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The nature and existence of numbers have been subjects of profound philosophical inquiry, leading to the development of three main perspectives: Mathematical Platonism, Mathematical Nominalism, and Mathematical Fictionalism. I will talk about these perspectives, delving into their core tenets, criticisms, and implications.
Mathematical Platonism posits that numbers are abstract objects existing independently of human thought or language. According to this view, mathematical claims are true, similar to claims about the concrete world. The abstract nature of numbers, existing outside space and time, presents a conceptual challenge. Critics question how humans can interact with these abstract entities. Platonists argue that, akin to claims about the concrete world, mathematical entities exist in their own right. This perspective maintains that numbers are discoverable rather than invented.
Contrary to Platonism, Mathematical Nominalism contends that numbers are not abstract objects but rather claims about concrete things in the world. This perspective suggests that mathematical discourse is about counting or comparing tangible objects. Teaching counting to children is often cited as an example, emphasizing a connection between numbers and physical entities. However, nominalism encounters challenges when confronted with more complex mathematical concepts, such as imaginary numbers or transcendent values like pi. The reduction of all mathematical talk to discussions about concrete objects poses limitations.
Mathematical Fictionalism takes an even more radical stance, asserting that mathematical discourse is systematically false but serves as a useful fiction. Despite the success and utility of mathematics, fictionalists deny the truth of mathematical claims. They argue that the effectiveness of mathematics in practical applications does not necessarily imply the truth of mathematical statements. Fictionalists draw parallels with other useful fictions, such as moral principles in religious texts, to illustrate that utility does not equate to truth.
In conclusion, the philosophy of mathematics offers three distinctive perspectives on the existence of numbers, each with its strengths and challenges. While Platonism posits the existence of abstract objects, Nominalism grounds numbers in the concrete world. Fictionalism, on the other hand, sees mathematical claims as useful fictions. Philosophical debates surrounding these perspectives continue to enrich our understanding of the nature of numbers and the role they play in human cognition and discourse. The ongoing exploration of these ideas underscores the dynamic and evolving nature of the philosophy of mathematics. I think I agree with Platonism. I do not think that mathematics exists in real life, let alone numbers. It is just what we refer to in our minds in order to make sense of things in the world. We just happened to notice that mathematics can be used to tel a lot about how the universe works, but mathematics this the only way? I can easily imagine aliens in our universe or in another one breaking the universe-code via music, for example.
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