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Alternatives to the :

Nonstandard Analysis and Smooth Analysis

A thesis presented to

the faculty of

the College of Arts and of Ohio University

In partial fulfillment

of the requirement for the degree

Master of Arts

Jesse P. Houchens

May 2013

© 2013 Jesse P. Houchens. All Rights Reserved. 2

This thesis titled

Alternatives to the Calculus:

Nonstandard Analysis and Smooth Infinitesimal Analysis

by

JESSE P. HOUCHENS

has been approved for

the Department of Philosophy

and the College of Arts and Sciences by

Philip W. Ehrlich

Professor of Philosophy

Robert Frank

Dean, College of Arts and Sciences 3 ABSTRACT

HOUCHENS, JESSE P., M.A., May 2013, Philosophy

Alternatives to the Calculus: Nonstandard Analysis and Smooth Infinitesimal Analysis

Director of Thesis: Philip W. Ehrlich

We attempt to clarify and evaluate what shall be called Mac Lane’s thesis—the

thesis that nonstandard analysis (NSA) and smooth infinitesimal analysis (SIA) are

alternatives to the standard approach to the calculus. In doing so, we outline the

historical approaches to the calculus, the standard approach to the calculus, and two

nonstandard approaches, namely NSA and SIA; we also attempt to clarify and evaluate a

of comparisons of NSA and SIA, namely Bell’s 5 mathematico-philosophical

contentions and Bell’s historical contention. 4

For my parents who continually remind me

“While it may not always be easy,

it will always be worth it.” 5 ACKNOWLEDGMENTS

First, I would like to thank the members of my thesis committee. Todd Eisworth provided insightful comments from the of a working mathematician. In addition, his course allowed me to clarify many of the ideas underlying this thesis. Stewart Shapiro travelled down to Athens for the defense as well as allowed me to participate in his in his seminar on continuity and at The Ohio State

University. His comments were very helpful in clarifying many of my arguments. In addition, it was in his seminar that I was first exposed to NSA and SIA and that many of the questions raised in section II.A were first voiced.

Second, I would like to express my sincere gratitude to Philip Ehrlich for advising this project. Each week, as I returned from Dr. Shapiro’s seminar, he would ask me what was discussed and my thoughts on the matter. When coming up with the topic of this thesis, he gave me invaluble advice for making the project manageable. In addition, he directed me to excellent sources on the history of the calculus as well as on each of the mathematical theories. Finally, his patience as I attempted to understand each of the theories as well as his willingness to always correct my mistakes and provide a surplus of comments on previous drafts was absolutely beneficial. Overall, one could say that

Phil’s guidance contributed significantly in improving the quality of this paper.

Finally, I would like to thank my family, former professors, and fellow graduate students as they listened to my struggles and constantly voiced their confidence in my abilities. 6 OF CONTENTS

Abstract...... 3

Dedication...... 4

Acknowledgements...... 5

List of Figures...... 7

Introduction...... 8

1. Historical and Theoretical Background...... 11 1.1. Historical Approaches to the Calculus...... 11 1.2. The Standard Approach to the Calculus...... 21 1.3. Nonstandard Analysis...... 30 1.4. Smooth Infinitesimal Analysis...... 47

2. Clarifications and Evaluations...... 57 2.1. Bell’s 5 Mathematico-Philosophical Contentions...... 57 2.2. Bell’s Historical Contention...... 70 2.3. Mac Lane’s Thesis...... 73

Conclusion...... 79

Works Cited...... 80 7 LIST OF FIGURES

Figure 1. Newton’s ………………….……………….……….13

Figure 2. Newton’s calculus...…………………………………………...... 14

Figure 3. Leibniz’ calculus………..………………………………………………..15

Figure 4. Toricelli’s trumpet………………………………………………………..21

Figure 5. Infinitesimal between dy /x and y /x …...…………...…..44

Figure 6. Motivation for nondegeneracy………….……………………….………..58

Figure 7. Local straightness SIA………………..…………………………………..62

Figure 8. Local straightness NSA 1………………………..………………………..63

Figure 9. Local straighntess NSA 2……………………………..…………………..64

8 INTRODUCTION

After a historically-sensitive discussion of integration and differentiation in

Mathematics, Form and , Saunders Mac Lane concludes with the following:

This discussion of alternative views of the calculus may serve to support our thesis that Mathematics is not just formalism and is not just empirically convenient ideas, but consists in formalizable intuitive or empirical ideas. Calculus fits this thesis as to the nature of Mathematics because it starts from problems (such as the calculation of or of rates of change) and from these problems develops suggestive ideas which ultimately can be fully formalized… Another thesis asserts that the same intuitive idea can be variously formalized. For the calculus, we know that this is the case. Indeed, the earlier and originally wholly speculative use of infinitesimals can be rigorously formalized in at least two different ways: By using ’s non-standard model of the reals, as presented for example in Keisler’s text [1976] on calculus, or by using Lawvere’s proposals to use ‘elementary topoi’ in which the real line R is presented not as a but as a in which there is a suitable infinitesimal neighborhood of 0 (see Kock [1981]). (Mac Lane 155)

The passage above contains two theses: A) The historical development of the calculus consisted of different attempts to formalize intuitive ideas suggested by basic problems;

B) There are at least two different ways to rigorously formalize the “originally wholly speculative use of infinitesimals”, specifically the ways employed in nonstandard analysis (NSA) and smooth infinitesimal analysis (SIA)1. These two theses, we believe,

motivate a third, namely that NSA and SIA are two alternatives to the calculus. Let this

third thesis be called Mac Lane’s thesis. John L. Bell has made a remark similar to Mac

Lane’s thesis restricted to SIA:

“As we show in this book, within [SIA] the basic calculus...can be developed along traditional ‘infinitesimal’ lines—with full rigor—using straightforward calculations with infinitesimals in place of the concept.” ([2008] 4)

1 While Mac Lane references Kock’s book on synthetic differential geometry (SDG), the fragment of SDG that addresses the calculus is SIA. 9 The purpose of this essay is to explicate and evaluate Mac Lane’s thesis. This will involve first discussing some of the differing historical approaches to the calculus as well as the problems that motivated its development. After doing so, the standard ε—δ approach to the calculus as well as the two nonstandard approaches, NSA and SIA, will be outlined to facilitate comparison among the theories.

According to John L. Bell, some of the differences between NSA and SIA are the following:

1. In models of [SIA], only smooth between objects are present. In models of nonstandard analysis, all set-theoretically definable maps (including discontinuous ones) appear. 2. The logic of [SIA] is intuitionistic, making possible the nondegeneracy of the microneighbourhoods ∆ and Mi, i = 1, 2, 3. The logic of nonstandard analysis is classical, causing all these microneighborhoods to collapse to zero. 3. In [SIA], the Principle of Microaffineness entails that all are ‘locally straight’. Nothing resembling this is possible in [NSA]. 4. The property of nilpotency of the microquantities of [SIA] enables the differential calculus to be reduced to simple . In [NSA] the use of infinitesimals is a disguised form of the classical limit method. 5. In any model of [NSA] R* has exactly the same set theoretically expressible properties as R does: in the sense of that model, therefore, R* is in particular an Archimedean . This means that the ‘infinitesimals’ and ‘infinite ’ of nonstandard analysis are so not in the sense of the model in which they ‘live’, but only relative to the ‘standard’ model with which the construction began. That is, speaking figuratively, a ‘denizen’ of a model of nonstandard analysis would be unable to detect the presence of infinitesimals or infinite elements in R*. This contrasts with [SIA] in two ways. First, in models of [SIA] containing invertible infinitesimals, the smooth line is non-Archimedean in the sense of that model. In other words, the presence of infinite elements and (invertible) infinitesimals would be perfectly detectable by a ‘denizen’ of that model. And secondly, the characteristic property of nilpotency possessed by the microquanities of any model of [SIA] (even those in which invertible infinitesimals are not present) is an intrinsic property, perfectly identifiable within that model.2 ([2008], 112)

2 In Bell [2006] some of the above contentions are slightly altered. I will take these altered versions into account in my evaluation. 10 Call the above Bell’s 5 mathematico-philosophical contentions. While these contentions are helpful for comparing NSA and SIA, some are in need of clarification. For example, in contention (3) Bell describes all curves in SIA as ‘locally straight’ and claims “nothing resembling this is possible in NSA.” Yet, no definition is given for ‘locally straight’, moreover nowhere else is this term used in [1998; 2008]. This unclarity makes it difficult to understand just what it is that is impossible in NSA.

Furthermore, Bell concludes the above contentions with the following:

The differences between [NSA] and [SIA] may be said to arise because the former is essentially a theory of infinitesimal numbers designed to provide a succinct formulation of the limit concept, while the latter is, by contrast, a theory of infinitesimal geometric objects, designed to provide an intrinsic formulation of the concept of differentiability. (Bell [2008] 112)

Call this Bell’s historical contention. Laugwitz, in the following passage, challenges

Bell’s historical contention:

It is a misleading simplification to say, as the author does, that NSA “is essentially a theory of infinitesimal numbers designed to provide a succinct formulation of the limit concept…” (Laugwitz [1999] 2).

Motivated by this challenge as well as the previously stated unclarities, we will attempt to clarify and evaluate Bell’s 5 mathematico-philosophical contentions and Bell’s historical contention before examining Mac Lane’s thesis, determining whether it is collectively illustrated by NSA and SIA.

11 1. HISTORICAL AND THEORETICAL BACKGROUND

This section is intended to provide the relevant historical and theoretical background for the clarification and evaluation of Bell’s 5 mathematico-philosophical contentions, Bell’s historical contention, and Mac Lane’s thesis. Section I.A presents some of the basic problems that motivated the development of the calculus as well as the historical approaches to solving such problems. Section I.B outlines the standard ε—δ approach to the calculus following Binmore [1982]. Section I.C outlines NSA following the approach presented in Keisler [2007]. And, section I.D outlines SIA following (for the most part) the approach presented in Bell [1998; 2008].

1.1. Historical Approaches

Some of the problems that motivated the development of the calculus include: (1) finding the instantaneous velocity of an object in continuous motion; (2) finding the under a continuous ; and (3) rigorously describing the continuity of functions.

While the ancients and medievals present historically interesting approaches to solving cases of (1) and (2) (c.f. Baron [1969]), we focus our discussion on the historical approaches beginning in the 17th century with Newton and Leibniz. We will first

overview the calculi of Newton and Leibniz. After which, we will discuss some of the

attempts in the 18th and 19th centuries to make these approaches more rigrorous.3

Before outlining the historical approaches of Newton and Leibniz, we note two

distinctions Guicciardini has made between their calculi and the contemporary calculus:

3 We note that if the reader is interested in more comprehensive accounts of the history of the calculus, it is recommended that they see, among other works, Boyer [1949], Edwards [1979], Guicciardini [2003], and Jahnke [2003]. 12 (a) “Neither Newton’s nor Leibniz’s calculi are about functions;” instead, they “talk in terms of ‘’... and they refer to these quantities, their rates of change, their differences, etc. related to specific geometric entities” (Guicciardini [2003] 73). (b) “While we are used to referring to calculus as the continuum of real numbers, the continuum to which Newton and Leibniz refer is geometrical or kinematical” (Guicciardini [2003] 73-74).

With these distinctions in mind, let us first look at the calculus of Newton.

Methods for finding the of a curve or the instantaneous velocity of an object in continuous motion were developed and employed by a of predecessors of Newton and Leibniz including Fermat and Barrow.4 Newton’s approach employs

what he referred to as ‘fluents’, ‘’, and ‘moments’; respectively, “the quantities

v,x,y,z generated by a ‘flow’,” the instantenous vÝ ,x Ý , yÝ ,z Ý , and “the infinitely small additions,o, by which those quantites increase during each infinitely small

of time” (Guicciardini [2003] 78). In the passage below, Guicciardini explains

the ways that these quantities were employed for differentiation. 5

Imagine a curve f (x,y)  0 to be generated by the continuous flow of a point P(t). If x,y are the Cartesian coordinates of the curve, yÝ / xÝ will be equal to tan , where  is the formed by the tangent in P(t) with the x-axis (see Fig. 3.3). According to Newton’s conception, the point will move during the “indefinitely small period of time” with uniform rectilinear motion from P(t) to Pt o . The infinitesimal triangle indicated in Fig. 3.3 has sides equal to yÝ o and xÝo and so tan  yÝo /xÝo  yÝ / xÝ . An extremal point will have yÝ / xÝ  tan  0. (Guicciardini [2003] 79)

4 C.f. Edwards pp. 122-141 5 We note that in this passage, Guicciardini uses functions as a convention to represent quantities. As stated in (a), neither Leibniz nor Newton speak of functions. 13

Figure 1.6 Newton’s differential calculus

Turning to integration, we note that ideas similar to “line segments, surfaces,

solids [being] made up of an actual infinite number of indivisible or infinitesimal

elements” were employed by Kepler, Cavalieri, and other predecessors of Newton and

Leibniz in “their determinations of areas and volumes” (Ehrlich [2006] 494). Newton’s

first treatment of integration contained the following set of three rules

Rule 1: If y  ax m / n , then the area under y is an /n  m x m / n 1 . Rule 2: If y is given by the sum of more terms (also an infinite number of terms),

y  y1  y2  ..., then the area under y is given by the sum of the areas of the corresponding terms.   Rule 3: In order to calculate the area under a curve f (x,y)  0, one must expand y as a sum of terms of the form ax m / n and apply Rule 1 and Rule 2. (Guicciardini  [2003] 76).

This version of ‘termwise integration’ was motivated by Newton’s heavy use of binomial  . Newton justified rule 1 by proving the fundamental theorem of the calculus

(hereafter the FTC; formally presented in section I.B). His proof that the integral of the

of z is equal to z rests on the “commonplace” assumption that the area under a

6 Taken from Guicciardini [2003] p. 79 14 cuve was considered equal to the sum of infinitely many infinitesimal strips.7 Going the

other way—i.e. proving that the derivative of the integral of z is equal to z—Newton’s

argument is outlined by Guicciardini as follows.

Newton considered a curve AD (see Fig. 3.2), where AB  x, BD  y and the area ABD  z. He defined B  o and BK  v such that “the rectangle BHK( ov) is equal to the space BD.”…With these definitions one has that A  x  o and the area A is equal to z  ov. At this point, Newton wrote:    “from any arbitrarily assumed relationship x and z I seek y.” He noted that the    increment of the area ov, divided by the increment of the abscissa o is equal to v.   But since one can assume “ B to be infinitely small, that is, o to be zero, v and y   will be equal.” Therefore, the rate of increase of the area is equal to the ordinate. (Guicciardini [2003] 77).

Figure 2.8 Newton’s integral calculus

In contrast, Guicciardini describes Leibniz’ approach to differentiation and

integration as follows:

Let us consider a curve C (see fig. 3.10) in a Cartesian coordinate system. Leibniz imagines a subdivision of the x-axis into infinitely many infinitesimal

7 C.f. Guicciardini [2003] pp. 76-77 8 Taken from Guicciardini [2003] p. 77 15

intervals with extremes x1,x2,x 3,etc. He further defines the differential

dx  xn1  xn . On the curve and on the y-axis one has the corresponding

successions s1,s2,s3,etc., and y1,y2,y3,etc. Therefore ds  sn1  sn and dy  y  y . The characteristic triangle has sides dx,ds,dy. The tangent to the n1 n curve C forms an angle  with the x-axis such that tan  dy /dx. The area  subtended to the curve is equal to the sum of infinitely many strips ydx…[Two]    aspects of Leibniz’s representation of the curve C in terms of differentials should  be noted.    1. The symbols d and applied to a finite x generate an infinitely   little and an infinitely great quantity, respectively… 2. Since geometrical is preserved, the symbols d and  can be iterated to obtain higher-order infinitesimals and higher-order … (Guicciardini [2003] 89) 

Figure 3.9 Leibniz calculus

In comparing the calculi of Newton and Leibniz, Edwards notes the following:

In regard to the calculus itself, discrete infinitesimal differences of geometric variables played the central role in Leibniz’ approach, while Newton’s fundamental concept was the or time rate of change, based on intuitive ideas of continuous motion. As a consequence, Leibniz’ notation and terminology effectively disguises the limit concept, which by contrast is fairly explicit in Newton’s calculus.

9 Taken from Guicciardini [2003] p. 89 16 For Leibniz, the separate differentials dx and dy are fundamental; their ratio dy/dx is ‘merely’ a geometrically significant quotient. For Newton, however, especially in his later work, the derivative itself—as a ratio of fluxions or an ‘ultimate ratio of evanescent quantities’—is the heart of the matter. (Edwards 266)

That said, as Guicciardini points out, both Newton and Leibniz’ infinitesimals were often said to be heuristic rather than actual. For example, Leibniz writes “instead of the infinite or infinitely small, one can take magnitudes that are so large or so small that the error will be less than the given error” (Leibniz [1701] 350; taken from Guicciardini [2003] 98). A similar idea is found in Book 1 of Newton’s Principia Mathematica.10 Guicciardini

warns that “the difference between the Leibnizian and the Newtonian calculi should not

be overstressed. In particular,…, the differences should not be looked for at the syntactic

or at the semantic level but rather at the pragmatic level11… there is not a single theorem which can be proved in one of the two calculi and which cannot have a counterpart in the other” (Guicciardini [2003] 96).

In the interest of our presentation of SIA, we briefly describe a version of the calculus proposed by Bernard Nieuwentijt. Suppose one lets the curve y  x 2 be a

variable quantity and differentiates this curve at some point x. For Leibniz, this can be

dy x  dx 2  x 2 2xdx  dx 2 done as follows:    2x  dx  2x . Yet, for Neuwentijt, dx dx dx

the computation can only be done as follows:

dy x  dx 2  x 2 2xdx  dx 2 2xdx     2x. The difference lies in the of dx dx dx dx

10 C.f. Guicciardini [2003] p. 82 11 That is, Guicciardini points out “Newton emphasized the use of infinite series. He expanded fluences into infinite series and ‘integrated’ termwise. Leibniz also employed this technique. However, Leibniz preferred integration in ‘closed’ form: He looked for quadratures expressed not by infinite series but by a finite combination of ‘functions’”(Guicciardini [2003] 100). 17 infinitesimal used. Leibniz’ infinitesimals might be thought of as ‘standard infinitesimals’, Nieuwentijt’s as ‘ infinitesimals’. Nilpotent infinitesimals can be set to zero only if they are raised to some power greater than one. In contrast, any standard infinitesimals can simply be set to zero when by themselves at the end of a computation. Nieuwentijt adopted the use of solely nilpotent infinitesimals in order to make the calculus more rigorous; however, he eventually gave up his position, admitting that his criticisms of Leibniz’ calculus were wrong.12

Having briefly overviewed Newton and Leibniz’ versions of the calculus, we now

turn to the 18th and 19th century developments that, among further uses for the calculus, brought forth more rigorous definitions of the derivative and the integral as well as concepts that played a central role in their calculations including functions, continuous functions, and limits of functions.

The first of these concepts to be made rigorous was a function. As Edwards explains,

Euler’s Introductio was the first work in which the function concept played a central and explicit role. It was the identification of functions, rather than curves, as the principal objects of study, that permitted the arithmetization of geometry, and the consequent separation of infinitesimal analysis from geometry proper (Edwards 270).

Euler’s definition can be summarized as follows: “a function is continuous if it is

characterized by a single analytic expression, and […] discontinuous if it lacks any

analytic expression” (Ehrlich [2006] 497). Among the problems with this definition,

12 Neuwentijt and his calculi are discussed in both Mancosu and Vermij. For a list of further differences between Leibniz and Neuwentijt’s calculi as well as a summary of the debate between the two authors and Jakob Hermann, see Mancosu pp. 159-164. 18 Edwards points out that it fails to describe the “connectedness” of the

(Edwards 302).

In 1791 Louis Arbogast presented the following condition for for the continuity of a function, making use of what would become the Intermediate value theorem (hereafter

IVT).

The consists in that a quantity cannot pass from one state to another without passing through all the intermediate states which are subject to the same law. Algebraic functions are regarded as continuous because the different values of these functions depend in the same manner on those of the variable; and, supposing that the variable increases continually, the function will receive corresponding variations; but it will not pass from one value to another without also passing through all the intermediate values. (cited by Jourdain [14], pp. 675-676; taken from Edwards 303)

The IVT would henceforth continue to play a major role in defining continuous functions.

For example, Andrè-Marie Ampère and Joseph-Louis Lagrange would accept “the [IVT] as a self-evident attribute of continuous functions;” moreover, Cauchy and Bolzano recognized the necessity of the IVT in their proofs of the convergence of bounded monotone (Grabiner 74-75, 81).

Jumping ahead to Cauchy, we see a number of definitions that are closer to our contemporary formulations. For example, in his Cours d’Analyse Cauchy gives the following condition for the continuity of a function.

“Let α be an infinitely small increment of x. Then f(x) is said to be continuous in this interval ‘if the of f(x+α)-f(x) decreases indefinitely together with that of  . In other words the function f(x) is continuous with respect to x within the given bounds if, within these bounds, an infinitely small increment of the variable always results in an infinitely small increase of the function itself.’” (Laugwitz [1997b] 655)13

13 It is significant to note that while Cauchy states that “infinitely small” may be intuitively understood as a “variable magnitude [that] decreases indefinitely, so that it 19

Lützen provides us with a number of Cauchy’s other significant definitions, which we list:

Limit: When the values successively attributed to the same variable approach a fixed value indefinitely, in such a way as to end up by differing from it as little as one could wish, this last value is called the limit of all the others. (Cauchy [1821] 19; taken from Lützen 158)

Convergence: A series is an indefinite of quantities u0,u1,u2,u3,etc... which succeed each other according to a fixed law. These quantities themselves

are the different terms of the series considered. Let sn  u0  u1  u2  ... un1 be

the sum of the first n terms, where n is an arbitrary . If the sum sn approaches a certain limit S indefinitely for increasing values of n, then the series is said to be convergent, and the limit in quest is called the sum of the series… (Cauchy [1821] 114; taken from Lützen 159)

Derivative: When the function y  f (x) is continuous between two given limits of the variable x, and one assigns a value between these limits to the variable, an infinitely small increment of the variable produces an infinitely small increment in the function itself. Consequently, if we then set x  i, the two terms of the x f (x  i)  f (x)  will be infinitesimals. But while these y i terms tend to zero simultaneously, the ratio itself may converge to another limit, either positive or negative. This limit, when it exists, has a definite value for each particular value of x; but it varies with x…(Cauchy [1823] 22; taken from Lützen 159)

Integral: Suppose that the function y  f (x) is continuous with respect to the

variable x between the two finite limits x  x0,x  X. We designate by

x1,x2,...,xn1 new values of x placed between these limits and suppose that they either always increase or always decrease between the first limit and the second.

We can use these values to divide the difference X  x0 into elements

x1  x0,x2  x1,x3  x2,...,X  xn1, which all have the same sign. Once this has been done, let us multiply each element by the value of f (x) corresponding to the

left-hand endpoint of that element: that is, the element x1  x0 will be multiplied

by f (x0), … and let S  x1  x0 f (x0 )  x2  x1f (x1)  ... X  xn 1 f (xn 1) be the sum of the products so obtained. The quantity S clearly will depend upon 1st: nd the number n of elements into which we have divided the difference X  x0; 2 : the values of these elements and therefore the mode of division adopted. converges on the limit zero,” Laugwitz has shown that “infinitely small” could not have been defined as such (c.f. Laugwitz [1997b] 654-657). 20 It is important to observe that if the numerical values of these elements become very small and the number n very large, the mod of division will have only an insignificant effect on the value of S… When the elements of the difference

X  x0 become infinitely small, the mode of division has only an imperceptible effect on the value of S; and, if we let the numerical values of these elements decrease while their number increases,, the value of S ultimately becomes, for all practical purposes,, . Or, in other words, it ultimately reaches a certain limit that depends uniquely on the form of the function f (x) and on the bounding

values x0,X of the variable x. The limit is what is called a definite integral. (Cauchy [1823] 122-124; Transl. Grabiner [1981] 171-174; taken from Lützen 159-160)

The definitions of a and the limit would later be replaced by

Weierstrass’ ε—δ condition (see section I.B), motivated by the nineteenth century move to rid analysis of the infinitely small. This modern formulation of the limit would eventually be employed in the definition of the derivative. The integral would also use the limit. In addition, it would develop into the Riemann integral which could be used for more than just continuous functions (see section I.B).

We close this section by noting that one final motivating problem of the calculus was to make sense of Torricelli’s discovery “that the volume of an infinitely long solid, obtained by revolving about its a portion of the equilateral hyperbola was finite” (see Figure 2) (Boyer [1949] 125). One of the tools that was employed by Euler,

Cauchy and other practitioners of the calculus to handle this problem and others was the improper integral (see section I.B).

21

Figure 4. Toricelli’s trumpet

In summary, we have overviewed some of the historical approaches to the calculus beginning in the 17th century as well as the attempts to make these approaches more rigorous. In the next section, we present the result of this rigorization, namely the standard approach to the calculus.

1.2. The Standard Approach

We begin by describing a few features of the system, R, relevant to our discussion. We will then present an overview of limits of functions, continuous functions, differentiation, and integration.

A subset S of R is bounded above if there exists a real number H such that for any x  S, x  H. Similarly, S is bounded below if there exists a real number h such that for any x  S, h  x . R satisfies, what Binmore calls, the continuum property in that

(a) “every non-empty set of real numbers which is bounded above has a smallest upper bound;” and, 22 (b) “every non-empy set of real numbers which is bounded below has a largest lower bound” (Binmore 14).

We note that (a) alone is usually referred to as the least upper bound principle and sometimes the completeness property.

R is said to be a totally ordered field in that14

i. R,0,,, is a where 0  R, +,· are binary functions, - is a

unary function on R, and for all a,b,c  R, the following laws hold:

a. Commutativity: a  b  b  a, a b  b a b. Associativity: a  (b  c)  (a  b)  c, a (b c)  (a b) c c. Identity: a  0  a d. Inverse (additive): a  (a)  0 e. Distributive: a (b  c)  a b  a c

ii. R,0,1,,,,1 is a field, where R,0,,, is a ring, 1 R, 1 is a function from

R \{0} into R, and for all a  0 in R the following laws hold:

a. Nontriviality: 1 0 b. Identity: 1 a  a c. Inverse (multiplicative): a a11

iii. R is ordered with a binary < where for all a,b,c  R, the following laws

hold:

a. Transitivity: If a  b and b  c then a  c . b. Trichotomy: Exactly one of the relations a  b, b  a, or a  b holds. c. Sum: If a  b and c  c then a  c  b  c. d. Product: If a  b and 0  c then a c  b c.

A sequence xn in R is said to be Cauchy if, given any   0, we can find an N1

such that, for any m  N1 and any n  N1, xm  xn  ; and, a sequence xn in R is said to converge to the limit l if and only if given any   0, we can find an N1 such that, for

14 c.f. Keisler [2007] pp.19-20. 23 any n  N1, xn  l   . R is said to be a Cauchy complete totally ordered field in that every in R is a convergent sequence.15 Furthermore, R is also said to be a Dedekind continuous totally ordered field in that whenever R “is partitioned into two nonempty subsets X and Y such that every member of X precedes every member of Y, then either X has a greatest member or Y has a least member, but not both” (Ehrlich

[2006] 498). Finally, R satisfies the in that the set of natural numbers, N, is unbounded above.16 Another way of stating the Archimedean property for

R is that “if a and b are real numbers with a  0, then there is a positive integer n such that na  b” (Kirkwood 29).

As Binmore tells us, “a function f from a set A to a set B (write f : A B) defines a rule which assigns to each x  A a unique element y  B. The element y is called the image of the element x and we write y  f (x)” (Binmore 65). Before defining the , the right and left limits of functions are defined.

Suppose that f is defined on an interval (a,b). We say that f (x) tends (or converges) to a limit l as x tends to b from the left and write f (x) l as x b  or, alternatively lim f (x)  l if the following criterion is satisfied. Given any x b    0, we can find a   0 such that f (x)  l   provided that b   x  b. [Likewise,] we say that f (x) tends (or converges) to a limit l as x tends to a from the right and write f (x) l as x a  or, alternatively, lim f (x)  l if the x a  following criteria is satisfied. Given any   0, we can find a   0 such that f (x)  l   provided that a  x  a . (Binmore 75-6)

Now, “suppose that f is defined on an interval (a,b) except possibly for some point  (a,b). We say that f (x) tends (or converges) to a limit l as x tends to ξ and write f (x) l as x  or, alternatively, lim f (x)  l if the following criterion is x

15 For the proof of this result, see Binmore p. 52. 16 C.f. Binmore p. 20 24 satisfied. Given any   0, we can find a   0 such that f (x)  l   provided that

0  x     ” (Binmore 76). The following properties hold for limits of functions.

Let f and g be defined on an interval (a,b) except possibly at  (a,b). Suppose that f (x) l as x  and g(x) m as x  and suppose that  and  are any real numbers. Then, as x , (i) f (x)  g(x) l  m, (ii) f (x)g(x) lm, and f (x) l (iii)  provided that m  0. (Binmore 80) g(x) m

In his discussion of continuous functions, Binmore first defines the continuity of a function at a point, on the left of a point, and on the right of a point. He then defines the continuity of a function over an open interval and over a closed interval. “Suppose that f is defined on an interval (a,b) and  (a,b). Then we say that f is continuous at the point  if and only if f (x)  f () as x ” (Binmore 78). As Binmore points out,

“roughly speaking, to say that f is continuous at the point  means that the graph of f does not have a ‘break’ at the point ” (Binmore 78). “If f is defined on an interval (a,b] and f (x)  f (b) as x b , we say that f is continuous on the left at the point b. If f is defined on an interval [a,b) and f (x)  f (a) as x a , then we say that f is continuous on the right at the point a” (Binmore 78).

“A function f is continuous on an open interval I if and only if it is continuous at each point of I” (Binmore 85). “A function f is continuous on a compact interval [a,b] if and only if it is continuous at each point of (a,b) and continuous on the right at a and continuous on the left at b” (Binmore 85). Generally speaking, f is continuous on I if and 25 only if, given any x  I and any   0, we can find a   0 such that f (x)  f (y)   provided that y  I and satisfies x  y   .17

Binmore further proves that if f is continuous on an interval I, then the image of I under f is also an interval; and, as a corollary to this theorem, he proves the IVT; i.e. if f is continuous on an interval I containing a and b and if  lies between f (a) and f (b), then we can find a  between a and b such that   f ().18

Having presented and discussed limits of functions and continuous functions,

Binmore next turns to the derivative. “Suppose that f is defined on an open interval I containing the point . Then f is said to be differentiable at the point  if and only if the

f (x)  f () limit lim exists. If the limit exists, it is called the derivative of f at the point x x  

 and denoted by f '() or Df (). For a function f which is differentiable at  we

f (x)  f () f (  h)  f () therefore have f '()  lim ,” equivalently f '()  lim x x   h  0 h

(Binmore 92). If f is differentiable at each point of an open interval I, f is differentiable

f '(x)  f '() on I.19 The of f at the point  is f ''()  lim , and the third x x   derivative (and so on) are defined similarly.20 Binmore proves the following properties hold for the derivative:

Suppose that f and g are functions defined on an open interval I containing the point . Let  and  be any real numbers. Then, if f and g are differentiable at , (i) D{f  g}  Df  Dg,

17 C.f. Binmore p. 86 18 C.f. Binmore p. 88 19 C.f. Binmore p. 93 20 C.f. Binmore p. 93 26 (ii) D{ fg}  fDg gDf f  gDf  fDg (iii) D  provided that g()  0. (Binmore 97) g  g2

In addition, the following property describing the differentiation of composite functions is proved: if g is differentiable at the point x and that f is differentiable at the point y  g(x), then ( f og)'(x)  f '(y)g'(x).21  As we turn to integration, we note that our discussion is restricted to the Riemann

Integral rather than the more general Lebesgue Integral.22 We further note that Binmore first defines the integral as the supremum of the sum of the areas under the curve (see below). This definition is valid only when the function being integrated is continuous.

Later, the definition of the Riemann integral is given. The Riemann integral can integrate

“any function which is bounded and has only a finite number of discontinuities on [a,b]” as well as “a function f which is monotone on [a,b]…although such a function may have an infinite number of discontinuities” (Binmore 129-130). With that in mind, we proceed to Binmore’s definitions.

Suppose that f is continuous on the compact interval [a,b]… We begin by

defining a partition P of the interval [a,b]. This is a finite set P  {y0,y1,...,yn}

of real numbers with the property that a  y0  y1  y2  ...  yn1 y n b. Since f is continuous on [a,b], it is bounded on [a,b]. It therefore makes sense, given a

partition P  {y0,y1,...,yn} of [a,b], to define m1  inf f (x), m2  inf f (x), y0 xy1 y1 xy2 and so on. For each partition P, we can then form the sum n

Sm (P)  mk (yk  yk1). [Now,] suppose that f is bounded above on [a,b] by H. k1

21 C.f. Binmore p. 99 22 For a discussion of the latter, c.f. Edwards 335-341. 27 [Then] f (t)  H for any t [a,b] [and] n n

Sm (P)  mk (yk  yk1)  H(yk  yk1)  k1 k1 . H{(y1  y0)  ... (yn  yn 1)}  H(yn  y0)  H(b  a)

It follows that the set of all numbers Sm (P), where P is a partition of [a,b], is bounded above by H(b  a). Hence it has a smallest upper bound and we may b define f (x)dx  supS (P) where the supremum extends over all partitions P a m P of [a,b]. (Binmore 121-122)

Call this the ‘supremum integral.’ What we shall refer to as the ‘infimum integral’ can be

defined similarly. Instead of m1,m2,...,mn , let M1  sup f (x), M2  sup f (x), and so y0 xy1 y1 xy2

n on. Then we can form the sum SM (P)   Mk (yk  yk1). Now, suppose that f is k1 bounded below on [a,b] by h. Then f (t)  h for any t [a,b], and

n n

SM (P)   Mk (yk  yk1)  h(yk  yk1)  k1 k1 . h{(y1  y0)  ... (yn  yn 1)}  h(yn  y0)  h(b  a)

It follows that the set of all numbers SM (P), where P is a partition of [a,b], is bounded below by h(b  a). Hence it has a greatest lower bound and we may define

b f (x)dx  inf SM (P) where the infimum extends over all partitions P of [a,b]. a P

“If f is not continuous on [a,b], these two definitions [namely the definitions of the ‘supremum integral’ and ‘infimum integral’] do not necessarily yield the same result.

But, if they do yield the same result, we say that f is Reimann integrable and call the common number obtained from the two definitions the Reimann integral of f on [a,b]”

(Binmore 128-129). 28 Assuming f is continuous on [a,b] and c is a constant, Binmore proves the

Reimann integral has the following basic properties:23

b (i) cdx  c(b  a) a b b (ii) { f (t)  c}dt  f (t)dt  c(b  a) a a b d b (iii) where a  d  b f (t)dt  f (t)dt  f (t)dt. a a d

Binmore also proves that if f and g are continuous on [a,b] and  and  are any real

b b b numbers, then f (t)  g(t) dt   f (t)dt   g(t)dt .24 Further, to integrate a a a composite functions, if has a derivative that is continuous on [a,b] and f is continuous on an open interval that contains the image of [a,b] under , then

 (b) b f (t)dt  f (u) '(u)du.25  (a ) a

With the integral defined, Binmore proves the FTC: if f is continuous on [a,b]

x and if F is defined on [a,b] by F(x)  f (t)dt for a  x  b, then F is [an a ] of f on [a,b]; i.e. F is continuous on [a,b], differentiable on (a,b) and

F'(x)  f (x) for each x  (a,b).26 From this theorem, it is proved that any function f which is continuous on [a,b] has an antiderivative on [a,b]. It is also proved that if G is

b any antiderivative of f, then f (t)dt  G(b)  G(a)  G(t) b .27 Making use of the a a antiderivative, the following theorem, namely ‘’, is proved: “Suppose

23 C.f. Binmore pp. 122-123 24 C.f. Binmore p. 131 25 C.f. Binmore p. 131 26 C.f. Binmore p. 126 27 C.f. Binmore p. 127 29 that f and g are continuous on [a,b] and have primitives F and G respectively on [a,b].

b b Then f (t)G(t)dt  F(t)G(t) b  F(t)g(t)dt ” (Binmore 131). a a a

Examples of the improper integral are given to handle cases when either the interval of integration is unbounded, open, or semi-open, or the function being integrated is unbounded for values in the interval (e.g. 1/x is unbounded for x  0). For example, if f is continuous on the unbounded interval [0, ), then we define

 X f (x)dx  lim f (x)dx provided that the limit exists. “Similarly, if f is 0 X  0

b b continuous on (a,b], then we define f (x)dx  lim f (x)dx provided that the limit  a y a  y exists.” (Binmore 134). From these examples, one can generalize and cover the cases mentioned above.

Before concluding, we note that one further concept of the calculus is the process of ‘approximating’ a curve given the of its tangent line. That is, as Binmore explains:

Suppose that f is differentiable at the point . Then the equation of the tangent to y  f (x) at the point  is y  f ()  x   f '(). Thus, for values of x which are ‘close’ to , it is reasonable to expect that f ()  x   f '() is a ‘good’ approximation to f (x). But how large may the error in this approximation be? Suppose now that f is n 1 times differentiable at the point . Then the first n 1 at the point  of the 1 1 2 1 n 1 P(x)  f ()  x   f '()  x   f ''()  ... x   f n 1 () are 1! 2! n 1 ! the same as those of f. We might therefore reasonably expect that P(x) will be a ‘very good’ approximation to f (x) for values of x which are ‘close’ to . But again, how large may the error be in this approximation? (Binmore 106)

To answer these questions, the following theorem is given.

Taylor’s Theorem: Suppose that f is n times differentiable on an open interval I which contains the point . Given any x  I, 30 1 1 2 1 n 1 f (x)  f ()  x   f '()  x   f ''()  ... x   f n 1 ()  E 1! 2! n 1 ! n 1 n where the error E satisfies E  x   f (n )() for some value  between  n n n! and x. (Binmore 106-107)

In summary, we have (for the most part) followed Binmore in presenting and discussing relevant properties of the real number system R, limits of functions, continuous functions, the derivative, the integral, the FTC, improper , and

Taylor’s theorem. This concludes our overview of the relevant theoretical background for the standard approach to the calculus. We now turn to the two nonstandard approaches to the calculus—first NSA and then SIA.

1.3. Nonstandard Analysis

NSA arose from the mathematico-logical investigations of Abraham Robinson.

While there are a number of different approaches to NSA, we limit our discussion to that presented in Keisler [2007]. We begin by presenting Keisler’s for a system R* and some of its basic properties. We then discuss limits of functions, continuous functions, differentiation, and integration.

Axiom 1: R is a complete ordered field.

Axiom 2: R* is an ordered field extension of R.

That is, R* satisfies the axioms of an ordered field and R is a subfield of R*. It is said that any element x  R* is either:

infinitesimal if x  r for all positive real r, finite if x  r for some real r, or infinite if x  r for all real r.

Note that all infinitesimal elements of R* are finite. 31 Axiom 3: R* has a positive infinitesimal, that is, an element  such that 0   and   r for every positive r  R.

Given the existence of a positive infinitesimal and the fact that R* is an ordered field, one can show that there are positive infinite, negative infinite, and negative infinitesimal elements of R*.

Two elements x,y  R* are said to be infinitely close, written x  y, if x  y is infinitesimal. Given a hyperreal number x  R*, one calls the monad of x the set monad(x)  y  R*:x  y and the galaxy of x the set galaxy(x)  y  R*:x  y is finite. Note that monad(0) is the set of all infinitesimal elements of R* and galaxy(0) is the set of all finite elements of R*.

Given a finite x  R*, the unique real r  x is called the standard part of x; written r  st(x). If x is infinite, st(x) is undefined. Letting x and y be finite elements of

R*, the following properties hold for the :28

a. x  y if and only if st(x)  st(y). b. x  st(x). c. If r  R then st(r)  r. d. If x  y then st(x)  st(y). e. st(x  y)  st(x)  st(y). f. st(x  y)  st(x)  st(y).

Axiom 4 (Function axiom): For each real function f of n variables there is a corresponding hyperreal function f * of n variables, called the natural extension of f . The field operations of R* are the natural extensions of the field operations of R.

Before giving the final axiom of R*, we define a system of formulas, a real solution and a hyperreal solution to a system of formulas, and the natural extension of a set. First, “a term is an expression which can be built up using the following rules:

28 C.f. [Keisler 2007] 5-6 for the proofs of these results. 32 (i) Every variable is a term. (ii) Every constant is a term.

(iii) If 1,...,n are terms and f is a real function of n variables, then f (1,...,n ) is a term.” (Keisler [2007] 8)

A formula is an equation or inequality between two terms. “A system of formulas is a nonempty finite set of formulas.” (Keisler [2007] 8). Letting S be a system of formulas, a

real solution of S whose variables are x1,...,xn is an n- c1,...,cn  of real constants

such that, when each xi is replaced by ci in S, each term within an equation or inequality in S is defined and each formula in S is true. Likewise, a hyperreal solution of S with the

variables x1,...,xn is an n-tuple c1,...,cn  of hyperreal numbers such that all the formulas

in S are true when each function is replaced by its natural extension and each xi is

29 replaced by ci.

“Given a set X  R, the natural extension of X is defined as follows. Consider a finite system F of formulas that has X as its set of real solutions. If F* is the system of natural extensions of the formulas in F, then the natural extension of X is the set of hyperreal solutions of the system F*” (Edwards 342). It can be shown that the natural extensions of sets preserve the usual set operations.

Axiom 5 (Transfer axiom): Given two systems of formulas S,T with the same variables, if every real solution of S is a solution of T, then every hyperreal solution of S is a solution of T.

The following is an example of the application of the Transfer axiom. “Let f and g be real-valued functions defined on the set D  R, with f (x)  g(x) for all x  D. Then the two formulas f (x)  g(x) and f (x)  g(x) have the same set of real solutions, namely D.

Hence the two formulas f *(x)  g*(x) and f *(x)  g*(x) have the same set D* of

29 C.f. Keisler [2007] pp. 8-9 33 hyperreal solutions. Thus it follows from the [transfer] axiom that f (x)  g(x) for all x  D implies f *(x)  g*(x) for all x  D*” (Edwards 342). Note the following is a

consequence of the transfer axiom: If f is a real function of n variables, c1,...,cn are real

30 constants, and f (c1,...,cn ) is defined, then f *(c1,...,cn )  f (c1,...,cn ).

As Keisler observes, “the Transfer axiom is equivalent to an apparently stronger statement called the Elementary Extension Principle” (EEP), which states intuitively

“that the real numbers (with all real functions and relations) satisfy the same sentences of first order logic as the hyperreal numbers (with the natural extensions of all real functions and relations)” (Keisler [2007] 175). To give a more exact formulation of the EEP, we follow Keisler in describing a formal language, L, for the real numbers using first order logic.

_ _ L has the following uncountable collection of symbols: c for each real constant, f

_ for each real function f of n variables, and P for each real relation P of n variables. For convenience, we refer to each as simply c, f, and P. In addition, L has the following logical symbols common to all first order languages: variablesv1,v2,v3,... , connectives

,,,,, quantifiers ,, parentheses, and commas. A term in L is a finite sequence of symbols built according to the rules given above.31 “When we replace each variable in

a term (v1,...,vn ) by a constant ci, we obtain a constant term, which is either equal to some real number or is undefined” (Keisler [2007] 176).

30 C.f. Keisler [2007] p. 10 31 C.f. Keisler [2007] pp. 175-176 34

“If P is a real relation of n variables and 1,...,n are terms, then P(1,...,n ) is called an atomic formula of L. The set of all (first order) formulas of L is defined as follows.

 Every atomic formula of L is a formula of L.  If , are formulas of L, so are ,   ,  ,   ,   .

 If  is a formula of L and vn is a variable, then vn, vn are formulas of L.” (Keisler [2007] 176).

We suppose the reader is familiar with the distinction between bound and free variables.

“A first order formula with no free variables is called a sentence” (Keisler [2007] 176).

We further suppose the reader is familiar with the truth values of sentences in L defined in the usual manner using recursion.32

As Keisler tells us, “given a formula  of L, the *-transform of  is the formula

 * of L * obtained by replacing each real function f and relation P occurring in  by its natural extension f * and P*. Given a term  of L, the *-transform  * is defined analogously” ([2007] 177). For example, since the continuity of a function f at some point x can be expressed by the following formula,

  0  0yx y    f (x)  f (y)   , its *-transform is expressed by the formula   *0   *0yx *y *  *  f *(x) * f *(y)*  *.33

Elemenetary Extension Principle: For every sentence  of L,  is true in R if and only if its *-transform  * is true in R*.34

32 C.f. Keisler [2007] pp. 176-177 33 C.f. Keisler [2007] p. 177 34 Note that the Transfer Axiom is just the special case of the EEP when  is a sentence of the form v1...vl 1  ... m  1  ...n , where  and  are atomic formulas, written P1,...,n  (Keisler [2007] 178). 35

By the EEP, assuming the formula above expressing the continuity of a function at a point is true in R, we can infer its *-transform is true in R*.

We have detailed the EEP in order to give an analogue of this principle for superstructure embeddings (to be defined) and eventually be able to distinguish between internal and external entities of a nonstandard . Keisler provides us with this analogue by first defining a superstructure embedding and then describing a language for the hyperreal numbers.

Letting the of a set X be defined as , (x)  Y :Y  X , we

define n-th cumulative power set of X recursively as V0(X)  X ,

 V (X)  V (X)  V (X) . The superstucture over X, is V (X)  V (X). n 1 n  n  U n  n 0

A superstructure embedding is a one-to-one mapping * of V(R) into another superstructure V(S) such that:

(i) R is a proper subset of S, r*  r for all r  R, and R*  S. (ii) For all x, y V (R), x  y if and only if x* y *.35

In virtue of (i), Keisler writes R* instead of S and denotes the superstructure embedding

36 by :V (R) V (R*) . “Notice that by (i), V (R)  V (R*) and * contains the identity on R, but * does not contain the identity map on V(R)” (Keisler [2007] 181).

Keisler forms a language L’, similar to L, for the hyperreal numbers using the first order predicate calculus with identity, a binary relation symbol , and a constant symbol

_ a for each element a V (R*).37

35 Keisler [2007] p. 181 36 C.f. Keisler [2007] p. 181 36 A bounded formula of L’ is an expression built according to the following rules.

 If x and y are variables or constants, x  y and x  y are bounded formulas, called atomic formulas.  If , are bounded formulas, so are ,   ,  ,   ,  .  If u is a variable, c is a constant, and  is a bounded formula, then u  c  and u c are bounded formulas.  If u,v are variables,  is a bounded formula, and v does not appear in  in the form u c  or u c, then u  c and u c  are bounded formulas. …A bounded sentence is a bounded formula in which each occurrence of a variable u is within the scope of a of the form u  x or u  x where x is another variable or constant. …Each bounded sentence is either true or false in the superstructure V (R*). The definition of ‘ is true in V (R*)’ is by induction on the complexity of the bounded sentence . The quantifier clauses are: v c (v) is true in V (R*) iff (b) is true in V (R*) for some b c. v c (v) is true in V (R*) iff (b) is true in V (R*) for all b c. …We call elements of R real numbers, the elements of V (R) \ R real sets, and arbitrary elements of V(R) real entities. Similarly, elements of R*, V (R*) \ R *, and V (R*) are called hyperreal numbers, hyperreal sets, and hyperreal entities, respectively. A real bounded formula is a bounded formula of L’ all of whose constants are real entities (Keisler [2007] 182).

“The superstructure embedding :V (R) V (R*) induces a mapping, called the

*-transform, from real bounded formulas to bounded formulas. The *-transform  * is obtained by replacing each constant c occurring in  by its image c*” (Keisler [2007]

183). For example, the *-transform of the real bounded sentence stating that every non- negative real number has a root, x  Rx  0 y  Ry y  x, is the

_ 37 Once again, “for simplicity we identify a with its constant symbol a” (Keisler [2007] 182). 37 bounded sentence that says every non-negative hyperreal number has a square root,

x  R* x  *0*y  R* y *y  x.38

A nonstandard universe is a superstructure embedding *:V (R) V (R*) which satisfies Leibniz’ Principle39: for each real bounded sentence   L',  is true if and only if  * is true. Such a nonstandard universe may be constructed using a bounded ultrapower; however, this will not be included in our presentation.40

Note that “the image of the power set of R will contain some but not all subsets of

R*, that is, (R) * will be a proper subset of R *. For example, the set R of real numbers and the set N of natural numbers are subsets of R* but do not belong to

(R) *” (Keisler [2007] 184-185).41

Keisler distinguishes between four kinds of entities in V (R*). An entity b V (R*) is said to be

real if b V (R) standard if b  a* for some a V(R) internal if b  a* for some a V (R) \ R, external if b is not internal.

Keisler lists the following as examples of entities that fit into each of these classifications:

38 C.f. Keisler [2007] p. 183 39 “Of course, Leibniz did not formulate his principle in anything like the present form…The name “Leibniz’ Principle” is used in the literature because Leibniz suggested that the real number should be extended to a larger system which has the same elementary properties but contains infinitesimals” (Keisler [2007] 183). 40 For details, see Keisler [2007] pp.189-194 41 To see this we note that the *-transform of the sentence “Every nonempty bounded set in (R) has a least upper bound” holds in V (R*). However R and N are bounded but have no least upper bound in R*, so R and N cannot belong to (R)*” (Keisler [2007] 184-185). 38 Standard and real: each r  R and each finite subset of R. external and real: N and R standard but not real: N* and R* Internal but not standard: each c  R* \R, [a,b]* where a,b  R * \R, h(x)  sin(Hx) where H is infinite. External but not real: monad(0), galaxy(0).

As Keisler notes, “several basic notions such as open set, continuity, and differentiability split into two separate notions when applied to internal functions and relations. Consider an internal function f on R* and a point c  R*” (Keisler [2007]

186). Where R is the set of positive real numbers, f is said to be continuous at c if it satisfies the following condition

  R   R x  R x  c    f (x)  f (c)  . We say that f is S- continuous at c if it satisfies the following real , condition

  R   R x  R* x  c    f (x)  f (c)   which is an external formula because it has the constant R+. In contrast, we say f is *-continuous at c if it satisfies the following hyperreal , condition

* *   R   R x  R* x  c    f (x)  f (c)  . This is an internal formula. There are internal functions that are everywhere *-continuous, but nowhere S- continuous and vice versa.42

Furthermore, “some properties of R are never preserved by the superstructure embedding” (Keisler [2007] 187). Keisler provides three examples. First is the

Archimedean property, x  R n  Nx  n. The *-transform of this formula is not

x  R * n  N x  n , but x  R *n  N *x  n, which carries a different

42 For an example, see Keisler [2007] pp. 186-187 39 meaning.43 The meaning changes because the formula is external. Another example is the least upper bound property: “where (x) is the bounded real formula stating ‘if x is nonempty and has an upper bound in R then x has a least upper bound in R,’”

x (R) (x) (Keisler [2007] 187). Once again, this formula is external and thus the

*-transform of this property has a different meaning. That is, it does not say that every nonempty set of R* that is bounded above has a least upper bound in R*—

x (R*)  *(x)— but that “every internal subset of R* which is nonempty and has an upper bound in R* has a least upper bound in R*,” formally x (R) * *(x)

(Keisler [2007] 187). Finally we look at induction over the natural numbers,

x (N) 0  x y  x y 1 x x  N. Note, the *-transform of this sentence is not x (N*) 0  x y  x y 1 x x  N *, but

x (N) * 0  x y  x y 1 x x  N *, which is satisfied by N* and restricts induction to just those internal subsets of N*.

Having presented Keisler’s axioms for the hyperreal number system R* as well as some of the basic properties of this system, we now turn to NSA’s characterization of limits of a functions, continuous functions, differentiation, and integration.

Letting L and c be real numbers, L is the limit of f(x) as x approaches c, written

L  lim f (x), if whenever x  c but x  c , we have f (x)  L. If there is no such L we x c say that the limit does not exist. L  lim f (x) is the limit of f(x) as x approaches c from x c the right when x  c but x  c. Likewise, L  lim f (x) is the limit of f(x) as x x  c 

43 This difference in meaning will come up again in section IIA, contention 5. 40 approaches c from the left when x  c but x  c. The lim f (x) exists if and only if x c lim f (x)  lim f (x).44 Supposing f is a standard function, Keisler notes that the x c  x c  following are equivalent:

(i) Whenever x  c but x  c , we have f (x)  L. (ii) Given any   0, we can find a   0 such that f (x)  L   provided that 0  x  c   .45

f is continuous at a real point c if f (c) is defined and whenever x  c , f (x)  f (c).46 Supposing f is a standard function, the following are proved to be equivalent:

(i) Whenever x  c , f (x)  f (c). (ii) For every real   0 there is a real   0 such that for all real x  (c  ,c  ) , we have f (x)  f (c)  .47

Let Y be a subset of the of f. f is continuous on Y if whenever c Y , x  c , and x Y *, we have f (x)  f (c).48 Supposing f is standard, this condition is equivalent to the following , condition: For every real   0 and c Y there is a real

  0 such that whenever x Y and 0  x  c   , we have f (x)  f (c)  .49 f is uniformly continuous on Y if whenever x,y Y * and x  y , we have f (x)  f (y).8

Once again, supposing f is standard, this condition is equivalent to the , condition: For every real   0 there is a real   0 such that whenever x, y Y , and 0  x  y   , we have f (x)  f (y)  .9

44 C.f. Keisler [2007] pp. 43-44 45 C.f. Keisler [2007] p. 71 46 C.f. Keisler [2007] p. 44 47 C.f. Keisler [2007] p. 72 48 C.f. Keisler [2007] p. 45 49 C.f. Keisler [2007] p. 73 41 Before discussing the IVT, we define the and, with this definition, an infinite partition of an interval. The natural extension Z* of Z, the , is called the set of hyperintegers. Defining the function [x] the greatest integer n  x , Keisler proves that Z* is the set of all hyperreal numbers y such that y  [x] for some hyperreal x.50

“A frequent construction in the calculus is the partition of a closed interval [a,b] into infinitely many subintervals of equal infinitesimal length. When x and y are hyperreal numbers, we call the set [x,y]*  z  R*:x  z  y a hyperreal closed interval. If a  x  y  b we call x, y* a hyperreal subinterval of a,b *. When

0 x and x  0, we call x, x  x * an infinitesimal interval” (Keisler [2007] 48).51

“Given a H  0, the closed hyperreal interval a,b * may be

b  a partitioned into subintervals of length   . The partition points are H a,a  ,a  2,...,a  K,...,a  H  b where K runs over the hyperintegers from 0 to H.

b  a If H is infinite, each subinterval will have infinitesimal length   , and the partition H is called an infinite partition of a,b*” (Keisler [2007] 48-49).

Keisler states the IVT as follows: “Suppose f is continuous on the closed interval

[a,b]. Then for each real number D between f (a) and f (b) there is a point c [a,b] such that f (c)  D” (Keisler [2007] 49). We note that the proof of this theorem involves only a finite partition over [a,b]. The IVT has the following useful consequence which involves hyperreal numbers: Hyperreal IVT—Suppose f (x) is a real function continuous

50 C.f. Keisler [2007] p. 47 51 Without ambiguity, we write a,b for a,b*. 42 on an interval I. Then, for each hyperreal x  y in I*, if u is a hyperreal number between f *(x) and f *(y), then there is a hyperreal z such that x  z  y and f *(z)  u.

We now turn to NSA’s characterization of differentiation.

A real number S is said to be the of a real function f at a real point a f (a x)  f (a) if S  st  for every nonzero infinitesimal x. The  x  derivative of a real function f is the real function f ' such that: f '(x)  the slope of f at x if it exists, f '(x) is undefined otherwise. (Keisler [2007] 33)

Supposing f is a real function and c is a real number, the definition of the derivative at c given above, S  f '(c), is equivalent to the following , condition: for every real   0 there is a real   0 such that whenever x is real and 0 x   , we

f (c x)  f (c) have  S   .52 f is differentiable at a real number a if and only if: x

(i) f (x) is defined for all x  a; and, f (a  x)  f (a) (ii) the quotient is finite and has the same standard part x for all nonzero x  0.53

y  f (x x)  f (x) is called the increment of y. When the derivative f '(x)

y exists, f '(x)  st . It is important to distinguish y from dy  f '(x)x. The latter x is called the differential of y. Since dx  x , Keisler shows that dy  f '(x)dx and

dy f '(x)  . Defining the tangent line to the curve y  f (x) at a real point (a,b) on the dx curve to be the line through (a,b) with slope f '(a), Keisler calls y the change in y along the curve and dy the change in y along the tangent line.

52 C.f. Keisler [2007] p. 72 53 C.f. Keisler [2007] p. 33 43 Letting x be a nonzero infinitesimal, u and v are said to be infinitely close

u v compared to x, written u  x (compared to x), if  .54 Using this definition, x x

Keisler proves the :

Suppose x is real, y  f (x), f '(x) exists, and x is a nonzero infinitesimal. Then y  f '(x)x x  dy x for some infinitesimal . In other words, y  dy (compared to x). (Keisler [2007] 35)

This idea is depicted by Keisler through the use of infinitesimal microscopes.55

Figure 5.56 Infinitesimal difference between dy /x and y /x

54 c.f. Keisler [2007] pp. 34-35 55 C.f. Keisler [2007] pp. 35-38 for details regarding the infinitesimal microscopes used below. 44

To keep the theory of integration as elementary as possible, we follow Keisler in restricting our presentation to only handle continuous real functions.57 First, we give

Keisler’s definitions of the Riemann sum and its natural extension, the infinite Riemann sum. For simplicity, Keisler considers “only partitions of [a,b] in which all subintervals except the last subinterval have the same length” (Keisler [2007] 60).

Let [a,b] be a subinterval of I and let x be a positive real number. The Riemann b sum  f (x)x is defined as the sum a b

 f (x)x  f (x0 )x  f (x1)x  ... f (xn 1)x  f (xn )(b  xn ) a where n is the largest integer such that a  nx  b, and

x0  a,x1  a x,...,xn  a  nx. … Given a continuous real function f on I and a subinterval [a,b] of I, let b S(x)   f (x)x be the finite Riemann sum. The infinite Riemann sum is the a b natural extension S *(dx)   f (x)dx where dx is a positive infinitesimal. a (Keisler [2007] 60)

b Keisler proves that the infinite Riemann sum, S *(dx)   f (x)dx, will be a finite a hyperreal number and defines the definite integral as follows:

Let a  b in I and let dx be positive infinitesimal. The definite integral of f from a to b with respect to dx is the standard part of the infinite Riemann sum, b b   f (x)dx  st f (x)dx. (Keisler [2007] 61) a   a 

56 Taken from Keisler [2012] p. 57. 57 To see a presentation of the Riemann integral developed so that it might handle functions with at most finitely many discontinuities, see Cutland pp. 64-67 45 We note that Keisler further proves that the definite integral is the only area function for f. “By an area function for f we mean a real function A(u,v), whose domain is the set of ordered pairs of elements of I, such that

i. A has the Addition property: A(a,c)  A(a,b)  A(b,c) for all a,b,c  I . ii. A has the Rectangle property: m(b  a)  A(a,b)  M(b  a)

whenever a  b in I and f has a minimum value m and a maximum value M on

[a,b]” (Keisler [2007] 59).

This is done by defining infinite lower and infinite upper Riemann sums with respect to dx and proving that these sums are infinitely close to each other.58

It is proved that the properties of the definite integral mimic those of the standard approach, e.g.

Let a  b in I, let c be a real constant, and let dx be positive infinitesimal. b  cdx  c(b  a) (i) a , b b  cf (x)dx  cf (x)dx (ii) a a , b b b  f (x)  g(x) dx   f (x)dx   g(x)dx (iii) a a a , b b (iv) if f (x)  g(x) for all x [a,b], then f (x)dx  g(x)dx .59 a a

With the definite integral defined, Keisler defines the antiderivative of a function and proves the first and second parts of the FTC:

Suppose the domain of f is an open interval of I. A function F is said to be the antiderivative of f on I if f is the derivative of F on I. FTC (1): Suppose I is an open interval, a  b in I, and F is the antiderivative of f b on I. Then f (x)dx  F(b)  F(a). a … x FTC (2): Let a  I and define F(x) for all x  I by F(x)  f (t)dt . a (i) F is continuous on I.

58 C.f. Keisler [2007] 64-66 59 C.f. Keisler [2007] pp. 61-63 46 (ii) F is an antiderivative of f on the interior of I. (Keisler [2007] 66-69).

We now turn to Keisler’s definitions of the improper integral in the ordinary fashion and using infinitesimals.

Suppose f is continuous on the half-open interval [a,b). The improper integral of b u f from a to b is defined as the limit  f (x)dx  lim  f (x)dx. If the limit exists, a u b  a the integral is said to converge. Otherwise, the integral is said to diverge… Other types of improper integrals are defined analogously. For example: If f is b b continuous on (a,b],  f (x)dx  lim  f (x)dx . If f is continuous on [a,), a u a  u  u f (x)dx  lim f (x)dx . (Keisler [2007] 87-88) a u a

Keisler rephrases the definition given above using infinitesimals for four cases.

Let f be continuous on [a,b). b b f (x)dx  L iff u  b but u  b, f (x)dx  L. a a b u f (x)dx diverges iff whenever u  b but u  b, f (x)dx is positive infinite. a a Let f be continuous on [a,).  H f (x)dx  L iff whenever H is positive infinite, f (x)dx  L . a a  H f (x)dx diverges iff whenever H is positive infinite, f (x)dx is positive a a infinite. (Keisler [2007] 88)

Before concluding, Taylor’s theorem is given as well as the following corrolary.

Taylor’s Theorem: Let f be a real function and c,x  R. If f n 1 exists between c n k n 1 f (c) k f (t) n 1 and x, then f (x)   x  c  x  c for some real t between k0 k! n 1 ! c and x. … Corrolary: Suppose c is real and f (n ) is continuous at c. Then whenever x  c n f (k)(x) and x is a nonzero infinitesimal, fxx   x k (compared to x n ). k0 k! (Keisler [2007] 110).

In summary, we have followed Keisler [2007] in presenting his axioms for the hyperreal number system R* as well as some of the properties of this system. We then 47 provided an overview of NSA’s characterization of limits of functions, continuous functions, differentiation, and integration using infinitesimal and infinite numbers. This concludes our discussion of the relevant theoretical background for NSA. Next, we present an overview of SIA.

1.4. Smooth Infinitesimal Analysis

SIA is a fragment of synthetic differential geometry—a category-theoretic60 approach to differential geometry developed by F.W. Lawvere and Anders Kock. While there are a number of different treatments of SIA (see Moerdijk and Reyes) we limit our discussion (for the most part) to that presented in Bell [1998; 2008]. First, we discuss some of the properties of the smooth real line, RS, (without going into great detail about the smooth world S). We then present overviews of differentiation and integration in

SIA.

61 RS is a commutative ring with the strict partial order relation < that satisfies the

following conditions: For any two elements a,b  RS ,

i. a  b and b  c implies a  c ii. (a  a) iii. a  b implies a  c  b  c for any c iv. a  b and 0  c implies ac  bc v. 0  a  a 1 vi. a  b implies a  b  b  a

60 “Roughly speaking, a category is a mathematical system whose basic constituents are not simply mathematical ‘objects’ (as sets are in set theory), but also ‘maps’ (‘functions’, ‘transformations’, ‘correlations’) between the said objects” (Bell [2008] 12). For further detail on , see Awodey [2006]. For further detail on synthetic differential geometry, see Kock [2006]. 61 Bell describes RS as a field (Bell [2008] 18). What he means is that RS is an intuitionistic field in that for any x  RS if x  0 (i.e. if x is provably not equal to zero), then x has a . 48

As Bell points out, (vi) does not imply that RS, satisfies the law of trichotomy. In fact, SIA rejects the law of trichotomy, instead adopting an .

2 Letting x  RS : x  0 , the fundamental assumption of SIA is the following.

62 Principle of Microaffineness : For any map g :  RS , there exists a unique b in

RS such that, for all  in  , we have g()  g(0)  b  .

The motivating intuition for the principle of microaffineness is that all curves in S

63 determined by functions from RS to RS satisfy the following principle.

Principle of Microstraightness64: For any smooth curve C and any point P on it, there is a (small) nondegenerate65 segment of C—a microsegment— around P which is straight, that is, C is microstraight around P.

It is because of the principle of microaffineness that RS is not a classical field, but a commutative ring. As Kock points out, if RS were a field,  would consist solely of 0.

Yet we know this is impossible, because if   {0} then   0 and b would no longer be unique. Further, we know SIA must adopt an intuitionistic logic because if there were some   0 that belonged to  we could define a function that would yield 1  0.66  is thus said to belong to the set of elements in RS that are not identical to 0, but

indistinguishable from 0, written  x  RS : x  0. This set coincides with the set of

62 C.f. Bell [2008] p. 21 63 C.f. Bell [2008] p. 21 64 C.f. Bell [2008] p. 9 65 In section II, I will address what Bell means by nondegenerate. 1,   0 66 Suppose g()   . If   0, then 1  g( )  0   b, which, when squared, 0,  0 0 0 0 yields 1  0. (Kock [2006] 5). 49 67 noninvertible elements of RS , , which includes the set of nilsquare infinitesimals,  .

68 A part A of RS is said to be microstable if a   A whenever and . In    addition, it can be proved that satisfies the following principle.

  69 Principle of Microcancellation: For any a,b  RS , if for all , then .

 A consequence of the principle of microaffineness is that every function f from RS

70 to RS is continuous—i.e. x  RS y  RS x  y  f (x)  f (y)  . That  said, what was historically taken to be “a self-evident attribute of continuous functions”

 (Grabiner 81), namely the IVT, is shown to fail in SIA. That is, it’s not the case that if f

is continuous on the closed interval [a,b], then for each smooth real number D between

f (a) and f (b) there is an element c [a,b] such that f (c)  D. This can be shown  using the following argument.    Suppose that the [IVT] were true in S for the polynomial function f (x)  x 3  tx  u. Then the value of x for which f (x)  0 would have to depend smoothly on the values of t and u. In other words there would have to 2 3 exist a smooth map g : RS RS such that g(t,u)  tg(t,u)  u  0. A geometric argument shows that no such smooth map can exist: for details, see Remark  VII.2.14 of Moerdijk and Reyes (1991).  (Bell [2008] 108)   The principle of microaffineness guarantees the possibility of nilsquare

infinitesimals. To guarantee the possibility of nilpotent infinitesimals that belong to

k  k x  RS : x  0 for each k 1, Bell adopts the following principle.

  67 C.f. Bell [2006] pp. 303-304 68 C.f. Bell [2008] p. 20 69 C.f. Bell [2008] p. 22 70 C.f. Bell [2008] p. 105 50 71 Principle of Micropolynomiality : For any and any g : k RS , there exist k n uniqueb1,...,bk in RS such that for all in  k we have g()  g(0)  bn . n 1  k We denote the set containing all nilpotent infinitesimals D  x  RS : k  0x  0 . D      is a subset of  and  coincides with [0,0].72

For the purposes of our discussion in secti on II.A, we point out that there are also

models of SIA that allow one to prove the existence of invertible infinitesimals

resembling those of NSA, i.e. infinitesimals that are smaller than any real number, but

greater than zero. As Bell explains73, this is done by constructing the smooth natural

* * numbers, NS , as to include infinite elements. Letting RS be the model of the smooth real

  * *  1 1 1  line that contains NS , one can prove x  RS (x  0)n  N   .    n 1 x n 1 These elements belong to the set of Robinsonian infinitesimals     1 1 1  RI  x  R* : (x  0)n  N   , a proper subset of the set of  S  n 1 x n 1

  *  1 1  infinitesiamls IN  x  RS : n  N  x  . The members of RI are    n 1 n 1

strictly larger than members of  and, unlike  \0, RI is said to be inhabited.74

 * * * 75 Further, we note that RS is S-Archimedean— x  RS n  NS x  n .  

  71 Note that this principle may be taken as asserting “that any RS-valued function on  behaves like a polynomial of degree k” (Bell [2008] 92-93). 72 C.f. Bell [2006] pp. 302-304 73 For details regarding the above, see Bell [2006] pp. 306-308 and Moerdijk & Reyes chapter VI.  74 That is, “while it is perfectly consistent to assert the presence of invertible infinitesimals…it is inconsistent to assert the ‘presence’ of nonzero noninvertible 51 Before proceeding to differentiation and integration, we look at two more properties of RS that do not hold for R nor R*, namely indecomposability and

nonpunctiformity. Let U be called a detachable part of RS if for any x  RS it is the case that either x is in U or x is not in U. RS is said to be indecomposable in that the only

76 detachable parts of RS are RS itself and its empty part. Note that the proof of this result requires either the constancy principle or the integration principle as presented below.

The nonpunctiformity of RS is made clear by looking at the relationship between

the space that contains RS , S, and the space that contains R, Set. A significant difference between S and Set is that all elements of the latter are ‘well-distinguished’; i.e.

x,y  Rx y  x  y , while S contains elements that are both ‘well-distinguished’ and not ‘well-distinguished’ (e.g. the potential elements of  \ {0}). With this in mind, Bell defines a point of a space S to be a map from a singleton (what he calls 1, a “terminal object”, or “one-point space”) to S, adding “we think of points as being the smallest possible nonempty spaces” (Bell [2006] 319). Because RS contains amorphous objects

(e.g. the potential elements of  \ {0}) that the point mapping does not apply to, it is said to be nonpunctiform (contrasting with R, all of whose elements satisfy the point mapping).77

We now turn to Bell’s presentation of differentiation. For a fixed x  RS , define

the function gx : RS by gx ()  f (x ). “By Microaffineness, there is a unique b in

infinitesimals, i.e. that  \{0} be inhabited” (Bell [2006] 307). In this sense, we say that the latter potentially exist. 75 C.f. Moerdijk & Reyes VI.2.5 76 C.f. Bell [2008] p. 29 77 C.f. Bell [2006] pp. 318-320 52

RS, whose dependence on x we will indicate by denoting it bx , such that, for all ,

f (x )  gx ()  gx (0)  bx    f (x)  bx   . Allowing x to vary then yields a function  x bx : RS RS , written f ' and called, as is customary, the derivative of f ” (Bell [2008]

 24). The equation above, “which may be written f (x )  f (x) f '(x) for an arbitrary   x  RS and , is called the fundamental equation of the differential calculus in S”

(Bell [2008] 24). The quantity f '(x) is the slope of the curve determined by f at x. The

 infinitesimal quantity f '(x)  f (x )  f (x) is the increment in the value of f on

passing from x to x .78 With the derivative defined, it is proved that every function in

S is “smooth in the technical sense of possessing derivatives of all orders” (Bell [2008]  25).

What is unique about SIA’s version of differentiation is that it requires no tools

outside of algebra to find the derivative of a function. For example, the derivative of

f (x)  x 2  3x at x can be found as follows79:

x  2  3x   x 2  3x f '(x) x 2  2x  2  3x  3  x 2  3x f '(x)  2x  3  f '(x) 2x  3  f '(x)

The derivative satisfies the following rules, mimicking those of the standard approach80:  Sum and scalar multiple rules: For any functions f ,g : J RS and any c in RS , f  g ' f 'g',cf ' cf ' where f  g, cf are functions xa f (x)  g(x), xa cf (x) respectively. : For any functions f ,g : J R , we have fg ' f 'g  fg' where fg  S    is the function xa f (x)g(x).   

 

78   C.f. Bell [2008] p. 24 79 Note the similarity between this method and the one used by Nieuwentijt. 80 C.f. Bell [2008] pp. 25-27 53 n n k1 Polynomial Rule: If f (x)  a0  a1x  ... an x , then f ' kak x . k1

Quotient Rule: If g : J RS satisfies g(x)  0 for all x  J, then for any f  f 'g  fg' f f (x) f : J RS ,  ' 2 where is the function xa . g  g g  g(x) Composite Rule: For any f,g : J R , g of ' g'of f ' where g o f is the  S   function xa gf(x) .   Inverse Function Rule: Suppose that f : J1 J2 admits an inverse g : J2 J1. Then f ' and g' are related by the equation f 'og g' g'of f '1.       In order to use the derivative to find of functions, one needs the      following definition. A point a  RS is said to be a , and f (a) a

stationary value, of a given function f : RS RS if for all , f (a )  f (a). With

this definition, Bell establishes 

Fermat’s Rule: A point a is a stationary point if and only if f '(a)  0 for all  ; i.e. f '(a)  0.

We now turn to Bell’s presentation of integration. To introduce integration Bell   posits the following principle.

Integration Principle: For any f :0,1RS there is a unique g :0,1RS such that g' f and g(0)  0.

This principle establishes the existence of a unique antiderivative g(x) for any x [0,1],  

 x  x written f (t)dt, or f , and called the definite integral of f over 0, x . To extend the 0 0   integral over arbitrary intervals, Bell first proves the following lemma.    Hadamard’s Lemma: For f : a,b RS and x, y  a,b we have 1 f (y)  f (x)  (y  x) f ' x  t(y  x) dt. 0 

   54

After doing so, he then proves that for any f : a,b RS there is a unique g : a,b RS

81 such that g' f and g(a)  0. For any f ,g : a,b RS , the following properties

(mimicking those of the standard approach) hold82:

b b b (a) f  g  f  g a a a b b (b) rf  rf for any r  R a a S b (c) f ' f (b)  f (a) a b b (d) f 'g  f (b)g(b)  f (a)g(a)  fg' (integration by parts) a a

Looking back at section I.B, we see that the Integration Principle is simply the first part of the FTC restricted to smooth functions—namely that for any smooth function

x f on a,b , if F is defined on a,b by F(x)  f (t)dt for a  x  b, then F is [an a antiderivative] of f on a,b . We further see that the second part of the FTC (once again restricted to smooth functions) is proved as a property (c) above. That said, Bell provides an independent argument for why F'(x)  f (x) (what he refers to as the FTC) assuming the integration principle and points out that “in order to be able to apply the

[FTC] we need to assume the following principle holds in S:

Constancy Principle: If f : J RS is such that f ' 0 identically, then f is constant” (Bell [2008] 28-29).

Before concluding, we turn to Taylor’s theorem in SIA. First, it’s proved that

n 1 83 whenever 1,...,n are in  , then1  ...n   0. With this in mind, Bell proves that

“if f : RS RS then for any x in RS and 1,...,k in  we have

81 C.f. Bell [2008] pp. 89-91 82 C.f. Bell [2008] p. 91 83 C.f. Bell [2008] exercise 1.12 55 k (n ) n f (x) fx1  ...k  f (x)  1  ...k ” (Bell [2008] 93). Using this result n 1 n! and the principle of micropolynomiality, Bell proves the following version of Taylor’s theorem.

If f : RS RS , then for any k 1, any x in RS and any  in  k , we have k x f (x )  f (x)   n f (n ) .84 n 1 n!

Notice that this version of Taylor’s theorem differs from that presented in sections I.B and I.C in that there is no determination of the error of the approximation.

In conclusion, we have followed (for the most part) Bell [1998; 2008] in presenting some of the basic properties RS, as well as SIA’s version of differentiation, integration, and Taylor’s theorem. This concludes the historical and theoretical background section of this thesis. In the next section, we will attempt to clarify and evaluate Bell’s 5 mathematico-philosophical contentions, Bell’s historical contention, and Mac Lane’s thesis.

84 C.f. Bell [2008] pp. 93-94 56 2. CLARIFICATIONS AND EVALUATIONS

Having outlined the relevant historical and theoretical background, we now turn to our clarification and evaluation of Bell’s 5 mathematico-philosophical contentions

(section II.A), Bell’s historical contention (section II.B), and Mac Lane’s thesis (section

II.C).

2.1. Bell’s 5 Mathematico-Philosophical Contentions

As mentioned in the introduction, while Bell’s 5 mathematico-philosophical contentions do point out interesting differences between NSA and SIA, some are in need of clarification. We address each contention individually.

1. In models of [SIA], only smooth maps between objects are present. In models of nonstandard analysis, all set-theoretically definable maps (including discontinuous ones) appear.

This claim is sufficiently clear. As discussed in section I.D, it is a consequence of the principle of microaffineness and the definition of the derivative that every function in S is smooth.

2. The logic of [SIA] is intuitionistic, making possible the nondegeneracy of the microneighbourhoods ∆ and Mi, i = 1, 2, 3. The logic of nonstandard analysis is classical, causing all these microneighborhoods to collapse to zero.

85 Note that M1  D, M2  , and M 3  0,0. The first unclarity of contention 2 is Bell’s use of nondegenerate. For Bell, an interval is what we shall call an interval set nondegenerate if it is “not identical to a single point” (Bell [2008] 8); i.e. any interval

a,b is set nondegenerate if, for any x a,b, a,b  {x}.86 In contrast with this notion, we shall call an interval a,b length nondegenerate if a  b. All length

85 C.f. Bell [2008] 95 86 C.f. Bell [2008] p. 8, footnote 11 57 nondegenerate in R, R*, and RS are set nondegenerate. In contrast, while all set

nondegenarate intervals in R and R* are length nondegenerate, this is not the case for RS

; e.g. Mi, i = 1, 2, 3 are set nondegenerate but not length nondegenerate.

Intuitively, the motivation for length nondegeneracy is that the length of a nondegenerate interval should be greater than 0. An intuitve motivation for set nondegeneracy could be that the set formed by a nondegenerate should contain more than a single element; however, it is clear that Mi, i = 1, 2, 3 do not satisfy this motivation— e.g. it is impossible to prove that there actually are nilpotent infinitesimals (other than 0) that belong to [0,0]. With that in mind, we look at Bell’s motivation for the set

nondegeneracy of Mi,i 1, 2, 3 given in the following passage:

Given a smooth curve AB, suppose we want to evaluate the area of the region ABCO by regarding it as the sum of thin rectangles XYRS.87

Figure 6. Motivation for nondegeneracy

87 We wait until section IIC to discuss whether it is even possible to think of the area under the curve AB as the “sum of thin rectangles” in SIA. 58 If X and S are distinguishable points then so are Y and R, so that the ‘area defect’ under the curve is nonzero; in this event the figure ABCO cannot literally be the sum of such rectangles as XYRS. On the other hand, if X and S coincide, then is zero but XYRS collapses into a straight line, thus failing altogether to contribute to the area of the figure. In order, therefore for ABCO to be the sum of rectangles like XYRS, we require that their base vertices X,S be indistinguishable without coinciding, and yet the area defect be zero. This desideratum (which is patently incompatible with the ) necessitates that the segment XS be a nondegenerate linear infinitesimal…(Bell [2008] 7-8)

From the above, we see that Bell’s motivation for the nondegeneracy of his linear infinitesimals is that their endpoints not coincide yet be indistinguishable, and that the area defect formed by the curve and these endpoints be zero. From our above

discussion, we see that Bell’s peculiar motivation is satisfied by Mi,i 1, 2, 3 .

With that in mind, it is worth asking further whether it is the fact that NSA’s logic is classical that keeps Mi, i = 1, 2, 3 in NSA from being set nondegenerate? That is, is belonging to Mi, i = 1, 2, 3, without being zero, “patently incompatible with the law of

excluded middle”? With regard to M2 , trivially yes. However with regard to M1, the

answer is no. A counterexample is Paolo Giordano’s recently developed Fermat Reals,

, an extension of R containing nilpotent infinitesimals that are strictly greater than

zero and that actually exist. 88 Note that, because the infinitesimals of R contain these

  two properties, M1  R satisfies both the intuitive motivations given above for set and 

88 Philip Ehrlich makes a similar point when he states: It is perhaps worth noting that these geometrical works [on geometries over rings containing nilpotent infinitesimals, non-Archimedean geometries over division rings that employ invertible infinitesimals, etc.], which are grounded in classical logic, along with the work of Giordano indicate how misleading Bell’s claim is that non-zero nilpotent infinitesimals are “possible” in SIA, unlike in NSA, because whereas “[t]he logic of SIA is intuitionistic…the logic of NSA is classical” (p. 308). (Ehrlich [2007] 363). For details regarding the construction of , it is recommended that the reader see Giordano [2010]. 59  length nondegeneracy, unlike M1  RS. Further note, that because the elements of M1 allow for the area defect to be zero and these elements are distinguishable from one another, one is led to question whether the indistinguishability of X and S needs to be part of Bell’s motivation given above.

3. In [SIA], the Principle of Microaffineness entails that all curves are ‘locally straight’. Nothing resembling this is possible in nonstandard analysis.

The unclarity of contention 3 lies in Bell’s use of ‘locally straight.’ Before clarifying this notion, however, we first make sense of what a straight line is in SIA.

That is, the classical notion of a straight line will not suffice to account for Bell’s microstraight lines. For example, take Hilbert’s first axiom of : “Two distinct points A and B always completely determine a straight line a” (Hilbert 2). The microstraight lines of SIA fail to completely determine a straight line for two reasons:

given any a  RS and   in a microstable part A of RS ,

(i) While a does not coincide with a  , a and a  will be

indistinguishable.

(ii) As emphasized in our discussion of nonpunctiformity, it is impossible

to prove that there actually is a point a  .

To account for both of these failings, one can revise Hilbert’s axiom as follows:

Given any point A, any set nondegenerate interval containing A determines a straight line.89

Note that the conditon that the interval be set nondegenerate allows this axiom to hold for not only SIA, but also the standard approach and NSA.

89 Note that this axiom moves beyond the primitive notions given by Hilbert and is not intended for his system. 60 With the notion of a straight line in SIA clarified, let us now turn to the property of ‘local straightness.’ The classical notion of ‘locality’ will not suffice for SIA. Take for example, a non-Eudlicdean space. Such a space is said to be locally Euclidean

(loosely speaking) when the theorems of the non- of that space coincide with their Euclidean counterparts in the limit (well-defined in the ordinary way); e.g. hyperbolic triangles are locally Euclidean. This notion of ‘locality’ does not suffice for SIA because there is no conception of the limit in this theory. Moreover, we believe

Bell would not choose to adopt this conception; for should he, all smooth curves of NSA and the standard approach would be said to be locally straight.

Instead, motivated by Bell’s principle of microstraightness, we believe Bell understands a curve y  f (x) to be ‘locally straight’ at a point x  a if, at that point, the curve coincides with an infinitesimal segment of y  f (x) which is straight. Bell interprets the principle of microaffineness as positing the ‘local straightness’ of the curves of SIA—“this [principle] says that the graph of g is a straight line passing through

0,g(0) with slope b” (Bell [2008] 21). Using infinitesimal microscopes, Dossena and

Magnani depict the situation for SIA as follows: 61

Figure 7.90 Local straightness SIA

Now, we ask whether there is anything “resembling [‘local straightness’] that is possible in” NSA? To answer this question, we overview and make use of Keisler’s presentation of infinitesimal microscopes in NSA.91

First, Keisler defines an -disc around a point in the hyperreal plane.

Given a point P(a,b) in the hyperreal plane and a positive hyperral number , the -disc around P is defined as the set of all hyperreal points x, y at distiance at most  from P. (Keisler [2007] 35)

Next, he defines a  -infinitesimal microscope.

Let P(a,b) be a point in the hyperreal plane and  be a positive infinitesimal. The  -infinitesimal microscope aimed at P is the mapping M from the 2 -disc around P onto the 2-disc around the origin given by the formula

90 Taken from Dossena and Magnani p. 210. For details regarding this construction, c.f. Dossena and Magnani pp. 203-212 91 “The following precise definition was suggested by Keith Stroyan [Stroyan 1997]” (Kesiler [2007] 35). 62 Max,b  y  x,y where x 2  y 2  4. Thus M maps a,b to 0,0 , magnifies distances 1/ , and preserves directions. (Keisler [2007] 35-36)

Finally, referring to the 2 -disc around P as the field of view of the microscocpe, Keisler points out the following:

A drawing of a  -infinitesimal microscope will distinguish two points if and only if the distance between them is not infinitesimal compared to  . Thus a point x, y is infinitely close to a,b if and only if it is in the field of view of some infinitesimal microscope aimed at a,b. (Keisler [2007] 36)

The following is an example of the use of infinitesimal microscopes.

By the [increment theorem], for each real point a we have f '(a)  S if and only if for every nonzero infinitesimal x, the curve at a  x is infinitely close to the tangent line at a x compared to x, that is, f (a  x)  f (a)  Sx (compared to x). It follows that f '(a)  S at a real point a if and only if the curve y  f (x) looks like a straight line with slope S in the field of view of any infinitesimal microscopes pointed at the point a, f (a). This is illustrated in figure 2a. (Keisler [2007] 36)

Figure 8.92 Local straightness NSA 1

92 Taken from Keisler [2007] p. 37 63 After giving this example, Keisler notes that “one would also like to use infinitesimal microscopes to illustrate the difference between the tangent line and the curve when the slope exists” (Keisler [2007] 36). This is done by using “a more powerful x 2- infinitesimal microscope within a x-infinitesimal microscope, as in figure 2c” (Kesiler

[2007] 36).

Figure 9.93 Local straightness NSA 2

Returning to our question as to whether there is anything “resembling [the ‘local straightness’ property] that is possible in” NSA, we propose that this depends on how one understands ‘resembling’. If one takes this term in the following restricted sense—that the smooth curves of NSA actually satisfy the ‘local straightness’ property (as defined above)—then Bell’s claim is correct. The infinitesimal difference between the curve and its tangent line show that the curve is not ‘locally straight.’ That said, if one adopts a

93 Taken from Keisler [2007] p. 37 64 more liberal interpretation of ‘resembling’ as meaning that the curves of NSA ‘appear’ to satisfy the ‘local straightness’ property in the field of view of an infinitesimal microscope, one can then say that, in some sense, is the ‘local straightness’ property is resembled in NSA.

4. The property of nilpotency of the microquantities of [SIA] enables the differential calculus to be reduced to simple algebra. In [NSA] the use of infinitesimals is a disguised form of the classical limit method.

The simple algebraic nature of the derivative in SIA was demonstrated in section I.D.

The reason the derivative in NSA is not ‘simply algebraic’ is because it relies on use of the standard part function. For example, let us differentiate the real function f (x)  x 2  3x at some point x. Where  is infinitesimal,

(x )2  3(x )  x 2  3x  2x  2  3  f '(x)  st  st  st2x  3  2x  3.      

The unclarity in contention 4 is what Bell means when he states that the use of infinitesimals in NSA is a ‘disguised form’ of the classical limit method. While we are uncertain what exactly is meant by Bell, we give three possible interpretations.

First, we note that Bell is not the first to use this term when discussing different versions of the calculus. For example, in comparing Leibniz and Newton’s distinct versions of the calculus, Edwards states the following:

In regard to the calculus itself, discrete infinitesimal differences of geometric variables played the central role in Leibniz’ approach, while Newton’s fundamental concept was the fluxion or time rate of change, based on intuitive ideas of continuous motion. As a consequence, Leibniz’ notation and terminology effectively disguises the limit concept, which by contrast is fairly explicit in Newton’s calculus. For Leibniz, the separate differentials dx and dy are fundamental; their ratio dy/dx is ‘merely’ a geometrically significant quotient. For Newton, however, especially in his later work, the derivative itself—as a ratio of fluxions or an ‘ultimate ratio of evanescent quantities’—is the heart of the matter. (Edwards 266) 65

Thus, because the infinitesimal difference is fundamental for Leibniz, rather than the ratio of such differences, it is said that his calculus is a ‘disguised form’ of the limit method.

In the same manner, Bell may be claiming that NSA’s use of infinitesimals ‘disguises’ the classical limit method in that the former takes as fundamental for the derivative the infinitesimal differences between quantities; whereas, the latter takes as fundamental the ratio of quantities both converging to zero and very close to zero. That said, we note the situation is somewhat different in that the infinitesimals of NSA are actual infinitesimals, while for Leibniz infinitesimals were often said to be heuristic tools that could be replaced with quanities “as small as you please”.94

A second possible interpretation is that NSA disguises the classical limit method in that when differentiating a function at a point, the syntactic relationship between the standard part function and infinitesimals is similar to that between the limit of a function and the quantity converging to zero. For example, differentiating the same function above using the standard approach one computes as follows:

(x x)2  3( x x)  x 2  3x 2xx  x 2  3x f '(x)  lim  lim  lim 2x  3 x  2x  3 x  0 x x  0 x x  0

That said, while the case is clear for the derivative, it is less so for the integral. Recall that in NSA the definite integral of some continuous function y  f (x) over the interval

 b  a,b  R * is defined as st f (x)dx. In contrast, recall in the standard approach that  a 

n the (supremum) integral is defined as the sup Sm (P) where Sm (P)  mk (yk  yk1), P k1

94 C.f. Edwards pp. 264-265 66 mk  inf f (x), and P  {y0,y1,...,yn} is some partition of [a,b]. If one uses an yk1 xyk infinite lower Riemann sum as the domain of the standard part function, one can in some sense claim that the infimum is ‘disguised’ by there being an infinite partition over the interval [a,b] and the supremum is ‘disguised’ by the standard part function. Once again, however, this is less clear than the case for the derivative.

Finally, a third possibility is that Bell is pointing out, as presented in section I.C, that assuming f is a standard function, the nonstandard definitions of limits of functions and continuous functions are equivalent to those given in the standard approach. Further, assuming f is a real function, the nonstandard definition of the derivative is equivalent to its definition in the standard approach.

5. In any model of [NSA] R* has exactly the same set theoretically expressible properties as R does: in the sense of that model, therefore, R* is in particular an Archimedean ordered field. This means that the ‘infinitesimals’ and ‘infinite numbers’ of nonstandard analysis are so not in the sense of the model in which they ‘live’, but only relative to the ‘standard’ model with which the construction began. That is, speaking figuratively, a ‘denizen’ of a model of nonstandard analysis would be unable to detect the presence of infinitesimals or infinite elements in R*. This contrasts with [SIA] in two ways. First, in models of [SIA] containing invertible infinitesimals, the smooth line is non- Archimedean in the sense of that model. In other words, the presence of infinite elements and (invertible) infinitesimals would be perfectly detectable by a ‘denizen’ of that model. And secondly, the characteristic property of nilpotency possessed by the microquanities of any model of [SIA] (even those in which invertible infinitesimals are not present) is an intrinsic property, perfectly identifiable within that model.

There are a few unclarities in contention 5. First, Bell’s statement that “R* is in particular an Archimedean ordered field” is misleading. Recall in section I.C that among the examples listed of properties of R that are not preserved by a superstructure 67 embedding due to their external nature was the Archimedean property. 95 In other words, the *-transform of the Archimedean property is not x  R *n  Nx  n, but

x  R * n  N * x  n, called the *-Archimedean property, which, as Robinson points out in the passage below, carries a different meaning than the Archimedean property:

And if we ask, for example, whether R* (like R) satisfies ’ axiom then the answer depends on our interpretation of the question. If by Archimedes’ axiom we mean the statement that from every positive number a we can obtain a number greater than 1 by repeated addition— a  a  ... a (n times) > 1—where n is an ordinary , then R* does not satisfy the axiom. But if we mean by it that for any a > 0 there exists a natural number n (which must be infinite) and that n a 1, then Archimedes’ axiom does hold in R*. (Robinson [1969] 34)

The second unclarity is Bell’s statement that “the ‘infinitesimals’ and ‘infinite numbers’ of nonstandard analysis are so not in the sense of the model in which they

‘live’, but only relative to the ‘standard’ model with which the construction began.”

While one can obtain R* by using the ultrapower construction on the real numbers, and hence ‘relativize’ infinite and infinitesimal numbers to the ‘standard model’, this is not the only way of obtaining the hyperreals. Among other ways is simply define the hyperreals as an ordered field extension of R. In this case, the infinite and infinitesimal numbers would be said to ‘live’ in that a denizen would need to be able to distinguish between infinite, finite but not infinitesimal, and infinitesimal numbers just as she needs to be able to distinguish between real numbers and rational numbers. For example,

95 We believe this unclarity is what motivates Laugwitz comment in his review of Bell [1998] that the distinction between the internal and the external is a very important part of NSA that seems to be largely ignored in Bell’s treatment (c.f. Laugwitz [1999]). As Nelson emphasizes, “Before applying transfer to an assertion, we must verify two things: that the assertion is internal and that all parameters in it have standard values. We call the violation of this rule illegal transfer. It is the most insidious pitfall awaiting the mathematician who wants to use nonstandard analysis” (Nelson 1166). 68 suppose a denizen of R* attempted to measure an object of finite length. If she only had a ruler of infinitesimal length, she would never be able to measure this object.96

The third unclarity is Bell’s statement that “the smooth line is non-Archimedean in the sense of that model. In other words the presence of infinite elements and

(invertible) infinitesimals would be perfectly detectable by a ‘denizen’ of” a model of the

* * smooth line with invertible infinitesimals”, call it RS . First, we point out that while RS is non-Archimedean, it satisfies an analogue of the *-Archimedean property, namely the S-

* * Archimedean property—x  RS n  NS x  n . Further, we point out that when the

* hyperreals are simply defined to be an ordered field extension of R, the denizen of RS would be in the same position as she is in R*.

Finally, the fourth unclarity is Bell’s statement that the characteristic property of

nilpotency of the microquantities of RS is perfectly identifiable within that model. Does

Bell mean identifiable to a denizen within the model? If so, it remains unclear how elements that are indistinguishable from 0 and only potentially exist, would be

identifiable to a denizen of RS ; whereas, actual infinitesimals would not be identifiable to a denizen of R*.

This concludes our attempt to clarify and evaluate Bell’s five mathematico- philosophical contentions. We take our most significant conclusion of this discussion to be that—positing that given any point A, any set nondegenerate interval containing A determines a straight line—the smooth curves of NSA resemble (in a liberal sense) the

96 We take this situation to be similar to that of a denizen of R attempting to measure the hypotenuse of a triangle with length sides with a ruler that was only able to measure rational lengths. 69 property of ‘local straightness’ satisfied by SIA; namely that any smooth curve y  f (x) will coincide with a straight infinitesimal segment of y  f (x) at every point on the curve. We will now turn to Bell’s historical contention for clarification and evaluation.

2.2. Bell’s Historical Contention

Once again, Bell’s historical contention states the following:

The differences between [NSA] and [SIA] may be said to arise because the former is essentially a theory of infinitesimal numbers designed to provide a succinct formulation of the limit concept, while the latter is, by contrast, a theory of infinitesimal geometric objects, designed to provide an intrinsic formulation of the concept of differentiability. ([3] 112)

Laugwitz challenges this contention as follows:

It is a misleading simplification to say, as the author does, that NSA “is essentially a theory of infinitesimal numbers designed to provide a succinct formulation of the limit concept…” (Laugwitz [1999] 2).

In support of Laugwitz’ challenge it is important to note the following:

i. “Any theorem which can be proved using [NSA] can be proved in Zermelo-

Fraenkel set theory with choice, ZFC, and thus is acceptable by contemporary

standards as a theorem in mathematics” (Henson and Keisler 377). Moreover, as

Henson and Keisler demonstrate, NSA (as practiced) is stronger than classical

analysis,97 and "in principle that there are theorems of which can be proved with

nonstandard analysis but cannot be proved by the usual standard methods"

(Henson and Keisler 377).

97 See Henson and Keisler for details. 70 ii. Robinson intended his work to be a vindication of the view of Leibniz, Bernoulli,

Euler, and Cauchy that an adequate foundation for the calculus could be based on

infinitesimals.98

iii. Robinson felt that NSA presented a natural way of performing the differential and

integral calculus (Dauben 184).

This is to be contrasted with SIA. SIA is a fragment of synthetic differential geometry—the purpose of the latter being to establish a foundation for synthetic reasoning in differential geometry (Kock [2007] ix). For Bell at least two of the purposes of SIA are:

iv. “to show how the traditional infinitesimal methods of can

be brought up to date—restored, so to speak” (Bell [2008] ix).

v. to capture some of the intuitive conceptions of a continuum as nonpunctiform

and indecomposable (Bell [2008] 3, 28-29).

With this in mind, it is worth discussing whether (ii)-(v) have actually been accomplished by NSA and SIA, respectively. With respect to (ii), as Ehrlich points out:

Robinson’s discoveries do not provide vindication of the Leibnizian formalism or of the seventeenth- and eighteenth-century preanalytic formalisms more generally. For example, whereas Leibniz conceived of differentiation and integration in terms of ratios of and infinite sums of infinitesimals, respectively, for Robinson they are real numbers that are infinitesimally close to such ratios and sums. ([Ehrlich 2006] 506)

That said, recent work has been done to reconstruct proofs in Euler’s Introductio using the tools of NSA.99 As will be discussed in section II.C, NSA does provide an alternative foundation for classical analysis using infinitesimals.

98 C.f. Robinson [1969] 71 With respect to (iii), while we feel it is a matter of aesthetic preference as to whether a way of differentiating and integrating is ‘natural’, it should be noted that

Kathleen Sullivan’s 1974 dissertation, which discusses the success that came from implementing Keisler’s Elementary Calculus: An Infinitesimal Approach in introductory calculus courses, in some sense provides support for Robinson’s claim.

Whether SIA actually accomplishes (iv) will be discussed in section II.C. With respect to (v), we first remind the reader of these notions. RS is said to be

100 indecomposable in that the only detachable parts of RS are RS itself and its empty part.

RS is said to be nonpunctiform because it contains amorphous objects that cannot be separated into terminal objects—namely the subsets of D excluding {0}.101

The indecomposability of RS is relatively uncontroversial; although, we note that there is a ‘stronger’ version of indecomposability that is proved to hold for the

102 intuitionistic continuum, but not RS . Whether RS is nonpunctiform is more controversial. For example, Hellman and Shapiro point out that “SIA has points galore, but it entertains (without asserting the existence of) nilsquare and nilpotent infinitesimals forming a ‘micro-neighborhood’… behaving as a mini-line (‘linelet’) that can be translated and rotated but not bent” (Hellman & Shapiro 2). We will examine whether

Bell’s motivation for the nonpunctiformity of RS is satisfied. Bell’s motivation can be found in the following passage.

99 This was in a recent presentation titled “A ‘Non-standard Analysis’ of Euler’s Introductio in Analysin Infinitorum” given by Pat Reeder at the Midwest Philosophy of Mathematics Workshop, 2012. 100 C.f. Bell [2008] p. 29 101 C.f. Bell [2006] pp. 318-320 102 For details, see Bell [2006] pp. 301-304 as well as Bell [2001]. 72 We observe that the ‘coherence’ of a genuine continuum entails that any of its (connected) parts is also a continuum, and accordingly, divisible. A point, on the other hand, is by its nature not divisible, and so…cannot be part of a continuum. Since an infinitesimal [as the ‘ultimate part of a continuum’] is a part of the continuum from which it has been extracted, it follows that it cannot be a point: to emphasize this we shall call such infinitesimals nonpunctiform. ([2] 3)

Thus, Bell’s motivation for the nonpunctiformity of RS is that the ‘ultimate part’ of a

continuum be divisible. Yet, Bell never demonstrates that the microneighborhoods of RS as ‘ultimate parts’, are divisible. Moreover, it seems more intuitive that an entity

“possessing (location and) direction without magnitude” (Bell [2008] 10) should not be

divisible. Bell’s condition for the nonpunctiformity of RS only demonstrates that there are amorphous elements. If he wishes to satisfy his motivation, he needs to demonstrate how such elements may be divided.

In summary, motivated by Laugwitz challenge, we have pointed out a number of further motivations behind NSA and SIA. We have further argued that the

nonpunctiformity of RS does not satisy Bell’s motivation for this property. Having done so, we now turn to Mac Lane’s thesis.

2.3. Mac Lane’s Thesis

We remind the reader that Mac Lane’s thesis is the claim that NSA and SIA provide two alternatives to the standard calculus. We note that this claim has been made

(or implied) for each theory independently in the mathematico-philosophical literature.

For example, in the preface of the second edition of Nonstandard Analysis, Robinson includes a statement of Kurt Gödel which mentions in closing “there are good reasons to believe that non-standard analysis, in some version or other, will be the analysis of the future” (Robinson [1974] x). Likewise, Hellman states that SIA “is intended as an 73 alternative to classical, ‘punctiform’ analysis’” (Hellman [2006b] 155). The purpose of this section will be to examine three interpretations of Mac Lane’s thesis based on different understandings of ‘the standard calculus’ and evaluate whether NSA and SIA satisfy each interpretation.

First, we give the following (common sense) definition for our use of alternative:

A: Let a mathematical theory T’ be called an alternative to theory T if T’ uses

different methods to solve the same problems103 T is intended to solve.

Now, suppose we let ‘the standard calculus’ be understood as

C1: Classical Analysis.

A number of reasons lead us to believe that NSA satisfies (A,C1).

 Any result of classical analysis can be proved in NSA.

 Assuming f is a real function, the nonstandard and the standard definitions of the

derivative are equivalent.

 The properties of the nonstandard definition of the integral mimic those of the

standard definition.

 One can define a hyperreal Riemann integral to handle curves with finitely many

discontinuities as well as the improper integral.

In contrast, some of the reasons we believe SIA fails to satisfy (A1,C1) are the following:

103 For example, with respect to the calculus, I mean those problems discussed in section I.A. It is not the case that T’ have the exact same answers to those problems as T does. For example, SIA has a different answer about to the problem of describing continua than the standard approach because the IVT fails in the former. This does not entail that SIA is not an alternative to the standard approach (given our definition A), but simply that its solutions to the problems of the calculus require a different type of continua.

74  There is no way to integrate functions over unbounded intervals as well as

intervals with finitely many discontinuities (i.e. no improper integral or Riemann

integral).

 Every function in SIA is smooth.

Thus, suppose, we restrict our definition of the ‘the standard calculus’ to

C2: The standard differential and integral calculus applied only to smooth functions

over bounded intervals.

The reasons given for NSA’s satisfaction of (A,C1) suffice for its satisfaction of (A,C2).

SIA may be said to satisfy (A,C2) in that the properties of the derivative and definite integral in SIA mimic those of their counterparts in the standard approach. That said, challenges to the claim that SIA satisfies (A,C2) may include:

a. SIA has no tools for looking at the convergence and of infinite

sequences.

b. SIA’s version of the Taylor’s theorem does not take into account approximative

error.104

With respect to (a), we note that while the IVT is a very important concept that drove the historical development of other notions of continuity, it is not necessary for solving the problems of C2 in SIA. With respect to (b), a similar statement can be made. That is, the convergence and divergence of infinite sequences plays an essential role in the

104 As stated below, Laugwitz points out: “For a kind of Taylor’s theorem, nilsquare infinitesimals are clearly not suitable, and there appears an ad-hoc introduction of higher- order nilpotent infinitesimals and of a new principle of micropolynomiality in place of microstraightness. Finite increments of the independent variable and remainder terms are outside the scope of the present approach” (Laugwitz [1999] 1).

75 development of the limit methods; further, it proves very helpful when evaluating integrals over unbounded intervals. That said, it is not essential in SIA for solving the problems of C2. Finally, with respect to (c), as motivated by Binmore’s explanation of

Taylor’s theorem (see Section I.B) we believe it is an important part of C2 that one be able to determine the approximative error of a polynomial from a curve. Thus, we make one final revision,

C3: The standard differential and integral calculus applied only to smooth functions

over bounded intervals and without approximative techniques. and say that NSA and SIA collectively illustrate Mac Lane’s thesis as restricted to

(A,C3).

Having established this, we return to (v) from section II.B, namely the claim that

SIA shows “how the traditional infinitesimal methods of mathematical analysis can be brought up to date—restored, so to speak” (Bell [2008] ix). While SIA can be granted an alternative to C3, Laugwitz challenges the claim that it is using “traditional infinitesimal methods.”

It appears that this claim is less justified… The integral is defined as antiderivative. Indeed, the definite integral cannot be obtained here as a sum of infinitely many infinitesimals since nilsquare infinitesimals are not invertible, and so no infinitely large natural numbers can be used at this stage. The intermediate value theorem is false even for third-degree , in contrast to Leibniz’ and Euler’s notions of continuity. A constancy principle must be postulated to have at hand that f ' 0 implies f  const. For a kind of Taylor’s theorem, nilsquare infinitesimals are clearly not suitable, and there appears an ad-hoc introduction of higher-order nilpotent infinitesimals and of a new principle of micropolynomiality in place of microstraightness. Finite increments of the independent variable and remainder terms are outside the scope of the present approach. (Laugwitz [1999] 1)

76 Recall that Laugwitz first challenge is justified by the fact that Newton and Leibniz as well as their successors, employed the idea that the area under a curve could be thought of as an infinite number of infinitesimal areas. We hinted at this challenge in Section I.D where we pointed out that the integration principle was simply the first part of the FTC.

Moreover, recall in Bell’s argument for the FTC, the integration principle is assumed.105

We also believe it is significant to note that Bell neglects the fact that one cannot take the sum of infinitely many infinitesimals when presenting his motivation for the

106 nondegeneracy of the microneighborhoods of RS . That is, Bell states:

If X and S are distinguishable points then so are Y and R, so that the ‘area defect’ under the curve is nonzero; in this event the figure ABCO cannot literally be the sum of such rectangles as XYRS. On the other hand, if X and S coincide, then is zero but XYRS collapses into a straight line, thus failing altogether to contribute to the area of the figure. In order, therefore for ABCO to be the sum of rectangles like XYRS, we require that their base vertices X,S be indistinguishable without coinciding, and yet the area defect be zero. (Bell [2008] 7-8).

Thus, this once again demonstrates why the above is an improper motivation for the set

nondegeneracy of Mi,i 1, 2, 3 .

In support of Laugwitz challenge regarding the IVT, the importance of this theorem for Ampere, Lagrange, Cauchy, and Bolzano when attempting to describe the continuity of functions was pointed out in section I.A. Furthermore, we believe one can safely claim that most (if not all) functions described as continuous by Leibniz and Euler would have satisfied the IVT.

105 See section I.D 106 See section II.A, contention 2 77 In addition, with respect to Laugwitz challenge regarding the constancy principle, we note that, in addition, in order to determine maxima or minima, one must define a stationary point and use Fermat’s rule.

Finally, it has been suggested that SIA has, in some sense, restored Neuwentijt’s calculus using nilsqure infinitesimals. Yet, this still remains controversial, for: (1)

Neuwentijt postulates the existence of an infinite number; (2) Neuwentijt defines an infinitesimal as the multiplicative inverse of such an infinite number; (3) Supposing an infinitesimal can be multiplied by a finite number, Neuwentijt’s first-order infinitesimals are all equal or one ends up with degrees of .107

In summary, we have argued that NSA and SIA collectively illustrate a restricted version of Mac Lane’s thesis, namely that NSA and SIA are two alternatives to the differential and integral calculus applied only to smooth functions over bounded intervals without approximative techniques. Furthermore, we have expanded on Laugwitz’ critique of Bell’s claim that SIA shows “how the traditional infinitesimal methods of mathematical analysis can be brought up to date—restored, so to speak” (Bell [2008] ix).

107 (3) is an objection raised by Jakob Hermann to Neuwentijt’s calculus. As Mancosu explains, “Let a,b,e, f be finite [and m infinite], and consider the quantities ab/em and ab/ fm with ab/em  ab/ fm. But then fm  bm and thus we end up with degrees of infinity” (Mancosu 164). For a more complete discussion of (1), (2), and (3) see Mancosu pp. 158-164. 78 CONCLUSION

In section I, we outlined the historical approaches to the calculus, the standard approach, NSA, and SIA. In section II, we attempted to clarify and evaluate Bell’s 5 mathematico-philosophical contentions, Bell’s historical contention, and Mac Lane’s thesis. With respect to our clarification and evalution of Bell’s 5 mathematico- philosophical contentions, one significant contribution made was that—positing that given any point A, any set nondegenerate interval containing A determines a straight line—the smooth curves of NSA resemble (in a liberal sense) the property of ‘local straightness’ satisfied by SIA; i.e. in NSA, it will appear in the field of view of an infinitesimal microscope that any smooth curve y  f (x) will coincide with a straight infinitesimal segment of y  f (x) at every point on the curve. In our clarification and evaluation of Bell’s historical contention we discussed different motivations of NSA and

SIA as well as argued that RS does not satisfy the version of nonpunctiformity it is motivated by. Finally, in our section on Mac Lane’s thesis, we argued that NSA and SIA are two alternatives to the differential and integral calculus applied only to smooth functions over bounded intervals without approximative techniques. We also argued that

SIA fails to show “how the traditional infinitesimal methods of mathematical analysis can be brought up to date—restored, so to speak” for a number of uses of traditional infinitesimals. 79 WORKS CITED

Baron, Margret. The Origin of the Infinitesimal Calculus. New York: Oxford University

Press, 1969.

Bell, John L. The Continuous and the Infinitesimal. Milano, Italy: Polimetrica, 2006.

------. "The Continuum in smooth infinitesimal analysis." Reuniting the Antipodes—

Constructive and Nonstandard Views of the Continuum. Ed. Peter Schuster, Ed.

Ulrich Berger and Ed. Horst Osswaldd. Dordrecht, The Netherlands: Kluwer

Academic Publishers, 2001. 19-24.

------. A Primer of Infinitesimal Analysis. Cambridge: Cambridge University Press,

1998.

------. A Primer of Infinitesimal Analysis. 2nd ed. New York: Cambridge University

Press, 2008.

Binmore, K.G. Mathematical Analysis: a straightforward approach. 2nd ed. Cambridge:

Cambridge University Press, 1982.

Boyer, Carl B. The History of the Calculus and its Conceptual Development. New York:

Dover Publications, 1959.

Dauben, Joseph W. “Abraham Robinson and Nonstandard Analysis: History,

Philosophy, and Foundations of Mathematics.” Minnesota Studies in the History

and Philosophy of vol. 11: History and Philosophy of Modern

Mathematics. Ed. William Aspray and Philip Kitcher. Minneapolis: University of

Minnesota Press, 1988. 80 Dossena, Riccardo and Magnani, Lorenzo. “Mathematics through :

Microscopes in Non-Standard and Smooth Analysis.” Studies in Computational

Intelligence (SCI). 64 (2007): 193-213.

Edwards, C. H., Jr. The Historical Development of the Calculus. New York: Springer-

Verlag, 1979.

Ehrlich, Philip. "Continuity." Encyclopedia of Philosophy. Ed. Donald M. Borchert. 2nd

ed. Detroit: Macmillan Reference USA , 2006. 490-491.

------. “The Continuous and the Infinitesimal in Mathematics and Philosophy by John L.

Bell.” The Bulletin of Symbolic Logic. 13.3 (2007): 361-63.

Giordano, Paolo. “The ring of Fermat reals.” Advances in Mathematics. 225 (2010):

2050-2075.

Grabiner, Judith V. A Historian Looks Back: The Calculus as Algebra and Selected

Writings. USA: The Mathematical Association of America (Incorporated), 2010.

------. “Newton’s Method and Leibniz’s Calculus.” A History of Analysis. Ed. Hans

Niels Jahnke. USA: The American Mathematical Society, 2003.

Hajnal, András and Hamburger, Peter. Set Theory. Trans. By Attila Máté. Cambridge:

Cambridge University Press, 1999.

Hellman, Geoffrey. "Mathematical Pluralism: The Case of Smooth Infinitesimal

Analysis." Journal of Philosophical Logic. 35.6 (2006): 621-651.

------“What is Categorical Structuralism.” The Age of Alternative Logics: Assessing

philosophy of logic and mathematics today. Ed. Johan van Benthem, Gerhard

Heinzmann, Manuel Rebuschi, and Henk Visser. Dordrecht, The Netherlands:

Springer, 2006. 81 Henson, C. Ward and Keisler, H. Jerome. “On the Strength of Nonstandard Analysis.”

The Journal of Symbolic Logic. 51.2 (1986): 377-386.

Hilbert, David. The Foundations of Geometry. Trans. E.J. Townsend. Chicago: The

Open Court Publishing Company, 1902.

Jahnke, Hans Niels. “Algebraic Analysis in the 18th Century.” A History of Analysis. Ed.

Hans Niels Jahnke. USA: The American Mathematical Society, 2003.

Keisler, H. Jerome. Foundations of Infinitesimal Calculus. Boston: Prindle, Weber &

Schmidt, Inc., 1976.

------. Foundations of Infinitesimal Calculus.

http://www.math.wisc.edu/~keisler/foundations.html. Creative Commons

Attribution-Noncommercial- Share Alike 3.0 Unported License, 2007.

------. Elementary Calculus: An Infinitesimal Approach. 3rd ed. New York: Dover

Publications, 2012.

Kirkwood, James R. An Introduction to Analysis. 2nd ed. Long Grove, Illinois:

Waveland Press, Inc., 2002.

Kock, Anders. Synthetic Differential Geometry. Cambridge: Cambridge University Press,

1981.

------. Synthetic Differential Geometry. 2nd ed. New York: Cambridge University Press,

2006.

Laugwitz, Detlef. “A Primer of Infinitesimal Analysis.” American Mathematical Society:

MathSciNet. 1999.

------.“On the Historical Development of Infinitesimal Mathematics: Part I”. The

American Mathematical Monthly. 104.5 (1997): 447-455. 82 ------. “On the Historical Development of Infinitesimal Mathematics: Part II”. The

American Mathematical Monthly. 104.7 (1997): 654-663.

Lützen, Jesper. “The Foundation of Analysis in the 19th Century.” A History of Analysis.

Ed. Hans Niels Jahnke. USA: The American Mathematical Society, 2003.

Mac Lane, Saunders. Mathematics: Form and Function. New York: Springer-Verlag,

1986.

Mancosu, Paolo. Philosophy of Mathematics and Mathematical Practice in the

Seventeenth Century. New York: Oxford University Press, 1996.

Moerdijk, Ieke and Reyes, Gonzalo E. Models for Smooth Infinitesimal Analysis.

Springer, 2010.

Robinson, Abraham. “The Metaphysics of the Calculus.” The Philosophy of

Mathematics. Ed. Jaakko Hintika. Oxford: Oxford University Press, 1969.

Shapiro, Stewart & Hellman, Geoffrey. “The Classical Continuum Without Points.”

2012, Philosophy of Science Assoc. 23rd Biennial Meeting, contributed papers.

Sullivan, Kathleen. “The Teaching of the Elementary Calculus Using the Nonstandard

Analysis Approach.” The American Mathematical Monthly. 83.5 (1976): 370-

375.

Stroyan, K.D. “Uniform Continuity and rates of growth of meromorphic functions.”

Contributions to Non-Standard Analysis. Eds. W.J. Luxemburg and A. Robinson.

Amsterdam: North-Holland (1972): 47-64.

Tall, D. “Elementary axioms and pictures for infinitesimal calculus.” Bulletin of the IMA.

18 (1982): 43-48. 83 Tall, D. “Natural and formal infinities.” Educational Studies in Mathematics. 48 (2001):

199-238.

Vermij, R. H. “Bernard Nieuwentijt and the Leibnizian Calculus”. Studia Leibnitiana.

21.1 (1989): 69-86.

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