- It does mean completely - though I'd say it's a little more emphatic.
- They can be used somewhat interchangeably without causing misunderstanding, but usually I think of "independent from" to be referring to individuals or groups.
"X being independent from Y" usually implies that neither X nor Y are subject to the other's decisions or orders.
While it might sound like a sub-group, the Southern Baptist Convention (widespread across the American south) is completely independent from the American Baptist convention.
On the other hand, "independent of" usually gets applied more to physical phenomena, and intangibles like ideas, facts, and processes.
"X being independent of Y" means generally that a change in X, or a different X entirely, won't be related to the value of Y. Or, that X wasn't taken into account in the generation of Y.
Our study showed that measurements of brain activity are completely independent of the subject's preferred bagel style, but very dependent on whether the subject was actually inside our MRI machine during the scan.
Frank's new success-guaranteed methods of cat-taming were intriguing, but clearly had been developed completely independent of any previous research, or indeed any first-hand experience with felines.
That sentence is pretty dense and quite abstract, as philosophy tends to be, but I'll try to break it down. I don't believe it's something easily understood, even without a language barrier. If you're curious and you haven't already, you may want to check out the philosophy stack exchange, and/or look up Platonic vs Aristotelian theories of knowledge/reality. The base of the discussion is trying to discover what a particular object 'really' is.
"Matter" here just means physical matter - atoms and such. It refers to the table, as an example of matter.
"as" could here be replaced with "the way"
One could simplify:
"Such an idea has the required permanence and independence of
ourselves, without being -- as matter would otherwise be -- something
quite unknowable, in the sense that we can only infer it, and can
never be directly and immediately aware of it."
This concept, (that our perceptions of something's physical form do not define whether or not it exists, but instead that things 'really' exist in some form that only God knows), implies the permanence and independence that we assume to be true (that objects exist whether we can see them or not, and that what they really are isn't related to us or the way we perceive them). The concept also doesn't mean that the table (like all objects/matter) isn't something we can actually know to exist, but merely something we can physically detect.
To venture into (amateur) philosophy, I'll try to give an analogy, that a table 'really' exists only in the mind of God is like saying that a group, like the American Congress, really exists as an idea in the mind of people.
So what is the American Congress? Is it "The set of elected officials that meet in the Capitol building in Washington D.C. to make America's laws"? According to that definition, if Congress stops making laws, or moved into a different building, it would cease to be Congress. So should the definition be something more general? More precise? To attempt to nail down a precise definition of what the congress is, using its attributes, is in line with Aristotelian philosophy - that something 'is' its description. Platonic philosophy objects, saying that every person could perceive the entity differently, and may not perceive it at all, and so what it really is, must not be dependent on its attributes - though its attributes should follow directly from whatever it 'really' is. A more platonic definition (I think) would be that "Congress exists as an idea in the mind of the people who know of it" - that no matter how its attributes change (like its members, actions, or location), Congress exists because people believe it to exist.
Hopefully that's more understandable than the original text - if any of the above language isn't clear/understandable, please let me know. If the words make sense but the concept doesn't, you may find people better versed on the subject in the philosophy Stack Exchange.
I've only tangentially studied philosophy, so if I've misrepresented something and someone more knowledgeable would like to correct my understanding, I'd appreciate it.