Notes for a History of the Calculus and its Metaphysics

Gil Morejón, 20171

I. Introduction: what is differential calculus?

Differential calculus attempts to establish the rate of change, also called the derivative, 𝑑𝑦

𝑑𝑥𝑓(𝑥),

for any given function 𝑓(𝑥). On a Cartesian coordinate plane, the value of a point on the x-axis,

or ordinate, is that which enters into the function 𝑓(𝑥), yielding a new value that is plotted on the

y-axis, or abscissa. Graphed this way on a plane, many functions can be geometrically represented

as curves, in which the rate at which the 𝑦 value changes relative to rate of change of the 𝑥 value

is not constant: the 𝑦 value may change at a greater or lesser rate given the same change in 𝑥.

Differential calculus seeks to establish this variable rate. Now, given any two points on a curve,

(𝑥1, 𝑦1) and (𝑥2, 𝑦2), the rate at which the 𝑦 value changes relative to the value of 𝑥 between these

two points, 𝑑𝑦

𝑑𝑥, can therefore be calculated as

𝑦2−𝑦1

𝑥2−𝑥1.

For linear equations, which can be expressed in the form 𝑦 = 𝑓(𝑥) = 𝑚𝑥 + 𝑏, where 𝑚

and 𝑏 are both constants, the derivative or differential is nothing other than the slope of the line or

the value of 𝑚, and the derivation is therefore entirely trivial, since the rate of change of y is the

same at every x. This is not the case for polynomial functions that involve variables with powers

1 Please let it be understood that what follows is necessarily cursory and, at times, surely oversimplifies

the complexities of the work discussed. As the author is not a trained mathematician, it is also possible

that some aspects of the exposition are incorrect or misleading. Any objections, clarifications,

suggestions, or complaints will be received with great appreciation at gdmorejon@gmail.com.

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greater than 1, such as the function we will take as our example throughout this appendix: 𝑓(𝑥) =

𝑥2 − 2𝑥 + 3. Here the function can be geometrically represented as a parabolic curve, and

therefore the rate at which 𝑦 changes relative to 𝑥 is different depending on the value of 𝑥. The

rate of change of a curve is equal to the gradient or slope of the line tangential to the curve at a

point, and that tangent is different at different points. Differential calculus thus tries to determine

the function that expresses this difference in the rate of change of 𝑦 relative to the change in 𝑥.

But at a given point on a curve, (𝑥1, 𝑦1), how do we calculate the rate at which y is changing

relative to x? The problem is that our method for determining the rate of change requires

calculating the difference between the rates at which y changes relative to x of two different points,

𝑦2−𝑦1

𝑥2−𝑥1; this actually determines the gradient of a secant of the curve, not a tangent. But, by

hypothesis, at a single point on a curve there is no second point, (𝑥2, 𝑦2), with which we could

make this calculation. How then can we determine the differential of a curve at a single point on

that curve? Attempting to answer this question is what motivated Leibniz’s approach, which

employed infinitesimals or infinitely small quantities.

II. From Leibniz to Hegel on mathematical infinities2

Leibniz and Newton are generally credited with having discovered the calculus independently,

although to claim that either of them is its ‘discoverer’ is problematic and reductive given the long

history of contributions to mathematical thought.3 Let us look at Leibniz’s method for determining

2 In these sections, I draw substantially from Gerdes 1985 and Duffy 2006. 3 Cf. Boyer, The History of the Calculus,177-8; consider also Anne McClintock’s reflections on the logic

of narratives of male birthing in Imperial Leather, e.g. in Chapter 6.

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the derivative, which relies on the notion of infinitesimals, or infinitely small quantities. We have

our function, 𝑓(𝑥) = 𝑥2 − 2𝑥 + 3. We want to determine 𝑑𝑦

𝑑𝑥, the rate at which 𝑦 changes relative

to the change of 𝑥. We will calculate 𝑦2−𝑦1

𝑥2−𝑥1 on the hypothesis that the difference beween 𝑦2 and

𝑦1, or 𝑑𝑦, and that between 𝑥2 and 𝑥1, or 𝑑𝑥, is infinitely small – that it infinitely approaches and

becomes functionally equivalent to zero.

First we note that 𝑑𝑦 = 𝑦2 − 𝑦1. In our function, 𝑥 is the independent variable, which

means that 𝑦 is a function of 𝑥; so, 𝑦2 is a function of 𝑥2 and 𝑦1 is a function of 𝑥1. Thus we can

now articulate 𝑑𝑦 strictly in terms of 𝑥:

1. 𝑑𝑦 = 𝑦2 − 𝑦1 = (𝑥22 − 2𝑥2 + 3) − (𝑥1

2 − 2𝑥1 + 3), and simplify:

2. 𝑑𝑦 = 𝑥22 − 2𝑥2 − 𝑥1

2 + 2𝑥1

But we also know that 𝑑𝑥 = 𝑥2 − 𝑥1 or, equivalently, 𝑥2 = 𝑑𝑥 + 𝑥1. So now:

3. 𝑑𝑦 = (𝑑𝑥 + 𝑥1)2 − 2(𝑑𝑥 + 𝑥1)−𝑥12 + 2𝑥1

Now we expand our terms using the binomial theorem and the distributive property:

4. 𝑑𝑦 = 𝑑𝑥2 + 2𝑑𝑥𝑥1 + 𝑥12 − 2𝑑𝑥 − 2𝑥1 − 𝑥1

2 + 2𝑥1, and simplify:

5. 𝑑𝑦 = 𝑑𝑥2 + 2𝑑𝑥𝑥1 − 2𝑑𝑥

Dividing both sides by 𝑑𝑥 gives us what we were after, a formula for 𝑑𝑦

𝑑𝑥:

6. 𝑑𝑦

𝑑𝑥= 𝑑𝑥 + 2𝑥1 − 2

Now, however, we seem to have a problem, for our equation literally begs the question: we are

defining the rate at which 𝑦 changes relative to the change of 𝑥 in terms of the rate of change of 𝑥;

that is, 𝑑𝑥 appears both on the left and right sides of the equation. Leibniz’s solution is, at this

point, to ‘suppress’ or ‘eliminate’ any terms on the right side of the equation that still include 𝑑𝑥.

Why is he able to do this? Because, as per our hypothesis, 𝑑𝑥 is infinitely small, or approximately

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and thus functionally equivalent to zero, and so can be ignored. As Leibniz writes in justification

of this procedure: “anyone who is not satisfied with this can be shown in the manner of Archimedes

that the error is less than any assignable quantity and cannot be given by any construction.”4 So,

we are left with the general formula that expresses the rate at which 𝑦 changes relative to the

change of 𝑥 for our function:

7. 𝑑𝑦

𝑑𝑥= 2𝑥 − 2

And indeed, for any point on the curve described by 𝑓(𝑥) = 𝑥2 − 2𝑥 + 3, the slope or gradient of

the line tangential to it at that point is given by our derivative function 𝑓′(𝑥) = 2𝑥 − 2, as

verification by experimental application makes apparent.

Leibniz provides a diagram in his justification for this infinitesimal method.5 We have two

straight lines, 𝐴𝑋 and 𝐸𝑌, which meet at point 𝐶. Lines 𝐴𝐸 and 𝑋𝑌 are perpendicular to the line

𝐴𝑋. Let 𝑥 be the length of line 𝐴𝑋, 𝑐 be the length of 𝐴𝐶, 𝑦 be the length of 𝑋𝑌, and 𝑒 be the

length of 𝐴𝐸. Now, 𝐶𝐴𝐸 and 𝐶𝑋𝑌 form two similar right triangles, meaning that the only

difference between them is one of scale – i.e., their

angles map onto one another. Due to this similarity, we

note that 𝑥−𝑐

𝑦=

𝑐

𝑒. If now we continuously slide the line

𝐸𝑌 rightward while maintaining its gradient (as in the

dotted line), the lengths 𝑐 and 𝑒 triangle 𝐶𝐴𝐸 will

shrink, while the ratio of 𝑐 to 𝑒 will remain constant. As

he writes:

4 Leibniz, “Justification of the Infinitesimal Calculus by That of Ordinary Algebra”, 546 5 Leibniz, "Justification", 545

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Now assume the case when the straight line 𝐸𝑌 passes through 𝐴 itself; it is obvious that

the points 𝐶 and 𝐸 will fall on 𝐴, that the straight lines 𝐴𝐶 and 𝐴𝐸, or 𝑐 and 𝑒, will vanish,

and that the proportion or equation 𝑥−𝑐

𝑦=

𝑐

𝑒 will become

𝑥

𝑦=

𝑐

𝑒. Then in the present case,

assuming that it falls under the general rule, 𝑥 − 𝑐 = 𝑥. Yet 𝑐 and 𝑒 [which now both = 0]

will not be absolutely nothing, since they still preserve the ratio of 𝐶𝑋 to 𝑋𝑌 [...] 𝑐 and 𝑒

still have an algebraic relation to each other. And so they are treated as infinitesimals,

exactly as are the elements which our differential calculus recognizes in the ordinates of

curves for momentary increments and decrements.6

The central insight here is that, although these quantities approach zero, they still maintain a

nonzero relationship with respect to one another even as they vanish. Or in other words, even if

𝑑𝑥 and 𝑑𝑦 both themselves = 0 at the terminal point of the continuous movement of the line, the

ratio 𝑑𝑦

𝑑𝑥 is not therefore nothing; rather, paradoxically, we obtain a

0

0 which expresses a specific

algebraic relationship: the function of the tangent of a curve at a given point. More precisely, we

can say that at this limit the gradient of secant of the curve and that of its tangent at a single point

become identical.

However, some serious issues trouble this method. First, in practically working through

our calculations, we must at first and for the most part treat 𝑑𝑥 and 𝑑𝑦 as if they were not equal to

zero, for if we immediately eliminate or suppress them we end up with nothing at all. If in step 3

above when we substituted 𝑑𝑥 + 𝑥1 for 𝑥2, we took 𝑑𝑥 to be functionally equivalent to zero, we

would get:

8. 𝑑𝑦 = (0 + 𝑥1)2 − 2(0 + 𝑥1)−𝑥12 + 2𝑥1

9. 𝑑𝑦 = 𝑥12 − 2𝑥1 − 𝑥1

2 + 2𝑥1

10. 𝑑𝑦 = 0

6 Leibniz, "Justification", 545

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This obviously doesn’t get us anywhere; as Marx emphatically notes in his mathematical

manuscripts, this “leads literally to nothing.”7 So what justifies ‘holding off’ on treating 𝑑𝑥 and

𝑑𝑦 as zero until the very end, in step 7 above? That the derivation ‘works’, in that our new function

𝑓′(𝑥) = 2𝑥 − 2 in fact yields the slope of the line tangent to the curve described by 𝑓(𝑥) = 𝑥2 −

2𝑥 + 3, is in no way a proof that the method of derivation is accurate or scientific. Marx refers to

this Leibnizian approach as the ‘mystical’ variant of differential calculus. Following Hegel, he

argues that its procedures, and in particular the suppression of 𝑑𝑥 and 𝑑𝑦 at the very end, were

justified not by rigorous mathematics but precisely because the experimental results were known

in advance.8

Leibniz was well acquainted with these charges, and struggled to demonstrate the

scientificity of this procedure.9 As Deleuze and Guattari write in A Thousand Plateaus, “for a long

time, [differential calculus] had only parascientific status and was labeled a ‘Gothic hypothesis’;

royal science only accorded it the value of a convenient convention or a well-founded fiction,”10

since its reliance on the ambiguous notion of the infinitesimal was considered unacceptable. As

infinitely small quantities, 𝑑𝑥 and 𝑑𝑦 are in this method sometimes treated as very small but

nonzero finite quantities (steps 1-6) and at other times treated as zero (step 7). This in fact was the

thrust of Berkeley’s influential response to Leibnizian and Newtonian calculus, The Analyst: Or a

Discourse Addressed to an Infidel Mathematician.11

7 Marx, “On the Concept of the Derived Function”, 3 8 Marx, “On the History of Differential Calculus”, 91-4; Hegel, Science of Logic, 273 9 The same can be said of Newton, whose method relied on ‘fluxions’ which were, despite his protests to

the contrary, functionally the same as infinitesimals, except that they represented infinitely small

increases rather than vanishingly small quantities. See Boyer, The History of the Calculus, 190-6 10 Deleuze and Guattari, A Thousand Plateaus, 363 11 Cf. Boyer, The History of the Calculus, 224-9

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Many of the subsequent attempts at providing a rigorous ground for differential calculus,

such as those of D’Alembert, Euler, and Lagrange, sought to eliminate the need to make use of

these infinitely small quantities, which were seen as ad hoc solutions at best, and metaphysical

absurdities compromising the rigor of mathematics at worst. However, it was not until Bolzano,

Cauchy, and Weierstrass that these fundamental issues were considered to be resolved.

Before turning to them, however, let us consider briefly Hegel’s reflection on the

differential calculus as he found it in the early nineteenth century, in the first Remark on

quantitative infinity in the Science of Logic. Hegel’s remarks are interesting in that, while he agreed

that the justifications of the infinitesimal calculus were inadequate and its methods ultimately

inconsistent, he nevertheless considered the calculus to have effected a genuine breakthrough in

the problem of the representation of the mathematical infinite. He tries to show that the true

mathematical infinite ‘erupts’, as though irrepressible, in the work of Euler and Lagrange, in spite

of their efforts to do away with the infinitesimals via limits and series. This tension was expressed

in the fact that while the methods employed in the calculus at this stage were not consistent, its

outcomes were anything but faulty:

The procedure of the infinitesimal calculus shows itself burdened with a seeming

inexactitude, namely, having increased finite magnitudes by an infinitely small quantity,

this quantity is in the subsequent operation in part retained and in part ignored. The

peculiarity of this procedure is that in spite of the admitted inexactitude, a result is obtained

which is not merely fairly close and such that the difference can be ignored, but is perfectly

exact. In the operation itself, however, which precedes the result, one cannot dispense with

the conception that a quantity is not equal to nothing, yet is so inconsiderable that it can be

left out of the account.12

12 Hegel, Science of Logic, 242

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Hegel’s distinction between ‘spurious’ and ‘true’ mathematical infinities turns on the way

in which the mathematical infinite is represented. Consider the equation: 2

7= 2.58714 … On the

right side, we have the ‘bad’ representation: an infinite series of numbers, which “exhibits the

contradiction of representing that which is a relation possessing a qualitative nature, as devoid of

relation, as a mere quantum, as an amount.”13 As compensation for this reduction of qualitative

relation to quantitative amount, the infinite series is always incomplete—the representation always

‘terminates’ with an ellipsis; in trying to represent the determinate finite quantity in this way, we

must therefore always add another number, and so the infinite is represented as an essential

incompleteness or the positing of a ‘beyond’. On the left side, we find the other side of the

contradiction: here there is no ‘beyond’, as its expression lacks nothing, but at the same time this

captures the infinite series by means of a relationship between finite quantities: “in the finite

expression which is a ratio, the negative is immanent as the reciprocal determining of the sides of

the ratio and this is an accomplished return-into-itself, a self-related unity as a negation of the

negation (both sides of the ratio are only moments), and consequently has within it the

determination of infinity.”14 Paradoxically, then, it is in fact the endless series which is the

deficient, finite expression of the infinite.

With the calculus, new modes of mathematical representation were developed, particularly

in the infinitesimals 𝑑𝑥 and 𝑑𝑦, and it here that Hegel’s real interest lies. In what he calls ‘power-

relations’, such as the function of a parabola, 𝑦2

𝑥= 𝑝, the quantitative relationship between 𝑥 and

𝑦 is not constant but variable; he argues that this indicates that the relationship of a magnitude to

a power is in fact fundamentally qualitative. Moreover, he suggests that this should have prompted

13 Hegel, Science of Logic, 247 14 Hegel, Science of Logic, 248

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the development of distinct symbols expressing the difference between a ‘variable’ understood as

a genuinely variable magnitude, on the one hand, and one understood as an unknown quantity

which is completely determined on the other (we will return to this).15 But whatever the nature of

this relationship, 𝑥 and 𝑦 here still signify quanta, or definite magnitudes. This is not the case in

the equations of the calculus:

𝐷𝑥, 𝑑𝑦 are no longer quanta, nor are they supposed to signify quanta; it is solely in their

relation to each other that they have any meaning, a meaning merely as moments. They are

no longer something (something taken as a quantum), not finite differences; but neither are

they nothing; not empty nullities. Apart from their relation they are pure nullities, but they

are intended to be taken only as moments of the relation, as determinations of the

differential co-efficient 𝑑𝑦

𝑑𝑥 .

In this concept of the infinite, the quantum is genuinely completed into a qualitative reality;

it is posited as actually infinite; it is sublated not merely as this or that quantum but as

quantum generally.16

In other words, differentials really do not express finite magnitudes; they are completely

undetermined in themselves, but reciprocally determining in relation to one another. Hegel argues

that the vanishing of the magnitudes 𝑑𝑥 and 𝑑𝑦, when they stand in this reciprocally determining

relationship, testifies to “the qualitative nature of what is quantitative, of a moment of a ratio as

such;”17 and that what remains is, as with Leibniz, not absolutely nothing but indeed “their

quantitative relation solely as qualitatively determined.”18

Hegel thus reproaches the mathematicians, not for their making use of mathematical

concepts of the infinitely small, but for their failing to draw out the radical conclusions of this

sense of the mathematical infinite; in their work, “the genuine Notion of the infinite is, in fact,

15 Hegel, Science of Logic, 252-3 16 Hegel, Science of Logic, 253 17 Hegel, Science of Logic, 267 18 Hegel, Science of Logic, 269

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implied in them, but […] the specific nature of that Notion has not been brought to notice and

grasped.”19 Indeed the persistence of infinitesimals, infinite series, and approaches toward limits

remained something of a scandal, and mathematicians sought to eliminate the need to refer to them

entirely. It was not until Abraham Robinson provided rigorous but controversial foundations for

the concept of the infinitesimal with the development of the axioms of non-standard analysis in

the late twentieth century that the Leibnizian approach found itself vindicated rather than

repudiated.20 But now let us turn to the other trajectory, which culminated in Weierstrass’ fully

arithmetized approach to the problems of calculus.

III. Bolzano, Cauchy and Weierstrass: toward non-geometric foundations

In his landmark The History of the Calculus and its Conceptual Development, Carl Boyer astutely

notes that the contradictions haunting the formulations of the calculus from Leibniz and Newton

until the nineteenth century “were in the last analysis equivalent to those which Zeno had raised

over two thousand years previously and were based on questions of infinity and continuity.”21 The

apparently insoluble paradoxes of the infinitely small in the calculus stem from a confusion

between geometric and arithmetic concepts, which do not have the same foundation or status.

Arithmetic concepts deal with the properties of numbers, and the relations between them.

Geometric concepts, on the other hand, deal with the properties of spatial figures. It became

apparent, by the beginning of the nineteenth century, that many of the concepts essential for the

method of calculus, in particular function and limit, were based on geometric intuition. As we have

19 Hegel, Science of Logic, 260 20 Duffy, “The mathematics of Deleuze’s differential logic and metaphysics”, 125 21 Boyer, The History of the Calculus, 267

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seen, such intuitions are metaphysically suspect insofar as they seem necessarily to involve

speculation as to whether or not continuous spatial magnitudes, such as any given section of a

curve, are infinitely divisible.22

Another aspect of the problem, however, was quite practical rather than metaphysical, and

lay in the fact that geometric intuition proved to be frequently unreliable when it came to problems

of differentiation. As Dan Smith writes, ‘geometric intuition’ refers not to empirical perception

but rather to “the ideal geometrical notion of continuous movement and space.”23 The trouble

began with Bernard Bolzano's construction, in 1837, of a function which, although continuous

everywhere, could not be differentiated at any point; this is to say that he proved that a function

might have no tangents at all even if it is entirely continuous—a claim unlikely to be intuitively

accepted, and which seems impossible to imagine geometrically or visually.24 Even more

strikingly, it was later shown that there are continuous curves defined by motion through space

that have no tangents, which again is clearly beyond geometric intuition.25 Hence in order to avoid

the metaphysically problematic consequences that otherwise threatened to condemn the calculus

to the status of mere unscientific conjecture, and in order to ground rigorously a method which

was not liable to be misled in being guided by a geometric intuition which had proven itself

unreliable, mathematicians by the early nineteenth century sought to provide arithmetic, non-

22 To provide another example of the kind of metaphysical speculation into which geometrical intuitions

can lead: when Descartes, in his Principles of Philosophy, argues for the infinite divisibility of matter, or

against the possibility of indivisible atoms, he does so on the basis of a geometric intuition of the nature

of extension, rather than on the basis of numerical properties of matter: "it is impossible that there should

be atoms, that is, pieces of matter that are by their nature indivisible. For if there were any atoms, then no

matter how small we imagined them to be, they would necessarily have to be extended; and hence we

could in our thought divide each of them into two or more smaller parts, and hence recognize their

divisibility." (Descartes, Principles of Philosophy, II §20) 23 Smith, “Axiomatics and Problematics”, 153 24 Boyer, The History of the Calculus, 269-70 25 Cf. Neikirk, “A Class of Continuous Curves”

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geometrical foundations and definitions for all the concepts involved in the methods of

differentiation and integration.

Let us take the concept of limit. In its geometrical articulation, which we saw above in

Leibniz, a limit is intuitively intelligible as a kind of threshold point at which differences tend to

vanish. A classical example of the geometric concept of limit is that of a polygon inscribed within

a circle: the more sides the polygon has, the closer it seems to come to being identical with the

circle in which it is inscribed. At the limit, if it had infinitely many sides, would the polygon

become actually identical with the circle? Boyer cites a late nineteenth century mathematician who

advances such a geometric conception: “whether one calls the circle the limit of a polygon as the

sides are indefinitely decreased, or whether one looks upon it as a polygon with an infinite number

of infinitesimal sides, is immaterial, inasmuch as in either case in the end 'the specific difference'

between the polygon and the circle is destroyed.”26 It is clear how this notion of the dynamic

disappearance of differences was central to the infinitesimal calculus: the real differences between

(𝑥1, 𝑦1) and (𝑥2, 𝑦2) ‘at the limit’, where the distance between them become infinitely small, are

taken to actually vanish, and we obtain the derivative by their elimination.

An arithmetic definition of the concept of limit was proposed by Augustin-Louis Cauchy

in the early nineteenth century: “When the successive values attributed to a variable approach

indefinitely a fixed value so as to end by differing from it by as little as one wishes, this last is

called the limit of all the others.”27 As Boyer writes, “Cauchy's definition appealed to the notions

of number, variable, and function, rather than to intuitions of geometry and dynamics.”28 That the

26 Boyer, The History of the Calculus, 272 (citing Giulio Vivanti, "Il concetto d'infinitesimo e la sua

applicazione alla matematica." Mantova, 1894.) 27 Cauchy, Œuvres (2), III, 19 (cited in Boyer, The History of the Calculus, 272; his translation) 28 Boyer, The History of the Calculus, 273

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variable can differ from the limit ‘as little as one wishes’ suggests that it is always possible to

determine a finite, nonzero quantity which is ‘closer to’ the limit than any other given quantity.

We might say that Cauchy reversed the order of definition: rather than defining the limit as that

into which infinitesimal differences vanish, he defined it in terms of the indefinite diminution of a

variable without ever vanishing; and on the basis of this new arithmetic definition of limit, he

redefined the infinitesimal arithmetically as a variable quantity whose indefinite diminution

converges toward zero.29

With this arithmetic conception of limit, Cauchy was able to articulate a non-geometric

method for differentiation. Still, Cauchy's formulations—‘indefinite approach’, ‘as little as one

likes’, and so on—risk suggesting precisely the geometric intuitions, the dynamics of infinitely

vanishing movements across continuous distances, the divisibility of spatial magnitudes, which it

was precisely his aim to avoid. Criticizing these formulations, Alfred North Whitehead wrote: “as

long as we retain anything like ‘ℎ tending to 𝑎,’ as a fundamental idea, we are really in the clutches

of the infinitely small; for we imply the notion of ℎ being infinitely near to 𝑎. This is just what we

want to get rid of.”30 It was Karl Weierstrass whose formulations in the middle of the nineteenth

century finally removed all remnants of geometric intuition and dynamism from the foundations

of the calculus.

29 Boyer, The History of the Calculus, 273-84 30 Whitehead, An Introduction to Mathematics, 229

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Weierstrass developed what is often referred to today as the ‘epsilon-delta’ definition of

the limit, whose elements are depicted in the following diagram.31 Boyer presents this definition

clearly: “The number 𝐿 is the limit of the function 𝑓(𝑥) for 𝑥 = 𝑥0 if, given any arbitrarily small

number 𝜖 [epsilon], another number 𝛿 [delta] can be found such that for all values of 𝑥 differing

from 𝑥0 [𝑎 in the diagram] by less than 𝛿, the value of 𝑓(𝑥) will differ from 𝐿 by less than 𝜖.”32

While this is basically a reiteration of Cauchy's definition of limit, it notably no longer has any

reference to the apparent dynamism of variable values ‘approaching’ it. This in fact involves a

redefinition of the concept of variable, which “does not represent a progressive passage through

all the values of an interval, but the disjunctive assumption of any one of the values in the

interval.”33 That is, there is no longer anything in the definition that might suggest that either 𝑥 or

𝑓(𝑥) are mobile; they do not move at all, but rather simply stand in for one value within a fixed

31 Diagram by Kouba, Duane. “Precise Limits of Functions as x Approaches a Constant.” UC Davis

Mathematics website.

<https://www.math.ucdavis.edu/~kouba/CalcOneDIRECTORY/preciselimdirectory/PreciseLimit.html> 32 Boyer, The History of the Calculus, 287 33 Boyer, The History of the Calculus, 288

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range of values. This, it is worth recalling, is precisely the distinction that Hegel suggested we

make between variable magnitudes and unknown but completely determined quanta; on

Weierstrass’ account, however, only the latter are legitimate.

For example, if we consider the difference between 𝑓(𝑥) where 𝑥 = 1 and again where

𝑥 = 3, we are quite clearly considering two different cases which are perfectly static, where the

function yields two distinct outputs; nothing about this difference indicates that somehow 𝑥 has

‘moved’ to 3 from 1, or that 𝑓(1) has ‘moved’ to 𝑓(3). We need not make any recourse to infinitely

small or vanishing quantities to establish the limit 𝐿 of a function 𝑓(𝑥) where 𝑥 = 𝑎. It suffices to

show that, for any given value between 𝑥 = 𝑎 − 𝛿 and 𝑥 = 𝑎 + 𝛿, some finite value 𝑥1 can always

be determined such that 𝑓(𝑥1) quantitatively differs from 𝑓(𝑎) less than both 𝑓(𝑎 − 𝛿) and 𝑓(𝑎 +

𝛿).

With this clarified concept of limit, we can now turn back to Cauchy’s method for finding

the derivative of a function, which was essentially the same as that given earlier by Bolzano,34

without fear of sliding back into conceiving of ‘approaching a limit’ as the continuous movement

of a variable. For a given function 𝑓(𝑥) = 𝑦, we want to know the rate at which 𝑦 varies relative

to the rate at which 𝑥 does, 𝛥𝑦

𝛥𝑥, at a given point, 𝑥. Let ∆𝑥 = 𝑖, that is, an incremental difference in

magnitude added to 𝑥.

𝛥𝑦

𝛥𝑥=

𝑓(𝑥 + 𝑖) − 𝑓(𝑥)

(𝑥 + 𝑖) − 𝑥

Simplifying the denominator, we obtain the general form of the derivative of a function as a ratio:

𝛥𝑦

𝛥𝑥=

𝑓(𝑥 + 𝑖) − 𝑓(𝑥)

𝑖

34 Boyer, The History of the Calculus, 275-9

—16—

Now, we establish the limit of this ratio as 𝑖 converges on zero:

𝑙𝑖𝑚𝑖→0 𝑓(𝑥 + 𝑖) − 𝑓(𝑥)

𝑖

The solution to which establishes the rate of change of 𝑥 relative to the rate of change of 𝑦, or the

gradient of the tangent to the curve described by 𝑓(𝑥) at a given point. This arithmetic definition

of the derivative of a function is entirely non-geometric, relying only on the concepts of function,

as a particular relationship between sets of ordered pairs; and limit, as defined above. Everything

is totally static: questions of geometric divisibility, of the potential identity of indiscernibles, and

of the status of infinitely small magnitudes do not arise.

As we have said, the insight in this reformulation of the method for finding the derivative

is that it does not rely on an understanding of the variables 𝑥, 𝑥0, 𝑖 and so on as traversing the

distance of an ideally continuous space. Rather, they represent fixed and discrete magnitudes,

which are in the last analysis each completely determined and static.

IV. Concluding remarks: on scientific images of thought

While the approach championed by Cauchy and Weierstrass has been generally accepted as

successfully having provided at long last a rigorous foundation for the calculus, a few final

comments are in order. First, not only did proceeding via infinitesimals persist as a practical

approach for deriving functions long after Weierstrass’ formulation, but also, as we mentioned,

these methods were eventually given a rigorous formulation in the work of Abraham Robinson,

who wrote in 1966: “Even now, there are many classical results in differential geometry which

have never been established in any other way [than through the use of infinitesimals], the

—17—

assumption being that somehow the rigorous but less intuitive 𝜖, 𝛿 method would lead to the same

result.”35 The crucial philosophical questions here, as in the case of the contemporary ‘victory’ of

the Copenhagen interpretation of quantum mechanics, are: what are the conditions under which a

particular scientific interpretation gains legitimacy and acceptance? On the basis of what, and by

whom, is this legitimacy delivered? These are emphatically political questions.

It indeed appears that the program of arithmetization and discretization advanced by

Weierstrass and his students, along with the related post-Cantorian impulse to reduce all

mathematical objects to set-theoretic constructions, has become something of an orthodoxy in

contemporary mathematics and its spontaneous philosophy. It is the implicit and explicit

presuppositions of such a scientific image of thought (e.g., that “only arithmetic is rigorous”36)

which stand in need of critique. Moreover, there are socio-political and historical reasons to be

suspicious of these axiomatic analytic approaches, as for instance Nicolas Bourbaki and more

recently Alex Galloway have argued.37 Deleuze and Guattari, in A Thousand Plateaus and

elsewhere, argue for a metascientific distinction between the ‘nomad’ or ‘minor’ sciences that

develop problematics, and the ‘major’ sciences that axiomatically pursue their formalization (and

often their neutralization) under conditions determined by state interests.38 The philosopher of

mathematics Fernando Zalamea has recently made compelling arguments against the hegemony

of axiomatic set-theory, suggesting that the diverse advances in twentieth-century mathematics

can furnish conceptual frameworks adequate to articulating transitory ontologies of irreducibly

35 Robinson, Non-Standard Analysis, 83; cited in Smith, “Axiomatics and Problematics”, 155 36 Smith, “Axiomatics and Problematics”, 154 37 Cf. for example Bourbaki, “The Architecture of Mathematics”, 31; Galloway, “The Poverty of

Philosophy” 38 Cf. Deleuze and Guattari, A Thousand Plateaus, Chapter 12 and 13; see also the critique of logicism

developed in Chapter 6 of What is Philosophy? On this distinction see Smith, “Axiomatics and

Problematics” and “Mathematics and the Theory of Multiplicities”

—18—

complex metastable objects.39 All of this is simply to suggest that what is meant by ‘rigor’ is not

always obvious; and that philosophical histories of science, if they are to avoid relapsing into

rehearsals of state-sanctioned ideologies of inimitable progress, require a sensitivity to the political

stakes of how rigor and legitimacy are conceived and established—that is, they must take seriously

the historical character of scientific practices. As Althusser said of the spontaneous philosophy of

scientists, which is to say their spontaneous ideology: not only is it inseparable from scientific

practice, but “it is ‘spontaneous’ because it is not.”40

39 Cf. Zalamea, Synthetic Philosophy of Contemporary Mathematics, especially Part Three 40 Althusser, “Philosophy and the Spontaneous Philosophy of the Scientists”, 88

—19—

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