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The Analyst: A Discourse Addressed to an Infidel Mathematician: Wherein It Is Examined Whether the Object, Principles, and Inferences of the Modern Analysis Are More Distinctly Conceived, or More Evidently Deduced, Than Religious Mysteries and Points of Faith, is a book by George Berkeley. It was first published in 1734, first by J. Tonson (London), then by S. Fuller (Dublin). The "infidel mathematician" is believed to have been Edmond Halley, though others have speculated that Isaac Newton was intended.[1]

The book contains a direct attack on the foundations of calculus, specifically on Isaac Newton's notion of fluxions and on Leibniz's notion of infinitesimal change.

Background and purpose

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From his earliest days as a writer, Berkeley had taken up his satirical pen to attack what were then called 'free-thinkers' (secularists, sceptics, agnostics, atheists, etc.—in short, anyone who doubted the truths of received Christian religion or called for a diminution of religion in public life).[2] In 1732, in the latest installment in this effort, Berkeley published his Alciphron, a series of dialogues directed at different types of 'free-thinkers'. One of the archetypes Berkeley addressed was the secular scientist, who discarded Christian mysteries as unnecessary superstitions, and declared his confidence in the certainty of human reason and science. Against his arguments, Berkeley mounted a subtle defense of the validity and usefulness of these elements of the Christian faith.

Alciphron was widely read and caused a bit of a stir. But it was an offhand comment mocking Berkeley's arguments by the 'free-thinking' royal astronomer Sir Edmund Halley that prompted Berkeley to pick up his pen again and try a new tack. The result was The Analyst, conceived as a satire attacking the foundations of mathematics with the same vigour and style as 'free-thinkers' routinely attacked religious truths.

Berkeley sought to take apart the then foundations of calculus, claimed to uncover numerous gaps in proof, attacked the use of infinitesimals, the diagonal of the unit square, the very existence of numbers, etc. The general point was not so much to mock mathematics or mathematicians, but rather to show that mathematicians, like the Christians they criticized, relied upon unknowable mysteries in the foundations of their reasoning. Moreover, the existence of these "superstitions" was not fatal to mathematical reasoning; indeed, it was an aid. So too with the Christian faithful and their mysteries. Berkeley concluded that the certainty of mathematics is no greater than the certainty of religion.

Content

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The Analyst was a direct attack on the foundations of calculus, specifically on Newton's notion of fluxions and on Leibniz's notion of infinitesimal change. In section 16, Berkeley criticises

...the fallacious way of proceeding to a certain Point on the Supposition of an Increment, and then at once shifting your Supposition to that of no Increment . . . Since if this second Supposition had been made before the common Division by o, all had vanished at once, and you must have got nothing by your Supposition. Whereas by this Artifice of first dividing, and then changing your Supposition, you retain 1 and nxn-1. But, notwithstanding all this address to cover it, the fallacy is still the same.[3]

It is a frequently quoted passage, particularly when he wrote:[4][5]

And what are these Fluxions? The Velocities of evanescent Increments? And what are these same evanescent Increments? They are neither finite Quantities nor Quantities infinitely small, nor yet nothing. May we not call them the ghosts of departed quantities?[6]

Berkeley did not dispute the results of calculus; he acknowledged the results were true. The thrust of his criticism was that Calculus was not more logically rigorous than religion. He instead questioned whether mathematicians "submit to Authority, take things upon Trust"[7] just as followers of religious tenets did. According to Burton, Berkeley introduced an ingenious theory of compensating errors that were meant to explain the correctness of the results of calculus. Berkeley contended that the practitioners of calculus introduced several errors which cancelled, leaving the correct answer. In his own words, "by virtue of a twofold mistake you arrive, though not at science, yet truth."[8]

Analysis

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The idea that Newton was the intended recipient of the discourse is put into doubt by a passage that appears toward the end of the book: "Query 58: Whether it be really an effect of Thinking, that the same Men admire the great author for his Fluxions, and deride him for his Religion?" [9]

Here Berkeley ridicules those who celebrate Newton (the inventor of "fluxions", roughly equivalent to the differentials of later versions of the differential calculus) as a genius while deriding his well-known religiosity. Since Berkeley is here explicitly calling attention to Newton's religious faith, that seems to indicate he did not mean his readers to identify the "infidel (i.e., lacking faith) mathematician" with Newton.

Mathematics historian Judith Grabiner comments, "Berkeley's criticisms of the rigor of the calculus were witty, unkind, and — with respect to the mathematical practices he was criticizing — essentially correct".[10] While his critiques of the mathematical practices were sound, his essay has been criticised on logical and philosophical grounds.

For example, David Sherry argues that Berkeley's criticism of infinitesimal calculus consists of a logical criticism and a metaphysical criticism. The logical criticism is that of a fallacia suppositionis, which means gaining points in an argument by means of one assumption and, while keeping those points, concluding the argument with a contradictory assumption. The metaphysical criticism is a challenge to the existence itself of concepts such as fluxions, moments, and infinitesimals, and is rooted in Berkeley's empiricist philosophy which tolerates no expression without a referent.[11] Andersen (2011) showed that Berkeley's doctrine of the compensation of errors contains a logical circularity. Namely, Berkeley's determination of the derivative of the quadratic function relies on Apollonius's determination of the tangent of the parabola.[12]

Influence

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Two years after this publication, Thomas Bayes published anonymously "An Introduction to the Doctrine of Fluxions, and a Defence of the Mathematicians Against the Objections of the Author of the Analyst" (1736), in which he defended the logical foundation of Isaac Newton's calculus against the criticism outlined in The Analyst. Colin Maclaurin's two-volume Treatise of Fluxions published in 1742 also began as a response to Berkeley attacks, intended to show that Newton's calculus was rigorous by reducing it to the methods of Greek geometry.[10]

Despite these attempts, calculus continued to be developed using non-rigorous methods until around 1830 when Augustin Cauchy, and later Bernhard Riemann and Karl Weierstrass, redefined the derivative and integral using a rigorous definition of the concept of limit. The idea of using limits as a foundation for calculus had been suggested by d'Alembert, but d'Alembert's definition was not rigorous by modern standards.[13] The concept of limits had already appeared in the work of Newton,[14] but was not stated with sufficient clarity to hold up to the criticism of Berkeley.[15]

In 1966, Abraham Robinson introduced Non-standard Analysis, which provided a rigorous foundation for working with infinitely small quantities. This provided another way of putting calculus on a mathematically rigorous foundation, the way it was done before the (ε, δ)-definition of limit had been fully developed.

Ghosts of departed quantities

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Towards the end of The Analyst, Berkeley addresses possible justifications for the foundations of calculus that mathematicians may put forward. In response to the idea fluxions could be defined using ultimate ratios of vanishing quantities,[16] Berkeley wrote:

It must, indeed, be acknowledged, that [Newton] used Fluxions, like the Scaffold of a building, as things to be laid aside or got rid of, as soon as finite Lines were found proportional to them. But then these finite Exponents are found by the help of Fluxions. Whatever therefore is got by such Exponents and Proportions is to be ascribed to Fluxions: which must therefore be previously understood. And what are these Fluxions? The Velocities of evanescent Increments? And what are these same evanescent Increments? They are neither finite Quantities nor Quantities infinitely small, nor yet nothing. May we not call them the Ghosts of departed Quantities?[6]

Edwards describes this as the most memorable point of the book.[15] Katz and Sherry argue that the expression was intended to address both infinitesimals and Newton's theory of fluxions.[17]

Today the phrase "ghosts of departed quantities" is also used when discussing Berkeley's attacks on other possible foundations of Calculus. In particular it is used when discussing infinitesimals,[18] but it is also used when discussing differentials,[19] and adequality.[20]

Text and commentary

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The full text of The Analyst can be read on Wikisource, as well as on David R. Wilkins' website,[21] which includes some commentary and links to responses by Berkeley's contemporaries.

The Analyst is also reproduced, with commentary, in recent works:

  • William Ewald's From Kant to Hilbert: A Source Book in the Foundations of Mathematics.[22]

Ewald concludes that Berkeley's objections to the calculus of his day were mostly well taken at the time.

  • D. M. Jesseph's overview in the 2005 "Landmark Writings in Western Mathematics".[23]

References

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Sources

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The Analyst

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The Analyst; or, A Discourse Addressed to an Infidel Mathematician is a 1734 philosophical and mathematical treatise by , the Anglo-Irish bishop and philosopher, in which he subjects the emerging methods of —known as fluxions in the Newtonian tradition and differentials in the Leibnizian—to a rigorous critique of their foundational principles and logical inferences. Berkeley contends that concepts such as infinitesimals and moments represent "the ghosts of departed quantities," lacking clear ontological status or precise definition, thereby rendering the 's applications more akin to assumption than demonstration. Addressed to an unnamed "infidel mathematician" presumed to be a skeptic like , the work draws a pointed between the perceived mysteries in and those dismissed by scientists in , urging greater humility and evidential standards in both domains. The treatise ignited the Analyst controversy, prompting defenses from figures such as James Jurin and , who sought to clarify the heuristic nature of fluxional methods while acknowledging Berkeley's challenge to articulate limits and continuity without circularity. Though Berkeley's immaterialist metaphysics—positing that reality consists in perceived ideas—underpinned his rejection of abstract mathematical entities independent of mind, his arguments exposed genuine foundational vulnerabilities in early , influencing later rigorization by analysts like and through epsilon-delta definitions. Published amid Berkeley's broader campaign against and , The Analyst exemplifies his strategy of leveraging empirical and logical scrutiny to defend orthodox Christianity, highlighting inconsistencies in the era's scientific establishment without denying the calculus's practical utility in physics and .

Historical and Intellectual Context

Emergence of Calculus in the Early 18th Century

Isaac Newton conceived the in the mid-1660s, during his of 1665–1666, treating mathematical quantities as "fluents" generated by continuous motion and their "fluxions" as instantaneous rates of change, denoted by a superimposed dot (e.g., x˙\dot{x}). He formalized this in the unpublished tract De methodis serierum et fluxionum, completed in 1671, which defined fluxions through limits of ratios of finite increments as those increments approached zero via the notation "o" for quantities of higher order vanishing faster. Although the tract remained unpublished until 1736, Newton incorporated fluxional ideas into (1687), using ultimate ratios in lemmas to derive results on motion and curves geometrically rather than algebraically. Independently, developed a parallel framework in the 1670s, publishing the first account of in October 1684 as "Nova methodus pro maximis et minimis, itemque tangentibus (pro quadratura, etc.)" in Acta Eruditorum. Leibniz introduced differentials dxdx and dydy as infinitesimally small increments, enabling characteristic triangles for tangents and rules for maxima/minima via algebraic expansion and neglect of higher-order terms like (dx)2(dx)^2. This notation emphasized static geometric and algebraic computation over Newton's dynamic, time-based fluxions, though both approaches derived , such as for products (d(xy)=x dy+y dxd(xy) = x\,dy + y\,dx), without deriving them from axioms. Newtonian and Leibnizian methods diverged notationally—dots versus dd—and conceptually, with Newton prioritizing limits to evade true infinitesimals and Leibniz embracing them as nonzero yet negligible quantities, yet both harbored ambiguities in treating vanishing increments as effectively zero in proofs while retaining finite effects. Core operations, including integration as inverse fluxions or summing infinitesimals, lacked demonstrative foundations beyond empirical success in problems like or curve quadrature. By the early 1730s, permeated mathematical practice: in Britain, fluxions informed treatises on and , as seen in works by followers like ; on the Continent, differentials fueled advances in by the Bernoulli brothers and Euler, solving transcendental equations and variational problems. Despite priority disputes erupting in 1711, the techniques proliferated for physical applications, such as celestial trajectories, underscoring their utility even amid unresolved logical gaps in handling "inassignable" quantities.

Berkeley's Philosophical and Mathematical Preparation

was born on 12 March 1685 near Thomastown in , . He received his early education at Kilkenny College before entering , in March 1700, where he earned a B.A. in 1704 and an M.A. in 1707, remaining associated with the institution as a fellow until 1724. During this period, Berkeley developed his philosophical framework through empiricist lenses, publishing An Essay Towards a New Theory of Vision in 1709, which analyzed as dependent on sensory experience rather than direct apprehension of external objects. In 1710, Berkeley articulated his doctrine of in A Treatise Concerning the Principles of Human Knowledge, positing that objects exist only insofar as they are (esse est percipi), rejecting material substance as an unnecessary abstraction and grounding reality in the mind's ideas sustained by divine . This empiricist idealism, emphasizing sensory data over innate or abstract entities, informed his later critiques of scientific concepts deemed ungrounded in phenomena. Berkeley's mathematical engagement was secondary to but evident in his familiarity with Newtonian methods, gained through travels in England and interactions with figures like Edmund Halley, who championed Newton's synthesis of and physics; Halley's influence exposed Berkeley to the fluxional techniques central to Newtonian analysis. Berkeley's 1721 Latin treatise De Motu (On Motion) demonstrated his mathematical preparation by questioning Newtonian notions of absolute motion, space, and time, as well as abstract "forces" like gravitation, which he viewed as metaphysical fictions unsupported by of relative motions alone. By 1734, having been appointed of Cloyne in and consecrated on 19 May, Berkeley resided in Ireland, where his role amplified his commitment to rational defenses against intellectual trends he saw as fostering , including unchecked reliance on mathematical innovations lacking demonstrative foundations.

Publication Details and Stated Aims

Circumstances of Publication in 1734

The Analyst was published anonymously in in 1734 by the bookseller Tonson in the form of a . A edition appeared the same year, printed by , indicating swift regional dissemination. The pamphlet circulated rapidly within mathematical and intellectual networks across Britain and beyond. Copies reached key figures such as and mathematician James Jurin, who penned a titled Geometry No Friend to Infidelity later that year under the pseudonym Philalethes Cantabrigiensis. Berkeley also shared the work with associates like , his American correspondent and future Yale president, facilitating transatlantic distribution amid his own ecclesiastical appointments in Ireland. This prompt spread, evidenced by early responses in print, underscored the tract's immediate impact on contemporary debates in .

Explicit Purpose: Addressing Infidel Mathematicians

The subtitle of Berkeley's 1734 , The Analyst: Or, a Discourse Addressed to an , frames the work as a pointed rhetorical confrontation with mathematicians who profess regarding divine truths while endorsing the principles of contemporary without sufficient justification. This dedication targets figures presumed to wield authority in rational inquiry, yet who extend their judgments presumptuously into religious matters, thereby revealing a selective application of critical standards. Berkeley explicitly states his objective in the initial sections to subject the object, principles, and methods of modern mathematicians to the same unfettered examination they employ against theological doctrines, aiming to expose their logical inconsistency. He asserts the need to "inquire into the Object, Principles, and Method of Demonstration admitted by the Mathematicians of the present Age, with the same freedom that you presume to treat the Principles and Mysteries of ," thereby claiming of free-thinking for his own critique. This approach underscores his intent to demonstrate that reliance on unproven mathematical hypotheses constitutes a parallel form of unexamined assent, akin to the credulity skeptics attribute to , and calls for mathematicians to adopt uniform rigor across domains. Through this linkage, Berkeley positions The Analyst as an instrument for upholding demonstrative reasoning from evident first principles—clear definitions, undeniable axioms, and methodical deduction—rather than mere opposition to analytical tools. His thus challenges the undue deference accorded to mathematical authority, particularly when it fosters , by insisting that true reason demands evidentiary foundations in all speculative pursuits, thereby defending intellectual and theological integrity against partial scrutiny.

Summary of Key Arguments

Fundamental Objections to Fluxions and Infinitesimals

In The Analyst, argued that the foundational concepts of fluxions, as introduced by to denote instantaneous rates of change or velocities of variable quantities, lacked precise definition, being described as neither finite quantities, nor infinitesimally small, nor nothing. This ambiguity extended to Leibniz's infinitesimals, which Berkeley viewed as similarly evasive, representing differences or increments that defied clear categorization and thus undermined the logical rigor of derivations. He contended that such terms, when employed without rigorous explication, introduced obscurity into mathematical reasoning, as the mind encountered "much Emptiness, Darkness, and Confusion" upon analysis. Berkeley further criticized the treatment of vanishing or evanescent quantities in limit processes, asserting that proofs assuming increments to approach zero while retaining proportions or consequences derived from their non-zero state committed logical inconsistencies. Specifically, he highlighted the error in destroying the initial supposition of nascent increments and yet preserving its effects, which rendered the method circular, as principles appeared to be validated by the conclusions they were meant to support, contrary to demonstrative logic. This approach, in his view, bypassed the need for clear premises, substituting technical rules for genuine proof. Despite these flaws, Berkeley acknowledged the practical efficacy of fluxional and methods in obtaining accurate results for problems like areas and maxima, attributing this not to sound foundations but to a "twofold mistake" where errors in assumption and retention canceled each other, yielding truth without achieving scientific demonstration. He emphasized that such compensatory mechanisms did not elevate the to the status of demonstrative knowledge, as it rested on unexamined hypotheses rather than evident principles.

The Concept of "Ghosts of Departed Quantities"

In The Analyst, employs the metaphor of "ghosts of departed quantities" to denote the evanescent increments central to Newtonian fluxions, portraying them as entities that momentarily influence computations—such as determining tangents or accumulating areas—before instantaneously vanishing without trace. These increments, described as the vanishing differences between successive positions of a fluent, are essential to deriving velocities or fluxions, yet Berkeley contends they lack substantive existence, operating instead as illusory remnants that mock the precision of Euclidean demonstration by effecting results akin to or multiplication by nothingness. The of lies in the temporal inconsistency of these quantities: they must "nasce" or emerge to generate finite effects, perform arithmetic operations like subdivision of curves into areas, and then "evanesce" into nullity, raising the question of when precisely they exert causal influence if no stable instant of being persists. Berkeley specifies that such increments "are neither finite Quantities, nor Quantities infinitely small, nor yet nothing," thereby evading categorization while underpinning procedures that presume their operational reality. This critique targets the foundational ambiguity in fluxional methods, where the vanishing process is invoked to resolve limits but introduces metaphysical obscurity incompatible with rigorous proof. Berkeley's thirty-fifth query intensifies this by interrogating how "infinitesimally small" quantities can divide finite ones—as in computing ordinates or fluxions—without clear ontological status: if existent, they contradict infinitesimal smallness by being assignable magnitudes; if nonexistent, they render division undefined. He demands demonstration of their actuality versus potentiality, aligning with Aristotelian strictures against void or actual infinity, which prohibit treating non-quantifiable "nothings" as divisors in geometric reasoning. This insistence underscores Berkeley's view that unproven assumptions about such ghosts erode mathematical certainty, reducing analysis to unsubstantiated supposition rather than evident .

Analysis of Specific Calculus Operations

In The Analyst, Berkeley dissected the Newtonian derivation of the fluxion of a product, such as xyxy, where the sides xx and yy are taken as flowing quantities with momentaneous increments oxox and oyoy. The method posits the augmented rectangle as (x+ox)(y+oy)=xy+x oy+y ox+ox oy(x + ox)(y + oy) = xy + x\,oy + y\,ox + ox\,oy, yielding the fluxion x˙y+xy˙\dot{x}y + x\dot{y} upon dividing by the infinitesimal time increment oo and letting increments vanish, while neglecting the ox oyox\,oy term as o2o^2. Berkeley contended this neglect is inconsistent, as the increments are initially supposed finite before vanishing; treating oo as negligible only after using it to generate the linear terms violates strict demonstrative reasoning, akin to admitting small errors intolerable in geometry: "Nor will it avail to say that abab is a quantity exceeding small: since we are told that in rebus mathematicis errores quam minimi non sunt contemnendi." Berkeley extended this scrutiny to higher-order fluxions, deeming the second fluxion—the fluxion of a fluxion, or —as already straining comprehension, with third, fourth, or higher fluxions utterly beyond human grasp. He described fluxions as velocities of evanescent increments, such that second fluxions represent "velocities of the velocities," and subsequent orders compound this obscurity: "But the velocities of the velocities, the second, third, fourth, and fifth velocities, &c. exceed, if I mistake not, all human understanding." This critique targeted rules for fluxions of powers, like xnx^n, where repeated applications presuppose vanishing increments that destroy prior assumptions, rendering the process a sequence of contradictory suppositions without rigorous foundation. Regarding the inverse method of fluxions—integration—Berkeley questioned how summing an infinity of these evanescent increments, or "moments," yields precise finite fluents, such as areas under curves. He portrayed the increments as neither finite, infinitely small, nor , but "ghosts of departed quantities," whose nullity should preclude any substantive sum, yet compensations among neglected higher-order terms (like o2o^2) purportedly ensure exactness: "the finite ... be equal to the remainder of the increment expressed by... equal, I say, to the finite remainder of a finite increment." This, he argued, relies on erasure of errors without justification, undermining the method's claim to demonstrative certainty despite empirical successes.

Philosophical Underpinnings

Reliance on First Principles and Demonstrative Reasoning

Berkeley insisted that qualify as a demonstrative science, comparable to , which succeeds through rigorous deduction from evident axioms and postulates. In geometry, he observed, clear definitions of sensible figures enable methodical reasoning, fostering habits of exact that extend beyond the field itself. By contrast, modern analysts, in his view, rely on hypotheses involving abstract entities whose existence cannot be distinctly conceived, substituting symbols for realities and thereby undermining demonstrative validity. He demanded that true proceed from principles clearly conceived, with rules demonstrated rather than merely computed, distinguishing the genuine from a mechanical operator. Obscure principles or incorrect reasonings, Berkeley argued, warrant no indulgence in any branch of demonstration, whether algebraic, geometric, or otherwise; yet analysts tolerate inconsistencies that would invalidate arguments elsewhere. Geometry's strength lies in its focus on assignable extensions—proportions derived from perceptible magnitudes—avoiding unverifiable approximations or evanescent notions that evade precise definition. This methodological rigor, Berkeley contended, requires rejecting artifice for genuine demonstration, ensuring conclusions follow inescapably from foundational truths rather than expedient manipulations. Analysts' complacency with paradoxes, he charged, erodes the precision expected in , where even minutest errors demand correction, paralleling the intolerance for in logical . Such standards, if applied consistently, would compel a return to verifiable foundations over speculative abstractions.

Integration of Theology and Mathematics

Berkeley critiques the "infidel mathematicians" for their selective skepticism, arguing that their acceptance of obscure concepts like fluxions and infinitesimals demands a degree of faith comparable to that required for religious doctrines, thereby exposing hypocrisy in deriding Christianity while relying on unproven mathematical mysteries. He asserts that those who tolerate "second or third Fluxions, a notation which is at least as hard to conceive as Transubstantiation," should not object to theological points on grounds of incomprehensibility. This analogy serves as a tu quoque argument, equating the "blind faith" in calculus's foundational principles with religious belief to defend the rationality of faith against Enlightenment rationalism. Underlying this critique is Berkeley's immaterialist philosophy, wherein mathematical objects—such as lines, surfaces, and quantities—exist not as independent abstractions but as ideas perceived by finite minds and ultimately sustained by the infinite mind of . He posits as the continuous perceiver and author of motion, ensuring the coherence of perceived mathematical relations without reliance on substrates or absolute space. This integration implies that human mathematical inventions, lacking self-evident rigor, must defer to divine order rather than claim autonomy from theological oversight. Berkeley thus calls for , urging mathematicians to apply the same demonstrative standards to their field as they demand of , while acknowledging the limits of reason before divinely ordained truths. In his queries, he questions whether mathematical repugnancies undermine claims to superior over , reinforcing that true certainty resides in submission to God's sustaining will rather than human abstraction.

Contemporary Reception and Debates

Outbreak of the Analyst Controversy

James Jurin, a physician and former of the Royal Society, published an anonymous pamphlet in 1734 under the Philalethes Cantabrigiensis, titled Geometry No Friend to , or a Defence of the Against the Objections of the Author of The Analyst. In this work, Jurin countered Berkeley's criticisms by approximating fluxions through geometric methods involving finite line segments, arguing that such constructions justified Newtonian procedures without relying on vanishing quantities. Jurin's response, appearing shortly after The Analyst's April 1734 release, marked the initial public defense of fluxions and escalated the debate by linking mathematical rigor to broader theological implications. Berkeley replied directly to Jurin in early 1735 with A Defence of Free-Thinking in Mathematics, maintaining that defenses like Jurin's evaded the core logical flaws in fluxional reasoning and failed to achieve demonstrative certainty. This exchange prompted further interventions, including Thomas Bayes's 1736 tract An Introduction to the Doctrine of Fluxions, and a Defence of the Mathematicians Against the Objections of the Author of The Analyst, which treated infinitesimals as heuristic fictions useful for deriving correct results when interpreted evanescently. Colin Maclaurin, a prominent Newtonian at the , began formulating a response in 1734, corresponding on the matter and initiating work that culminated in his comprehensive Treatise of Fluxions (1742), aimed at demonstrating fluxions' algebraic foundations independent of infinitesimals. These early reactions from 1734 to 1736 highlighted divisions among British mathematicians, with defenders seeking to preserve fluxions' practical efficacy amid Berkeley's challenge to their foundational status.

Principal Mathematical and Philosophical Responses

James Jurin responded to Berkeley's in his 1734 pamphlet Geometry No Friend to , employing geometric constructions to demonstrate that Newtonian fluxions could approximate results akin to Euclidean demonstrations without relying on actual infinitesimals or vanishing quantities. Jurin's approach involved finite polygonal approximations that converge to curvilinear figures, thereby evading the "ghosts of departed quantities" derided, while conceding the need for methods demonstrable without indeterminate forms. Colin Maclaurin, in his 1742 Treatise of Fluxions, further addressed foundational concerns by reformulating calculus operations using Newton's concept of prime and ultimate ratios, which describe limits of ratios as quantities approach evanescence without assuming the quantities themselves exist momentarily. Maclaurin supplemented this with extensive proofs derived from ancient Greek geometric methods, such as those of Archimedes, to derive fluxional results rigorously, thereby yielding to Berkeley's demand for demonstrative reasoning while defending the method's validity through tangible, finite constructions. Philosophically, defenders countered that calculus's predictive power in empirical physics—evident in applications to and since Newton's Principia (1687)—outweighed demands for Euclidean purity, as successful outcomes verified causal over abstract foundational completeness. This pragmatic stance conceded Berkeley's point on logical gaps but prioritized causal realism in , arguing that Berkeley's immaterialist , which denied independent abstract entities, unduly biased his insistence on sensory-derived ideas for all mathematical reasoning.

Enduring Influence and Criticisms

Contributions to the Rigorization of

The publication of The Analyst in 1734 highlighted foundational ambiguities in Newtonian fluxions, eliciting defensive works that advanced efforts toward greater logical precision in , even if initial responses retained informal elements. Colin Maclaurin's A Treatise of Fluxions (), composed partly in direct rejoinder to Berkeley's objections, derived fluxional results through limits of finite ratios and akin to Archimedean methods, thereby sidestepping overt appeals to vanishing quantities while demonstrating the method's validity for practical computations. Similarly, Jean le Rond d'Alembert's 1748 Traité de dynamique and his 1754 entry on "Différentiel" reconceived differentials as limits wherein one quantity approaches another without attainment, providing a finite-based alternative to infinitesimals that addressed Berkeley's charge of obscurity. These 18th-century endeavors, though not fully eliminating intuitive appeals, exposed persistent gaps in rigor and stimulated subsequent refinements. By the early , Augustin-Louis Cauchy's Cours d'analyse de l'École Royale Polytechnique (1821) marked a decisive step, defining the via a limit process bounded by inequalities—specifically, the approaching a finite value as the increment tends to zero—thus furnishing with a demonstrative framework independent of metaphysical infinitesimals. Cauchy extended this to integrals as limits of sums, emphasizing precursors to ensure logical coherence. , in Berlin lectures from the 1850s to 1860s, formalized Cauchy's ideas through the ε-δ definition, stipulating that for every ε > 0 there exists δ > 0 such that the function difference remains below ε when the argument difference is below δ, thereby eradicating residual vagueness and establishing analysis on arithmetic foundations alone. Historians of , including Florian Cajori, have characterized The Analyst as precipitating this trajectory toward epsilon-based rigor, terming it "the most spectacular event of the century in the of British mathematics" for galvanizing scrutiny of unproven assumptions and fostering proof-centric instruction over application. While direct causation remains debated, the controversy undeniably amplified demands for demonstrative certainty, paving the way from fluxional approximations to modern .

Impact on Philosophy of Mathematics

Berkeley's The Analyst (1734) intensified philosophical debates on the ontological commitments of by exposing the reliance of on notions like infinitesimals, which he deemed metaphysically incoherent "ghosts of departed quantities." This critique challenged the prevailing assumption that mathematical infinities possess intuitive reality, thereby fueling arguments for anti-realist interpretations of that prioritize symbolic manipulation over discovery of independent abstract entities. Philosophers influenced by such skepticism, including those in formalist traditions, have cited Berkeley's analysis to advocate for viewing as a consistent game of symbols devoid of extrinsic reference, rather than a realist exploration of platonic forms. The work's insistence on clear, finitary conceptions prefigured constructivist philosophies that reject non-constructive proofs and actual infinities, as seen in L.E.J. Brouwer's , which demands mental constructions for mathematical objects. While Brouwer's framework emerged independently in the early twentieth century amid set-theoretic paradoxes, Berkeley's earlier demolition of intuitive fluxions contributed to a broader of questioning classical ' acceptance of completed infinities without evidential warrant, thereby sustaining debates on realism versus in mathematical . By juxtaposing mathematical ambiguity with demands for rigor in other domains, The Analyst highlighted epistemic tensions between and its applications, anticipating inquiries into why empirically successful theories built on dubious mathematical foundations warrant realist interpretations of unobservables. Detractors of Berkeley's argue that this overlooks mathematics' demonstrated predictive potency, as calculus-derived models have yielded precise forecasts—from planetary orbits to —suggesting foundational opacity need not undermine practical validity, a point reinforced in responses emphasizing utility over metaphysics.

Modern Interpretations and Rehabilitations

In the mid-20th century, Abraham Robinson's development of non-standard analysis provided a rigorous framework for , utilizing to formalize intuitive methods akin to those critiqued in The Analyst. Robinson's approach, first outlined in papers from 1961 and elaborated in his 1966 book Non-Standard Analysis, constructs an extension of the reals containing genuine infinitesimals and infinitely large numbers, enabling precise definitions of and integrals without the "ghosts of departed quantities" Berkeley derided. This rehabilitation demonstrates that Berkeley's logical objections to infinitesimal reasoning could be resolved through model-theoretic semantics, vindicating the power of early while confirming the need for foundational clarification he demanded. Twentieth- and twenty-first-century historians of mathematics have increasingly credited Berkeley with exposing substantive pre-rigor flaws in fluxional calculus, particularly his of "compensating errors," where approximations like neglecting higher-order terms in series expansions coincidentally balance out to yield accurate results. Otto Becker, in early 20th-century German , highlighted Berkeley's role in anticipating demands for deductive rigor, influencing later foundational crises. More recent examinations, such as Kirsti Andersen's study in Historia Mathematica, dissect Berkeley's specific arguments on error compensation in Leibnizian differentials, affirming their insight into why intuitive methods succeeded despite logical gaps, though not always endorsing his dismissal of the entire enterprise. These reassessments portray Berkeley as prescient in privileging proof over empirical success, spurring the 19th-century arithmetization of by Cauchy and Weierstrass, albeit without overlapping contemporaneous debates. Counterinterpretations emphasize that Berkeley's critiques often hinged on semantic misunderstandings of fluxions as literal quantities rather than provisional symbols, undervaluing their utility in discovery before rigorization. Scholars argue his insistence on absolute certainty overlooked the provisional nature of early as a tool for generating verifiable results, where compensating errors functioned as an unrecognized rather than mere . Furthermore, Berkeley's theological framing—positing mathematicians' "infidel" reliance on unproven methods as akin to blind faith—has been deemed extraneous to the core mathematical issues, serving rhetorical rather than analytical purposes, thus limiting the enduring philosophical bite of his beyond its call for precision.

Original Editions and Berkeley's Follow-Ups

The Analyst was first published in in 1734 by J. Tonson, with the full title The Analyst: Or, a Discourse Addressed to an Infidel Mathematician. Wherein It Is Examined Whether the Object, Principles, and Inferences of the Modern Analysis Are More Distinctly Conceived, or More Evidently Deduced, Than Those of the Ancients. A edition appeared the same year, printed by S. Fuller, serving as an early reprint with minor variations such as differences but retaining the original content. In 1735, Berkeley published A Defence of Free-Thinking in , a direct sequel responding to James Jurin's anonymous pamphlet Geometry No Friend to (1734), which had accused Berkeley of undermining Newtonian fluxions and promoting . In this work, printed in by M. Rhames, Berkeley clarified that his critique in The Analyst targeted the lack of rigorous demonstration in fluxional methods rather than rejecting their utility, while defending the role of free inquiry in against charges of infidelity. He emphasized that true mathematical validity requires evident principles and deductive steps akin to ancient , not reliance on unproven infinitesimals or "ghosts of departed quantities." Berkeley also issued Reasons for Not Replying to Mr. Walton's Full Answer in 1735, a brief letter addressed to "P.T.P." that explained his decision to forgo further engagement with critic Thomas Walton's responses, which primarily concerned Berkeley's The Minute Philosopher but intersected with themes of demonstrative reasoning raised in The Analyst. This pamphlet, again published anonymously in , underscored Berkeley's view that exhaustive rebuttals were unnecessary when core positions on and proof remained unaddressed by opponents. These follow-ups formed Berkeley's immediate bibliographic extensions to The Analyst, maintaining focus on foundational critiques without altering the original text.

Scholarly Commentaries and Analyses

Alexander Campbell Fraser's edition of Berkeley's works, published from 1871 to 1901, incorporated The Analyst with extensive annotations that highlighted its philosophical underpinnings, portraying the critique of fluxions as an extension of Berkeley's immaterialist toward abstract entities in both and . Fraser's commentary underscored the text's aim to expose inconsistencies in Newtonian analysis, arguing that Berkeley's objections stemmed from a demand for evidential clarity akin to that required in religious discourse. Twentieth-century analyses often contrasted Berkeley's logical rigor with prevailing mathematical practices. In The Development of Logic (1962), William and Martha Kneale offered a sympathetic reading of Berkeley's arguments, interpreting his rejection of infinitesimals as a valid challenge to ambiguous quantifiers, which anticipated later developments in formal logic by emphasizing precise definitions over intuitive appeals. The Kneales contended that Berkeley's method exposed flaws in probabilistic reasoning derived from fluxions, aligning it with broader inductive critiques. Recent philosophical examinations have focused on Berkeley's treatment of "compensating errors," where apparent inaccuracies in methods cancel out. Kirsti Andersen (2011) dissected one such argument, demonstrating that Berkeley correctly identified non-compensatory discrepancies in certain Leibnizian differentials, though she noted limitations in his generalization to all fluxional computations; this analysis reframes The Analyst as a targeted logical probe rather than a wholesale rejection of utility. Scholars like those in contemporary interpret the work's as advancing a prioritizing foundational transparency over instrumental efficacy, influencing debates on mathematical without resolving tensions between proof standards and applied success.

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