Under the assumption that the study of mathematical practice is meant to focus on the actual practices of mathematicians, see Van Bendegem (2018), the use of the ellipsis dots, as in N = {1, 2, 3, …} or a_i for i = 1, 2, 3, …, should be studied in its own right and not be “eliminated” away, replacing, e.g., the second example above by a_i for i in N. Following Wittgenstein’s remark in Wittgenstein (1978: 114) that the three dots should be seen as a proper sign, next to other signs such as variables and function symbols, I will investigate how such a sign functions within mathematics and show that it is actually a quite complicated practice that is one of the ingredients to distinguish the mathematician from the non-mathematician. Compare the above example of N with these two sets: {1, 4, 9, 16, …} and {1, 2, 4, 8, 16, 31, 57, 99, 163, …}. The first case will eliminate some non-mathematicians, but the second case no doubt requires a trained mathematician to be able to see how the row of numbers is to be continued. What seems to be present here is a form of tacit socially shared knowledge, embedded in particular practices. This leads to the interesting question whether this knowledge can be made explicit and perhaps even formalized. An exploration into computer science will show that such attempts have been made, e.g. Brāzma (1991), and that a formalization is indeed (at least partially) possible. Finally, the examples mentioned here all involve infinity in some sense and thus the connection with infinity needs to be investigated as well. Reading this abstract backwards, the exploration undertaken here can also be seen as a contribution to the visualisation of the infinite.

References

Alvis BRĀZMA (1991): “Inductive synthesis of dot expressions”. In: Janis Bārzdinš & Dines Bjørner (eds.), Baltic Computer Science. Selected Papers. Springer: New York, pp. 156-212.
Jean Paul VAN BENDEGEM (2018): “The Who and What of the Philosophy of Mathematical Practices”. In: Paul Ernest (ed.), The Philosophy of Mathematics Education Today. New York: Springer, pp. 39-59.
Ludwig WITTGENSTEIN (1978³): Remarks on the Foundations of Mathematics. Oxford: Blackwell.