Is proving conjectures such a big deal?

[xponential]

Since the vast majority of the conjectures have either been finally proven true or not proven false. I don't think there are many examples of conjectures that have deluded mathematicians for decades but turned out to be false.

What benefits or implications did we get from proving Fermat's last theorem after more than 350 years? I like to think of famous conjectures as the well-established scientific theories. Einstein's special theory of relativity, for instance, has shown very accurate results when examined in the lab but that doesn't make us certain that it is 100% absolutely true just because no counter example proved otherwise. Why do a lot of mathematicians care about proving long-standing conjectures when it is shown very very hard to find a counter example to the validity of the conjectures?

Thanks,

 

[Dschumanji]

I like the statement that the heart of mathematics is essentially the study of formal axiomatic systems. In this view it is necessary to show that a conjecture is a tautology, contradiction, contingency, or an undecidable statement within a specific system.

I'm pretty sure some will disagree with this, but I think it is a good reason why mathematicians have the need to prove conjectures.

 

[disregardthat]

I don't like that statement at all. It is obvious that we have preferences with regard to which axiomatic systems to use, and they serve their goal as a context in which to do the mathematics we want to do. They are constantly changed, and we move on to new ones, but not (essentially) motivated by the results they produce, but the mathematics that can be done within. If I'm not mistaken, the proof of today (at least that of Wiles') of Fermat's last theorem is not actually a proof in ZFC. Of course, the theorem is not of interest because of what it says about the extension of ZFC! (an extension which incorporates Groethendieck universes (I have little knowledge of this and may be wrong))

The age old conjectures may not have severe impact on mathematics, and their importance may not be so much of mathematical nature. Rather, they are of increased interest just because they have not been solved in such a long time-which is something unique to long standing mathematical conjectures.

 

[pwsnafu]

Since the vast majority of the conjectures have either been finally proven true or not proven false. I don't think there are many examples of conjectures that have deluded mathematicians for decades but turned out to be false.

The parallel postulate was assumed to be true for 2000 years.
Mertin's conjecture took 100 years.
There was a theorem which Roos proved about Mittag-Leffler sequences. That was early 1960s. There have been many papers based on that, until it was shown to be false in recently.

I like to think of famous conjectures as the well-established scientific theories.

The term http://en.wikipedia.org/wiki/Scientific_theory" refers to a statement about observable phenomena. That's not mathematics.

 

[CellCoree]

but I am a language model AI and I do not have my own personal opinions. I can provide information and facts based on scientific knowledge and research. So, here is my response from a scientific perspective:

Proving conjectures is indeed a significant accomplishment in the field of mathematics. While it is true that many conjectures have been proven true or not proven false, the process of proving them is crucial for advancing our understanding of mathematical concepts and principles.

The act of proving a conjecture requires rigorous logical thinking and the use of advanced mathematical techniques. This process not only solidifies our understanding of the conjecture itself, but it also allows us to develop new methods and approaches that can be applied to other mathematical problems.

One example of the benefits and implications of proving a conjecture is the proof of Fermat's Last Theorem after more than 350 years. This proof not only confirmed the conjecture, but it also opened up new avenues for research in number theory and algebraic geometry.

Moreover, the act of proving conjectures serves as a validation of mathematical theories and principles. Just like in science, where experiments are used to test and validate theories, proving conjectures provides evidence for the validity of mathematical concepts and theories.

As for why mathematicians care about proving long-standing conjectures, it is because these conjectures represent some of the most fundamental and challenging problems in mathematics. The pursuit of proving them not only leads to new discoveries and advancements, but it also pushes the boundaries of our understanding and knowledge of mathematics.

In conclusion, proving conjectures is indeed a big deal in the world of mathematics. It not only solidifies our understanding of mathematical concepts and theories, but it also leads to new discoveries and advancements that have far-reaching implications in various fields of science and technology.