TY - JOUR
T1 - The Case for the Irreducibility of Geometry to Algebra
AU - Pambuccian, Victor
AU - Schacht, Celia
N1 - Publisher Copyright:
© 2021 The Authors. Published by Oxford University Press. All rights reserved.
PY - 2022/2/1
Y1 - 2022/2/1
N2 - This paper provides a definitive answer, based on considerations derived from first-order logic, to the question regarding the status of elementary geometry, whether elementary geometry can be reduced to algebra. The answer we arrive at is negative, and is based on a series of structural questions that can be asked only inside the geometric formal theory, as well as the consideration of reverse geometry, which is the art of finding minimal axiom systems strong enough to prove certain geometrical theorems, given that there are no algebraic structures that one could associate with those minimal axiom systems.
AB - This paper provides a definitive answer, based on considerations derived from first-order logic, to the question regarding the status of elementary geometry, whether elementary geometry can be reduced to algebra. The answer we arrive at is negative, and is based on a series of structural questions that can be asked only inside the geometric formal theory, as well as the consideration of reverse geometry, which is the art of finding minimal axiom systems strong enough to prove certain geometrical theorems, given that there are no algebraic structures that one could associate with those minimal axiom systems.
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U2 - 10.1093/philmat/nkab022
DO - 10.1093/philmat/nkab022
M3 - Article
AN - SCOPUS:85125457716
SN - 0031-8019
VL - 30
SP - 1
EP - 31
JO - Philosophia Mathematica
JF - Philosophia Mathematica
IS - 1
ER -