2.     Poincaré and others commonly describe the deductive and inductive process of logic as movement up or down stairs. Deduction moves down, step by step, from the general to the particular; whereas induction moves up, step by step, from the particular to the general. I argue, in a forthcoming book, Why The Readymades Aren't Readymades: Decoding Duchamp's Paradigm Shift, that Duchamp's famous work, Nude Descending a Staircase (1912), is a schematic diagram of deduction; whereas his earlier work, Nude Ascending (1911) depicts induction.

3.     I cannot help mentioning how elementary this logic is. In a children's cable program, called "Blues Clues," a blue dog gives clues to the host who has been "trained" to immediately sit in a "thinking chair" once he receives "3 clues." The man then sits in a chair and tries a variety of logical sequences of the 3 facts until a right combination is found to solve the puzzle every time!

4.     Essentially, Duchamp has made a discovery about the discovery of generalizations. He then makes his own generalization and its attendant machine (Large Glass), initial conditions (notes), and tools (3 Standard Stoppages) -- for making further generalizations!

5.     Poincaré said that among the greatest discoveries are those that borrow laws from one field and unexpectedly apply them to another -- as in applying the laws of gaseous molecules to the Milky Way. Similarly, Charles Darwin deductively applied Adam Smith's theory of economics to arrive at his original and revolutionary theory of evolution. Therefore, Duchamp taking Poincaré's law (from science) and applying it in a new discipline (art) counts as an original discovery.

6.     Craig Adcock's (1984) paper first brought this point to my attention and was pivotal to my discovery.

7.     I am grateful to Camfield for his discussion on the readymades.

8.     Duchamp emphasized the importance of scale. Duchamp stated, "Scale...is highly important" (as cited in Bonk, 1989, p. 184). I believe that the Fountain urinal, his Rebus drawing (1961) (where a toilet is shown emptying into the Seine River), and his L.H.O.O.Q., (1919) (his Mona Lisa readymade, where Leonardo's concept of the universal cycling water system of the earth is famously depicted in the background) all appear as isolated 3-D facts but are invisibly linked parts of a larger 4-D macrocosmic whole that we were meant to deduce. Therefore, the urinal represents the macrocosmic scale in this readymade series of the 3 scales of nature. I am grateful to Jeffrey Epstein for pointing out that the H.O.O. in the title of L.H.O.O.Q. might be read as , the chemical formula of water (personal communication, September 1997).

9.     Poincaré explains that the unity of nature expresses itself in the similar relation among the "Milky Way," "dust in fluid," and "gaseous molecules": "if the different parts of the universe were not organs of the same body, they would not re-act one upon the other...We need not, therefore, ask if Nature is one, but how she is one" (Poincaré, 1902/1952, p. 145).

10.     Poincaré said that the greatest act of chance is the birth of a genius when, by chance, the right ovum and sperm meet (after chance orgasm, of course). Poincaré also said that not all mathematicians are cast in the same mold; some are more intuitive; others more logical - but intuitive mathematicians make more discoveries! (See Poincaré, 1908, chapter on Chance.)

11.     In his Green Box notes, Duchamp's "Pendu Bride" (a pendulum that "swings to and fro") generates movements and letters that are captured both as "snapshots," maps of these varied movements (3 draft piston nets in the Milky way cloud), and a Triple cipher (see Sanouillet & Peterson, 1973, pp. 36-37).

12.     This is precisely why Duchamp could say that the readymades "lack uniqueness" and that "reproductions" [as representations] "deliver the same message" (see Sanouillet & Peterson, 1973, p. 142).

13.     Duchamp stated "I'm not at all sure that the concept of the Readymade isn't the most important single idea to come out of my work" (as cited in Kuh, 1905, p. 92; emphasis mine). The Large Glass was his master work. Duchamp categorizes all of his readymades as a "single idea coming out of his work." I believe that his one idea-readymades-came in a "limited series" out of his Large Glass, just as Poincaré's "readymades" came out of his creativity machine as one idea (in a series of revelations).

14.     Duchamp was amused by "the various interpretations" scholars and artists offered, but, importantly, he did not treat them all as equal and distinctly praised only a few. For example, he inscribed a book to Richard H. Hamilton "mon grand dechiffreur" or "my great decipherer" (Maharaj, 1996, p. 72).