Fair enough on your first paragraph. (Though what distinguishes mathematical facts which are "coincidence" from mathematical facts forced true by mathematical structure?)
Actually, I'd say it is odd to find this sort of analysis to give great confidence about the results, not just because it ignores the possibility of structure, but also because it ignores the possibility of coincidence!
After all, there are some things which we probably want to call mathematical coincidences which heuristic argument would tell us are bogglingly unlikely and yet which nonetheless happen. For example, another probabilistic argument: Consider the basic arithmetic 2-ary operations +, -, * , and ^. There are 20!/(10! * 11!) full binary trees with 11 leaf nodes, 10^4 ways to label their internal nodes with one of these 4 operations, and 11! ways to assign the values 0 through 10 to those leaf nodes in some permutation. Thus, there are at most 20!/10! * 10^4 (overall, less than 10^16) values which can be generated using these operators and the natural numbers up through 10 once each (and this is an overestimate, ignoring the structure of, e.g., commutativity of + and * which causes less distinct values to be produced).
We would expect, therefore, that the closest we could get one of those values to line up with a particular unrelated mathematical constant, even focusing only on the fractional part, is no more than about 16 or so decimal digits. If there are thousands of mathematical constants we might compare it to, perhaps we'd get a match to about 20 digits on one.
And yet! And yet (1 + 9^(0 - 4^(6 * 7)))^(3^(2^(10 * 8 + 5))) lines up with e to 18457734525360901453873570 digits.
Now, what's happening here is that we have this other nice lining up of 9^(4^(6 * 7)) = 3^(2^(10 * 8 + 5)), which we can then plug into e ~= (1 + 1/n)^n with huge n. And this, in turn, is because 9 = 3^2 and 4 = 2^2 and 1 + 2 * 6 * 7 = 10 * 8 + 5. There's a bit of an explanation. And yet... surely if anything is a coincidence, this is?
So... I guess what I'm saying is, it is odd to use probabilistic arguments to rule out coincidence. The whole thing that makes a coincidence remarkable is that it is the sort of thing we would consider unlikely, and yet such things do occur, even in mathematics.
I don't consider the example you gave to be a coincidence, because if you know the definition of e then you know why one of those numbers could line up really, really well with e.
I'm basically saying that Feynman's argument rules out seemingly random counterexamples like 27^5 + 84^5 + 110^5 + 133^5 = 144^5, which disproved the sum of powers conjecture.
It seems to Feynman's argument works via the opposite of ruling out coincidences of that sort. It shows "If nothing surprising happens, then we expect very nearly zero counterexamples to Fermat's Last Theorem"... But the "surprising" in "If nothing surprising happens" encompasses both surprising structure (things which are true for some deep, clean reason) AND surprising numeric coincidences (things which are true for no good reason, but just happen, surprisingly, to line up in a nice, unexpected way).
Actually, I'd say it is odd to find this sort of analysis to give great confidence about the results, not just because it ignores the possibility of structure, but also because it ignores the possibility of coincidence!
After all, there are some things which we probably want to call mathematical coincidences which heuristic argument would tell us are bogglingly unlikely and yet which nonetheless happen. For example, another probabilistic argument: Consider the basic arithmetic 2-ary operations +, -, * , and ^. There are 20!/(10! * 11!) full binary trees with 11 leaf nodes, 10^4 ways to label their internal nodes with one of these 4 operations, and 11! ways to assign the values 0 through 10 to those leaf nodes in some permutation. Thus, there are at most 20!/10! * 10^4 (overall, less than 10^16) values which can be generated using these operators and the natural numbers up through 10 once each (and this is an overestimate, ignoring the structure of, e.g., commutativity of + and * which causes less distinct values to be produced).
We would expect, therefore, that the closest we could get one of those values to line up with a particular unrelated mathematical constant, even focusing only on the fractional part, is no more than about 16 or so decimal digits. If there are thousands of mathematical constants we might compare it to, perhaps we'd get a match to about 20 digits on one.
And yet! And yet (1 + 9^(0 - 4^(6 * 7)))^(3^(2^(10 * 8 + 5))) lines up with e to 18457734525360901453873570 digits.
Now, what's happening here is that we have this other nice lining up of 9^(4^(6 * 7)) = 3^(2^(10 * 8 + 5)), which we can then plug into e ~= (1 + 1/n)^n with huge n. And this, in turn, is because 9 = 3^2 and 4 = 2^2 and 1 + 2 * 6 * 7 = 10 * 8 + 5. There's a bit of an explanation. And yet... surely if anything is a coincidence, this is?
So... I guess what I'm saying is, it is odd to use probabilistic arguments to rule out coincidence. The whole thing that makes a coincidence remarkable is that it is the sort of thing we would consider unlikely, and yet such things do occur, even in mathematics.