The passage you quoted was:
Science seeks to explain natural events with natural causes. The Turing hypothesis does this. Beyond the bounds of science, there is no objective argument for anything really, just philosophical ones. To take nothing away from philosophical arguments, they fundamentally rely on unprovable premises.
is this analysis of philosophy a science?
is it the "best" science of philosophy?
if not, then can it be proven anyway?
and if it was, would it be viciously circular anyway?
The passage you quoted is bad philosophy. Since he is talking about unprovable premises he thinks that it is possible to prove stuff. This would mean that there is a process that shows that a particular idea is true or more probable than its competitors - let's call it justification. In reality, you can't prove any position or show it is probable. Any argument requires premises and rules of inference and it doesn't prove (or make probable) those premises or rules of inference. If you're going to say they're self evident then you are acting in a dogmatic manner that will prevent you from spotting some mistakes. If you don't say they are self evident then you would have to prove those premises and rules of inference by another argument that would bring up a similar problem with respect to its premises and rules of inference.
In reality all knowledge is created by conjecture and criticism. You notice a problem with your current ideas, propose solutions, criticise the solutions until only one is left and then find a new problem. Experiments are useful only as criticism. Ideas can't be derived from experiment any more than from any other set of premises. Rather, the idea is that you work out how the consequences of one theory differ from those of another and then look for problems with those consequences.
In science the criticism involves experimental tests: you conjecture ideas about experimental setups that would enable you to see the relevant consequences and criticise them. Once you have a setup that works about as well as you can make it work you use it to do the test. If the results are compatible with one theory and not the others then you may have successfully refuted some false ideas. Sometimes a purported successful experimental test will be successfully criticised because a test is a conjecture about something that happened and that conjecture may be wrong, so experiments don't prove anything.
An idea about standards is not scientific. It does not say that it is impossible to break the standards, just that breaking the standards would be a bad idea. So no experiment can distinguish between different standards. By invoking unprovability, the author of the comment you quoted makes a substantive philosophical and unscientific claim about standards. He also claims there are no objective standards for assessing philosophical claims. So if we were to take his position seriously we have to conclude that it is completely arbitrary and no more of less worthwhile than any other position. In addition, because scientific theories can't be proven his adoption of that standard also implies that all scientific positions are completely arbitrary and no more of less worthwhile than any other position. So the commenter's position is inconsistent and so it is false. But it is worse than false, for as long as he holds it he will not be looking for criticisms of the sort that he can only get from good philosophy books, which brings me to some recommended reading:
"Realism and the Aim of Science" by Karl Popper
"The Beginning of Infinity" by David Deutsch.