But people often write papers anyway assuming X and Y and Z are true just to see what happens because of it, it's not like you really need it to be true to at least publish things.
And for your example that touches on applications of math, it approaches scientific research from the wrong perspective. If some math follows reality, (agrees with experiment and even better makes predictions that turn out to be right), there is has to be some backing to it, even if the rigor isn't fully there. Physics often runs with ideas before Math has caught up (see QM in the early days, HEP in general, etc).
So, if we back off from applications who, quite frankly, only care about whether X is true in only a few, really important number of conjectures I can think of--big questions like P=NP for example--then math itself I imagine is, well, a study for its own sake. So, what above exhaustive proofs of your run-of-the-mill conjecture? Who would want to read a paper from a math person that merely iterates through a 200TB cache of data?
People forget that science is a social phenomenon as much as it is systematic process.
I don't want a math paper detailing 200TB of data. I want a paper telling me the generation of the digital input, the method that was used to actually prove it if it's novel. I want the authors to show how they verified the result and if it's important how they verified the automatic verification. The paper will end with "Oh and by the way, we used this conjecture because it was difficult, it's true. You can verify it yourself."
I want this all so I can do this myself, so I can prove a run-of-the-mill conjecture and not waste my time with it. I want this because proving is difficult and there is more to SAT than mathematical proofs, it is used in the industry. I want this because I trust a verified proof that the airplane won't crash more than a human proof.
The availability of computer proofs does not devalue the study of mathematics.
And for your example that touches on applications of math, it approaches scientific research from the wrong perspective. If some math follows reality, (agrees with experiment and even better makes predictions that turn out to be right), there is has to be some backing to it, even if the rigor isn't fully there. Physics often runs with ideas before Math has caught up (see QM in the early days, HEP in general, etc).
So, if we back off from applications who, quite frankly, only care about whether X is true in only a few, really important number of conjectures I can think of--big questions like P=NP for example--then math itself I imagine is, well, a study for its own sake. So, what above exhaustive proofs of your run-of-the-mill conjecture? Who would want to read a paper from a math person that merely iterates through a 200TB cache of data?
People forget that science is a social phenomenon as much as it is systematic process.