Editing Arithmetic (section)

=== Philosophy ===
{{main|Philosophy of mathematics}}
The philosophy of arithmetic studies the fundamental concepts and principles underlying numbers and arithmetic operations. It explores the nature and [[Ontology|ontological status]] of numbers, the relation of arithmetic to language and [[logic]], and how it is possible to acquire arithmetic [[knowledge]].<ref>{{multiref | {{harvnb|Hofweber|2016|pp=153–154, 162–163}} | {{harvnb|Oliver|2005|p=58}} | {{harvnb|Sierpinska|Lerman|1996|p=827}} }}</ref>

According to [[Platonism]], numbers have mind-independent existence: they exist as [[abstract objects]] outside spacetime and without causal powers.<ref>{{multiref | {{harvnb|Oliver|2005|p=58}} | {{harvnb|Horsten|2023|loc=§ 3. Platonism}} }}</ref>{{efn|An [[Quine–Putnam indispensability argument|influential argument]] for Platonism, first formulated by [[Willard Van Orman Quine]] and [[Hilary Putnam]], states that numbers exist because they are indispensable to the best scientific theories.{{sfn|Colyvan|2023|loc=Lead Section}}}} This view is rejected by [[Intuitionism|intuitionists]], who claim that mathematical objects are mental constructions.<ref>{{harvnb|Horsten|2023|loc=§ 2.2 Intuitionism}}</ref> Further theories are [[logicism]], which holds that mathematical truths are reducible to [[logical truth]]s,<ref>{{multiref | {{harvnb|Horsten|2023|loc=§ 2.1 Logicism}} | {{harvnb|Hofweber|2016|pp=174–175}} }}</ref> and [[Formalism (philosophy of mathematics)|formalism]], which states that mathematical principles are rules of how symbols are manipulated without claiming that they correspond to entities outside the rule-governed activity.<ref>{{harvnb|Weir|2022|loc=Lead Section}}</ref>

The traditionally dominant view in the [[epistemology]] of arithmetic is that arithmetic truths are knowable [[a priori]]. This means that they can be known by thinking alone without the need to rely on [[sensory experience]].<ref>{{multiref | {{harvnb|Oliver|2005|p=58}} | {{harvnb|Sierpinska|Lerman|1996|p=830}} }}</ref> Some proponents of this view state that arithmetic knowledge is innate while others claim that there is some form of [[rational intuition]] through which mathematical truths can be apprehended.<ref>{{multiref | {{harvnb|Oliver|2005|p=58}} | {{harvnb|Sierpinska|Lerman|1996|pp=827–876}} }}</ref> A more recent alternative view was suggested by [[Naturalism (philosophy)|naturalist]] philosophers like [[Willard Van Orman Quine]], who argue that mathematical principles are high-level generalizations that are ultimately grounded in the sensory world as described by the empirical sciences.<ref>{{multiref | {{harvnb|Horsten|2023|loc=§ 3.2 Naturalism and Indispensability}} | {{harvnb|Sierpinska|Lerman|1996|p=830}} }}</ref>