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Mathematical realism is a philosophical position that asserts the existence of mathematical objects and structures as independent entities that exist objectively, regardless of human thought or perception. According to mathematical realism, mathematical truths are discovered rather than invented, and they have an existence that is separate from the physical world.
Mathematical realists argue that mathematical objects, such as numbers, sets, and geometric shapes, exist in a realm of their own, often referred to as the "Platonic realm." They believe that these objects have properties and relationships that can be discovered through mathematical reasoning and investigation.
One of the key arguments for mathematical realism is the indispensability argument. This argument suggests that mathematics plays an essential role in scientific theories and explanations, and it is difficult to explain the success and applicability of mathematics in science if mathematical objects are merely human inventions or mental constructs.
Mathematical realists also emphasize the objectivity and universality of mathematical truths. They argue that mathematical statements, such as "2+2=4," hold true regardless of cultural, historical, or individual perspectives. These truths are seen as timeless and independent of human beliefs or experiences.
However, mathematical realism is not without its challenges. One of the main criticisms is the problem of access to mathematical objects. Unlike physical objects that can be observed or measured, mathematical objects are not directly accessible through our senses. Critics argue that if mathematical objects exist independently, it is unclear how we can have knowledge or understanding of them.
Another challenge is the ontological commitment of mathematical realism. Accepting the existence of a separate realm of mathematical objects raises questions about the nature of this realm and its relationship to the physical world. Some philosophers propose alternative positions, such as mathematical fictionalism or nominalism, which deny the existence of mathematical objects or interpret them as useful fictions or linguistic constructs.
In conclusion, mathematical realism is a philosophical position that asserts the existence of mathematical objects as independent entities with objective properties. It argues that mathematical truths are discovered and have a separate existence from the physical world. However, this position faces challenges regarding access to mathematical objects and the ontological commitment it entails.