According to fictionalism, mathematics is a collection of useful fictions whose statements are, despite their usefulness, actually all false. In these fictions there are recurring "characters" like numbers, straight lines, graphs and many others, all entirely fictitious. Nevertheless, the fictions are useful because they convey (or rather, reflect) truths about our world. Furthermore, discussing our experiences in terms of carefully chosen, representative fictional characters greatly facilitates communication.

Although I agree that mathematics is a collection of stories, I still think that the ideas (i.e. the characters) in those stories are real, in one way or the other, simply because we do sense them. The assumptions the stories make about the ideas, however, may well (all) be fictional. It's like writing a guide about an existing city without knowing it well.

Also, in my view the ideas exist right here with us (just like the city in the previous analogy), not only in a separate "world of ideas" as platonism would suggest. E.g. in a computer network, there is a graph right there belonging to the network; where else could it be? Putting it another way, I don't think the network is more real than the graph.

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