Nelson Goodman introduced a problem with human intuition through
induction in his book
"Fact, Fiction and Forecast" published in 1965. The problem has to do with examining the color of an
emerald at different times. First Goodman defines the word
grue. The definition of
grue has two parts. Objects that are assigned the property of
grue are:
Objects FIRST examined before January 1, 2003 that are green.
Objects not examined before January 1, 2003 that are blue.
The time of January 1, 2003 is completely arbitrary. The date serves to distiguish between objects examined before a certain time and objects examined after that time.
This definition of grue can be difficult to understand initially. Assigning the property of grue to an object depends on the time it was FIRST seen and its color. A green object FIRST seen before the date is called grue. Likewise, a blue object FIRST seen after the date can be called grue as well. However, green objects FIRST seen after the date and blue objects FIRST seen before the date are not grue. Once an object is described as having the property of grue, the object can always be described as having the property of grue. For example, a green object seen before the date is given the property of grue. After the date, that object can still be described as being grue. This is because the object was first seen before the date.
With the definition of grue out of the way, Goodman continues the riddle with the examination of emeralds. One examines a set of emeralds before the date. The emeralds are seen to be green. As it is before the date that the emeralds are examined, they are also described as being grue. One will induce that all emeralds are green and grue.
After the date one picks up a new emerald, one that has not been previously examined. The expectation, based on induction, is that the emerald will be grue and green. For the emerald to be grue, it has to have the color of blue. Grue objects first examined after the date must be blue by definition. A blue emerald directly violates the other induction, that emeralds are green. This contradiction is Goodman's proof for showing that induction is totally useless.
Note: In response to Rose Thorn's write up: Goodman's New Riddle of Induction does not mention mathematical induction, in fact, it has nothing to do with mathematical induction. Goodman discusses philosophic induction, the generalization about the whole based on a part (See Webster's write up in Induction, specifically the part on philosophy). In other words, although we witness gravity everyday, there is no guarantee that gravity will be around forever.