I'll try to keep this in order, but there are challenges. And I'm a dilletante.
@AbnormalPsych or dozens of other professors could correct me or explain it better.
My read of Gelman is that he believes that publishers should not require someone has to show statistical significance as a prerequisite, before allowing authors to move to more advanced statistical analysis (i.e. why require statistical significance before allowing other metrics, why not just allow all metrics to be used). This stance is consistent with the ASA publications about this matter. This argument is about the error in how the concept of the null hypothesis is administered. It is not an argument stating that the whole concept is erroneous. Which you disagree with in your initial point, and then contradict yourself in your latter points.
That leaves out Gelman's points about the error of dichotomous decisions vs. decisions along a continuum. This is also a non-issue. Science has established how to statistically handle this matter for more than a century. Any of the physiology literature has continuous decisions, using pretty fun methods. So does economic literature. So does fluid dynamics. etc.
In regards to his proposition to use Bayes most powerful: I get it. You have advanced education in respiratory physiology. Of course you love anything written by Laplace. Gelman assumes that problems have many potential answers, and a priori assumptions are inevitable, which makes Bayes most powerful a given. However, there is considerable debate as to whether Bayes most power represents an inductive model of reasoning or deductive model of reasoning. Because setting determines appropriateness, this is exceptionally important. For example, if I showed up to a neurological examination with a sledge hammer, you might point out that this is the wrong type of hammer. Mathematical logic has several different divisions. Broad strokes, there are two or more gross options as I see it:
1) Our professions are based upon theories (e.g., materialism, every variable is known or knowable). Then we are operating on a priori assumptions, which requires deductive reasoning (i.e., starting at some general theory/hypothesis, and determining if the data "fit" with these hypotheses). This would be model theory and the opposite of how Bayes most powerful operates. So Bayes would be inappropriate for this form of deductive reasoning.
2) Our professions are limiting themselves to generally agreed upon sets of data or study, the a priori assumptions offered by Bayes are somehow exempt as evidence of a hypothesis and deductive reasoning, and I still don't understand how set theory is not a branch of model theory. That being the case, we are blind men attempting to figure out the elephant through induction. If this is the case, then the inductive model used by Bayes is appropriate, and then null hypothesis is useless. However, if this is true then Bayes cannot prove or support anything. It can only describe. Which would then require the user to go back, create some theory or model by which to make the required a priori assumptions, which would subsequently be again subject to testing of the null hypothesis and other methods of deductive reasoning.
Or maybe I have no idea what I'm talking about. I dunno.
