Infinitely Big and Infinitely Small
"Speaking philosophically, I no more support infinitely small magnitudes than infinitely large ones, or no more infinitesimals than infinituples. For I consider both to be fictions of the mind, suitable for calculating by an abbreviated way of speaking, such as imaginary roots are in algebra." --Leibniz to Des Bosses, 1706
Our interest is primarily in the infinitely small, but the above quote shows Leibniz to have similar reservations about the infinitely large. Drawing on an interpretation defended by Hidé Ishiguro, Ric Arthur ("A Complete Denial of the Continuous?") argues that in both cases Leibniz understands the relevant quantity terms "syncategorematically." By this he means that 'infinitesimal' and 'infinite' are not to be taken as ordinary referring terms, designating infinitely small or infinitely large quantities, but rather as shorthand for the properties of certain infinite series.
With respect to the infinitely large, Leibniz holds that any portion of matter is infinitely divided into parts; however, he denies that it follows from this that there is a determinate number of parts (an "infinity") into which it is divided. Arthur has a nice way of drawing this distinction (p. 7):
To assert an infinity of things categorematically is "to assert that there exists some one number of parts y which is greater than any finite number x, i.e. that (∃y)(∀x)Fx → y > x," with Fx = x is finite, and x and y numbers.
This is what Leibniz denies. Instead he understands talk of an "infinity of parts" syncategorematically, which means, "for any finite number x that you choose to number the parts, there is a number of parts y greater than this: (∀x)(∃y)Fx → y > x."
It is important to recognize that Leibniz does not claim merely a potential infinity of parts, i.e., that matter can in principle be divided without limit. He believes that matter is actually infinitely divided. What he denies is that there is a determinate set or totality of parts whose size is given by an infinite number. In the wake of Cantor, this may strike us as wrong-headed, since the first transfinite cardinal, designated ℵ0 (aleph-null), specifies the size of a countably infinite set--a set equinumerous with the set of natural numbers. Arthur and Greg Brown have debated in print whether Leibniz simply missed the boat on this point (see Arthur's paper for citations). Arthur believes that he didn't, and that Leibniz in fact offers a defensible alternative to Cantor on the infinite.
Now how does all of this relate to infinitesimals? In the case of matter, the flip-side of the claim that a continuum is not divided into an infinite totality of parts, but only into parts ad infinitum, is that in its division we never reach infinitely small parts, but only finite parts as small as one chooses. The more important case concerns the properties of an ideal mathematical continuum, such as a line or a plane. Here the crucial innovation of the calculus lies in its giving us a precise way to handle infinite series of vanishingly small, finite quantities. Consider the series 1/2 + 1/4 + 1/8 + 1/16 + ..., whose sum converges on 1. For any finite number of terms, the sum of this series is less than 1. One might imagine then that to reach 1, there must exist an actual infinity of terms, the last of which would be the infinitesimal element (1/∞), which is the difference between the closest possible finite approximation and 1. This is the kind of reasoning that the syncategorematic interpretation rejects. On this account, to say that the sum of the series converges on 1 is simply to say that for any pre-assigned error ε, there is an n, such that 1 - ∑ (1/2n) < ε. Much more on this can be found in Arthur's paper, which we'll discuss in week 6.
Our interest is primarily in the infinitely small, but the above quote shows Leibniz to have similar reservations about the infinitely large. Drawing on an interpretation defended by Hidé Ishiguro, Ric Arthur ("A Complete Denial of the Continuous?") argues that in both cases Leibniz understands the relevant quantity terms "syncategorematically." By this he means that 'infinitesimal' and 'infinite' are not to be taken as ordinary referring terms, designating infinitely small or infinitely large quantities, but rather as shorthand for the properties of certain infinite series.
With respect to the infinitely large, Leibniz holds that any portion of matter is infinitely divided into parts; however, he denies that it follows from this that there is a determinate number of parts (an "infinity") into which it is divided. Arthur has a nice way of drawing this distinction (p. 7):
To assert an infinity of things categorematically is "to assert that there exists some one number of parts y which is greater than any finite number x, i.e. that (∃y)(∀x)Fx → y > x," with Fx = x is finite, and x and y numbers.
This is what Leibniz denies. Instead he understands talk of an "infinity of parts" syncategorematically, which means, "for any finite number x that you choose to number the parts, there is a number of parts y greater than this: (∀x)(∃y)Fx → y > x."
It is important to recognize that Leibniz does not claim merely a potential infinity of parts, i.e., that matter can in principle be divided without limit. He believes that matter is actually infinitely divided. What he denies is that there is a determinate set or totality of parts whose size is given by an infinite number. In the wake of Cantor, this may strike us as wrong-headed, since the first transfinite cardinal, designated ℵ0 (aleph-null), specifies the size of a countably infinite set--a set equinumerous with the set of natural numbers. Arthur and Greg Brown have debated in print whether Leibniz simply missed the boat on this point (see Arthur's paper for citations). Arthur believes that he didn't, and that Leibniz in fact offers a defensible alternative to Cantor on the infinite.
Now how does all of this relate to infinitesimals? In the case of matter, the flip-side of the claim that a continuum is not divided into an infinite totality of parts, but only into parts ad infinitum, is that in its division we never reach infinitely small parts, but only finite parts as small as one chooses. The more important case concerns the properties of an ideal mathematical continuum, such as a line or a plane. Here the crucial innovation of the calculus lies in its giving us a precise way to handle infinite series of vanishingly small, finite quantities. Consider the series 1/2 + 1/4 + 1/8 + 1/16 + ..., whose sum converges on 1. For any finite number of terms, the sum of this series is less than 1. One might imagine then that to reach 1, there must exist an actual infinity of terms, the last of which would be the infinitesimal element (1/∞), which is the difference between the closest possible finite approximation and 1. This is the kind of reasoning that the syncategorematic interpretation rejects. On this account, to say that the sum of the series converges on 1 is simply to say that for any pre-assigned error ε, there is an n, such that 1 - ∑ (1/2n) < ε. Much more on this can be found in Arthur's paper, which we'll discuss in week 6.
posted by Don at 9:19 AM
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