Do Mathematical Objects Exist, or Are They Just Linguistic Constructs

Since the beginning of time, mathematics has had a special place within human knowledge. It connects to all academic fields, including biology, chemistry, physics, and many other STEM-related fields. The tools mathematics provides, including sets, objects, formulas, and proofs, are indispensable to all fields of study. However, in the field of philosophy, rather than the utilization of these abstracts, philosophers are concerned with the nature of these abstracts themselves. Are these abstract formulas and constants discovered or invented? The main contending positions on this matter include Platonism and Nominalism, where Platonism says that these abstracts exist in a higher realm in their perfected state, while Nominalists want to say that these abstracts are simply linguistic conventions and tools. This debate has predominantly been explored within the realm of analytical philosophy, especially through the dialogue of W.V.O Quine and Rudolf Carnap; their conversations help us understand this debate and the different positions of Platonism and Nominalism. Carnap, taking the view of the Nominalist, wants to explicitly state his view through the fact that his linguistic frameworks would lead us to believe that mathematical objects and their existence depend on the language and linguistic framework an individual adopts, which aligns with pragmatic and language-relativist views. Quine, by contrast, wants to say that we ought to be ontologically committed to entities that are indispensable to our theories. Thus leading him to believe that mathematics is not simply a byproduct of our language. In this essay, I will go on to argue that the existence of mathematical objects is best understood as framework-dependent entities, where the objectivity of these framework-dependent systems depends on the pragmatic success of language and the predictive power that these systems provide us.