For example, even if one day we can go below Planck scale 10 (-35) meters, there are still 10 (-350) meters, 10 (-3500) meters and so on. We don't know whether the whole universe is limited, but the universe we know is limited.
All human knowledge comes partly from the internal structure of the human brain determined by evolution and genes, and partly from the physical interaction between the brain and the environment, including the light and sound waves emitted by things in the environment acting on the sensory organs, and then acting on the brain through neural channels, including the control of things in the environment by the brain, and so on.
In short, modern natural science has described a limited and material universe for us, including ourselves.
On the other hand, modern mathematics depicts a completely different mathematical world for us. There are infinite natural numbers, uncountable real numbers, infinite sets with larger infinite cardinality, and even so-called large cardinality in modern set theory.
Real infinity, even different levels of real infinity, has become an indispensable thing in modern mathematics. In this mathematical world, there are abstract mathematical objects or structures such as topological (geometric) spaces with arbitrary dimensions, various types of algebraic structures and function spaces. These things in the mathematical world are immaterial, do not exist in the space-time of the universe, and are not simple abstractions of any limited things in the space-time of the universe.
Is there really such an abstract mathematical world independent of the material world? If it really exists, how can the limited, material and human brain that exists in this physical universe recognize the intangible and abstract things in that mathematical world, especially the infinite things in that world?
For example, the brain's understanding of atoms and electrons in the material world will eventually indirectly affect the brain, but there is no causal connection between the abstract things in that mathematical world and the brain (in the material world), and there is a gap between them. How does the brain know them?
For another example, the range of human activities is limited, and our understanding of things far away from us in the material world, such as microscopic particles, distant stars, the origin of the universe and so on. , can only be uncertain speculation, and in mathematics, how can we be so sure to achieve infinite, even different levels of reality infinite?
On the other hand, if this abstract mathematical world does not really exist, for example, if it is only our imagination or our "thought creation", are mathematical axioms and theorems still objective truths? As we know, people generally believe that mathematical theorems are the most reliable truth and the model of human knowledge.
These questions are to be answered by contemporary philosophy of mathematics. 1At the beginning of modern mathematics at the end of the 9th century and the beginning of the 20th century, due to the discovery of the paradox of set theory, the basic problems of mathematics once puzzled the most outstanding mathematicians at that time, such as Poincare, Hilbert, Brouwer, Herman Will and von Neumann.
At that time, the research of mathematical philosophy was basically the basic research of mathematics, and there were three basic schools of mathematics: logicism, intuitionism and formalism. After entering the thirties and forties, the debate on the basis of mathematics has basically settled among mathematicians, and the norms of modern mathematics have been firmly established. This is called classical mathematics.
Today, if a mathematician or scientist is only interested in developing mathematical theories, proving mathematical theorems or applying mathematics to science, then he or she does not need to care about any basic mathematical problems. Most mathematicians think that the basic problems of mathematics no longer exist. But this does not mean that there is no problem with mathematical philosophy. The questions mentioned above should be of interest to anyone who has a little knowledge of modern mathematics and a little philosophical curiosity. These questions are a challenge to philosophers.
If we can't answer these questions well, it means that, philosophically speaking, we still lack a very thorough and clear understanding of the essence of mathematics, or the essence of our own mathematical knowledge.
Therefore, since the mid-20th century, most researches on the philosophy of mathematics no longer try to provide a foundation for mathematics, but reflect and analyze the practice of modern mathematics from a philosophical perspective. This kind of reflection and analysis may not directly affect mathematicians' mathematical practice, but it is an attempt to describe and understand our own mathematical cognitive activities, and it is an analysis and research based on human mathematical practice.
It is in this sense that the above problems are puzzling: it is the foundation of science to admit that human beings really have the knowledge of modern mathematics and its wide application in science, and then ask, as a part of the material world, how can limited human beings know those infinite and immaterial mathematical objects independent of the material world?
And if human mathematical theory is not describing an abstract mathematical world independent of the material world, what is mathematical knowledge about? Can mathematics still provide objective truth? How to apply it to science? Explaining these problems is the task of contemporary mathematical philosophy. It requires us to have a clearer and more thorough understanding of the essence of human mathematical practice. Of course, this may in turn affect the mathematical practice of mathematicians and scientists, or affect the methods of mathematics education.
Philosophers in the past, from Plato to Kant and Mill, have tried various answers to this question. However, modern mathematics developed since the end of19th century has gone far beyond the previous mathematics in content, which makes these traditional answers to philosophical questions of mathematics fall behind. This is also one of the reasons why contemporary mathematical philosophy can continue to exist.
On the other hand, whether we can answer philosophical questions about human mathematical knowledge is also the touchstone of whether philosophy can stand on its feet. From Plato to Kant and Mill, many philosophers in history took it as their responsibility to answer questions about mathematical philosophy. Kant, in particular, regards answering how mathematical knowledge is possible as one of the main purposes of his philosophy.
-Ye Feng's philosophy of mathematics in the 20th century.