History of mathematical conjuctures

[Werg22]

I have to preapre a small oral presentation on the history of mathematical conjectures, highlighting the most important. One of the main question this revolves around is if a mathematical conjecture ever slowed down the advancement of mathematics. If anyone could give me pointers on where to look and some basic information, I'd be greatly thankfull.

 

[mathman]

if mathematical ever slowed down the advancement of mathematics.

Could you clarify?

 

[matt grime]

I don't understand the sentence

"if mathematical ever slowed down the advancement of mathematics."If you want to know about the most important (in some obvious sense) conjectures then look at the Clay Institute Millennium Problems.

 

[Werg22]

Sorry about the mistake, I edited the post.

 

[CRGreathouse]

if a mathematical conjecture ever slowed down the advancement of mathematics. If anyone could give me pointers on where to look and some basic information, I'd be greatly thankfull.

Hmm. I think the conjecture that Euclid's 4th postulate can be derived from other accepted axioms slowed down the advancement of non-Euclidian (elliptic and hyperbolic) geometries. No other conjectures immediately come to mind.

 

[mathwonk]

i do not believe a conjecture can really slow down progress. (i believe that was the 5th postulate by the way.)

the stupidity of people in sticking to a conjecture's truth rather than exploring other possibilities may slow their own progress in deciding it, but this should not be blamed on the conjecture.

If you look into it, I believe many conjectures were actually just questions until they were solved, and afterwards their askers decided they had been conjectures.

other famous conjectures were actually mistakes, and when they were exposed thye became conjectures.


the only case I know of where progress was actually slowed, was not by a conjecture, but a false theorem, the case when pontrjagin announced an erroneous calcuation of a certain homotopy group.

his announcement contradicted bott's conjectural belief as to the periodicity of stable homotopy groups. pontrjagins fame and reputation, caused bott to delay work on his own correct conjecture for some time.

at length the error was revealed and bott immediately began his successful proof.

 

[CRGreathouse]

(i believe that was the 5th postulate by the way.)

You're quite right, of course. The fact that I haven't taken but a single geometry course since high school must be showing...!

 

[mathwonk]

another example was severi's erroneous proof that the variety parametrizing nodal plane curves with a given degree, and given number of nodes (ordinary double points), is irreducible, became called severi's conjecture. until it became joe harris's theorem.

also zariski seems to have played a role in this scenario of erroneous arguments.

according to a famous physicist, what matters is not whether a conjecture is right or wrong, just that it give rise to useful work. this must be examined in each case individually.

shafarevich made a conjecture about the structure of universal coverings of algebraic varieties some years back which may be false but has insipred a lot of interestin work.

the poincare conjecture has inspired good work in topology for over 100 years it seems. the weil conjectures inspired whole areas to be developed.