P-value has always been controversial. It is required for certain publications, banned from some journals, hated by many yet quoted widely. Not all p-values are loved equally. Because what someone popularized some 90 years ago, the small values below 0.05 have been the crowd's favorite.
When we teach hypothesis testing, we explain that the entire spectrum of p-value is to serve a single purpose: quantifying the "agreement" between an observed set of data and a statement (or claim) in the null hypothesis. Why do we single out the small values then? Why can't we talk about any specific p-value the same way we talk about today's temperature? i.e., as a measure of something.
First of all, the scale of p-value is hard to talk about, which is different from temperature. The difference between 0.21 and 0.20 is not the same as 0.02 and 0.01. It almost feels like we should use the reciprocal of the p-values to discuss the likeliness of the corresponding observed statistics assuming the null hypothesis is true. If the null hypothesis is true, it takes, on average, 100 independent tests to observe a p-value below 0.01. The occurrence of a p-value under 0.02 is twice as likely, taking only about 50 tests to observe. Therefore 0.01 is twice as unlikely as 0.02. Using similar calculation, 0.21 and 0.20 are almost identical in terms of likeliness under the null.
In introductory statistics, it is said a test of significance has four steps: stating the hypotheses and a desired level of significance, computing the test statistics, finding the p-value, concluding given the p-value. It is step 4 here requires us to draw a line somewhere on the spectrum of p-value between 0 and 1. That line is called the level of significance. I never enjoyed explaining how one should choose the level of significance. Many of my students felt confused. Technically, if a student derived a p-value of 0.28, she can claim it is significant at a significance level of 0.30. The reason why this is silly is because the significance level should convey a certain sense of rare occurrence, so rare that it is deemed contradictory with the null hypothesis. No one of common sense would argue a chance that is close to 1 out of 3 represents rarity.
What common sense fails to deliver is how rare is contradictory enough. Why 1/20 needs to be a universal choice? It doesn't. Statisticians are not quite bothered by "insignificant results" as we think 0.051 is just as interesting as 0.049. We, whenever possible, always just want to report the actual p-value instead of stating that we reject/accept the null hypothesis at a certain level. We use p-value to compare the strength of evidence between variables and studies. However, sometimes we don't have a choice so we got creative.
For any particular test between a null hypothesis and an alternative, a representative (i.e., not with selection bias) sample of p-values will offer a much better picture than the current published record of a handful of p-values under 0.05 out of who-knows-how-many trials. There have been suggestions on publishing insignificant results to avoid the so-called "cherry-picking" based on p-values. Despite the apparent appeal of such a reform, I cannot imagine it being practically possible. First of all, if we can assume that most people have been following the 0.05 "rule", publishing all the insignificant results will result in a 20-fold increase in the number of published studies. Yet it probably will create a very interesting data set for data mining. What would be useful is to have a public database of p-values on repeated studies of the same test (not just the null hypothesis as often the test depends on what is the alternative as well). In this database, p-value can finally be just what it is, a measure of agreement between a data set and a claim.