In a nice and witty lecture titled "how to write mathematics badly" (available on YouTube at <https://www.youtube.com/watch?v=ECQyFzzBHlo&t=23s>), Jean-Pierre Serre describes various ways in which a paper can be poorly/confusingly/inaccurately written.


Around min 34:00 in the previous link, he criticizes the use of the word "constant", in particular in inequalities. The example he provides is of the type:



> 
> $$\|Af\|\le C\|f\|$$ for some constant $C$
> 
> 
> 


where $A$ is a complicated operator depending on many parameters. In this case, he says, usually the only thing that the writer means is that $C$ does not depend on "some of the data" of the problem. He adds that this attitude "caused lots of mistakes".


What are examples of these mistakes? Has any significant piece of mathematics been rewritten or erased altogether because of some problem with proofs invoking "constants" too nonchalantly?



How to invoke constants badly
**Edit:** The original answer below refers to Nelson's attempt from 2011. Upon a cursory look at the afterword by Sam Buss and Terence Tao to [Nelson's paper](https://arxiv.org/abs/1509.09209) placed in arxiv in 2015 (after his death), it seems he later attempted to address the error referred to in the original answer below; it would be interesting to know what the experts think on how successful his efforts were or potentially can be.


**Original Answer:** Edward Nelson's recent project on finding inconsistency of arithmetic (which was the subject of [a MathOverflow Question](https://mathoverflow.net/questions/36693/nelsons-program-to-show-inconsistency-of-zf)) might be pertinent. [The error](https://golem.ph.utexas.edu/category/2011/09/the_inconsistency_of_arithmeti.html#c039547), discovered by Terence Tao, seems to be the dependence of a constant on the underlying theory that Nelson did not account for.


