If we need proof for a belief, then we will need proof for the proof, and then proof for the proof, and then proof for the proof for the proof, ad infinitum ad nauseum. The demand for evidence has to stop somewhere. Take your belief that there is a physical world. Suppose I told you that only minds exist, and the you need to prove that there is something physical in existence. Lots of people think they don't have to prove the existence of the physical world, but belief in the physical world is a belief just like any other belief. If we must accept NO belief without supporting evidence, then we ought not to believe in the existence of the physical world without evidence.
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The way to judge basic foundational assumptions is not by looking at their causes, but by looking at their results.
Russell's distinction between knowledge by description (stuff others tell us in speech or writing) and knowledge by acquaintance (stuff known through our senses) is useful here. We normally demonstrate the existence of physical objects to others by exhibiting them to the senses. We know this is not absolutely reliable---the senses can make mistakes---but we know when this can happen (poor light, ambient noise, inattention, hallucination, etc), to some extent it is predictable, and we can guard against it when required. False belief through the senses is rare. We accept that the the physical world is not trying to deceive us. Belief through description is wholly another matter. We know there are many ways in which we can form false beliefs from innocent linguistic mistakes to deliberate deception. It's not rare at all and not generally predictable. So we are constantly on guard---even trusted sources can pass on false belief. If the default setting for acquaintance is acceptance, the default for description is more a corrigible scepticism. Corrigible in the sense that we tend to require a lesser (possibly none) or greater corroboration depending on the source and the egregiousness of what we are being told. So the upshot of this is that beliefs are quite naturally not all equal in respect of the degree of supporting evidence we tend to require for them.
Occam's Razor ought to come into play here. A purely mental reality requires a great many more assumptions to support it than does a physical universe.
OP
" If we need proof for a belief,"... " If we must accept NO belief without supporting evidence, then we ought not to believe in the existence of the physical world without evidence."
Your terminology is inconsistent.
You are conflating proof with evidence.
There is evidence the real physical world exists as I sense it. All my senses provide vivid evidence of a real physical existence.
There is no absolute proof that the nature of my existence is fairly represented by my senses. But something real must exist, else nothing would exist, and nothing cannot contemplate its own non-existence.
The mind must be a process of the physical, what else could it be? What is doing the thinking? Absolutely nothing at all? Nothing cannot do something, such as think thoughts.
Therefore, even if all that exists are minds those minds require a real existent stuff because the alternative, absolutely noting at all, cannot possibly think thoughts.
"The mind must be a process of the physical, what else could it be? What is doing the thinking? Absolutely nothing at all? Nothing cannot do something, such as think thoughts."
You are assuming that the only alternatives are "physical" and "nothing". But there may exist a third alternative, the supernatural. And just as you have evidence for, but not proof of, the physical, there is also evidence for (but not proof of) the supernatural.
"Supernatural" is an incoherent nonsense term that means nothing.
No, there is no evidence for the supernatural.
"Immaterial" is a self contradictory term.
You are just imagining nonsense.
Your reply sounds like nothing so much as a stubborn child sticking his fingers in his ears, yelling "Nya, Nya, Nya!" to drown out the sound of anyone contradicting him.
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