This is part of why I wanted to clarify. When I wrote "Could we stick to asking whether numbers have mind-independent existence, or whether platonism is true, or something?", I clearly meant that we should stick with asking whether or not numbers exist outside of the mind. That is what mind-independent means. And platonism contends that numbers are mind-independent (and also that they exist in some "other realm").
Point taken.
Further, when one reasons about numbers, one does not reason about the concept of a number. When we say that a system capable of representing arithmetic truths contains a statement of its own consistency if and only if it is inconsistent, we are not reasoning about concepts in the mind of a mathematician, we are reasoning about abstract objects. But do the abstract objects exist? I say not. They are but ideas in a mathematician's head. Very useful ideas, sure. Subjective ideas, of course not. mathematics is not subjective, yet numbers are not mind independent.
Again, this comes down to the definition of existence. I can tentatively derive that from the above, your definition of existence involves "major" physical manifestation (major meaning, in this case, at a higher level than inforamtion-carrying physicality, such as chemical or electrical states).
Do the abstract objects exist outside of a particular persons mind?
Yes, the concepts have been written down many times, and are often coded into computer systems. It could be debatable whether the "understanding" of the concept creates the concept, but this term is debatable as well - for example, the computer "understands" the concept to the extent that it can use the concept meaningfully within limited parameters - most people are about the same.
Are these concepts physically manifest above the level of informational physicality? No.
So it all depends.
For example, if someone told me a story about a dancing chicken, this story would exist, because the story is the words I heard or read - regardless of whether or not the words were in physical, verbal, or electronic form.
One representation of it would exist, but if we consider the story to be more than its mental representation, it does not exist. Numbers, as humans reason about them, are not conceived of as mental representations but as, well, numbers. And when you say that you have the idea of the number 2317 in your mind, that does not mean that you have the number 2317 in your mind.
When someone says, "It's only in your head", they don't mean that "it" exists. When a paranoid person thinks that a man is after him with a gun, but the man is imaginary, we do not say that the man with the gun exists, just because a representation of such a man exists in the paranoid person's head.
So we cannot say that numbers exist just because we as humans have representations of numbers in our heads.
This assumes that the nature of numbers is not an idea or a representation - I believe that this is exactly what a number is. The number 5 may be used a representation of, for example, 5 cows - but this doesn't mean that the number 5 becomes something other than the number 5 as a result. In the same way, the letter c may be used as a representation of the speed of light, but this doesn't stop it from also being the letter c.
In the example of the gunman above, the gunman doesn't exist, as the idea of the gunman is based on false data.
If you were to ask "Does the idea of the gunman exist?", the answer would have to be yes.
Further, it leads to bizarre consequences like numbers popping in and out of existence as we think of them. If no one was thinking of the number 345678729435 while I typed this, that must mean the number popped into existence as I wrote it and flickers out of existence when I forget it until someone else comes along and imagines it.
I don't see that as so bizarre - for example, if an airtight room exists, you could say that it "flickers in and out of existence" every time someone opens or closes a door.
In reality it doesn't - it just becomes something you'd have to use different words to describe.
With numbers, the concept of numbering exists, and it briefly becomes something you might describe as "the number 345678729435" when this number is used, then becomes something else.
All spiritual qualities are mental - spirituality is a mental concept. For example, if a god or gods existed, then they would be literal entities, and wouldn't have any inherent spirituality. Any spirituality directed towards these creatures would be entirely mentally based.
Religious people give spirituality metaphysical qualities. They believe that gods literally have some property called spiritualness, or some such.
Point taken again - there's a few different meanings of the word.
I usually take the word as meaning "Thing to which great significance or symbolism is ascribed", but it can also mean such things as "non-physical existance", or even "form of religious 'energy'" in some cases.
Thought I'd throw a few more questions out there, just to clarify the definition of existence.
Does thought exist?
Does conflict exist?
Does happiness, sadness, or any other emotional state exist?
Do groups exist?
Does propoganda or rhetoric exist?
Does prejudice exist?