Frege and Hume at Thanksgiving

It’s almost Thanksgiving here in Canada, so here’s a thanksgiving themed post about concepts from Frege and Neo-logicism.

In his Grundlagen (1884), Frege proposes that the number that belongs to two concepts is the same just in case the objects falling under those concepts can be correlated one-to-one (i.e. they’re equinumerous). The formalization of that claim is known as Hume’s Principle, as is the basis of the program is philosophy of mathematics known as neo-logicism. Part of the idea is that we can tell if two collections are the ‘same size’ if we can correlate their members one-to-one, without resorting to counting or otherwise using numbers.

Frege illustrates the idea with his well-known waiter example. What follows is an adaption of that example from my dissertation. I have expanded it to incorporate discussion of content carving, another important concept for neo-logicisists.

If Hera is hosting thanksgiving for her family and asks Hero to set the table, he might set a plate in front of each chair, a fork to the left and a knife to the right of each plate, and so on. So long as Hera has put out one chair for each guest (and one for herself), Hero will have set the right number of places, and if Hera asks Hero whether he has put out the correct number of dessert spoons, Hero needs only to check that there is a (small) spoon above every plate. He need not count spoons, or indeed place settings. Hero and Hera are able to go back and forth between questions about numbers of spoons and statements about on-to-one correspondences between guests and spoons seamlessly, and without loss or gain of relevant information.

In other words, Hera was able to recarve the content of Hero’s statement about a one-to-one correspondence between dessert spoons and chairs (via plates, which is possible because equinumerosity is an equivalence relation), into a judgment about whether there are the same number of spoons as guests. In other words again, Hero and Hera were warranted in moving from the content of statements about correspondences between the concepts spoons (on Hera’s table), plates (on Hera’s table), chairs (at Hera’s table) and guests (invited by or identical to Hera), to judgments about the equality of the numbers belonging to those same concepts.

(Of course, had Hera instead been cooking for the guests at Hilbert’s hotel, asking about the number of spoons would have been ambiguous, though the answer provided by HP would have been the correct one in this context. Luckily Hera didn’t have to entertain such a large (and likely quite annoyed due to all the room switching) crowd.)

If you want to know more about Hera and Hero, I’ve written about them before.