TY - JOUR

T1 - On the constructive axiomatic method

AU - Родин, Андрей Вячеславович

PY - 2018/6

Y1 - 2018/6

N2 - The received notion of axiomatic method stemming from Hilbert is not fully adequate to the recent successful practice of axiomatizing mathematical theories. The axiomatic architecture of Homotopy type theory (HoTT) does not fit the pattern of formal axiomatic theory in the standard sense of the word. However this theory falls under a more general and in some respects more traditional notion of axiomatic theory, which I call after Hilbert and Bernays constructive and demonstrate using the Classical example of the First Book of Euclid's Elements. I also argue that HoTT is not unique in the respect but represents a wider trend in today's mathematics, which also includes Topos theory and some other developments. On the basis of these modern and ancient examples I claim that the received semantic-oriented formal axiomatic method defended recently by Hintikka is not self-sustained but requires a support of constructive method. Finally I provide an epistemological argument showing that the constructive axiomatic method is more apt to present scientific theories than the received axiomatic method.

AB - The received notion of axiomatic method stemming from Hilbert is not fully adequate to the recent successful practice of axiomatizing mathematical theories. The axiomatic architecture of Homotopy type theory (HoTT) does not fit the pattern of formal axiomatic theory in the standard sense of the word. However this theory falls under a more general and in some respects more traditional notion of axiomatic theory, which I call after Hilbert and Bernays constructive and demonstrate using the Classical example of the First Book of Euclid's Elements. I also argue that HoTT is not unique in the respect but represents a wider trend in today's mathematics, which also includes Topos theory and some other developments. On the basis of these modern and ancient examples I claim that the received semantic-oriented formal axiomatic method defended recently by Hintikka is not self-sustained but requires a support of constructive method. Finally I provide an epistemological argument showing that the constructive axiomatic method is more apt to present scientific theories than the received axiomatic method.

KW - Axiomatic method

KW - Constructive mathematics

KW - Euclid

KW - Homotopy type theory

KW - Topos theory

KW - Homotopy Type theory

KW - Constructive Mathematics

KW - LOGIC

UR - http://www.scopus.com/inward/record.url?scp=85048557587&partnerID=8YFLogxK

UR - http://arxiv.org/abs/1408.3591

UR - http://www.mendeley.com/research/constructive-axiomatic-method

U2 - 10.2143/LEA.242.0.3284751

DO - 10.2143/LEA.242.0.3284751

M3 - Review article

VL - 61

SP - 201

EP - 231

JO - Logique et Analyse

JF - Logique et Analyse

SN - 0024-5836

IS - 242

ER -