There's a tendency to get into all sorts of philosophical quicksand when talking about existence – does it mean physical existence, or space-time existence, is there a sense in which Harry Potter or Frodo have a fictional existence, etc. But let's keep it simple. When I ask whether the square root of 2 exists, all I am asking is whether there is a positive number such that when you multiply it by itself, the product is the number 2. In that way I haven't used the word 'existence' or 'exist' at all. Of course, I used the expression 'there is', but people don't seem to find the verb 'is' problematic in the same way as the verb 'exists'. So, is there a square root of 2?
This brought us to a discussion of sequences of decimals. We agreed that such sequences exist, i.e. that some things are sequences of decimals, or some sequences are of decimals, and we agreed that these sequences can be multiplied. Then we agreed that these sequences can be infinite, and that infinite sequences can also be multiplied. Finally, we agreed that (if we bought the other stuff), there is at least one infinite sequence such that when multiplied by itself, the product is 2 (or rather, the product is 1.999999… which you can verify on an Excel spreadsheet). So, there is a square root of 2. Or, if you like, the square root of 2 'exists'.
Is that it? Of course, we had to buy a couple of ideas. First, that some things are numbers. Some things are chairs, some things are tables, some things are stars or planets, some things are or may be angels (pace Anthony, who does not believe that any things are angels). And some things are numbers. Note my avoidance of the word 'exists'. Second, some numbers correspond to finite sequences of decimals, others to infinite sequences. Do we buy that? Time to read some more Ockham. More later.