TY - JOUR ID - 88099 TI - Philosophical Thinking about Abstract Algebra JO - Philosophy JA - JOP LA - en SN - 2008-1553 AU - ِYousefi, Saeed AU - Mahmood Robati, Sajjad AD - Graduate Master of Philosophy, Imam Khomeini International University AD - Associate Professor of Pure Mathematics, Imam Khomeini International University Y1 - 2022 PY - 2022 VL - 20 IS - 1 SP - 189 EP - 204 KW - Frege KW - Logicism KW - Hilbert KW - formalism KW - Wittgenstein DO - 10.22059/jop.2022.331717.1006653 N2 - In this paper, we intend to examine the controversy between Frege and Hilbert over geometry about abstract algebra; whether Frege's thinking about abstract algebra works out or Hilbert's? To this end, we first study the times when Frege and Hilbert were engaged in philosophy, to have an idea of the dominant intellectual atmosphere of that time and its influence on these two mathematicians. With this in mind, then we have pointed out some differences between Frege and Hilbert's approaches. We have concluded that their most important differences are the role of the subject in mathematics and the discussion about compatibility. After presenting the normative definition of algebra, we have investigated the points made by Hilbert and Frege, using the later Wittgenstein theories expressed in the books On Certainty and Philosophical Investigation. Finally, by considering all the aspects, we have concluded that what seems to be more useful in the philosophical study of algebra is a third approach, which is Wittgenstein's approach with some modifications, since it guarantees some important features. These features are meaningfulness, intersubjectivity, and rule-orientation, all of which are important features for mathematicians. UR - https://jop.ut.ac.ir/article_88099.html L1 - https://jop.ut.ac.ir/article_88099_ff5d279c52faea5c96828653cb07cf80.pdf ER -