Cats can't see with their whiskers. But they do use their whiskers to expand their area of perception into places their sight can't go.
Cats can't see right under their noses, so their whiskers feel minute differences in whatever is there, and the cat gets the information.
And, to get to the science of it, cats use the same part of their brains for vision as they do for their whiskers. This is probably where the belief comes from.
The whiskers on the cat's paws work the same way; passing on vital information our cat can't see directly. But the cat will process this with visualization, and act accordingly.
Cats think with geometric precision about where stuff is and where their stuff is. One reason they are so good at it is because they have a great feedback system through several senses. Their paws are especially good at picking up tiny vibrations because of those whiskers on their feet.
The next time we can't figure out what a cat is looking at, we should remember that they can see certain things much better than we can.
So the joke is really on us.
For more about cat vision, find out What Cats See.