Katz and Sherry’s [Pt.4.5] “Leibniz’ Infinitesimals: Their Fictionality, Their Modern Implementations, And Their Foes From Berkeley To Russell And Beyond,” 4.5 ‘Mathematical Implementation of Status Transitus,’ summary

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Mikhail G. Katz  and David Sherry

“Leibniz’s Infinitesimals: Their Fictionality, Their Modern Implementations, And Their Foes From Berkeley To Russell And Beyond”

4. Cum Prodiisset

 

4.5 Mathematical Implementation of Status Transitus

Brief Summary:

In Cum Prodiisset Leibniz discusses his Law of Continuity, and he provides some examples. Katz and Sherry will examine some of the mathematical procedures involved in these examples. What we find is that the same formulations and procedures are used regardless of whether we are dealing with finite or infinite values. This is because the law of continuity postulates that what holds for the finite holds for the infinite (and infinitesimal), [because there is a continuous transition from one to the other.]

Summary

 

In section 4.2 and 4.4, Katz and Sherry (KS) discuss three applications of Leibniz’ law of continuity and infinitesimals in his Cum Prodiisset. Now they will examine them more mathematically.

Leibniz distinguishes assignable finite quantity d(x) from infinitesimal value dx.

The assignable quantity (d)x passes via infinitesimal dx on its way to absolute 0. Then the infinitesimal dx is the terminus, or the status transitus. Zero is merely the shadow of the infinitesimal. This particular status transitus is the foundation rock of the Leibnizian definition of the differential quotient.
[KS 581]

[Recall Leibniz’ example of finding the tangent to a parabola. He writes:

“let ₁X₂X, the difference between A₁X and A₂X, be called dx;”

“and similarly, let D₂Y, the difference between ₁X₁Y and ₂X₂Y, be called dy.” (Leibniz 151)

Then, we shrink dx and dy down to find the tangent.

(Animated diagram by Corry Shores, using OpenOffice Draw and Unfreez)

Here Leibniz does not notate a difference between dx and d(x), but we would think of the finitely large line as being d(x) and the infinitesimally small one as dx.

KS also refer to Z as being the shadow of the infinitesimal. In the appendix they explain the shadow as being a real number assigned to hyperreal infinitesimals. It is the next closest value, only infinitesimally different. In these cases below, the shadow is like rounding the infinitesimal value to the nearest real number.]]

Let’s recall the first example Katz and Sherry examine:

(1) In the context of a discussion of parallel lines, he writes: when the straight line BP ultimately becomes parallel to the straight line VA, even then it converges toward it or makes an angle with it, only that the angle is then infinitely small (Child 1920, p. 148).
[[KS579]]

[Image from Leibniz/Child 148]

[We animated it thus:

(Animated diagram by Corry Shores, using OpenOffice Draw and Unfreez)

]

In KS’s rendition, they have us think of this diagram on its side with the moving line intercepting the x axis. In their formulation, H is the x-intercept value.

Example (1) of parallel lines can be elaborated as follows. Let’s follow Leibniz in building the line through the point (0,1) parallel to the x-axis in the plane. Line L_H with y-intercept 1 and x-intercept H is given by y = x / H .
[KS 581]

[Referring back to Leibniz’ diagram, the y-intercept here is like point P, which remains the same, all while the x-intercept, equivalent to point C, moves to greater distances.]

We will move the x-intercept to infinity. [KS later refer to the ‘finite portion of L_H.’ Perhaps one way to picture this is to think that the lines meet way off into infinity, but they ‘drag’ the finite part of the line all the way up such that it is only infinitesimally away from being y = 1. Yet, somewhere the lines must meet, which means it cannot stay at y = 1 everywhere. So does that mean that in the infinite part, it is no longer infinitely close to being parallel? I am not sure, but I think it is always infinitely close to being parallel, but just not always being infinitely close to y = 1. So far off into infinity, it will maybe be infinitely close to being parallel to y = 0.5, and so on diminishing. For every y position of the line in its infinite part, there would be an infinite stretch both ways, making every position be nearly parallel to the line.]

Now let H be infinite. The resulting line LH has negative infinitesimal slope, meets the x-axis at an infinite point, and forms an infinitesimal angle with the x-axis at the point where they meet. We will denote by st(x) the assignable (i.e., real) shadow of a finite x. [KS 581]

[They now will formulate for finite values for the x and y locations included in Line H.]

Then every finite point (x,y) ∈ L_H satisfies

[by substituting the formula for y. Now, because we are supposing H to be infinite, that means you have a finite value divided by an infinite one, which means it is nearly zero, which means that the y value is very close to 1, and thus:]

Hence the finite portion of L_H is infinitely close to the line y = 1. The line y = 1 is parallel to the x-axis, and is merely the shadow of the inassignable L_H. Thus, the parallel line is constructed by varying the oblique line depending on a parameter. Such variation comprises the status transitus L_H defined by an infinite value of H.
[KS 582]

Now recall example 3:

(3) Finally, a conception of a parabola expressed by means of an ellipse with an infinitely removed focal point is articulated in the following terms: a parabola is the ultimate form of an ellipse, in which the second focus is at an infinite distance from the given focus nearest to the given vertex (Child 1920, p. 148).
[KS 579]

[[Recall from our prior discussion that by moving one focus of the ellipse infinitely away from the other, the ellipse transforms into a parabola.

(Animated diagram by Corry Shores, using OpenOffice Draw and Unfreez)

]

KS will provide a formulation to illustrate this better.

To implement example (3), let’s follow Leibniz in deforming an ellipse, via a status transitus, into a parabola. The ellipse with vertex (apex) at (0,-1) and with foci at the origin and at (0,H) is given by

[They then square both sides and expand the values:]

Then they kept the radical on the left side, moving the rest over:

Then they square both sides, and reduce to get:

[We notice that even though H is an infinite number, they still apply operations on the figures that normally apply to finite numbers. But whether or not that is mathematically admissible is uncertain, hence it the ‘postulate’ of the law of continuity that we can treat them in like manner.]

The calculation (4.1) through (4.4) depends on the following habits of general reasoning (to echo Child’s translation) with assignable quantities, which are generalized to apply to inassignable quantities (such as the terminus/status transitus) in accordance with the law of continuity:

• squaring undoes a radical;

• the binomial formula;

• terms in an equation can be transferred to the other side; etc.

General reasoning of this type is familiar in the realm of ordinary finite real numbers, but why does it remain valid when applied to the realm of infinite or infinitesimal numbers? The validity of transfering such general reasoning originally instituted in the finite realm, to the realm of the infinite is postulated by Leibniz’s law of continuity.
[KS 582]

[Because the same general reasoning is used for finite and infinite, that might give a ‘between’ status or ambiguous status to the status transitus. Notice that we are still using a formulation for an ellipse. And yet, the resulting figure is in transit to being a parabola. Off in infinity, it might be still elliptical. But it draws out the finite part such that only parabolic parts remain within the realm of assignable values.]

We therefore apply Leibniz’s law of continuity to Eq. (4.4) for an infinite H. The resulting entity is still an ellipse of sorts, to the extent that it satisfies all of the Eqs. (4.1) to (4.4). However, this entity is no longer finite. It represents a Leibnizian status transitus between ellipse and parabola. This status transitus has foci at the | origin and at an infinitely distant point (0, H).
[KS 582|583, boldface mine]

[[Recall that we left off before with:

It seems in the following that KS will assign real numbers so to find the “real shadow” of this figure, that is, the finite assignable values it is infinitely close to.]]

Assuming x and y are finite, we set x₀ = st(x) and y₀ = st(y), to obtain a real shadow of this entity:

then simplifying:

Thus, the finite portion of the status transitus (4.4) is infinitely close to its shadow (4.5), namely the real parabola y ¼ x2 4 1 (in Leibniz’s terminology as translated by Child, ‘‘it is really true’’ that this parabola has no focus at infinity—see Sect. 4.4). This is the kind of payoff Leibniz is seeking with his law of continuity.
[KS 583]

Bibliography:

Katz, M.; Sherry, D. Leibniz's Infinitesimals: Their Fictionality, Their Modern Implementations, And Their Foes From Berkeley To Russell And Beyond. Erkenntnis 78 (2013), no. 3, 571-625. See http://dx.doi.org/10.1007/s10670-012-9370-y, http://www.ams.org/mathscinet-getitem?mr=3053644, and http://arxiv.org/abs/1205.0174

The above bibliography material taken from the following source, a page by Mikhail Katz, which links to many other recent publications on infinitesimals.

http://u.cs.biu.ac.il/~katzmik/infinitesimals.html

Katz and Sherry’s [Pt.4.2] “Leibniz’s Infinitesimals: Their Fictionality, Their Modern Implementations, And Their Foes From Berkeley To Russell And Beyond,” 4.2 ‘Law of Continuity, with Examples’, summary

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[The following is summary. My own comments and citations are placed in double brackets. All boldface and underlying are mine.]

 

Mikhail G. Katz  and David Sherry

“Leibniz’s Infinitesimals: Their Fictionality, Their Modern Implementations, And Their Foes From Berkeley To Russell And Beyond”

4. Cum Prodiisset

 

4.2 Law of Continuity, with Examples

Brief Summary:

In Cum Prodiisset Leibniz discusses his Law of Continuity. According to one formulation, in a continuous transition, the final ending (the terminus) of the transition may be included with that transition.

Summary

 

The basis of the calculus that Leibniz formulates in Cum prodiisset is his Law of Continuity (LC). It takes a variety of forms. Here is one formulation:

In any supposed continuous transition, ending in any terminus, it is permissible to institute a general reasoning, in which the final terminus may also be included.9 [[ft 9: Boyer claims that Leibniz used this formulation of LC in ‘‘a letter to [Pierre] Bayle in 1687’’ (Boyer 1959, p. 217). Boyer’s claim contains two errors. First, the work in question is not a letter to Bayle but | rather the Letter of Mr. Leibniz on a general principle useful in explaining the laws of nature, etc. (Leibniz 1687). Second, while this letter does deal with Leibniz’ continuity principle, it does not contain the formulation In any supposed continuous transition, ending in any terminus, etc.; instead, it postulates that an infinitesimal change of input should result in an infinitesimal change in the output (this principle was popularized by Cauchy in 1821 as the definition of continuity in Cauchy 1821, p. 34). Boyer’s erroneous claims have been reproduced by numerous authors, including Kline (1972, p. 385).]]
[[KS 577. footnote, 577-578]]

The final terminus in this explanation is an ending of a transition. KS then give five reasons that the terminus “encompasses inassignable quantities” [[infinitesimal quantities, see KS ‘Leibniz’s Laws of Continuity and Homogeneity” http://arxiv.org/pdf/1211.7188.pdf. See Page 578 of KS ‘Leibniz’s Infinitesimal’ for the five reasons.]]

In Cum Prodiisset, Leibniz offers a number of examples for how the Law of Continuity can be applied. KS will focus on three of them [[quoting]]:

(1) In the context of a discussion of parallel lines, he writes: when the straight line BP ultimately becomes parallel to the straight line VA, even then it converges toward it or makes an angle with it, only that the angle is then infinitely small (Child 1920, p. 148).

(2) Invoking the idea that the term equality may refer to equality up to an infinitesimal error, Leibniz writes: when one straight line is equal to another, it is said to be unequal to it, but that the difference is infinitely small (Child 1920, p. 148).

(3) Finally, a conception of a parabola expressed by means of an ellipse with an infinitely removed focal point is articulated in the following terms: a parabola is the ultimate form of an ellipse, in which the second focus is at an infinite distance from the given focus nearest to the given vertex (Child 1920, p. 148).

[[KS 579, for more on the ellipse example and on the law of continuity, see this entry on Leibniz’ letter to Malebranche.]

Bibliography:

Katz, M.; Sherry, D. Leibniz's Infinitesimals: Their Fictionality, Their Modern Implementations, And Their Foes From Berkeley To Russell And Beyond. Erkenntnis 78 (2013), no. 3, 571-625. See http://dx.doi.org/10.1007/s10670-012-9370-y, http://www.ams.org/mathscinet-getitem?mr=3053644, and http://arxiv.org/abs/1205.0174

The above bibliography material taken from the following source, a page by Mikhail Katz, which links to many other recent publications on infinitesimals.

http://u.cs.biu.ac.il/~katzmik/infinitesimals.html

Katz and Sherry’s [Pt.4.1] “Leibniz’s Infinitesimals: Their Fictionality, Their Modern Implementations, And Their Foes From Berkeley To Russell And Beyond,” 4.1 ‘Critique of Nieuwentijt’, summary

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Mikhail G. Katz  and David Sherry

“Leibniz’s Infinitesimals: Their Fictionality, Their Modern Implementations, And Their Foes From Berkeley To Russell And Beyond”

4. Cum Prodiisset

 

4.1 Critique of Nieuwentijt

Brief Summary:

Leibniz begins Cum Prodiisset by criticizing Niewentijt’s position that the product of two infinitesimals is zero.

Summary

Niewentijt “defended a conception of infinitesimal according to which the product of two infinitesimals is always zero.” [KS 577] In his Cum Prodiisset, Leibniz begins by criticizing this argument. Regarding Niewentijt’s positions, “Leibniz rejects nilsquare and nilcube infinitesimals, which are altogether incompatible with his approach to differential calculus, as we will see in Sect. 4.6.” [577]

Bibliography:

Katz, M.; Sherry, D. Leibniz's Infinitesimals: Their Fictionality, Their Modern Implementations, And Their Foes From Berkeley To Russell And Beyond. Erkenntnis 78 (2013), no. 3, 571-625. See http://dx.doi.org/10.1007/s10670-012-9370-y, http://www.ams.org/mathscinet-getitem?mr=3053644, and http://arxiv.org/abs/1205.0174

The above bibliography material taken from the following source, a page by Mikhail Katz, which links to many other recent publications on infinitesimals.

http://u.cs.biu.ac.il/~katzmik/infinitesimals.html

 

Katz and Sherry’s [Pt.4] “Leibniz’s Infinitesimals: Their Fictionality, Their Modern Implementations, And Their Foes From Berkeley To Russell And Beyond,” 4 ‘Cum Prodiisset’, summary

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Mikhail G. Katz  and David Sherry

“Leibniz’s Infinitesimals: Their Fictionality, Their Modern Implementations, And Their Foes From Berkeley To Russell And Beyond”

4. Cum Prodiisset

Brief Summary:

In the following subsections, the authors will focus on Leibniz’ Cum Prodiisset, because it is of crucial importance for understanding Leibniz’ fundamental stance.

Summary

 

Around 1701, Leibniz published, Cum Prodiisset, which is “of crucial importance in understanding Leibniz’s foundational stance”. [KS 577] The authors will focus on it in the following subsections.

Bibliography:

Katz, M.; Sherry, D. Leibniz's Infinitesimals: Their Fictionality, Their Modern Implementations, And Their Foes From Berkeley To Russell And Beyond. Erkenntnis 78 (2013), no. 3, 571-625. See http://dx.doi.org/10.1007/s10670-012-9370-y, http://www.ams.org/mathscinet-getitem?mr=3053644, and http://arxiv.org/abs/1205.0174

The above bibliography material taken from the following source, a page by Mikhail Katz, which links to many other recent publications on infinitesimals.

http://u.cs.biu.ac.il/~katzmik/infinitesimals.html

Katz and Sherry’s [Pt.3] “Leibniz’s Infinitesimals: Their Fictionality, Their Modern Implementations, And Their Foes From Berkeley To Russell And Beyond,” 3 ‘A Pair of Leibnizian Methodologies, summary

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Mikhail G. Katz  and David Sherry

“Leibniz’s Infinitesimals: Their Fictionality, Their Modern Implementations, And Their Foes From Berkeley To Russell And Beyond”

3. A Pair of Leibnizian Methodologies

Brief Summary:

Leibniz had two methodologies, which the authors call the A-methodology (using Archimedes’ exhaustion) and the B-methodology (using infinitesimals). Recent Leibniz scholars either acknowledge both of Leibniz’ methodologies or just the first type. The authors believe those in the second camp are misreading Leibniz’ notion of the infinitesimal’s fictionality.

Summary

Leibniz had two infinitesimal calculus methodologies: one by exhaustion and one using the law of continuity. (KS 575) The first relies on Archimedes’ exhaustion method, and the authors call it the ‘A-methodology’. The second uses infinitesimals and is called the ‘B-methodology’.

Leibniz considered infinitesimals as fictions. In his time, this was a controversial position, especially for some of his disciples, like Bernoulli, l’Hôpital, and Varignon. Accordiing to Ferraro, “Leibniz’s infinitesimals enjoy an ideal ontological status similar to that of the complex numbers, surd (irrational) exponents, and other ideal quantities.” (576)

The authors will now examine how commentators attribute either both A and B methodologies or just the A-methodology. They first quote from Leibniz’ 1702 letter to Varignon.

Here Leibniz outlines a geometrical argument involving quantities c and e described as ‘‘not absolutely nothing’’, and goes on to comment that c and e [KS quoting Leibniz:]

are treated as infinitesimals, exactly as are the elements which our differential calculus recognizes in the ordinates of curves for momentary increments and decrements (Leibniz et al. 1702, pp. 104–105). [KS 576]

Jesseph argues that Leibniz proposes both A and B methodologies. Like Bos, Jesseph emphasizes Leibniz’ law of continuity and regards it not as a mathematical principle but rather as a “a general methodological rule with applications in mathematics, physics, metaphysics, and other sciences’’ (KS 576 quoting Jesseph ibid p.21).

Recent work on Leibniz’ calculus is divided into two camps: 1) Those who recognize both methodologies (Bos, Ferraro, Horváth, Jesseph, and Laugwitz), and 2) those who have a syncategorematic interpretation that only recognizes the A-methodology. The authors believe that the second reading “is due to an incorrect analysis of Leibniz’s fictionalism.” (KS 577)

Bibliography:

Katz, M.; Sherry, D. Leibniz's Infinitesimals: Their Fictionality, Their Modern Implementations, And Their Foes From Berkeley To Russell And Beyond. Erkenntnis 78 (2013), no. 3, 571-625. See http://dx.doi.org/10.1007/s10670-012-9370-y, http://www.ams.org/mathscinet-getitem?mr=3053644, and http://arxiv.org/abs/1205.0174

The above bibliography material taken from the following source, a page by Mikhail Katz, which links to many other recent publications on infinitesimals.

http://u.cs.biu.ac.il/~katzmik/infinitesimals.html

Katz and Sherry’s [Pt.2] “Leibniz's Infinitesimals: Their Fictionality, Their Modern Implementations, And Their Foes From Berkeley To Russell And Beyond,” 2 ‘Preliminary Developments’, summary

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Mikhail G. Katz  and David Sherry

“Leibniz's Infinitesimals: Their Fictionality, Their Modern Implementations, And Their Foes From Berkeley To Russell And Beyond”

2. Preliminary Developments

Brief Summary:

Indivisibles are not infinitesimals, although they are often confused. Indivisibles have one dimension less than what they divide. Infinitesimals have the same dimensions as what they are a part of.

Summary

 

We should first distinguish indivisibles from infinitesimals.

Leibniz first uses the term ‘infinitesimal’ in 1673, but he credits the coinage to Mercator.

There is more than one conception of the infinitely small. Many commentators do not differentiate them. For example, Boyer seems to imply that Archimedes infinitesimal and kinematic methods provided the basis for Leibniz’ differential calculus. (Boyer, The concepts of the calculus, p.59.) (KS 573d).  However, Archimedes’ infinitesimal method uses indivisibles and not infinitesimals. His indivisibles are the limits of division, and thus they have one dimension less than the areas they are dividing (they are one dimensional while what they divide is two dimensional).

Archimedes’ infinitesimal method employs indivisibles. For example, in his heuristic proof that the area of a parabolic segment is 4/3 the area of the inscribed triangle with the same base and vertex, he imagines both figures to consist of perpendiculars of various heights erected on the base (ibid., 49–50). The perpendiculars are indivisibles in the sense that they are limits of division and so one dimension less than the area. Qua areas, they are not divisible, even if, qua lines they are divisible. In the same sense, the indivisibles of which a line consists are points, and the indivisibles of which a solid consists are planes. We will discuss the term ‘‘consist of’’ shortly.
(KS 574a, boldface and underlining mine)

[See Boyer’s treatment of Archimedes here, and see the original Archimedes’ text] However, Leibniz’ infinitesimals have the same dimension as the figures they make-up, and thus they are not like Archimedes’ indivisibles.

Leibniz’s infinitesimals are not indivisibles, for they have the same dimension as the figures that consist of them. Thus, he treats curves as composed of infinitesimal lines rather than indivisible points. Likewise, the infinitesimal parts of a plane figure are parallelograms. The strategy of treating infinitesimals as dimensionally homogeneous with the objects they compose seems to have originated with Roberval or Torricelli, Cavalieri’s student, and to have been explicitly arithmetized by Wallis (Beeley 2008, [Infinity, Infinitesimals, and the Reform of Cavalieri: John Wallis and his Critics. In Goldenbaum and Jesseph Infinitesimal Differences: Controversies between Leibniz and his Contemporaries] p. 36ff).
(KS 574)

Democritus used both indivisibles and infinitesimals. (KS 574)

Plutarch notes a puzzle of Democritus’ that would arise for indivisibles but not for infinitesimals. The puzzle is this. A cone can be thought as being made of many surfaces parallel to the base. If they are all equally, it would be a cylinder. But if they were all different, it would look like a staircase. However, we do not encounter this problem when we use the concept of the infinitesimal.  “This puzzle need not arise for infinitesimals of the same dimension, with an infinitesimal viewed as a frustum of a cone rather than a plane section.” (KS 574) So we are to think of each slice as being infinitesimally thin, but the outer part is at an angle. Thus one slice picks up where the prior leaves off, and we have an infinity of slices all making a smoothly tapering cone. (see diagrams of frustums below)

(thanks wikimedia commons)

(thanks wikimedia commons)

[We give a more detailed treatment of the Plutarch text here. From that entry, here is a moving diagram for the puzzle, including the distinction between the indivisibles and infinitesimals:]

[Moving diagram by Corry Shores, made with Open Office Draw and Unfreez]

Zeno’s “metrical paradox proposes a dilemma: If the indivisibles have no magnitude, then a figure which consists of them has no magnitude; but if the indivisibles have some (finite) magnitude, then a figure which consists of them will be infinite.” (574) There is a further problem for indivisibles. They are boundaries of what they limit. But this means they are not immediately up against one another. So we seem unable to concatenate them in order to increase a magnitude.

If a magnitude consists of indivisibles, then we ought to be able to add or concatenate | them in order to produce or increase a magnitude. But indivisibles are not next to one another; as limits or boundaries, any pair of indivisibles is separated by what they limit. Thus, the concepts of addition or concatenation seem not to apply to indivisibles.
(574-575)

These problems do not apply to Leibniz’ infinitesimals. They do not have a zero magnitude, so they do not have the problem of being unable to add up to a larger magnitude. However, their magnitude is not finite, so an infinity of them is not infinitely large. Infinitely many infinitely small magnitudes make up finite magnitudes. This also allows us to perform arithmetic operations on them, which distinguishes them from Archimedes’ methods.

The paradox may not apply to infinitesimals in Leibniz’s sense, however. For, having neither zero nor finite magnitude, infinitely many of them may be just what is needed to produce a finite magnitude. And in any case, the addition or concatenation of infinitesimals (of the same dimension) is no more difficult to conceive of than adding or concatenating finite magnitudes. This is especially important, because it allows one to represent infinitesimals by means of numbers and so apply arithmetic operations to them. This is the fundamental difference between the infinitary methods of Archimedes (and later Cavalieri) and the infinitary methods of Leibniz and his followers.
(575)

Not rigorously making this distinction has led to misleading claims being made about 17th century calculus. In the following, the authors will “say that a magnitude consists of infinitesimals just in case the infinitesimals and the original magnitude have the same dimension. Otherwise, we shall use the term indivisible.” (575)

 

Bibliography:

Katz, M.; Sherry, D. Leibniz's Infinitesimals: Their Fictionality, Their Modern Implementations, And Their Foes From Berkeley To Russell And Beyond. Erkenntnis 78 (2013), no. 3, 571-625. See http://dx.doi.org/10.1007/s10670-012-9370-y, http://www.ams.org/mathscinet-getitem?mr=3053644, and http://arxiv.org/abs/1205.0174

The above bibliography material taken from the following source, a page by Mikhail Katz, which links to many other recent publications on infinitesimals.

http://u.cs.biu.ac.il/~katzmik/infinitesimals.html

 

 

Image credits:

Frustrum:

http://commons.wikimedia.org/wiki/File:Frustum_of_a_cone.jpg

http://commons.wikimedia.org/wiki/File:Frustum_of_a_Decagonal_Pyramid.svg

Katz and Sherry [ED] “Leibniz’s Infinitesimals: Their Fictionality, Their Modern Implementations, And Their Foes From Berkeley To Russell And Beyond”, entry directory

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Mikhail G. Katz  and David Sherry

 

“Leibniz’s Infinitesimals: Their Fictionality, Their Modern Implementations, And Their Foes From Berkeley To Russell And Beyond”

Abstract

1. Introduction

2. Preliminary Developments

3. A Pair of Leibnizian Methodologies

4. Cum Prodiisset

4.1 Critique of Nieuwentijt

4.2 Law of Continuity, with Examples

4.3 Souverain Principe

4.4 Status Transitus

4.5 Mathematical Implementation of Status Transitus

4.6 Assignable Versus Unassignable

Bibliography:

 

Katz, M.; Sherry, D. Leibniz's Infinitesimals: Their Fictionality, Their Modern Implementations, And Their Foes From Berkeley To Russell And Beyond. Erkenntnis 78 (2013), no. 3, 571-625. See http://dx.doi.org/10.1007/s10670-012-9370-y, http://www.ams.org/mathscinet-getitem?mr=3053644, and http://arxiv.org/abs/1205.0174

 

The above bibliography material taken from the following source, a page by Mikhail Katz, which links to many other recent publications on infinitesimals.

http://u.cs.biu.ac.il/~katzmik/infinitesimals.html

Katz and Sherry’s “Leibniz’s Infinitesimals: Their Fictionality, Their Modern Implementations, And Their Foes From Berkeley To Russell And Beyond,” Introduction, summary

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Mikhail G. Katz  and David Sherry

“Leibniz’s Infinitesimals: Their Fictionality, Their Modern Implementations, And Their Foes From Berkeley To Russell And Beyond”

1. Introduction

Brief Summary:

Many calculus historians deny a historical continuity from Leibniz’ concept of infinitesimals to Robinson’s. Part of the reasoning behind their conclusion is that Berkeley successfully proved the flaws of this concept, and Robinson built his notion of the infinitesimal using modern logical tools that were unavailable to Leibniz. The authors will show that Berkeley’s critique does not hold so strongly, because Leibniz’s infinitesimal is rigorous even to today’s standards, and thus there may in fact be a historical continuity from Leibniz to Robinson.

Summary

 

The calculus Leibniz invented was based on infinitesimal (infinitely small) quantities. This concept of the infinitesimal was seen as being not mathematically rigorous. However, in the 20th century, Abraham Robinson invented non-standard analysis, which provides a consistent theory of infinitesimals. But since Robinson’s non-standard analysis makes use of modern logic, many commentators say that there is no historical continuity from Leibniz’ to Robinson’s infinitesimals. (572)

Robinson himself suggests a link to Leibniz via Hilbert, as all three regard infinitesimals as not real entities. (572) For Leibniz, infinitesimals are mental fictions. Nonetheless, this does not prevent them from being meaningful:

Leibniz’s was a remarkably modern insight that mathematical expressions need not have a referent, empirical or otherwise, in order to be meaningful. (Katz and Sherry [henceforth KS, or ‘the authors’] 572)

KS will argue primarily two things.

1) “Leibniz’ system for the calculus was free of contradiction, and incorporated versatile heuristic principles such as the law of continuity and the transcendental law of homogeneity” [see Fig 1, reproduced below] (572d). And

2) they will “undermine the view that Berkeley’s objections to the | infinitesimal calculus were so decisive that an entirely different approach to infinitesimals was required.” (572|573)

Jesseph (in his Berkeley’s philosophy of mathematics. Science and its Conceptual Foundations) argues that this early concept of the infinitesimal was “a conceptual dead-end, and a consistent theory of infinitesimals required a fresh start.” (KS 573) The authors, however, will argue that “Leibniz’s defense of the infinitesimal calculus—both philosophical and mathematical—guided his successors toward an infinitesimal analysis that is rigorous by today’s standards.” (573) To do this, they will show that Berkeley’s critiques “stem from philosophical presuppositions which are neither necessary nor desirable from Leibniz’s perspective.” (573)

Bibliography:

Katz, M.; Sherry, D. Leibniz's Infinitesimals: Their Fictionality, Their Modern Implementations, And Their Foes From Berkeley To Russell And Beyond. Erkenntnis 78 (2013), no. 3, 571-625. See http://dx.doi.org/10.1007/s10670-012-9370-y, http://www.ams.org/mathscinet-getitem?mr=3053644, and http://arxiv.org/abs/1205.0174

The above bibliography material taken from the following source, a page by Mikhail Katz, which links to many other recent publications on infinitesimals.

http://u.cs.biu.ac.il/~katzmik/infinitesimals.html

Katz and Sherry’s “Infinitesimals: Their Fictionality, Their Modern Implementations, And Their Foes From Berkeley To Russell And Beyond,” abstract, summary

summary by Corry Shores
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Mikhail G. Katz  and David Sherry

“Infinitesimals: Their Fictionality, Their Modern Implementations, And Their Foes From Berkeley To Russell And Beyond”

Abstract

Summary of Abstract:

Many historians of calculus deny a strong continuity between Leibniz’ infinitesimal calculus and Robinson’s non-standard analysis. They (including Robinson) base this judgment largely on Berkeley’s criticism of Leibniz’ infinitesimal. The authors will show instead that Leibniz’ defense of infinitesimals has a stronger basis than Berkeley’s critique, thereby supporting the notion that non-standard analysis develops more directly from Leibniz’ calculus.

Abstract [quoting the text]:

Many historians of the calculus deny significant continuity between infinitesimal calculus of the seventeenth century and twentieth century developments such as Robinson’s theory. Robinson’s hyperreals, while providing a consistent theory of infinitesimals, require the resources of modern logic; thus many commentators are comfortable denying a historical continuity. A notable exception is Robinson himself, whose identification with the Leibnizian tradition inspired Lakatos, Laugwitz, and others to consider the history of the infinitesimal in a more favorable light. Inspite of his Leibnizian sympathies, Robinson regards Berkeley’s criticisms of the infinitesimal calculus as aptly demonstrating the inconsistency of reasoning with historical infinitesimal magnitudes. We argue that Robinson, among others, overestimates the force of Berkeley’s criticisms, by underestimating the mathematical and philosophical resources available to Leibniz. Leibniz’s infinitesimals are fictions, not logical fictions, as Ishiguro proposed, but rather pure fictions, like imaginaries, which are not eliminable by some syncategorematic paraphrase. We argue that Leibniz’s defense of infinitesimals is more firmly grounded than Berkeley’s criticism thereof. We show, moreover, that Leibniz’s system for differential calculus was free of logical fallacies. Our argument strengthens the conception of modern infinitesimals as a development of Leibniz’s strategy of relating inassignable to assignable quantities by means of his transcendental law of homogeneity. (Katz and Sherry, 571)

Bibliography:

Katz, M.; Sherry, D. Leibniz's Infinitesimals: Their Fictionality, Their Modern Implementations, And Their Foes From Berkeley To Russell And Beyond. Erkenntnis 78 (2013), no. 3, 571-625. See http://dx.doi.org/10.1007/s10670-012-9370-y, http://www.ams.org/mathscinet-getitem?mr=3053644, and http://arxiv.org/abs/1205.0174

The above bibliography material taken from the following source, a page by Mikhail Katz, which links to many other recent publications on infinitesimals.

http://u.cs.biu.ac.il/~katzmik/infinitesimals.html