hypothesis

Because statistical "testing" involves the decision to reject the randomness of an event. It can say nothing about the cause of an event.

Sorry.
I should have started that answer with a "No"...followed by "Because....etc".

You certainly can test the alternative hypothesis, but the outcome of the test will not say the same thing about the null hypothesis as testing the null hypothesis would. That's because no test can ever prove a hypothesis. It can only refute it or fail to refute it.

For example, suppose your null hypothesis is that "pharmaceutical P has about the same effect as a placebo" and your alternative hypothesis is that "pharmaceutical P makes a difference compared to the placebo". Now, if you test the the alternative hypothesis and fail to refute it, that says nothing at all about the null hypothesis. You haven't refuted it, but you haven't failed to refute it either because you haven't tried. On the other hand, if you succeed at refuting the alternative hypothesis, that is a stronger statement about the null hypothesis than a test of the null hypothesis could have yielded: Rather than just failing to refute it, you have affirmed it.

So yes, of course you can test the alternative hypothesis, but the outcome of testing it doesn't say the same thing as the outcome of testing the null hypothesis would have.

I am just registering that I do not agree with most of that with respect to statistical testing. For example, in your example the "null hypothesis" is not about "having the same effect as a placebo"...its about there being no difference attributable to either course of "treatment". Nor is statistical testing about the absolute concept of "refutation". That concept resides in the realm of universalistic statements rather than statistical ones. Statistical testing is confined to "significance levels" for acceptance or rejection of the null hypothesis and those significance levels are either conventional or negotiable.

But that's not what prevents testing the alternative hypothesis from yielding the same information as testing the null hypothesis. If you wish, replace "about the same as placebo" with "no difference attributable to either treatment"; also replace "refute" with "obtain a p-value smaller than 0.05". It's still the same argument, and still correct.

The OP clearly does not understand that theoretically any number of alternative hypotheses can be cited for a departure from chance. That is why I assert that only the "hypothesis of chance" i.e. "the null hypothesis" can be tested statistically.