I saw an interesting maths problem the other day. It was about finding intelligible rules behind a seemingly random arrangement of digits.
So, what digit is supposed to take the question mark’s place?
I thought right away of the possibility that the different digits each could represent some hidden value that added together would make up the result. I saw that the expressions that only contained prime numbers, or in this case, more specifically, numbers divisible only by one and by itself had a compound value of zero.
I realized that the key to solving this problem is to look upon how the factorization of every integer can be represented geometrically.
You can see for example that the numbers only divisible by one and by themselves only have ONE single geometrical form. Those are of course 1,2,3,5 and 7.
All the other integers, except zero, have more than one geometrical representation.
Zero has of course a kind of special status. The only possible geometrical representation is the “void”, or an empty representation. This is analogous to the empty set in set theory. The empty set is still a set.
The solution to the problem is to value each integer in terms of the number of factorizations that does NOT contain the factor ONE.
With this interpretation the answer to the problem is easy…
2581 = ?
0+0+2+0 = 2
There’s supposed to be another method of solving the problem but that method is dependent on the digits’ graphical form and only applicable if they are represented by Arabian numerals.
What it’s really about
Ok. I admit. I concocted this just to prove a philosophical point.
The really interesting question to answer here is of course: “IS there a true solution?”
And the answer should be no.
Reflecting Quine’s idea of the indeterminacy of translation, the deeper epistemological truth is that there will always be an infinite number of explanatory theories compatible with any set of empirical data
Like Quine’s anti-realism about the meanings of linguistic expressions and like Wittgenstein’s rejection of the possibility of private meaning, there is nothing there to be really right or wrong about.