highlight an entirely different distinction, that the choice of such a label may not be totally inappropriate is suggested by Tarski’s point of view^{[6]} to the effect:

“… that Hilbert’s alleged hope that meta-mathematics would usher in a `feeling of absolute security’ was a `kind of theology’ that `lay far beyond the reach of any normal human science’ …”.

**Brouwerian Atheism**

We note second that, in sharp contrast, constructive approaches to mathematics—such as Intuitionism^{[6a]}—can be labelled `atheistic’ since they *deny* both that:

FOL is consistent (since they deny the Law of The Excluded Middle^{[7]};

and that:

The standard interpretation of FOL is sound (since they deny Aristotle’s particularisation).

The significance of the label `atheistic’ is that whereas constructive approaches to mathematics deny the faith-based belief in the unqualified validity of Aristotle’s particularisation over infinite domains, their denial of the Law of the Excluded Middle is itself a belief—in the inconsistency of FOL—that is also not unequivocally self-evident, and must also be appealed to as an article of faith since it does not take into consideration the objective criteria for the consistency of FOL that follows from the Birmingham paper.

Although Brouwer’s explicitly stated objection appeared to be to the Law of the Excluded Middle as expressed and interpreted at the time^{[8]}, some of Kleene’s remarks^{[9]}, some of Hilbert’s remarks^{[10]} and, more particularly, Kolmogorov’s remarks^{[11]} suggest that the intent of Brouwer’s fundamental objection (see this post) can also be viewed today as being limited only to the yet prevailing belief—as an article of faith—that the validity of Aristotle’s particularisation can be extended without qualification to infinite domains.

**Finitary Agnosticism**

In our investigations, however, we adopt what may be labelled a finitarily `agnostic’ perspective^{[12]} by noting that although, if Aristotle’s particularisation holds in an interpretation then the Law of the Excluded Middle must also hold in the interpretation, the converse is not true.

We thus follow a middle path by explicitly assuming on the basis of the Birmingham paper (reproduced in this post) that:

FOL is finitarily consistent;

and by explicitly stating when an argument appeals to the postulation that:

The standard interpretation of FOL is sound.

The significance of the label `agnostic’ is that we neither hold FOL to be inconsistent, nor hold that Aristotle’s particularisation can be applied—without qualification—over infinite domains.

**The two seemingly contradictory but perhaps complementary perspectives**

It follows from the above that the Brouwerian Atheistic perspective is merely a restricted perspective within the Finitary Agnostic perspective, whilst the Hilbertian Theistic perspective contradicts the Finitary Agnostic perspective.

Curiously, a case can be made (as is attempted in this post, and in this paper that I presented on 10th June 2015 at the Epsilon 2015 workshop on Hilberts Epsilon and Tau in Logic, Informatics and Linguistics, University of Montpellier, France, June 10-11-12) that the Hilbertian perspective formalises the concept of `algorithmically verifiable truth’ in the non-finitary—hence essentially subjective—reasoning that circumscribes human intelligences (which, by the Anthropic principle, can be taken to reflect the ‘truths’ of natural laws), whilst the Finitary perspective formalises the concept of ‘algorithmically computable truth’ in the finitary—hence essentially objective—reasoning that circumscribes mechanical intelligences; and that the two are complementary, not contradictory, perspectives on the nature and scope of quantification.

**References**

**BBJ03** George S. Boolos, John P. Burgess, Richard C. Jeffrey. 2003. *Computability and Logic* (4th ed). Cambridge University Press, Cambridge.

**Be59** Evert W. Beth. 1959. *The Foundations of Mathematics.* Studies in Logic and the Foundations of Mathematics. Edited by L. E. J. Brouwer, E. W. Beth, A. Heyting. 1959. North Holland Publishing Company, Amsterdam.

**BF58** Paul Bernays and Abraham A. Fraenkel. 1958. *Axiomatic Set Theory.* Studies in Logic and the Foundations of Mathematics. Edited by L. E. J. Brouwer, E. W. Beth, A. Heyting. 1959. North Holland Publishing Company, Amsterdam.

**Br08** L. E. J. Brouwer. 1908. *The Unreliability of the Logical Principles.* English translation in A. Heyting, Ed. *L. E. J. Brouwer: Collected Works 1: Philosophy and Foundations of Mathematics.* Amsterdam: North Holland / New York: American Elsevier (1975): pp.107-111.

**Br13** L. E. J. Brouwer. 1913. *Intuitionism and Formalism.* Inaugural address at the University of Amsterdam, October 14, 1912. Translated by Professor Arnold Dresden for the Bulletin of the American Mathematical Society, Volume 20 (1913), pp.81-96. 1999. Electronically published in Bulletin (New Series) of the American Mathematical Society, Volume 37, Number 1, pp.55-64.

**Br23** L. E. J. Brouwer. 1923. *On the significance of the principle of the excluded middle in mathematics, especially in function theory.* Address delivered on 21 September 1923 at the annual convention of the Deutsche Mathematiker-Vereinigung in Marburg an der Lahn. In Jean van Heijenoort. 1967. Ed. *From Frege to Gödel: A source book in Mathematical Logic, 1878 – 1931.* Harvard University Press, Cambridge, Massachusetts.

**Co66** Paul J. Cohen. 1966. *Set Theory and the Continuum Hypothesis.* (Lecture notes given at Harvard University, Spring 1965) W. A. Benjamin, Inc., New York.

**Cr05** John N. Crossley. 2005. *What is Mathematical Logic? A Survey.* Address at the *First Indian Conference on Logic and its Relationship with Other Disciplines* held at the Indian Institute of Technology, Powai, Mumbai from January 8 to 12. Reprinted in *Logic at the Crossroads: An Interdisciplinary View – Volume I* (pp.3-18). ed. Amitabha Gupta, Rohit Parikh and Johan van Bentham. 2007. Allied Publishers Private Limited, Mumbai.

**Da82** Martin Davis. 1958. *Computability and Unsolvability.* 1982 ed. Dover Publications, Inc., New York.

**EC89** Richard L. Epstein, Walter A. Carnielli. 1989. *Computability: Computable Functions, Logic, and the Foundations of Mathematics.* Wadsworth & Brooks, California.

**Fr09** Curtis Franks. 2009. *The Autonomy of Mathematical Knowledge: Hilbert’s Program Revisited}.* Cambridge University Press, New York.

**Go31** Kurt Gödel. 1931. *On formally undecidable propositions of Principia Mathematica and related systems I.* Translated by Elliott Mendelson. In M. Davis (ed.). 1965. *The Undecidable.* Raven Press, New York.

**HA28** David Hilbert & Wilhelm Ackermann. 1928. *Principles of Mathematical Logic.* Translation of the second edition of the *Grundzüge Der Theoretischen Logik.* 1928. Springer, Berlin. 1950. Chelsea Publishing Company, New York.

**Hi25** David Hilbert. 1925. *On the Infinite.* Text of an address delivered in Münster on 4th June 1925 at a meeting of the Westphalian Mathematical Society. In Jean van Heijenoort. 1967.Ed. *From Frege to Gödel: A source book in Mathematical Logic, 1878 – 1931.* Harvard University Press, Cambridge, Massachusetts.

**Hi27** David Hilbert. 1927. *The Foundations of Mathematics.* Text of an address delivered in July 1927 at the Hamburg Mathematical Seminar. In Jean van Heijenoort. 1967.Ed. *From Frege to Gödel: A source book in Mathematical Logic, 1878 – 1931.* Harvard University Press, Cambridge, Massachusetts.

**Kl52** Stephen Cole Kleene. 1952. *Introduction to Metamathematics.* North Holland Publishing Company, Amsterdam.

**Kn63** G. T. Kneebone. 1963. *Mathematical Logic and the Foundations of Mathematics: An Introductory Survey.* D. Van Norstrand Company Limited, London.

**Ko25** Andrei Nikolaevich Kolmogorov. 1925. *On the principle of excluded middle.* In Jean van Heijenoort. 1967. Ed. *From Frege to Gödel: A source book in Mathematical Logic, 1878 – 1931.* Harvard University Press, Cambridge, Massachusetts.

**Li64** A. H. Lightstone. 1964. *The Axiomatic Method.* Prentice Hall, NJ.

**Me64** Elliott Mendelson. 1964. *Introduction to Mathematical Logic.* Van Norstrand, Princeton.

**Ma09** Maria Emilia Maietti. 2009. *A minimalist two-level foundation for constructive mathematics.* Dipartimento di Matematica Pura ed Applicata, University of Padova.

**MS05** Maria Emilia Maietti and Giovanni Sambin. 2005. *Toward a minimalist foundation for constructive mathematics.* Clarendon Press, Oxford.

**Mu91** Chetan R. Murthy. 1991. *An Evaluation Semantics for Classical Proofs.* Proceedings of Sixth IEEE Symposium on Logic in Computer Science, pp. 96-109, (also Cornell TR 91-1213), 1991.

**Nv64** P. S. Novikov. 1964. *Elements of Mathematical Logic.* Oliver & Boyd, Edinburgh and London.

**Qu63** Willard Van Orman Quine. 1963. *Set Theory and its Logic. Harvard University Press, Cambridge, Massachusette.*

**Rg87** Hartley Rogers Jr. 1987. *Theory of Recursive Functions and Effective Computability.<i.* MIT Press, Cambridge, Massachusetts.

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**Sh67** Joseph R. Shoenfield. 1967. *Mathematical Logic.* Reprinted 2001. A. K. Peters Ltd., Massachusetts.

**Sk28** Thoralf Skolem. 1928. *On Mathematical Logic* Text of a lecture delivered on 22nd October 1928 before the Norwegian Mathematical Association. In Jean van Heijenoort. 1967. Ed. *From Frege to Gödel: A source book in Mathematical Logic, 1878 – 1931.* Harvard University Press, Cambridge, Massachusetts.

**Sm92** Raymond M. Smullyan. 1992. *Gödel’s Incompleteness Theorems.* Oxford University Press, Inc., New York.

**Su60** Patrick Suppes. 1960. *Axiomatic Set Theory.* Van Norstrand, Princeton.

**Ta33** Alfred Tarski. 1933. *The concept of truth in the languages of the deductive sciences.* In *Logic, Semantics, Metamathematics, papers from 1923 to 1938* (p152-278). ed. John Corcoran. 1983. Hackett Publishing Company, Indianapolis.

**Tu36** Alan Turing. 1936. *On computable numbers, with an application to the Entscheidungsproblem.* In M. Davis (ed.). 1965. *The Undecidable.* Raven Press, New York. Reprinted from the Proceedings of the London Mathematical Society, ser. 2. vol. 42 (1936-7), pp.230-265; corrections, Ibid, vol 43 (1937) pp.\ 544-546.

**Wa63** Hao Wang. 1963. *A survey of Mathematical Logic.* North Holland Publishing Company, Amsterdam.

**An12** Bhupinder Singh Anand. 2012. *Evidence-Based Interpretations of PA.* In Proceedings of the Symposium on Computational Philosophy at the AISB/IACAP World Congress 2012-Alan Turing 2012, 2-6 July 2012, University of Birmingham, Birmingham, UK.

**Notes**

Return to 1: Hi25.

Return to 2: Hi27, pp.465-466; see also this post.

Return to 3: Hi25, p.382.

Return to 4: Hi25, pp.382-383; Hi27, p.466(1).

Return to 5: See for instance: Hi25, p.382; HA28, p.48; Sk28, p.515; Go31, p.32.; Kl52, p.169; Ro53, p.90; BF58, p.46; Be59, pp.178 & 218; Su60, p.3; Wa63, p.314-315; Qu63, pp.12-13; Kn63, p.60; Co66, p.4; Me64, p.52(ii); Nv64, p.92; Li64, p.33; Sh67, p.13; Da82, p.xxv; Rg87, p.xvii; EC89, p.174; Mu91; Sm92, p.18, Ex.3; BBJ03, p.102; Cr05, p.6.

Return to 6: As commented upon in Fr09, p.3.

Return to 6a: But see also Ma09 and MS05.

Return to 7: “The formula is classically provable, and hence under classical interpretation true. But it is unrealizable. So if realizability is accepted as a necessary condition for intuitionistic truth, it is untrue intuitionistically, and therefore unprovable not only in the present intuitionistic formal system, but by any intuitionistic methods whatsoever”. Kl52, p.513.

Return to 8: Br23, p.335-336; Kl52, p.47; Hi27, p.475.

Return to 9: Kl52, p.49.

Return to 10: For instance in Hi27, p.474.

Return to 11: In Ko25, fn. p.419; p.432.

Return to 12: See, for instance An12 (reproduced in this post).

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