Ben Franklin.

Thomas Edison and his light bulb.

Discovered or Invented - Who Can Say?

Schoolchildren are taught that Benjamin Franklin discovered lightning and Thomas Edison invented the light bulb. But what does it mean when we 'classify' a new idea or object or thought or concept (or continent) as a discovery or an invention?

Concerning mathematics, Barry Mazur, the author of both forewords to our recent Lillian Lieber books on mathematics, reminds us that the baldly formulated query: "'Is mathematics discovered or invented?' will never just go away." Referring to this puzzlement as The Question, he considers it "more a token expression of the general perplexity that we have about the status of mathematical objects, and the nature of the mathematical imagination than an answerable question. If we wish to pay homage to the passionate felt experience that makes it so wonderful to think mathematics, we had better pay attention to it."

Here's one way Mazur differentiates the two dispositions:

There are at least two standard ways of—if not exactly answering, at least—fielding The Question by offering a vocabulary of location. The colloquial tags for these locations are In Here and Out There (which seems to me to cover the field).

For those in the "In Here" camp invention in Mathematics occurs because something goes on in the intellect of the mathematician, whose faculties and understanding make a new object of thought. And for those members of the "Out There" camp, new Mathematical ideas come into sight because the mathematician has learned the structure of a Something that is not ultimately grounded in human intelligence, or human abilities of comprehension—a Something "out there" in short.

What does this have to do with Paul Dry Books? Well, our two most recent books, Lillian Lieber's Infinity and Peter Kalkavage's The Logic of Desire: An Introduction to Hegel's Phenomenology, can make you wonder about "The Question" with respect to Transfinite numbers and Spirit in History. Were the Transfinites there always and only waiting until Georg Cantor invited them to join our human conversation and was Geist's presence and progress in history there for anyone to see who came along at the right moment and who possessed the sight/insight equal to Hegel's? Or is one or both of these concepts an invention of an enormously capable mind? To read more about either book, click on their covers below.

If you would like to read more about Barry Mazur's thoughts on invention vs. discovery in Mathematics, click here and open the PDF entitled "mathematical Platonism and its opposites."

In case this musing has been too abstract to delight (we do hope our books delight), I'm going to end, where Barry Mazur begins his piece, by quoting Huck as he and Jim float down the Mississippi.

We had the sky up there, all speckled with stars, and we used to lay on our backs and look up at them, and discuss about whether they was made or only just happened—Jim he allowed they was made, but I allowed they happened; I judged it would have took too long to make so many.

In this reverie of pure delight, Huck seems to place himself in the "Discovered" camp, while Jim is a member of the "Invented" one, albeit Jim probably would say the stars were made by God. As Mark Twain hints, even in our humble musings The Question abides and is full of significance to us.