How exactly Arnauld's Paradox is solved in modern mathematics?

173 Views Asked by At

Wells, David Graham, The Penguin dictionary of curious and interesting geometry, New York, NY: Penguin Books. xiv, 285 p. (1991). ZBL0856.00005.

A friend of Pascal, Antoine Arnauld, argued that if negative numbers exist, then $$\frac{-1}{1} = \frac{1}{-1}$$ which seems to assert that the ratio of a smaller to a larger quantity is equal to the ratio of the same larger quantity to the same smaller.

Most educated adults today would reject this idea after a moment's thought. No wonder this paradox was discussed at length.

1

There are 1 best solutions below

0
On

If one understands the fraction $\frac{m}{n}$ as $m$ resources distributed among $n$ users then we can make the following allegory.

You (1) and your antipode (-1), let's call it wife, share the same bank account with zero balance and saving/kredit features. If you log in once to the account and see that it is a debt of 1 million for you, that is $$ \frac{-1}{1}. $$ It means that your wife has taken 1 million for shopping (positive amount distributed over your antipode) $$ \frac{1}{-1}. $$