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Rigor and formalization

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Abstract

This paper critically examines and evaluates Yacin Hamami’s reconstruction of the standard view of mathematical rigor. We will argue that the reconstruction offered by Hamami is premised on a strong and controversial epistemological thesis and a strong and controversial thesis in the philosophy of mind. Secondly, we will argue that Hamami’s reconstruction of the standard view robs it of its original philosophical rationale, i.e. making sense of the notion of rigor in mathematical practice.

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Notes

  1. For an overview of the main lines of criticism and responses, see Hamami (2022, pp. 435–441).

  2. This is an assumption of the standard view. This infallibist conception of mathematics is not necessarily close to mathematical practice. For a fallibist account of mathematics that incorporates the objectivity of mathematics see e.g. Toffoli (2021).

  3. To say that an inference is valid in the community \({\mathcal {M}}\) is to say that the inference can be validated using the validation procedures used in the community.

  4. The consequences of the existence of open-textured mathematical concepts for the notion of rigor are discussed in Zayton (2022).

  5. A formal deduction or a derivation in a formal system is a finite list of formulas, such that each formula in the list is either an axiom, a logical truth, or follows from formulas appearing earlier in the proof by using one of the rules of deduction of the system.

  6. The axioms of ZFCU are the usual axioms of Zermelo-Fraenkel augmented with axiom of choice but with the Axiom of Foundation replaced the Axiom of Universality.

  7. Azzouni refers to algorithmic systems rather than formal systems. Algorithmic systems are systems in which “ [...] ‘proofs’, however they are understood, are (in principle) mechanically recognizable” (Azzouni, 2004, p. 86). It is clear that a formal system with a recursive language and a set of recursive inference rules is an algorithmic system in Azzouni’s sense.

  8. It is axiomatized as an extension of the modal logic K with the so-called Löb’s axiom:

    $$\begin{aligned} \Box (\Box \varphi \rightarrow \varphi ) \rightarrow \Box \varphi \end{aligned}$$

    This logic also validates \(\Box \varphi \rightarrow \Box \Box \varphi \).

  9. A modal logic K is axiomatized by the following set of axioms and rules:

    • Propositional tautologies.

    • Axiom K: \(\Box (\varphi \rightarrow \psi ) \rightarrow (\Box \varphi \rightarrow \Box \psi )\).

    • NEC: if \(\varphi \) is a theorem so is \(\Box \varphi \).

  10. A good analogy for this situation is to think about formal vs informal provability as a case that is analogous to formal and informal computability and Church-Turing thesis.

  11. This would adhere to the intuition that formal provability is more stringent than informal provability.

  12. Assume for the moment that there is such a map \({\mathcal {T}}\) (meeting the necessary desiderata) between rigorous proofs and formal deductions in S. We will denote the class of formal deductions in S by \(\mathbb{F}\mathbb{D}\). The pre-image of \(\mathbb{F}\mathbb{D}\) under \({\mathcal {T}}\) are those informal arguments that can be translated into formal deductions. The standard view states that the pre-image of \(\mathbb{F}\mathbb{D}\) under \({\mathcal {T}}\) is \(\mathbb {RIG}\) i.e. the set of all rigorous proofs.

  13. To be clear, this does not imply that making a proof gapless or formalizing a proof never has any epistemic benefits. What it does mean is that whether these activities have positive epistemic benefits, depends on the context in which they take place. As similar idea is defended by Wagner (2022). He argues that formalization is a way to resolve dissensus in mathematics. One of the functions of formalization is thus to forge consensus within a mathematical community. When there is no dissensus about whether to accept a given proof, then formalization is neither required nor does it seem particularly valued. When there is dissensus partial formalization is used to forge consensus.

  14. We would like to thank the referee for pressing us to make the dialectical role of the assumption concerning the Turing-computability of human mathematical cognition more precise. The referee’s comments on our discussion of this issue in previous versions have helped us to clarify this important issue.

  15. This raises the question as to whether “neural computation” is still computation if it differs in fundamental aspects from the paradigm cases of digital and analog computation. For criticisms of general theories of computation (i.e. theories subsuming digital, analog and neural computation) like Piccini’s, that try to develop a general mechanistic theory of computation (see e.g. Hutto & Myin, 2018; Myin & Zahidi, 2015).

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Acknowledgements

The research of P. Pawlowski has been supported by the BOF (Bijzonder Onderzoeksfonds) post-doctoral mandate. The late stage of this reasearch has been supported by FWO grant. This research has been supported by the FWO senior post-doctoral grant 1255724N. The research of K. Zahidi was supported by the FWO-project G005321N Wiskundige verklaringen: waarom zijn ze belangrijk? The authors are listed alphabetically. The authors would like to thank two anonymous referees for their insightful critical remarks on a previous version of this paper. These comments have helped a great deal to improve this paper.

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  1. Centre for Logic and Philosophy of Science, Department of Philosophy and Moral Sciences, Ghent University, Ghent, Belgium

    Pawel Pawlowski & Karim Zahidi

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Pawlowski, P., Zahidi, K. Rigor and formalization. Synthese 203, 87 (2024). https://doi.org/10.1007/s11229-024-04521-2

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