Credentials don’t make an incorrect argument right, and the lack of them can’t make a correct argument wrong. The track record tells you more – George Green and Srinivasa Ramanujan ( and Freeman Dyson) did what they did. In that sense, degrees don’t matter.
But they can give you signals of greater or lesser utility. Ph.Ds in math or the hard sciences prove you have some brains – not necessarily that you will make good use of them, or that you’ll be useful, but sure, you probably have some brains. Or at least you once did. Doesn’t necessarily mean that you know much outside your specialization, or have much sense. Although you might.
What about a Ph.D. in psychology? it doesn’t mean that you can’t have some brains, but its predictive value isn’t very high.
An M.D.? Again, doesn’t mean that you can’t be smart, but, usually, not born puzzle solvers. Significantly overrated by both the general public and holders as an indicator of general omnicompetence.
Ph.D. in education? On average, it predicts that you’re dumber than someone with a B.A in education, already below the general college average.