From Frege to Godelby Palle Yourgrau
A World Without Time
(Excerpt from page 172 of Palle Yourgrau's A World Without Time)
Indeed, one can ask how exactly Frege came up with his own analysis of the concept of number in his path-breaking study The Foundations of Arithmetic. A natural number, Frege begins, is what answers the question "How many?" ("How much?", is answered by a real number.) But how many what? First we must determine what it is that is really numbered. His answer: not the nine planets themselves (each by itself one, and the group as such is also one), but rather the concept is a planet. To assign a number to a concept, then, is to detemine how many instances it has. The number nine numbers the planets because the concept is a planet has nine instances. Such are the opening moves in Frege's deep and beautiful analysis of the concept of number, at the conclusion of which he offers an explicit definition: the number assigned to the concept F is the extension of the concept equinumerous with F. (Concepts are equinumerous when there is a one-to-one correlation between their extensions, i.e., to the set of objects falling under each. In effect, then for Frege, the number three is the set of all trios, i.e., of all things that are three in number.) Nothing like this had ever been proposed before. As Frege himself put it, his discussion raised the subject to an entirely new level.
What does it take to make a great scientific discovery? Two elements are crucial. One must have an insight into which problems are ripe for resolution, and one must then have the craft -- or invent it -- to solve the problem one has had the audaciy to recognize as solvable.
As Einstein wrote later, "the antipathy of these scholars towards atomic theory can indubitably be traced back to their positivistic philosophical attitude. This is an interesting example of the fact that even scholars of audacious spirit and fine instinct can be obstructed in the interpretation of facts by philosophical principles."
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