Modern mathematical proofs are spread over hundreds of pages, and only a small number of people understand them. Can they be considered true?
Such a small number of verifiers leads to the fact that there is a chance that no one has noticed any error
As I can tell you, in my dissertation I had 4 pages of calculations on the convergence of the Stokes integral in C^n, which sets the action of a continuous linear functional on the main space of analytic functions in a convex domain C^n. Henkin skipped the paper in DAN ( which simply claimed convergence under the 6 mp page limit). Zakharyuta looked at the formula, asked me to be silent for 10 minutes, then said yes, there is convergence. On the other hand, in Krasnoyarsk, I was forced to write several boards from bottom to top. This does not mean that Eisenberg knew the Martineau-Eisenberg integral representation worse than Henkin – it's just a somewhat weighted Stokes calculation, which he was the first to do, and not Henkin or Zakharyuta. It was just that each of the three Mathematicians looked at the formula from their own perspective. For me, each of these positions was like heaven. Experts do not always check how you are looking for a differential of any form, they can simply go to what you think is very difficult for completely different reasons and they are not going to tell you the details of their thinking at all.
Palamodov and Hermander presented rather similar questions in different ways. Why have almost all specialists in partial differential operators read Hermander's monographs ? Why everyone uses the Hermander technique. Because it is transparent and understandable.
The author of the question is right. In fact, specialization in mathematics has reached such an extent that even mathematicians can no longer understand each other. In addition, the presence of a mathematical apparatus is sometimes perceived as a kind of “litmus test” of the scientific nature of the work, its truth. Elena Ventzel, a specialist in the field of probability theory, wrote about the fact that this may be an illusion (her pen name is “Irina Grekova”). Here is an excerpt from her article:
«…The mere presence of a mathematical apparatus does not give accuracy and reliability to scientific research….With the help of mathematical symbols, you can write so much nonsense, so many empty, pseudoscientific inventions, that sometimes you are surprised. There is a mathematical apparatus, but there is no science, because this apparatus is applied to solving an absurd, far-fetched, ugly problem that has nothing to do with anything… There is an abundance of work that uses mathematical tools, but there is no sign of science. The flaw of these works is the lack of a pre-mathematical, qualitative analysis of the phenomenon, a genuine statement of the problem.”
Really, but how to check the truth of a” newborn “mathematical theory, “calculus” or geometry? Let us recall the history of mathematical analysis. The truth of its foundations has been criticized by many: Bishop Berkeley, Michel Rolle, and others. Scientists explain that any newborn scientific idea looks like an “ugly duckling” – its “evidence base”is too weak. The “newborn “”Infinitesimal Analysis” justified the criterion of practice – with the help of the new calculus, it was possible to solve many” unsolvable ” (for elementary mathematics) practical problems. Over time, the problems of justifying the new calculus also became clear.
Unfortunately, not all areas of mathematics can be reliably tested in practice. It is very difficult (and I may be wrong) to test in practice the truth of some conclusions of probability theory, mathematical statistics, etc. Developing a reliable “truth criterion” for mathematical papers seems to be an urgent task. In the meantime, the internal consistency of the theory (“internal perfection”- according to A. Einstein) and the persuasiveness of the logic of its proofs serve as such criteria. Let's remember the story: his “Pangeometry” by N. I. Lobachevsky decided to publish it only because he clearly saw its internal consistency (although he did not understand where his new geometry was being implemented).
There is also another (“external”) “truth criterion” – the “citation index”. Elena Wentzel (“Irina Grekova”)also wrote about him:
“Any attempts to objectively evaluate a scientific result by some formal, calculated criterion (for example, by the “citation index” or “economic benefit”), as a rule, do not justify themselves. The integral effect of human opinions remains the only basis for distinguishing the original from the ersatz, both in science and in art. The authentic sooner or later makes its way (often, unfortunately, too late).”
It is said strongly, but whether Irina Grekova is right, I don't know. The topic of finding a reliable truth criterion for mathematical papers is a very complex philosophical problem that requires the efforts of many people: philosophers, logicians, and mathematicians themselves. Or am I wrong? Don't know…
I apologize for being so boring. There can be no a priori errors in the proof!
Only the recognition of a “proof” in natural or formal language by a proof can be erroneous; a formal proof cannot be erroneous by definition.
What is striking is the growing number of people in society who are overthrowing the rank of research scientists, saying that there is less and less truth in evidence, in vaccinations, flights to the moon, and other truths that they do not perceive…
It can be expanded to the root question: What is the criterion for the scientific character of modern knowledge? In conditions when 99% of what each of us knows or has heard, he can not personally verify (verify, put an experiment, deduce a complete proof,…), what do we consider scientific knowledge? We ourselves can neither verify nor prove the vast majority of statements of serious physics and mathematics.
We trust Bayes ' theorem and indirect confirmation.
In a sense, this can be called trust in authorities (peer-reviewed journals,…). But it should not be equated with an unconditional attitude to the authorities of antiquity (Aristotle,…) or religious dogmas. Any fact that comes up that contradicts the authorities is put on the scales of Bayes ' Theorem.
the criterion of truth is practice.This also applies to mathematics. If mathematical theories find convincing confirmation in practice,they can be recognized as true (even if relative). This was the case with Euclidean geometry, mathematical analysis, etc. It is difficult to confirm those mathematical theories that are difficult (purely technologically) to test in practice (some conclusions of probability theory, Bayes ' hypothesis, etc.), You also need to remember about logical sophisms that even professional mathematicians can allow in their reasoning. And also remember about unjustified extrapolations (a manifestation of the method of incomplete mathematical induction), which can lead to both significant and insignificant errors (a striking example is the detection of the divergence of a series, which was successfully used in practice by astronomers,but which (in principle) was discovered only by Henri Poincare)
AI can and should solve problems and prove theorems, check solutions and proofs, and much more similar things….
There are already good prerequisites for this.
In mathematics, there are basic premises (axioms, postulates…). These are “first-class bricks”, based on which the mathematics building is built. The main method of construction is logic. If there are no PARALOGISMS (unintentional violations of logic) or SOPHISTRY (intentional violations) in the proof, then the proof is considered to beCORRECT (true).
You can replace some of the original premises with others, then you will get a NEW section of mathematics.
This is not the case in physics. There are quantitative connections (mathematics) and technical explanation (philosophy). This is where falsification begins. Even if the theory and experiment are compatible, the physical interpretation may be wrong.
This truth is consumed exclusively by those who produce it. Nothing is infallible in our world. Einstein used to say that mathematics is the only perfect way to fool yourself.
The only criterion for proof is practice. Unfortunately, 99% of all mathematical proofs are virtual nonsense, so you can only check them with other mathematical calculations. That is, pure bullshit on a stick. And only 1% of all mathematics is of applied value and can be tested by practice. For example, launching spacecraft. But there is one caveat: it is not known what will have an applied value. That is, what was pure theory suddenly becomes in demand for practical purposes.
In describing physical processes, formulas only create the illusion of correctness. For example, molecular kinetic theory, in which formulas are used to calculate the number and force of impacts of molecules against walls and among themselves. Naturally, this is definitely not realistic to consider. Actually, MKT is a stupid and false theory, and the energy from tapping does not arise. The space of an atom is filled with matter that can be emitted in the form of photons, and the energy of the body is equal to the energy of this matter, and its quantity and density increases with the absorption of photons, and decreases with the emission of photons.
mathematics,as I understand it,is a tool that is both flawed and at the same time questionable-it is unprovable-whether it is correct to use it in many cases of cognition of reality…
If multi-page proofs are created, then someone needs it. The truth of a proof, as an abstraction, is unlikely to have much value. It is the consequences of the truth or falsity of the proof that are important for their application to any significant tasks.
Therefore, it is quite possible to consider the proofs conditionally true, as long as no one has refuted them or found facts that contradict them. I believe that this is exactly the case.
True ones…FOR WHOM?
For those who don't understand it?
Or for those who understand it?
In our world, everything is relative.
And everything “true” in our world…also relatively.
What used to be true in various scientific and non-scientific fields…
Today, it is either refuted or simply ceases to be true.
Flat land on three elephants and a turtle…he won't let you lie.
My proof of an infinite set of prime numbers in arithmetic progression on 1 page. based on an unknown property of natural numbers, which I proved in 1997. Have YOU seen a similar Dirichlet proof? And that he's guilty of something? Are they not true?
BBK 16.2.3. K 683-V. I. Korolchuk. “The solution of famous mathematical problems”. Simferopol: Tavriya,2oo4, – with ISBN 966-572-468-1
It seems to me that it's not just about mistakes. People study proofs of statements given by other people, not in order to detect errors, but in order to understand how the proof was given, what techniques were used, and on the basis of this understanding to go further. But the trend seems to be that the proof of some statements is sometime (soon?)away. it will require not hundreds, but thousands of pages for presentation. And then not only will there be no people who can read this proof, but there will be no people who can write it.
Gentlemen, don't you think we're close to a dead end?
By the way, it is possible that in physics the situation is similar, although from the other side – from the experimental side. I read somewhere (I don't remember where, but the source is quite decent) that in order to test some hypotheses, such energies are needed that the accelerator providing them must be the size of a small galaxy. And how do we move forward?
The question concerns not only mathematics, but also all other sciences, without exception. Any programmer knows that there are no programs without errors, but nevertheless we use these programs.
You can not accept as true formulas invented by a person who knows only part of the information about the laws of the universe in a certain period of time.