The question is to understand how it can be potential.
The number three exists right now (according to the Platonists and all their followers, including you), or it doesn’t exist (according to the others).
If it exists, then so do the numbers nine and ten, and all the numbers. Which of them doesn’t exist? Does it only rise from nothingness into existence the moment someone thinks about it? That’s an actual infinity. How is it not?

Each number exists actually, not potentially. But regarding the set of all numbers, one can say that there is no greatest number that exists, or that the number of existing numbers is as large as you like.

I’m not managing to understand, and it seems this isn’t the place for it.
If you’ve written about this somewhere else and can give me a reference, then please do. If not, then too bad.

As far as I remember, I haven’t written about this, and I don’t see what there is to write. What’s unclear here? Even if all the numbers are existing entities, it is still possible to say, instead of “there are infinitely many existing numbers” (which is seemingly a concrete statement and not a potential one), the potential statement: “there exists a number of objects as large as you want.”

I don’t know how to explain any better what I don’t understand. I feel dumb. It just seems to me like a contradiction in the same breath that ‘all’ the numbers are existing entities, but we still don’t ‘say’ that there are infinitely many existing numbers. I don’t understand why the concrete statement (which creates problems for the view that there is no actual infinity) isn’t correct.

I didn’t say it isn’t correct. What I said was that if you object to the existence of an actual infinity and assume that it cannot exist (as I too tend to think), then even under the Platonic assumption there is no need to assume that. There too, you can express everything in a potential way.

Can’t one say that the world of ideas works in decimal? Or in some other form?

One could also say that the idea of number exists, and the divisions are on our side (or that every division exists as an idea). There are lots of possibilities, but I answered even on the assumption that each number in itself exists, and even then I don’t see a problem.

How can one decide, or think, that numbers “exist,” and how can one think otherwise? I don’t see what could convince me or make me lean to one side or the other. What am I missing here?

How can one decide whether parallel lines meet or not? There is intuition. The same applies here. This is not a decision based on arguments. But read about Platonism. That’s not the topic here.

If it’s only direct intuition, then why is there a correlation between engaging in mathematics and Platonism, as written above: “Most mathematicians are Platonists, and so am I”?

Because someone who works in the field has more well-grounded intuition. He encounters this material every day and knows it well.

But most people in the world encounter numbers all the time. Why, from a Platonist standpoint, are numbers less visible to intuitive perception than all sorts of monstrous mathematical constructions? Why isn’t this like saying that a delivery driver encounters and knows gravity better than a person who works at a computer (and delivery drivers would therefore have a more shared and better-founded opinion on whether gravity is an object or not)?

They don’t really encounter them; they just pass by them. It isn’t the same someone who lives with a partner as someone who knows him and meets him from time to time.
But I have no arguments on this matter. If you don’t agree, then fine. To each his own impression.

Nicanor, regarding your question about the correlation, it may be something psychological—it’s more fun for them to think they’re dealing with something objective and not just with human concepts.