The concept of mathematical realism is an eschatological philosophy that believes that all objects in mathematics are real, causally inert, and eternal. This view raises the question of how humans can know anything about mathematical objects. Physical objects require a causal relation between the knower and the object. Unlike physical objects, mathematical objects do not have causal relation with their knowners. As such, mathematicians believe that they have true beliefs about their mathematical objects.
This cosmology posits that physical objects exist, but that the mathematical cloud encapsulates them is a separate entity. For example, elementary particles have a Lie group corresponding to them, which is a fundamental principle of realism. Whether the object is real or not is up for debate, but mathematical realism has its advocates. This is the most controversial theory, and it has been called a “scientific” religion.
However, some philosophers disagree with this belief. Those who believe in mathematical realism say that mathematics is an independent entity that contributes to the scientific explanation of the world. If they were not, then mathematics would be merely an abstraction. For instance, if we were to have a ‘cluster’ of interacting objects, there would be no way to know which object is which. Then, we could have a cloud of geometric objects that corresponds to human brain activity.
Mathematicians claim that the truths of mathematics are incomprehensible, and they are independent of human activities. In fact, they are facts, and mathematics is based on them. To be truly meaningful, mathematicians must discover these facts. In other words, a mathematical statement is true if it describes these mathematical facts. Therefore, it is impossible to deny that the reality of mathematical structures is indistinguishable from the reality of the world.
Why Is Mathematical Realism So Important?
Despite the opposition of mathematical realism with scientific realism, both types of realism entail an insistence on the existence of mathematical entities. In this sense, they are incompatible with the scientific realism theory of science. In fact, it is an idealistic idea, and the mathematical world is a rational construct. This is the only way to know the truth. So, why is it so important?
In mathematics, realism is a way of looking at the world from the outside. In this view, mathematics is objective and independent of mathematicians. Hence, mathematical realism is a necessary condition of science. It is impossible to create a mathematical object, without a world. For that reason, it is a mistake to say that the mathematics of realism is true. It does not have any objective existence.
Moreover, there are many arguments that support the mathematical realism theory. For example, the indispensability argument is a strong argument for platonism. It is the only argument for platonism. If it fails, it would leave platonism in a state of bankruptcy. And since the arguments of the realism theory are incommensurable, this view cannot be argued against. It is not possible for any form of realism to be true without proofs.
In addition to these arguments, there is another major type of mathematical realism. In this case, the realism is a kind of realism based on the fact that mathematics is the study of objects, not of people. The existence of the mathematical object is also a matter of faith. Although this is a fundamental philosophical view, it is a mistaken belief. It is not the only form of realism, but it is the most popular form of the philosophy of mathematics.
Mathematical objects and properties
In the modern era, the idea of mathematical objects and their properties has been the subject of controversy for centuries. The idea that we are able to perceive things using our minds is problematic, but many modern working mathematicians and philosophers of mathematics are adamant that the truth is not observable. In this case, mathematical realists view the world as an unobservable, and therefore cannot be proven. As a result, they reject the notion of realism.
In addition to its philosophical appeal, mathematical realism is often linked to anti-realism. This belief, which says that the world is not real, but does not have any immaterial properties, is a common philosophy. While the concept of realism is a logically consistent concept, it cannot be proven by any other means. Thus, it is impossible to define mathematics in terms of its properties. Rather, it is a set of principles that relate to reality.