Let's look at the definition of Wikipedia: What is science? What is mathematics?
It can be concluded from the above definition that both are knowledge. The difference is that scientific knowledge is related to the specific universe and is an experimental explanation and prediction; Mathematical knowledge is about abstract things, which is obtained through deduction on the basis of axioms and definitions.
Whether mathematical research is purely abstract is also questionable in philosophy. For example, can we get 5+7= 12 without any experience (this formula was mentioned by Kant in Critique of Pure Reason)? Russell and Whitehead tried to deduce the whole arithmetic system only by logic, resulting in 10 volume, 1800 pages. Finally, they got close.
But as far as teleology is concerned, mathematics can indeed be divorced from concrete applications and can be carried out purely in the abstract field, without experiments, just thinking, that is, logical reasoning. Many mathematical achievements in history, such as irrational numbers, imaginary numbers and even non-Euclidean geometry, have been idle for a long time, and people don't know how to use them.
On the other hand, science is different. Since its birth, science has been related to application, relying on experiments or specific verification. Although science will also use some abstract concepts, such as gravity, supergravity and even quarks, which no one has ever seen, they will eventually make realistic predictions and then test them with experience.
Physics, chemistry, biology, psychology and economics are all linked to concrete things.
Mathematics uses deduction, while scientific research relies more on induction and deduction.
Deduction is syllogism. If the major premise and minor premise are correct, then the result will always be true.
The research method of mathematics is to make axiomatic assumptions (without proof) first, and then deduce them step by step according to deductive method, so as to obtain deterministic knowledge.
If a thing can be deduced layer by layer, then we can trace it back layer by layer. Who can guarantee that the original assumption is correct? This is the first question of cause put forward by Aristotle.
In mathematics, this problem is solved by axioms, which are self-evident and do not need to be proved, such as the shortest straight line between two points. In other words, axiom itself is not the product of logical reasoning, it is the starting point of logical reasoning, and we just assume that it is correct. It's not necessarily right. For example, we now know that in the large-scale universe, the straight line between two points is not the shortest, because space will be distorted.
What about induction? The simplest example is the swan. One swan is white, and two are white. We see thousands of swans are white, so we come to a conclusion that all swans are white. We all know the result. Later, people found the black swan in Australia. So induction is unreliable.
Deduction can ensure the correctness of the conclusion, but it does not produce new knowledge. Its correctness is only based on the correctness of the premise hypothesis, and deduction cannot guarantee the correctness of the original hypothesis. Although induction can't guarantee the correct result, it can gain new knowledge, because you can always get a hypothesis from observation, such as the swan is white, which can't be deduced by deduction.
Of course, science will also use deduction, but it is at the level of secondary reasoning. If we go back to the most primitive knowledge, that is, the first factor level, the difference between mathematics and science is enormous. Mathematics depends on axioms and science depends on induction.
The essence of scientific research method is to put forward a hypothesis on the basis of observation, and then verify this hypothesis with experiments and facts. The credibility of the hypothesis will change. When the evidence supporting this hypothesis increases, we say that its credibility increases. Once counterexamples appear and are falsified, we will revise them or put forward new hypotheses. Historically, the birth and death of concepts such as ether and phlogiston are such a process.
Of course, in the latest universe and microphysics, there are also many assumptions beyond human experience, such as the beginning of the Big Bang. At this time, how to ensure that the theory is correct? It depends on the prediction given by this theory and then test it in reality. The Big Bang theory predicted the red shift of the universe and observed it in reality, so the verification was successful and the theory was correct.
Then, is it possible to carry out scientific research only through deduction, so as to gain some knowledge?
This problem is equivalent to that if we can find the correct "axiom" without proof in nature and in these specific things and phenomena, then we can deduce it by deduction, thus gaining some knowledge.
In fact, in human history, many studies have done this. To be precise, people once thought that there was transcendental knowledge, which naturally solidified in people's brains and everyone had it, so it was certain. On this basis, we can gain some knowledge.
The metaphysical system of philosophy is almost based on transcendental knowledge, but it is no longer used by people and is only studied as history.
At present, the general view in academic circles is that we have transcendental abilities, such as thinking ability, language learning ability and perception of the outside world, but we have no transcendental knowledge. If a person has never seen red, then he won't know what red is. If a person has never been exposed to human language, then he will not learn to speak and write.
All human knowledge comes from observation and perception, so the first cause of science can only come from induction.
This is easy to understand in physics and chemistry, but it is still vague in some social sciences, such as economics. Because economics studies concrete human society, it is difficult to do experiments to verify it, so many methods of economic research are actually closer to deduction. The Austrian school pushed this method to the extreme. They think it can't be verified, so they can only rely on deduction. The logical starting point must be transcendental knowledge, that is, "human actions have a purpose." Based on this, we can deduce deterministic knowledge layer by layer, that is, the whole economics building.
Transcendental knowledge is not recognized in mainstream scientific circles at present, and it can also be studied in non-scientific fields.
In fact, even with transcendental knowledge, we still can't learn science like mathematics. Our logical deduction ability is bounded, and there are at least three obstacles: 1) We can't exhaust all preconditions; 2) emergence; 3) Calculating irreducibility.
1, prerequisite
The prerequisite of mathematics is limited. For example, Euclid's geometry has only five axioms, because mathematics studies abstract things, so it doesn't matter how many assumptions you make, as long as the process is correct.
But science can't. Scientifically study specific things. For example, if you study the relationship between the earth and the sun, whether it will rain tomorrow and whether the economy will rebound after tax cuts, there are many decisive factors, which are theoretically infinite. We can't exhaust all the preconditions to study anything, even if there are too many conditions, so we invented theories, especially scientific theories, to describe a law in a simplified way.
Then the question is, since you have simplified and only used a limited prerequisite, even under this limited prerequisite, the knowledge is correct. Once applied to specific things, when the preconditions are inconsistent or more preconditions are not covered by the theory, how can you ensure that the conclusion is necessarily correct? I can't.
This situation is very common in weather forecast and even in the whole social science field. No matter how to study the theory and put forward more mathematical models, the final prediction result is still inaccurate.
Economics has been practiced for hundreds of years, but it can't accurately predict the economic crisis, and even the explanation can't reach an agreement. It is because human society is too complicated and there are too many influencing factors that any theory is studied on the premise of simplification and is not accurate enough.
Step 2 show up
Simply put, the whole is not equal to the sum of the individuals.
When we make logical deduction layer by layer according to individual laws, we may not get the overall result. There is an unpredictable "logical jump".
This phenomenon generally exists in the whole nature, from micro to macro, from material to human beings.
For example, we can't simply summarize the laws of motion of molecules through the laws of motion of atoms. The uncertainty in quantum mechanics cannot deduce the certainty of the macro world. Even if we know every molecular structure of the human body like the palm of our hand, we can't deduce our specific thinking and actions. Everyone is rational, but the group will show irrational madness.
Because of this, we need chemistry, biology, psychology and sociology, microeconomics and macroeconomics based on physics. In Hawking's words, we always need some "effective models", and it is impossible to deduce everything only physically.
These logical jumps, or effective models, cannot be obtained by mathematics or logic, and can only be solved by putting forward assumptions based on observation and induction.
3. The calculation cannot be simplified.
The above picture is generated by a computer game called Cellular Automata, which has eight rules, as shown below. If we ask what is the number in line 100 of the program? Can you answer quickly according to those eight rules? No, under this game rule and this combination, you can only go through the program from beginning to end until you get to100000 lines.
This is the so-called computational irreducibility, which was put forward by stephen wolfram. Its core idea is that truly complex systems cannot be simplified. Even if you can find its basic mathematical structure, you can't simplify it, you can only do it honestly.
In other words, complex systems are unpredictable.
As early as 100 years ago, the mathematician Poincare discussed the famous "three-body problem". Even if the initial conditions of three interacting objects are known, there is only gravitational interaction between them, and it is impossible to get the exact solution (analytical solution) of their future trajectories.
Mathematics comes from deduction, and we can get some knowledge, but only if everyone agrees with its axioms. Scientific knowledge comes from induction and summary, which is a hypothesis, not determined by 100%, and its credibility is affected by confirmed or falsified evidence.
Due to emergent properties, inexhaustible preconditions, irreducible calculation and other reasons, it is impossible for us to acquire all the new knowledge through deduction and layer-by-layer reasoning. There is always a logical leap in nature, which makes new knowledge beyond the logical reasoning ability of human beings.