Founding Mathematics on Semantic Conventions

This book presents a new nominalistic philosophy of mathematics: semantic conventionalism. Its central thesis is that mathematics should be founded on the human ability to create language - and specifically, the ability to institute conventions for the truth conditions of sentences.

This philosophical stance leads to an alternative way of practicing mathematics: instead of "building" objects out of sets, a mathematician should introduce new syntactical sentence types, together with their truth conditions, as he or she develops a theory.

Semantic conventionalism is justified first through criticism of Cantorian set theory, intuitionism, logicism, and predicativism; then on its own terms; and finally, exemplified by a detailed reconstruction of arithmetic and real analysis.

Also included is a simple solution to the liar paradox and the other paradoxes that have traditionally been recognized as semantic. And since it is argued that mathematics is semantics, this solution also applies to Russell's paradox and the other mathematical paradoxes of self-reference.

In addition to philosophers who care about the metaphysics and epistemology of mathematics or the paradoxes of self-reference, this book should appeal to mathematicians interested in alternative approaches.

這本書提出了一種新的數學名義主義哲學：語義傳統主義。其核心論點是數學應該建立在人類創造語言的能力上 - 具體而言，是建立在為句子的真值條件制定慣例的能力上。

這種哲學立場導致了一種替代的數學實踐方式：數學家不應該通過從集合中“構建”對象，而應該在發展理論的過程中引入新的句法句型，以及它們的真值條件。

語義傳統主義首先通過批評康托爾集合論、直覺主義、邏輯主義和預言主義來進行證明；然後根據自身的條件進行證明；最後，通過對算術和實分析的詳細重構來予以實例證明。

書中還包括了對語義傳統主義所認識的謊言悖論和其他傳統上被認為是語義的悖論的簡單解決方案。並且由於主張數學是語義學，這個解決方案也適用於羅素悖論和其他與自我參照有關的數學悖論。

除了關心數學的形而上學和認識論，或者自我參照悖論的哲學家外，這本書還應該吸引對替代方法感興趣的數學家。