As early as the 1900s, people discovered that when they were studying math, they didn't know what they were learning. When we learn biology, we are studying organisms that exist in the real world. When we learn physics, we are studying phenomena in the real world, and we can verify it experimentally. But what about mathematics? Is it an object that studies the real world or an abstract entity? Does it exist only in our minds or in some other field?
Ancient Greek philosopher Platonic believes that mathematical objects, shapes, numbers and their relationships belong to their own ideal world. Unlike our world, he calls it the formal world. But Hilbert is a modern person, and the perspective of Platonism is too mysterious to it. Hilbert is interested in the actual question: what can be proved and what can't be proved? German philosopher Emmanuel Kant proposed another popular view that mathematics is a psychological structure based on our intuition. But Hilbert thinks this is also a problem, after all, intuition often leads us astray.
has multiple perspectives in other disciplines. For example, literature is praised for its ability to have multiple interpretations of the same thing, and even the laws of physics are modified over time. But for mathematics, this doesn't work, and the fact that two plus two equals four will never be modified and changed.
Different mathematicians have different explanations of mathematics. In addition to making the reputation of mathematics questioned, these different views also lead to the existence of practical problems. Those who agree with intuitiveism do not believe that the concept of infinity is suitable for mathematics. Those who think logic is the foundation of mathematics encounter logical paradoxes and contradictions. We can't let different mathematicians study according to different rules, which is unacceptable to Hilbert, and he intends to do something about it.
He proposed an ambitious plan: to axiomatically all mathematics. Hilbert believes that if we regard mathematics as a formal system, there will be no more differences on what is allowed and does not.
So what is a formal system? It consists of a bunch of formal symbols that can then be operated according to a fixed set of rules. Chess is a formal system, where chess pieces are formal symbols that move according to the rules of the game. Importantly, chess pieces and rules make no sense outside of the game. Hilbert thinks that's how we look at mathematics.
About 2,000 years ago, Euclid did something similar. By proposing five axioms and some rules, the Euclid geometry system was born. With these axioms, we don't need to know whether the triangle is in the formal world or in our minds, we know that the sum of its angles is always 180 degrees. So in this way, Euclidean geometry can be regarded as a formal system.
Hilbert wants to do something similar to find the basic axioms that can build a formal system, which will remove any disagreement about what is allowed and not allowed. First, Hilbert considered three main contents in the basic mathematical system, namely consistency, integrity and determinability.
consistency means that no contradiction can be proved in the system. For example, if it is not possible to prove that 2+2=4 and 2+2≠4, inconsistency will make the entire system useless. Integrity means that all real mathematical statements present in the system can be proven within the system. Finally, there is determinability, and there should be a valid procedure to determine the authenticity of any mathematical statement.
22 years old Alan Turing developed a strong interest in the last content - determinability: Are there any valid procedures to determine the authenticity of any mathematical statement? Here Turing found a small problem, what exactly is an effective program? Since the word "effective program" is too vague to make any strict definition, he decided to define it himself.
His definition is based on a "human computer".He believes that anything that "human computers" can do by unconsciously following a series of instructions is an effective program. He considered the basic actions that "human computers" do when computing: first they read instructions, second they read and write symbols on a piece of paper according to the instructions, then they occasionally erase the symbols and replace them with new symbols, and finally stop when they finish the calculation.
Turing realized that every step the "human computer" can be copied by a very simple theoretical machine. He conceived a theoretical machine that could perform the same tasks as any other machine. Although this theoretical machine is very simple, it can in principle do everything that a "human computer" can do. This abstract machine is now called Turing machine , and is the definition of a valid program. An effective program is anything that can be calculated by a Turing machine in a limited time.
This gave birth to the entire field of computer science. Turing machines are a blueprint for modern computers, from desktops, laptops to smartphones to computers on space stations, all based on Turing's model. Anything that any of these machines can do can in principle be done by a Turing machine.
