But this is contradicted by many Putnam and IMO winners, who obv have as much math "cleverness" as anybody at least in the sense of you being able to measure, going into stuff that you would also say is abstract nonsense. No reason to think Gauss might not do the same thing. That is my only point....you are def wrong on other stuff but I don't care enough to go through it.

After Bob Beamon jumped 29 feet I bet a lot of prospective broad jumpers decided to take up the hop skip and jump instead. But not the very very best.

When there's a change in a complex of incentives it's correct to point out one effect from a change in one of the incentives. However, it's a mistake to treat it as the only effect from the change in the complex of incentives. The particular incentive may have many effects and there may be many other incentives changing in the complex as well.


PairTheBoard

#45: and billions of other questions and answers yet once this generation realizes that math and physics matter more than ever.

I think I have improved on my technique of explaining it.

1. Godel proved that some math statements are unprovable. (I am using that word rather than "undecidable" because I don't think number theory conjectures that involve finding a counterexample can be undecidable, since undecidable would seem to imply you can't find a counterexample so the conjecture would be true.)

2. Someone here said that it was proven that number theory conjectures have been proven to never be unprovable if they are indeed true. I am not sure he is right or if he might have been talking about only Diophantine equations rather than other types of number theory conjectures such as those involving prime numbers. If he is right about all number theory conjectures there is no need to read further.

3. There are number theory conjectures that would likely have no counterexample if the numbers behaved like random numbers and you only took their sparseness into account. It wouldn't matter that there is an infinite number of opportunities to find a counterexample because it gets sufficiently harder and harder after each failure. I believe Goldbach's Conjecture meets this criteria. Call them Sparse conjectures.

4. There are also number theory conjectures where the density of the numbers involved would make the probability essentially 100% of finding a counterexample if those numbers behaved randomly. I believe Euler's (incorrect) conjecture that three cubes can't sum to a cube is an example. Call these Dense conjectures.

5. My contention is this: If there are actually any unprovable (but true) number theory conjectures, they will all be sparse. Conversely any true dense conjecture has an essentially zero probability of being unprovable.

This is definitely wrong. Its been shown that Diophantine equations are equivalent to Turing machines. So there exists a Godel like encoding of statements like "ZFC is consistent" which amount to saying that a certain Diophantine equation has no solution. This statement would be true (provided ZFC is consistent) but cannot be proven in ZFC.

^ if I'm thinking of the same thread that DS is then whoever it was who said that (I cant remember who it was but it was someone worth listening to) also went on to say they were mistaken.

If the Flynn effect is real CF Gauss should be no more intelligent than the average Joe by todays standards.

https://en.wikipedia.org/wiki/Flynn_effect

If by average you mean the average top 100 of our century (past 100 years running say)! Because Gauss was clearly a very bright brain anyway no matter what era you choose to study him being born. The further back in time you study him the more singular he will appear.