> I'm getting quite frustrated with this kind of ontological debate. Exist or not exist, why does it matter?
One's philosophical / metaphysical worldview determines how one views reality.
As an example, some variations of Islam follow(ed) occasionalism:
> Occasionalism is a philosophical doctrine about causation which says that created substances cannot be efficient causes of events. Instead, all events are taken to be caused directly by God. […] The doctrine states that the illusion of efficient causation between mundane events arises out of God's causing of one event after another. However, there is no necessary connection between the two: it is not that the first event causes God to cause the second event: rather, God first causes one and then causes the other.
Whereas Christianity rejected it and went with secondary causation:
> Secondary causation[1][2][3] is the philosophical proposition that all material and corporeal objects, having been created by God with their own intrinsic potentialities, are subsequently empowered to evolve independently in accordance with natural law.
So in in the first case asking "Why did X happen?" you answer "God willed it.", while the second case you would say "There is something in Object A that interacted with Object B." The latter then leads you down the path of examining objects and their relationships, as opposed to chalking events to spirits, gods, or God exclusively. Plants/crops growth because of something with-in themselves and not because of Ceres / Demeter willed it.
Without this worldview, you don't operate under (e.g.) the zeitgeist of being able to investigate Nature:
> That this objective reality is governed by natural laws;[35][36]
> That reality can be discovered by means of systematic observation and experimentation.[35][36]
> That Nature has uniformity of laws and most if not all things in nature must have at least a natural cause.[36]
"One's philosophical / metaphysical worldview determines how one views reality." I fully agree but I fail to see how it is true in the case of ontology of mathematics. All ontological frameworks will have to explain the existence of the same entities so what is left is the difference of vocabulary and maybe cultural practices embedded in each view, but the reality will be the same. You won't meaningfully deny the existence of triangle only because you changed your ontology.
No, but it might change your perspective on things and lead you to explore a different set of ideas (something that may end up being quite important when doing scientific research?).
One's philosophical / metaphysical worldview determines how one views reality.
As an example, some variations of Islam follow(ed) occasionalism:
> Occasionalism is a philosophical doctrine about causation which says that created substances cannot be efficient causes of events. Instead, all events are taken to be caused directly by God. […] The doctrine states that the illusion of efficient causation between mundane events arises out of God's causing of one event after another. However, there is no necessary connection between the two: it is not that the first event causes God to cause the second event: rather, God first causes one and then causes the other.
* https://en.wikipedia.org/wiki/Occasionalism
Whereas Christianity rejected it and went with secondary causation:
> Secondary causation[1][2][3] is the philosophical proposition that all material and corporeal objects, having been created by God with their own intrinsic potentialities, are subsequently empowered to evolve independently in accordance with natural law.
* https://en.wikipedia.org/wiki/Secondary_causation
So in in the first case asking "Why did X happen?" you answer "God willed it.", while the second case you would say "There is something in Object A that interacted with Object B." The latter then leads you down the path of examining objects and their relationships, as opposed to chalking events to spirits, gods, or God exclusively. Plants/crops growth because of something with-in themselves and not because of Ceres / Demeter willed it.
Without this worldview, you don't operate under (e.g.) the zeitgeist of being able to investigate Nature:
> That this objective reality is governed by natural laws;[35][36]
> That reality can be discovered by means of systematic observation and experimentation.[35][36]
> That Nature has uniformity of laws and most if not all things in nature must have at least a natural cause.[36]
* https://en.wikipedia.org/wiki/Naturalism_(philosophy)#Provid...
So on a day-to-day basis you may not deal with these axioms/beliefs, but they are there nonetheless.