This write-up is coming from a rather intriguing situation within the philosophy of mathematics. Thousands of years have been spent to discussing whether or not infinite is ever actualized or simply a potential that exists within the application of a given function. The literature is voluminous on the subject, but for the most part, everyone seems to agree that infinite is at a minimum not conceivable within finite minds like ours. If someone asks you to think about the set containing all natural numbers, you might visualize a wall where in the farthest left hand corner there are numbers 1, 2, 3, 4... and they are increasing infinitely rightward. But now we have to visualize a wall that will never end, which doesn't seem like something we can easily, or at least readily, conceive of.
The consequences this bears on the philosophy of mathematics are quite substantial. Take for instance the discussion of what kind of a thing is a number. In the ontology of mathematics, there are a number of ways we could go about trying to answer this question. I'm claiming that the first paragraph has already set out an answer to this question before it was posed. In other words, admitting the existence of actual infinite makes it easy to suggest that the kinds of things that actualize infinite include the set of natural numbers. There is the possibility that someone who believes the infinite is actualized believes that its done only in concrete objects, that is, abstract objects like numbers are non-existing things, whereas the infinite number of atoms in the universe is. It's an interesting view, but it would still suggest that the individual who believes there are infinitely many atoms has to believe in the sensibility of the lowest cardinality of infinite, which opens them to further discussion on how that concept of infinite can be squared within mathematical analysis. More discussion in the paragraphs below.
The obvious question is why couldn't there be a philosopher who believed that numbers were real, existent abstract objects and yet infinite would ever be actualized?
It's not an attractive position, because it appears to commit you to the following claims:
1) Being that infinite is never actualized, and being that there are numbers existing mind-independently, there are only a finite amount of numbers in existence.
The only way I can make sense of this view is believing that numbers come into existence in an emergent manner. For example, the number two comes into existence when I am discussing the two pieces of cherry pie left on the kitchen counter. Given that no one is thinking about an infinitely many numbers, there are only a potentially infinite amount of numbers, therefore there are not an actually infinitely many of them at any given time. But there's an equivocation between numbers being real and numbers being realized. If numbers are real, existing abstract objects, then it follows that they do not require a cognizing subject in order to bring them into existence. They are in existence first, and the cognizing subject is merely apprehending them in a way similar to apprehending objects with our sense perception. It follows from that the the numbers that exist exist permanently, and their number is whatever the set containing all the numbers contains. If that number is not infinite, in the case of natural numbers, then it has some finite value. But why should we believe its value is finite if there is no uttermost limit we can conceive for natural numbers? Therefore, there have to be infinitely many numbers if they exist.
Alternatively, one could say that infinite is realized, but not by numbers, because those entities are in some way mind dependent, but in the number of particles there are in the universe, et cetera as discussed above. What I find difficult to comprehend about this possible view is that the number of particles in the universe correspond to some natural number n. That number represents some fact about the world, but it is not about that number, it is about the number of particles. Therefore, no one has to be committed to the existence of numbers.
I am endeavoring to uncover the relationship between actual and potential infinite and the ontology of numbers, and why we should take one question as prior to the other. For this entry, I have assumed the questions over actual versus potential infinite are the ones that come before our concepts of number, but that might not necessarily be the case. If I was motivated to believe that numbers were mind-independent, for example, then I would be forced to believe in the actuality of numbers. On the other hand, if I believed that numbers were mental constructs, then I would be inclined to believe in only potential infinities, rather than actual ones. What I have argued, however, is that it's not conceivable that these questions could become relevant prior to the discussion of infinite. Suppose that numbers top out at some number p. Further suppose we know the number, and we live in a world in which everyone agrees that it's just not conceivable that there is something larger than p. Analogously, we would have the same intuitions about p that we do about physical objects and the number of atoms that compose them. Nobody in their right mind believes that discrete physical entities can an infinitely many atoms (even though they may hold infinitely many fundamental entities, but that is a separate matter). Numbers would then occupy the same realm as amounts of things that make up other things. Following from that, every number corresponds to some amount, and for that reason they are always finite, and that there is only some large number p at which numbers end.
I think you would agree that this is not something reasonable people at this world would believe, in the main on pains of their being properties of numbers that simply do not obey the theory of correspondence to amounts in the universe above. Even a fish within a constrained fish bowl can talk about numbers in such a way that it is not determined by the limits of their worldview.