1. Logical and mathematical truths cannot be proven.
False, because every mathematical theorem has to be proven to be accepted. The same with logic. Math is proven with math, not physics. The fact that you cannot test math by throwing a ball and calculating the distance doesn't mean mathematical propositions cannot be proven. The Pythagorean theorem is proven, noone is suspicious of it.
2. Metaphysical truths.
- There are other minds, other than my own.
If I conduct a conversation with someone isn't that proof of a second mind?
- The external world is real.
What does "real" mean? In what way might it be not be real? Like in the Matrix? I would say it can either be real (in whatever sense of the word) or it can be an imitation so good as to be perceived as real. In either case, what does it matter? Something not real indistinguishable from real is what? Well, real.
- The past was not created five minutes ago with an appearance of age.
Well, I was here 10 minutes ago, and the history books said back then what they say right now. I've no idea what he's trying to prove with this one.
3. Ethical beliefs about statements of value. You can't show whether the Nazi scientists in the camps did anything evil versus the scientists in Western democracies.
Ethical judgments are measured in suffering. If you make someone suffer, it is not ethical (how could it be?). So it is easy to show that someone is ethical or not.
4. Aesthetic judgments cannot be accessed by the scientific method, because the beautiful, like the good, cannot be scientifically proven.
First, there is no objective truth about what is beautiful or not. That alone is enough to dismiss it, it's not a universal truth. But going further, music is basically mathematical patterns in sound. And musical scientists have studied music, considering the various possibilities of chords and harmonies possible. It turns out that the human ear only "likes" some of these, and these have been reused over and over again in the course of musical history. We may not yet know
why we find those harmonies soothing, but perhaps one day we will, when we've learned more about the brain. So to "prove" that something is beautiful I don't know, but certainly it is "accessible to the scientific method".
5. Science itself. Science itself cannot be justified. The speed of light in the theory of relativity.
I don't know what point he's making. Is it about science or about the theory of relativity?
Science itself can be proven. The principle is you observe a pattern, you perform experiment, and you find out something. There is no difficulty in testing the method of science to see whether it produces results that are true of the world. No better test, in fact, to observe that buildings stay up, bridges hold, airplanes don't crash. What better proof could you possibly want?
As for scientific propositions that cannot seem to be proven, that's not a flaw
of science, it's our lack of advancement
in science. Science is based on finding formulas that time after time give the right results. The fact that you cannot strictly test everything in those formulas is a flaw, but it does not change the fact that these calculations, performed constantly, produce the right results, even with some measure of internal uncertainty.