Ramble, Part 20: Powers to the People!
Apr. 5th, 2007 12:08 pm![[personal profile]](https://www.dreamwidth.org/img/silk/identity/user.png){kind=small}
stoutfellowRené Descartes is well-known as a seminal figure in the history of philosophy. It is less well-known that he holds a comparable position in the history of mathematics. Though a great mathematician, he is not, in my judgement, among the very greatest; he may be in the top ten all-time, but not in the top five. However, his influence on mathematical methodology and notation is second only to that of Euclid, and the next few Ramble posts will explore that influence.
Let's begin with a quote from Descartes' La Géométrie.
But there is more going on than convenience and extensibility. Good notation has what might be called an emergent quality: it contains within itself the possibility of generalization, of continued innovation. We've seen this before, in the step from decimal notation to decimal fractions. Exponents provide a textbook case of this potentiality; their meaning has been extended repeatedly, in multiple directions.
The first step comes with the recognition of structure in the notation, in the "Laws of Exponents": aman=am+n, (am)n=amn, and anbn=(ab)n. Note that these laws cannot even be enunciated without the appropriate notation; how could, say, Viète have spoken thus? This is the first fruit of the new notation.
The second fruit... At some point after being introduced to exponents, the student of mathematics is told that a1=a, and a0=1. These are often given as if they were edicts from on high, but this is a misrepresentation, both historically and as a model of how mathematics works. The question is not "what does a1 mean?", but rather "what should it mean?" - what meaning can we give it that will be most useful? Our guideline lies in the structure of the previously-defined exponents: it is good and valuable structure, and ought to be preserved, if possible. Whatever a1 is to mean, it ought to fit the first of the laws given above; that is, we ought to have that ana1=an+1. To make this happen, we must set a1=a. I want to emphasize, though, that this is indeed a choice, made for our convenience. Most often, we do not make our definitions and derive our results from them; rather, we frame our results and shape our definitions to make the results true.
Similar reasoning leads to the decision to make a1/2 refer to the square root of a, and ultimately to give meaning to expressions like ap/q, for any choice of integers p,q. (Care needs to be taken, here; the extension of the laws of exponents to fractional exponents contains a few booby-traps, which I may discuss at a later point.) The notation can be stretched yet further, to allow exponents to be irrational numbers, or even complex numbers; the process of generalization takes a detour through logarithms, but it works satisfactorily (with the same caveats as for fractional exponents).
In the course of the twentieth century, exponents were generalized in another direction. It has become conventional to write BA to represent the set of all functions from the set A to the set B. Part of the prompting for this generalization lies in the fact that, if A and B are finite sets, with m and n elements respectively, then the set BA has nm elements. Another part, though, lies in the realization that many of the familiar laws of arithmetic - the associative, commutative, and distributive laws, as well as the laws of exponents - reflect fundamental properties of sets, and - in some contexts, and by some authors - some parts of set notation have been modified to make this relationship clear. This extension of exponential notation to things that are not even numbers has gone yet further, as I discussed here and here.
All of these generalizations have made the task of thinking about mathematics easier, by making clear what other notation obscures. None of them, I daresay, were in the mind of Descartes when he introduced the notation, but they lay implicit in the new symbolism; and that is one of the hallmarks of good notation.
This is only one of the innovations that Descartes introduced, and frankly it is one of the lesser ones. We'll be looking at some of the others in the next couple of Ramble posts.
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Let's begin with a quote from Descartes' La Géométrie.
...a-b will indicate that b is subtracted from a; ab that a is multiplied by b; a/b that a is divided by b; aa or a2 that a is multiplied by itself; a3 that this result is multiplied by a, and so on, indefinitely.In those last two clauses - offhand, in the midst of doing something else - Descartes introduces to mathematics the concept of exponents. Viète had taken a step in this direction, but his notation, using such abbreviations as quad and cub, could be extended only by the constant coinage of new abbreviations. Descartes' notation, by contrast, is indefinitely extensible, using the same principles that Leonardo of Pisa had introduced to Europe for the representation of numbers themselves.
But there is more going on than convenience and extensibility. Good notation has what might be called an emergent quality: it contains within itself the possibility of generalization, of continued innovation. We've seen this before, in the step from decimal notation to decimal fractions. Exponents provide a textbook case of this potentiality; their meaning has been extended repeatedly, in multiple directions.
The first step comes with the recognition of structure in the notation, in the "Laws of Exponents": aman=am+n, (am)n=amn, and anbn=(ab)n. Note that these laws cannot even be enunciated without the appropriate notation; how could, say, Viète have spoken thus? This is the first fruit of the new notation.
The second fruit... At some point after being introduced to exponents, the student of mathematics is told that a1=a, and a0=1. These are often given as if they were edicts from on high, but this is a misrepresentation, both historically and as a model of how mathematics works. The question is not "what does a1 mean?", but rather "what should it mean?" - what meaning can we give it that will be most useful? Our guideline lies in the structure of the previously-defined exponents: it is good and valuable structure, and ought to be preserved, if possible. Whatever a1 is to mean, it ought to fit the first of the laws given above; that is, we ought to have that ana1=an+1. To make this happen, we must set a1=a. I want to emphasize, though, that this is indeed a choice, made for our convenience. Most often, we do not make our definitions and derive our results from them; rather, we frame our results and shape our definitions to make the results true.
Similar reasoning leads to the decision to make a1/2 refer to the square root of a, and ultimately to give meaning to expressions like ap/q, for any choice of integers p,q. (Care needs to be taken, here; the extension of the laws of exponents to fractional exponents contains a few booby-traps, which I may discuss at a later point.) The notation can be stretched yet further, to allow exponents to be irrational numbers, or even complex numbers; the process of generalization takes a detour through logarithms, but it works satisfactorily (with the same caveats as for fractional exponents).
In the course of the twentieth century, exponents were generalized in another direction. It has become conventional to write BA to represent the set of all functions from the set A to the set B. Part of the prompting for this generalization lies in the fact that, if A and B are finite sets, with m and n elements respectively, then the set BA has nm elements. Another part, though, lies in the realization that many of the familiar laws of arithmetic - the associative, commutative, and distributive laws, as well as the laws of exponents - reflect fundamental properties of sets, and - in some contexts, and by some authors - some parts of set notation have been modified to make this relationship clear. This extension of exponential notation to things that are not even numbers has gone yet further, as I discussed here and here.
All of these generalizations have made the task of thinking about mathematics easier, by making clear what other notation obscures. None of them, I daresay, were in the mind of Descartes when he introduced the notation, but they lay implicit in the new symbolism; and that is one of the hallmarks of good notation.
This is only one of the innovations that Descartes introduced, and frankly it is one of the lesser ones. We'll be looking at some of the others in the next couple of Ramble posts.
Previous Next
Ramble Contents
no subject
Date: 2007-04-11 05:07 pm (UTC)[pedant hat] If you're going to put in accents, there's another one after the m in Geometrie.
:)
no subject
Date: 2007-04-11 06:04 pm (UTC)