Most certainly it would suffice. The 10 years old is awesome… only once I enjoyed the 18 years old version when my wife surprised me. It was a birthday present (cost her 80 bucks a VERY long time ago) and it lasted well over a year… but it was something to behold. And we can conduct a good conversation with some “virtual” malt Scotch and “virtual” Napoleon cognac.
Yes, we did.
Let’s concentrate on this concept. I agree that math is a pure abstraction of certain parts of reality.
But, we agreed that the notion “abstract objects” is nonsensical - and that is the crucial point. So these abstractions cannot exist without a mind which can “make” the abstractions! However, I agree that the parts of reality which can be “abstracted” do exist without our existence. As a result, I differentiate between the “attributes” and the abstractions which are “abstracted” from them. (A cumbersome sentence, I am afraid.)
An example: it is an objective fact of reality that a water molecule contains one oxygen atom and two hydrogen atoms. We do discover this fact. But the mental image (or abstraction) of
H2O does not exist if there are minds to perceive it. We discover the physical fact, but we create the abstractions of it. Do I make myself clear? I hope so.
I think you did. Riemann’s geometry is just as true as Euclid’s and Bolyai’s. They are all mental constructs, with their axioms.
Certainly! The selection of axioms is arbitrary, in the sense that we can select any axioms we want to, as long as they are internally consistent. They are not necessarily applicable to anything. Euclid’s geometery is perfect for a flat surface, and it fails on the sphere, and also fails of the pseudo-sphere
en.wikipedia.org/wiki/Pseudosphere On the other hand, Riemann’s is “true” (applicable) on the surface of a sphere and “fails” (false) on a flat surface (also on the pseudo-sphere). And on the surface of an irregular shape there is “no geometry” at all!
I am pretty sure about the nature of our “disagreement”.

There is no disagreement, just a slight difference of words. If you agree that there is no abstract object of “one” which exists independently from our minds, we are home, scot free. Suppose, that humans are the only beings in the universe who are able to conceptualize. Before the first human was able to come to the concept of “one apple” versus “two apples” - and it was a huge, incredible breakthrough - there simply was no “one” and no “two” - even though the physical attributes and properties which make these concepts possible were all present. This is the important part I want to emphasize…
As you say, if the sun would go nova, in a few billion years a new set of “beings” would come to the same “result” of “one” and “two”, and I do agree with you. But in the meantime, there would be no “one” and no “two”. The point is that if you start to state the “mind-independent” existence of “numbers”, you will HAVE to accept that the Hamlet, and the Ninth Symphony also have a “mind-indpendent” existence, and they are also “discovered” and not created, which is sheer nonsense.
The numbers seem to lend themselves to be designated as “existing” apart from the mind, but stipulating their “mind-independant” existence will lead to the acceptance of “abstract objects”. Numbers are very unique in the sense that they “seem” to be “obvious”. It seems obvious to stipulate “one apple” and “two apples”… and it is obvious. But there are humans who are incapable of conceptualizing “minus two apples”. Or a distance of “minus three meters”. Zero used to be a huge problem, too. One apple is fine, but “zero” apples??? Or “minus one apple”? The natural numbers (1, 2, 3, etc…) seem to exist independently from our mind, but everything else is problematic. As such it makes much more sense to accept that even the simple “one” is an awesome creative abstraction and not a “discovery” of a nebuolus “abstract” one, than the opposite.
Be as it may, I enjoy this conversation tremendously, which is based solely on reason and not “faith”. I am sick and tired of the illogical and irrational “non-arguments” put forth by the plethora of nincompoops around here.