Abstract: The relationship between mathematics and philosophy: This paper first introduces Plato's philosophy of mathematics, then tells the philosophy of mathematics, then introduces inevitability and innate knowledge, then introduces the three principles and the modern development of philosophy of mathematics, and finally briefly summarizes the philosophy of mathematics. Keywords: Plato's philosophy of mathematics, three principles of innate inevitability knowledge
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One: Plato's philosophy of mathematics
Plato's mathematical philosophy is mainly embodied in the problem of mathematical ontology. He takes a realistic position on the problem of mathematical ontology, that is, he thinks that the object of mathematics is what he says? The world of ideas? The real existence. Plato's understanding is based on his belief in the absolute truth of mathematics. He believes that mathematical objects are independent and objective existence independent of human thinking.
In addition to realism, Plato also emphasized the congenital nature of mathematical cognitive activities. According to Plato's point of view, the world of ideas is the object of rational understanding, and this understanding can only be achieved through? Memories of nature? Be realized; Because objects also exist in the world of ideas, in Plato's view, mathematics belongs to the science of studying ideas? Dialectical method? , that is, an innate understanding.
Besides the innate nature of mathematics, Plato also emphasized the role of mathematical cognition in general rational cognition: because mathematical objects are said to be between perceptual things and ideas? Intermediary object? Therefore, there is also a kind of understanding of mathematics? Bridge? Function, it can stimulate people, thus causing the soul? Innate knowledge? The memory of. Plato said: geometry will lead the soul to truth and produce philosophical spirit. ?
Two: philosophy of mathematics
The development of mathematics in the direction of formalization and abstraction, the progress of mathematical logic and basic research of mathematics, and the discovery of paradox have created a new period of mathematical philosophy research.
Mathematicians believe that mathematics is based on a series of self-evident principles. The duty of mathematicians is to discover the conclusions drawn from these principles as completely as possible. He should frankly admit that these principles themselves are some obvious insights, so they form an unbreakable and eternal foundation. On the contrary, philosophers will let mathematicians explore the conclusions drawn from these principles; He is not interested in these conclusions. However, he must explain the fact that we have some insights for us to use, and he also needs to explain the objects related to these insights. They agree that the object of mathematics does not belong to the material world, and the insight of mathematics cannot be based on experience, because this self-evident thing that is suitable for mathematical principles does not belong to our empirical knowledge but is unique to mathematical principles.
Three: inevitability and innate knowledge
Philosophy of mathematics is a branch of epistemology, which deals with cognition and knowledge in philosophy to a great extent. However, mathematics is different from other learning efforts at least on the surface. In particular, it is different from other scientific pursuits. Mathematical propositions, such as 7+5= 12, are sometimes regarded as examples of inevitable truth, and it is almost impossible to have other situations.
Scientists will be happy to admit that her more basic argument may be wrong. The history of the scientific revolution confirmed this humility, in which the long-held belief was overthrown. Can mathematics really support this humility? Can you doubt that mathematical induction holds true for natural numbers? Can you doubt that 5+7= 12? Is there a mathematical revolution, the result of which is to overthrow the long-standing core mathematical concepts? On the contrary, mathematical methodology does not seem as necessary or inevitable as science. Unlike science, mathematics is developed through proof. A successful and correct proof sweeps away all doubts based on reason, not only
Are reasonable doubts. A mathematical proof should show that its premise logically contains its conclusion. The premise cannot be true, and the conclusion cannot be false.
? Innate? What does this word mean? Before experience? Or? Independent of experience? . This is an epistemological concept, if knowledge is not based on any? Experience about the special process of events in the real world? That proposition is defined as innate knowledge.
Some philosophers think that there is no innate knowledge, but for others, typical innate knowledge includes? All red objects are colored? And then what? Nothing can be completely red and green at the same time. . Mathematics doesn't seem to be based on observation like science, but on proof.
Therefore, any complete mathematical philosophy has the obligation to explain the inevitability and congenital nature of mathematics at least on the surface. Four: Three Basic Principles
The basic problems about the logic and epistemology of mathematics have not been completely solved so far. This question is very important for mathematicians and philosophers, because in? The most reliable science? On this basis, any uncertainty will be extremely disturbing. Of all the efforts made so far to solve this problem. None of them can be said to have solved all the difficulties. These efforts mainly follow three directions: logicism with Russell as the main advocate, intuitionism with Brouwer as the representative, and formalism with Hilbert.
One of the most important problems in the foundation of mathematics is the relationship between mathematics and logic. The theory of logicism is that mathematics can be reduced to logic, and accordingly, mathematics is only a part of logic. The argument of logicism can be divided into two parts. First, mathematical concepts can be derived from logical concepts through clear definitions. The other part is that mathematical theorems can be deduced from logical axioms through pure logical deduction.
Intuitionistic mathematicians believe that mathematical work should be regarded as the natural function of his intelligence and a free and vivid thinking activity. In his view, mathematics is the product of human spirit. He uses language, whether natural or formal, just to exchange ideas, that is, to let others or himself understand his own mathematical ideas. This language partner is not a representative of mathematics, let alone mathematics itself.
Dealing with the structure of mathematics immediately is probably the most in line with the positive attitude of intuitionists. The most important cornerstone of this construction is the concept of unit, which is the construction principle on which integer sequences depend. Integers must be treated as a unit, and the only difference between them is their position in this sequence.
The leading idea of Hilbert's proof theory is that even if the statement of classical mathematics is actually wrong in content, classical mathematics contains an internal closed program, which runs according to fixed rules known to all mathematicians. It basically includes the continuous construction of some combinations of original symbols, which are considered as? Is that correct? Or? Proved it? . What is this constructor? Limited? And direct and constructive.
Five: the modern development of mathematical philosophy
Since 1950s, the philosophy of mathematics has entered a new period of development. Compared with the basic research of mathematics, this new development shows some remarkable different characteristics.
(1) the change of research position, that is, from being seriously divorced from actual mathematical activities to being closely integrated with them. Specifically, in the basic research of mathematics, although logicism and other schools put forward different views, what they actually do is a kind of work that tends to be standardized. Modern philosophy of mathematics thinks that philosophy of mathematics should be in the work of mathematicians? Philosophy of life? That is, the philosophical views of mathematics researchers, teachers and users on their work.
The shift of research position directly leads to new mathematical concepts. For example, it is based on the investigation of mathematicians' actual words and deeds and examples in the history of mathematics that empiricism can be used in modern mathematical philosophy? Revival? .
(2) The contents and methods of the research show obvious openness, especially absorbing many important research questions and beneficial ideas from the general philosophy of science, which is very different from the closed basic research of mathematics in the past.
For example, lakatos's quasi-empirical view of mathematics actually extended Popper's falsificationist philosophy of science to the field of mathematics. For another example, under the influence of Kuhn's philosophy of science research, the social and cultural research on mathematics appeared. Obviously, this dynamic study of mathematics is different from the previous research tradition, that is, simple.
The static analysis of the logical structure of mathematical knowledge is quite different.
In addition, another important feature of the new research is that it highlights the sociality of mathematical research. Finally, the close connection with actual mathematical activities can also be regarded as an important embodiment of the openness of modern mathematical philosophy research. Especially as the research of thinking method, the research of mathematical methodology has made new progress in modern times.
Six: Summary
Mathematics and philosophy are different from each other and silently connected. If you knock on one door, another door will open the window for you immediately. The philosophy of mathematics is constantly changing. With the development of the times, there will be different performances, and people's research will be different from before.
refer to
(1) Xia He wrote Western Philosophy of Mathematics.
People's Publishing House 1986 65438+ 10/0/3 published in October.
(2) Lin Xiashui, editor-in-chief of Translation Series of Mathematical Philosophy.
P24-p25 Knowledge Publishing House published in July 1986.
(3) The Philosophy of Mathematics-Reflections on Mathematics, edited by Wang Ning, is a translation series of western mathematical cultural concepts.
Beautiful Stuart? Sha Boli's Translation of Hao Zhaokuan and Yang Du
Fudan University Press published p20-p23 in February 2009.
(4) Philosophy of Mathematics is a beautiful Paul? Benacerraf Hillary? Putnam
The Commercial Press published p47-p76 in February 2003.
(5) Xu Lizhi read Xu Lizhi's Philosophy of Mathematics.
Dalian University of Technology Press published p73-p82 in June 2008.
April 6, 1965 438+02
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