TY - JOUR

T1 - A defense of Isaacson’s thesis, or how to make sense of the boundaries of finite mathematics

AU - Dopico, Pablo

N1 - Funding Information:
I would like to thank Carlo Nicolai for his support and constant feedback; and Daniel Isaacson, Luca Incurvati, Johannes Stern, Maciej Glowacki, Luca Dondoni and Amedeo Robiolio for reading and commenting on this paper. I would also like to thank two anonymous reviewers for this journal, since their detailed comments and multiple suggestions helped me improve this paper substantially. Earlier versions of the paper were presented at the Konstanz Summer School in the Philosophy of Mathematics, the Proof Theory Winter School 2021, the 2022 POLIS conference, and the Humboldt and KCL’s research seminars, so I am really thankful to all respective audiences for their pertinent comments. This work was made possible with the support of an LAHP-AHRC studentship, and of PLEXUS (grant agreement no. 101086295), a Marie Skłodowska-Curie action funded by the EU under the Horizon Europe Research and Innovation Programme.
Publisher Copyright:
© 2024, The Author(s).

PY - 2024/2/1

Y1 - 2024/2/1

N2 - Daniel Isaacson has advanced an epistemic notion of arithmetical truth according to which the latter is the set of truths that we grasp on the basis of our understanding of the structure of natural numbers alone. Isaacson’s thesis is then the claim that Peano Arithmetic (PA) is the theory of finite mathematics, in the sense that it proves all and only arithmetical truths thus understood. In this paper, we raise a challenge for the thesis and show how it can be overcome. We introduce the concept of purity for theories of arithmetic: a theory of arithmetic is pure when it only proves arithmetical truths. Then, we argue that, under Isaacson’s thesis, some PA-provable truths—including transfinite induction claims for infinite ordinals and some consistency statements—are seemingly not arithmetical in Isaacson’s sense, and hence that Isaacson’s thesis might entail the impurity of PA. Nonetheless, we conjecture that the advocate of Isaacson’s thesis can avoid this undesirable consequence: the arithmetical nature, as understood by Isaacson, of all contentious PA-provable statements can be justified. As a case study, we explore how this is done for transfinite induction claims with infinite ordinals below ε . To this end, we show that the PA-proof of such claims employs exclusively resources from finite mathematics, and that ordinals below ε are finitary objects despite being infinite.

AB - Daniel Isaacson has advanced an epistemic notion of arithmetical truth according to which the latter is the set of truths that we grasp on the basis of our understanding of the structure of natural numbers alone. Isaacson’s thesis is then the claim that Peano Arithmetic (PA) is the theory of finite mathematics, in the sense that it proves all and only arithmetical truths thus understood. In this paper, we raise a challenge for the thesis and show how it can be overcome. We introduce the concept of purity for theories of arithmetic: a theory of arithmetic is pure when it only proves arithmetical truths. Then, we argue that, under Isaacson’s thesis, some PA-provable truths—including transfinite induction claims for infinite ordinals and some consistency statements—are seemingly not arithmetical in Isaacson’s sense, and hence that Isaacson’s thesis might entail the impurity of PA. Nonetheless, we conjecture that the advocate of Isaacson’s thesis can avoid this undesirable consequence: the arithmetical nature, as understood by Isaacson, of all contentious PA-provable statements can be justified. As a case study, we explore how this is done for transfinite induction claims with infinite ordinals below ε . To this end, we show that the PA-proof of such claims employs exclusively resources from finite mathematics, and that ordinals below ε are finitary objects despite being infinite.

UR - http://www.scopus.com/inward/record.url?scp=85183724423&partnerID=8YFLogxK

U2 - 10.1007/s11229-024-04488-0

DO - 10.1007/s11229-024-04488-0

M3 - Article

SN - 0039-7857

VL - 203

JO - SYNTHESE

JF - SYNTHESE

IS - 2

M1 - 54

ER -