Speaking as a current researcher in pure math -- you're right, but I don't think this is easily resolved.
Math research papers are written for other specialists in the field. Sometimes too few details are provided; indeed I commonly gripe about this when asked to do peer review; but to truly provide all the details would make papers far, far longer.
Here is an elementary example, which could be worked out by anyone with a good background in high school math. Prove the following statement: there exist constants C, X > 0 such that for every real number x > X, we have
log(x^2 + 1) + sqrt(x) + x/exp(sqrt(4x + 3)) < Cx.
i.e. the left-side is big-O of the right.
This flavor of statement comes up all the time in analytic number theory (my subject), would be regarded as obvious to any specialist, and in a paper would be invariably stated without proof.
To come up with a complete, rigorous proof would be long, and not interesting. No professional would want to read one.
I agree this attitude comes with some tradeoffs, but they do seem manageable.
Math research papers are written for other specialists in the field. Sometimes too few details are provided; indeed I commonly gripe about this when asked to do peer review; but to truly provide all the details would make papers far, far longer.
Here is an elementary example, which could be worked out by anyone with a good background in high school math. Prove the following statement: there exist constants C, X > 0 such that for every real number x > X, we have
log(x^2 + 1) + sqrt(x) + x/exp(sqrt(4x + 3)) < Cx.
i.e. the left-side is big-O of the right.
This flavor of statement comes up all the time in analytic number theory (my subject), would be regarded as obvious to any specialist, and in a paper would be invariably stated without proof.
To come up with a complete, rigorous proof would be long, and not interesting. No professional would want to read one.
I agree this attitude comes with some tradeoffs, but they do seem manageable.