Matematic Evidence and "Maturity": WHAT REASONS THE TRUTH OF MATEMATIC

INTRODUCTION

Mathematical evidence has a special role in public debate. In experimental science, a new device can change the measured value. In law, a new witness or a new document may change the conclusion of the court. In mathematics, it seems that once proven Pythagorean theorem will remain true forever.

In a way, that's true. The proof is the final logical path from definitions and accepted axioms to conclusion. If every step is correct, the new measurement cannot disprove it. But 20. The century has shown that this certainty has deeper layers: it depends on axiomech, formal system and our belief that there is no error in the evidence. Gödel has also proved that a sufficiently strong consistent formal system cannot decide all arithmetic truths within itself.

1. EVIDENCE OF NON-MEASUREMENT

So let's imagine the argument that the sum of the first n odd numbers is n ². We can test a thousand cases. That's a strong record, but still only a thousand cases.

The proof will show why the relationship applies to all natural n. It can use algebraic transformation, geometric design or mathematical induction. The number of possible n is infinite, but the proof is finite.

It is this compression of infinitely many cases into the logical structure that is one of the greatest forces of mathematics.

2. AXIOMS: WHERE THE EVIDENCE STARTS

No evidence begins with absolute nothing. Mathematical theory works with definitions, logical rules and axioms - statements accepted as default.

For example, in Eucleid geometry, certain postulates are used for points, lines and parallel lines. 19. Centuries of mathematicians found that by changing the postulate on parallel lines, consistent Neeukleid geometry can be created.

That didn't mean Euclid was "disproved." It meant that his geometry describes one axiomatic system. Another set of axioms creates a different geometry.

3. WHEN IS THE REAL

It is useful to distinguish between syntactic and semantic views.

Syntax is a verb that can be proved if there is a formal sequence of steps according to system rules. Semantically, it is true in a particular model if it matches the meaning of symbols in that model.

In common mathematical practice, the difference often hides because we work with stable structures. But in mathematical logic, it is crucial.

4. HILBERT'S SEN

At turning 19. and 20. There was a strong hope that mathematics could be formalized completely: to choose precise axioms and mechanical rules, to prove the consistency of the system and in principle to decide any well-formulated mathematical question.

David Hilbert was one of the main representatives of this program. His famous optimism is summed up by the motto "We need to know. We'll know. "

Then Kurt Gödel came.

5. GÖDEL'S FIRST NOTICE

In 1931 Gödel showed that each sufficiently expressive, effectively axiomatted and consistent formal system able to describe basic arithmetic contains statements that cannot be proved or refuted within this system.

Intuitively he created a way of arithmetic to allow the talk of his own evidence, and he compiled an allegation of an analogous sentence "this claim is not verifiable in this system."

If a consistent system could do that, it would get in conflict. If the system is consistent, the sentence will therefore remain inconclusive - and that is precisely what is true in the intended sense.

6. SECOND REALITY: THE SYSTEM CAN'T REALIZE OWN CONSISTENCE

Gödel's second sentence goes on. A sufficiently strong consistent system cannot demonstrate its own consistency by standard means.

This does not mean that mathematics is unreliable or that "nothing can be known." It means that the absolute, closed base that would expose itself to the ultimate certificate of indisputability is running into a logical limit.

We can prove the consistency of one system in a stronger system. But then naturally we ask for the consistency of the stronger one.

7. INFULTISATION IS NOT THE WHAT PARADOX

A common mistake says, "Gödel proved that mathematics contradicts itself." On the contrary. Its famous result is conditional on consistency: if the system is consistent, there are undecided sentences in it.

Incompleteness means the limits of demonstrability, not universal chaos.

8. INDEPENDENT CONTROL HYPOTHESIS

A significant example is the continuum hypothesis. Gödel and later Paul Cohen showed that in the commonly used theory of sets of ZFC sets the hypothesis of continuity cannot be proved nor refuted if ZFC is consistent.

Mathematicians can then examine the extension of the axiomatic system in which the hypothesis is true, and systems in which it is false.

And it's a good way to see that "what's the right answer?" can depend on which axioms we want to think of as the basis.

9. FERMAT: 350 YEARS BETWEEN TRAINING AND EVIDENCE

Pierre de Fermat you are at 17. The century noted that the equation x ^ n + y ^ n = z ^ n does not have positive integer solutions for n > 2 and claimed that it has "truly remarkable proof," for which the edge of the book is too small.

For over three centuries, no one knew the general evidence. Andrew Wiles in 90. years proved Fermat's last theorem through a deep combination of elliptic curves and modular forms.

Interestingly, the first version of Wiles' evidence contained a gap. This was discovered during the expert inspection and subsequently repaired. The story shows that "published evidence" is not automatically the same as flawless evidence. The human community must check the logical structure.

10. THE EVIDENCE SUPPORT FACT

The four-colour sentence was proven in 1976 by computer-testing a large number of cases. This has triggered a philosophical debate: is there evidence that no man can walk through by hand the same kind of mathematical knowledge?

Today, the use of computer technology is more common. However, we distinguish between a program that tests something numerically and formally verified evidence, whose every step is controlled by a small trusted core of the proof assist.

11. PROOF ASSISTANTS: LEAN, COQ, ISABELLE

Systems such as Lean, Coq and Isabelle allow you to write definitions and evidence in an exact formal language. The computer checks that every step is consistent with logic.

That's an important shift. For very long or technically complex evidence, part of human uncertainty can be transferred to a small verification kernel. If kernel is working properly and the formal statement really corresponds to the mathematical theorem, we will get extremely strong control.

12. FORMAL-VERIFIED SOFTWARE

The same logic has a practical impact on computer science. Instead of testing a program for millions of inputs, it can be shown by some properties that the program will never break the given invariant.

Formal verification is used for microcores of operating systems, cryptographic protocols, compilers and hardware. For example, the well-known project L4 produced machine-controlled evidence of important micronucleus properties.

The evidence, of course, does not cover everything. If the hardware is defective, the specification or physical implementation is incorrect, the formal correctness of the source model may not save the whole system. Mathematics guarantees exactly what we formulated - nothing more.

13. _

In the common language we use the same word for two different concepts.

In court, there is evidence that increases or reduces the likelihood of a factual claim. The witness may lie, DNA may be contaminated, and the document may be forged. The Court of First Instance shall act according to the standard of evidence and facts available.

Mathematical evidence is not statistical evidence. It is a deduction: if the assumptions apply, the conclusion must be valid. Therefore, in mathematics it makes no sense to say that the sentence after the correct proof "to 99.9% true."

14. PROHIBITION IN PHYSICATION

Physics uses mathematics, but physics is proven by experiment. Equations can be impeccable derived from theory and yet theory can be a bad model of nature.

Newton's mechanics are mathematically consistent and extremely useful, but at high speeds and strong gravity, it is replaced by a more accurate relativistic description. It was not to refute algebraic evidence, but to find that the physical world did not meet Newton's preconditions in all regimes.

15. IS THE MATEMATIC PROOF "IMPORTANT"?

If we have correct proof of a sentence from explicitly given axioms and rules, its validity will not change by measuring a different value of the physical constant tomorrow. In this sense, the mathematical necessity is extremely stable.

However, our assessment of a particular text as evidence may change if we discover a hidden error. We can also go into another axiomatic system, where the analogous question has different properties.

"Eternal certainty" is not mystical absolutes. It is a conditional logical necessity: if the assumptions and rules are correctly established and the derivation is correct, the conclusion necessarily follows.

16. WHAT GEDEL DIDN'T VISIT US

Gödel's incompleteness did not destroy mathematics. Mathematicians continue to prove millions of sentences, build theories and use them in technology. Incompleteness has only shown that no sufficiently rich mechanical formal system is at the same time a complete university of all arithmetic truths and a definitive certificate.

It's more of an extension of our understanding of mathematics than its defeat.

CONCLUSION

Mathematical evidence is probably the most rigorous form of justification mankind has created. Its strength is that the conclusion is not dependent on future measurement, the opinion of authority or majority voting.

At the same time, mathematical logic has shown its own limits to formal systems. And modern proof assistants remind us of one more practical boundary: perfect logic must be correctly written and controlled by imperfect people and machines.

The most accurate wording of this thesis is therefore: the correct mathematical evidence provides a lasting logical certainty with respect to the given axioms and rules - but mathematics itself teaches us to be very precise in what we have done exactly.

17. THE FOUR-FOUR SETTLEMENT AND QUESTION "DID MAN SEE THE WHOLE EVIDENCE?"

The four-colour sentence states that each planar map can be coloured in no more than four colours so that the adjacent areas are not of the same colour. The first generally accepted evidence of Apple and Haken from 1976 was used by the computer to verify a large number of configurations.

A part of the mathematicians were initially unhappy because an individual could not go through all the calculations manually. The dispute revealed that the mathematical certainty has a social layer: we trust not only logic, but compiler, processor, program and control procedure.

The later formal verification of the four-colour sentence in the proof assistantu Coq significantly reduced this problem. The large calculation is reduced to an object that controls a small logic core.

18. POINTING DOCUMENT: THE EVIDENCE THAT THE COMMISSION CHECKS ON THE YEAR

Grigory Perelman published in 2002- 2003 preprints solving Poincarý's conjecture using Ricci's flow. The text was not immediately adopted simply because it was written by a brilliant mathematician. Several teams spent years completing the details and checking the argument.

The example shows that the mathematical community has its own equivalent of replication: other experts must be able to understand and independently confirm reconstruction. The difference from the experiment is that they do not replicate the natural phenomenon, but the logical derivation.

19. WHEN THE COMPUTER FINDS THE PROFESSIVE

Automated theory provers can search for evidence without a person knowing the way in advance. Modern systems combine symbolic logic, SAT / SMT solvers and increasingly machine learning.

This raises a new question: if the machine finds formally the right proof of a million steps that no person intuitively understands, do we have mathematical knowledge, or just a certificate of truth? From the point of view of formal correctness, the certificate can be perfect. From the point of view of mathematics as an understanding can still lack an explanatory idea.

20. PROFIT AND EXPLANATION ARE ALWAYS NOT ALWAYS THE SAME

Two evidence may be equally correct, while having a completely different intellectual value. One can be a long calculation, the other a short construction that reveals why a sentence must apply.

The mathematicians are therefore looking not only for "Q.E.D." but also for elegant evidence, new connections and generalization. The eternal certificate of truth is only one part of mathematical work; the other is the structure of understanding that the proof creates.

21. WHAT DOES THE FISH IN AXIOM

Axiom is not an empirical claim that would "fail" by remeasuring. It's part of the definition of theory. However, it may prove that the chosen system is inconsistent or is not an appropriate model of the phenomenon we wanted to describe.

Therefore, it is always important to state in which system the sentence applies. In common mathematics the context is often assumed, but for the foundations of mathematics is explicit the introduction of axiomatic framework components of accuracy.

RESOURCES AND OTHER READING

[1] Stanford Encyclopedia of Philosophy - Gödel's Incomplete Theories.

plato.stanford.edu
https://plato.stanford.edu/entries/goedel-incompleteness/

[2] Lean theorom prover.

Lean-lang.org
https://lean-lang.org/

[3] Coq Proof Assistant.

coq.inria.fr
https://coq.inria.fr/

[4] Isabelle theorom prover.

isabelle.in.tum.de
https://isabelle.in.tum.de/

[5] seL4 - formally verified microkernel.

sel4.systems
https://sel4.systems/

[6] MacTutter History of Mathematics - Andrew Wiles and Fermat's Last Theor.

mathshistory.st-andrews.ac.uk
https://mathshistory.st-andrews.ac.uk/Biographies/Wiles/