The Philosophical Divide: Abstract Realism vs. Mathematical Realism

Introduction

While the terms abstract realism and mathematical realism might seem straightforward, they delve into complex areas of philosophy that help us understand the nature of existence and knowledge. This article aims to elucidate the significant metaphysical differences between these two perspectives, providing a deeper understanding of their implications for our understanding of reality.

Abstract Realism

Abstract realism is a philosophical theory that posits the existence of abstract objects, which are non-physical and non-mental. These objects, such as mathematical objects, properties, and propositions, are considered to exist independently of the mind and the material world. Abstract realists argue that abstract objects are not dependent on the physical world for their existence; they exist in a realm that is distinct from physical and mental reality.

One key aspect of abstract realism is the idea that abstract objects are perfectly known by the mind. This means that mathematicians, for instance, can engage with abstract mathematical concepts without relying on empirical evidence. The concept of zero in mathematics, for instance, is a quintessential example of an abstract object. While zero cannot be physically manifested or directly observed, its existence is crucial for the functioning of mathematics, and mathematicians can interact with it as a fully formed and known entity.

Mathematical Realism

Mathematical realism, on the other hand, deals specifically with the ontological status of mathematical objects. This perspective holds that mathematical entities, such as numbers and sets, exist independently of the human mind and the physical world. Mathematical realists believe that mathematical truths are objective and are discovered rather than invented. This means that mathematical objects have a reality that is independent of human thought and physical reality.

According to mathematical realists, mathematical objects have a real and objective existence, much like physical objects. This can be seen in statements like, “There exist infinitely many prime numbers,” which are taken to be true regardless of whether anyone has found or can find such numbers. Mathematical realists often use the concept of mathematical platonism to support their view, which suggests that mathematical entities exist in a non-physical realm, akin to a perfect form or ideal.

Metaphysical Differences

The core metaphysical differences between abstract realism and mathematical realism lie in the nature of the abstract entities they consider.

Abstract Objects: Abstract realism includes a broader range of entities, such as properties, propositions, and relations, which are not necessarily mathematical. For example, moral properties, aesthetic qualities, and logical structures are all considered by abstract realists to be real and independent of physical or mental realities.

Mathematical Objects: Mathematical realism, while sharing the belief in the existence of abstract entities, focuses specifically on objects that can be quantified and manipulated through mathematics. This includes numbers, sets, and other mathematical structures. The key contention is that these mathematical objects have a real existence that is discoverable and independent of human thought.

Empirical Evidence: One of the main points of contention between these two perspectives is the role of empirical evidence and discovery. While abstract realists may argue that knowledge of abstract entities is innate or purely theoretical, mathematical realists argue that mathematical knowledge is discovered through rigorous proof, evidence, and logical deduction. This means that mathematical realists would claim that mathematical truths are objective and can be verified through rigorous methods, whereas abstract realists might argue that abstract knowledge is more a matter of conceptual insight and theoretical understanding.

Conclusion

The metaphysical differences between abstract realism and mathematical realism are fundamental to how we understand the nature of reality. While both perspectives acknowledge the existence of abstract entities, their views on how these entities are known and their relationship to physical and mental realities diverge. Understanding these differences helps us to navigate complex philosophical questions about the nature of knowledge, existence, and the limits of human understanding.

By delving into the intricacies of these perspectives, we gain valuable insights into the broader landscape of metaphysics and the ways in which we conceptualize and understand the world around us. Whether you adopt an abstract realist or a mathematical realist viewpoint, these concepts continue to shape our understanding of reality and the nature of knowledge.