Realism admits that mathematical objects exist independently of our thoughts, while anti-realism holds that mathematical objects do not exist or exist independently of our thoughts, while nominalism asserts that mathematical objects do not exist at all. This classification seems to be accepted by most authors.
The debate between mathematical realism and mathematical anti-realism is the focus of mathematical philosophy debate in the twentieth century. Historically, the dispute between contemporary mathematical realism and mathematical anti-realism is related to and different from the dispute between realism and nominalism in western traditional philosophy. First of all, abstract mathematical objects are different from the so-called * * * phase or concept in western traditional philosophy. * * * stages and concepts are related to a kind of concrete things, which can be regarded as the representation or abstraction of a kind of concrete things.
For example, Plato imagined that perfect circles really existed, while those concrete imperfect circles in the material world were just the shadows of perfect circles. Conversely, we can also say that the so-called perfect circle is, in a sense, the representation or abstraction of a concrete and imperfect circle. However, if the universe is finite and discrete, then the exact circle in Euclid's geometry actually has no "shadow" in the material world. Similarly, those very large numbers may not be the representation or abstraction of any real concrete thing or its physical quantity in the universe. Not to mention those abstract function spaces, topological spaces, infinite cardinality and even large cardinality. They have no "shadow" in the material world, nor are they the direct abstraction of any specific thing or its attributes.
They are different from the * * * phase or concept in traditional philosophy. Platonism, realism and nominalism in traditional philosophy refer to the theory of affirming or denying the independent existence of * * * and ideas. Therefore, realism or Platonism in modern mathematical philosophy transcends realism and Platonism in the traditional sense. Realism in modern mathematical philosophy is to assert the objective reality of an abstract mathematical world that is completely independent of the material world and has no similarity with the material world. This is due to some unprecedented characteristics of modern mathematics, that is, the objects that modern mathematics talks about are not simple abstractions of concrete things, and their eyes are far beyond any concrete things in the material world, and there can be no "shadows" in the material world.
The characteristics of modern mathematics make the debate between realism and anti-realism sharper and more meaningful.
This feature of modern mathematics makes the conflict between realism and anti-realism more acute, and also makes the problems faced by the above-mentioned simple mathematical realism and anti-realism more prominent.
On the one hand, if an abstract thing, such as * * * phase or concept, is only a representative of a corresponding concrete thing, or the corresponding concrete thing is "abstract" in a sense, then realists may say that it is not so incredible to assert the existence of abstract things. They may say that we can know those abstract things through concrete things, and then the epistemological problem of abstract things can be solved.
On the contrary, anti-realists can also say that when we talk about so-called abstract things, we just adopt a certain way to talk about those corresponding concrete things, instead of really asserting the existence of those so-called abstract things. In other words, Plato's so-called perfect circle is just an illusion. We just imagine such a perfect circle as many imperfect circles in the real world, and the so-called perfect circle in our geometry should be understood as the corresponding (approximate) conclusions about various imperfect circles.
According to this, anti-realists can think that it is meaningless to assert that the concept of * * * exists independently of concrete things, and it can not increase our real knowledge, but is only the theory of some philosophers, which is a complicated philosophy. In other words, the question about whether the concept of * * * is independent of those specific things is based on the misuse of language, which is a much ado about nothing.
However, modern mathematics does seem to be talking about abstract mathematical objects that are completely independent of concrete things and have no similarity with concrete things, and mathematics is considered to provide the most reliable knowledge and scientific basis.
Mathematical truth is regarded as the most reliable truth. Many mathematicians and scientists whom we respect seem to hold this belief. All these seem to support realism, which shows that realism seems to be a belief confirmed by scientific practice, which is not only the result of some philosophers' delusions, but also a belief based on language misunderstanding.
On the other hand, the objects studied by modern mathematics, especially infinite objects and abstract mathematical structures, go far beyond the simple abstraction of concrete things in the universe and have no similarity with concrete things. It is hard to say that we can know those infinitely abstract mathematical objects through limited concrete things. At least, it needs to construct a complex philosophical theory to explain.
Moreover, we can't avoid talking about real numbers, functions, topological spaces and other mathematical objects like avoiding talking about perfect circles (but only imperfect circles in the material world) (because there is nothing similar enough to them in the material world to replace them).
Therefore, these two characteristics of modern mathematics, that is, it is the basis of science and its transcendence (at least on the surface), not only strengthen the reasons for supporting realism, but also increase the difficulty of solving the problems of realism epistemology. This makes us believe that there is really a mystery in it, not just a problem that some confused minds imagine to be much ado about nothing.
-Ye Feng's philosophy of mathematics in the 20th century.