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Work #1594 · Late period

The Analyst

Berkeley's 1734 'The Analyst, or a Discourse Addressed to an Infidel Mathematician' — critique of Newton's calculus and free-thinking mathematicians

George Berkeley · 1734 · English · Philosophical-mathematical critique

Tradition: Anglo-Irish idealism / philosophy of mathematics / Anglican polemic

Berkeley's 1734 'The Analyst' — 'ghosts of departed quantities' — the philosophical critique of Newton's infinitesimal calculus

Published in 1734, 'The Analyst, or a Discourse Addressed to an Infidel Mathematician' is Berkeley's philosophical-mathematical critique of Newton's calculus of fluxions and Leibniz's differential calculus, addressed to Edmond Halley ('the Infidel Mathematician' — Berkeley believed that Halley's mathematics-confident-atheism had influenced the young Joseph Addison's deathbed disbelief). Berkeley argues that the foundations of the calculus — Newton's 'fluxions' (instantaneous rates of change), Leibniz's 'differentials' (infinitesimally small differences), and the use of 'evanescent increments' that vanish at the moment of being divided — are no more rigorous than the religious mysteries free-thinking mathematicians ridicule. Berkeley's most-quoted passage: '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?' (§35). The argument is rhetorically devastating and philosophically substantial: the standard practice of the calculus in 1734 did rely on systematically setting infinitesimals first equal to a non-zero quantity (for purposes of cancelling) and then equal to zero (for purposes of arriving at a finite answer). Berkeley exposed real foundational difficulties that would not be properly resolved until the nineteenth-century arithmetisation of analysis by Cauchy, Weierstrass, and Dedekind. The book is credited with motivating that arithmetisation programme by the rigour-demand it placed on subsequent mathematical foundations.

Author

Editions cited

  • The Analyst; or, A Discourse Addressed to an Infidel Mathematician (J. Tonson, London, 1734)
  • In The Works of George Berkeley, Bishop of Cloyne, ed. A. A. Luce and T. E. Jessop (Nelson, 1948-57, 9 vols), vol. 4
  • Modern edition: D. M. Jesseph (ed.), George Berkeley: Philosophical Writings (Cambridge, 2009)
  • Commentary: Douglas M. Jesseph, Berkeley's Philosophy of Mathematics (Chicago, 1993); Carl B. Boyer, The History of the Calculus and Its Conceptual Development (Dover, 1959)
Compare this work

School Embodiments

Logicism · 30%
Idealism · 14%
Rationalism · 14%
Anglican Broad-Church · 16%
Realism · 12%
Philosophy of Science · 14%
Logicism 30%

Defining eighteenth-century philosophy-of-mathematics polemic.

"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?" (The Analyst, §35)
Idealism 14%

Berkeleian-idealist anti-abstractionist framework.

"The same abstractionist confusion plagues mathematics and metaphysics." (The Analyst, §6)
Rationalism 14%

Rationalist-critical methodology.

"The foundations of mathematics must withstand reason's examination." (The Analyst, preface)
Anglican Broad-Church 16%

Anti-free-thinking apologetic framework.

"Free-thinking mathematicians have no warrant for their religious scepticism." (The Analyst, opening)
Realism 12%

Realism — but here against pretended-rigorous mathematical objects.

"Mathematical entities must be intelligible or be rejected." (The Analyst, §38)
Philosophy of Science 14%

Philosophy of science applied to Newton's mathematical methodology.

"The methods of the calculus are no better grounded than religious mystery." (The Analyst, §50)

Internal Tensions

Exposed real foundational difficulties in the calculus — credited with motivating the nineteenth-century arithmetisation of analysis. Berkeley's argument was philosophically substantial as well as rhetorically devastating; subsequent mathematicians (Lagrange, Cauchy, Weierstrass) had to work out the rigorous foundations the calculus had originally lacked.

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I. Time

1734. Berkeley was 49 and had just returned to England from his three-year residence in Rhode Island (1729-31); he became Bishop of Cloyne later in 1734.

Attributes
Extent: Infinite Ontological Status: Emergent Grain: Continuous Freedom: Non-Deterministic Traversability: Linear Direction: Uni-directional Dimensionality: One
✦

II. Space

London publication; Berkeley's Anglo-Irish bishop-philosopher context. The Royal Society debate over the foundations of the calculus (Newtonian fluxions vs. Leibnizian differentials) was at its height in 1734.

Attributes
Extent: Infinite Ontological Status: Emergent Curvature: Flat Dimensionality: Three Locality: Non-local
◉

III. Matter

Single critical-philosophical pamphlet (~80 pages). Form is treatise-essayistic with numbered paragraphs (§§1-67) running through the argument.

Attributes
Extent: Finite Ontological Status: Emergent Conservation: Conserved Dimensionality: Three Locality: Non-local
◎

IV. Observer

Late Berkeley. The observer-philosopher is the Anglo-Irish bishop-philosopher (Berkeley would die in 1753) at the height of his anti-free-thinking apologetic work.

Attributes
Time Instance: Single Space Instance: Single Knowledge Extent: Total Knowledge Retainment: Total Physicality: Disembodied Agency: Active Number: Plural Metaphysical Agency: Personal
⚡

V. Energy

Polemical-philosophical-mathematical energies. The pamphlet's rhetorical force was substantial — it was widely read and replied to (Jurin's 'Geometry No Friend to Infidelity', 1734; Robins's 'Discourse Concerning the Nature and Certainty of Sir Isaac Newton's Methods of Fluxions', 1735) and stimulated a substantial mathematical-foundational literature.

Attributes
Extent: Finite Ontological Status: Emergent Conservation: Conserved Dispersibility: Irreversible
⧉

VI. Information

Single pamphlet of 67 numbered paragraphs. The 'ghosts of departed quantities' phrase (§35) is the most-cited passage.

Attributes
Ontological Status: Emergent Cosmic Conservation: Conserved Personal Conservation: Conserved Granularity: Continuous

Personas that cite this work

George Berkeley Sir Isaac Newton

Personas with the nearest attribute fingerprint

Historical figures whose own classification on the same six-dimensional grid lands closest to this work's. Computed by attribute-agreement on coordinates both address.

97%
George Berkeley
33/34
88%
Socrates
22/25
82%
Immanuel Kant
28/34
79%
Edmund Husserl
27/34
79%
Jonathan Edwards
27/34
79%
Isaac ben Solomon Luria (the Ari)
26/33

Computed school proximity

The work's attribute fingerprint scored against all schools using the same quiz scorer. Useful as a sanity check on the hand-curated embodiments above.

85%
Baha'i Faith
22/26
80%
Sikhism
20/26
79%
Hermeticism
20/26
79%
Christian Platonism
22/26
77%
Neo-Platonism
20/26
76%
Hinduism (Generic)
19/26
74%
Kabbalah (Lurianic)
19/26
74%
Idealism
19/26

How The Analyst resolves each dilemma

42 resolved positions across 4 dimensions, including 3 distinctive where the majority of schools go the other way · 15 unaligned.

Each dimension is sorted so minority positions come first. Mainstream positions are folded into an expandable list.

Time · 9 dilemmas, all mainstream
Do you really choose? The future is open and you are a genuine origin of it. 69% Are addicts responsible for their addiction? The addict could have chosen otherwise — that's why recovery is real. 69% Should we hold AI systems responsible for what they do? An AI without a free will is not the kind of thing that can be responsible. 69% Is the universe running out of usable energy? Time is unbounded but matter is finite; usable energy can fail without time failing. 47% Are natural resources fundamentally finite, or only practically so? Time goes on but matter is bounded; we are eventually constrained even with infinite time. 47% Could we owe future generations more than is materially possible to provide? Time is unbounded but matter is not; we can owe more across long time than the matter can provide. 47% How much weight do future people deserve? Time arises from events or from a deeper substrate — the future is not yet. 32% Is regret rational? The past is not a thing now; regret is the present holding what is no longer. 32% Do we owe extinct species something we cannot give them? Past species no longer exist; what we owe is to the present and the future. 32%

Matter · 7 dilemmas · 3 distinctive

What stuff is — fundamental, relational, or appearance.

Distinctive · only 23% of schools agree (47/202)
Is the world created from nothing?
Creatio ex nihilo is one of the most distinctive Western-theological claims. Whether matter was created from nothing, eternally exists, or is sustained moment-by-moment turns on what kind of thing matter is.
Matter is real but emerges from something deeper — neither bedrock nor created-from-nothing.
On this view, matter is genuinely there, but it isn't the floor of reality. It depends on something more fundamental — dependent origination, mind, divine sustaining act, computational substrate, or the structure of conditions — and is conserved only at its own level of description. …
Roads not taken Yes — matter was created and is conserved as a real substance. (55%) · Matter is constituted by relations; the question of 'from what?' presupposes substance. (16%) · Matter arises and dissolves through cosmic rounds; neither created from nothing nor eternal. (4%)
Distinctive · only 23% of schools agree (47/202)
Is the physical world fully real?
Realists, idealists, and relationalists divide on whether matter exists mind-independently, derivatively, or as a pattern of relations. The split runs deeper than any single scientific question.
Real but sustained — not mind-independent in the strict realist sense.
On this view, the physical world is real enough — it has its own laws, its own conservation principles, its own resistance to wish — but it is not the floor of being. It is sustained by something else: mind, divine attention, computational substrate, or …
Roads not taken Yes — the physical world is fully real, mind-independent, persisting. (55%) · Real as relations — neither pure substance nor pure construction. (16%) · Real for this cycle — the deepest reality cycles through creation and dissolution. (4%)
Distinctive · only 23% of schools agree (47/202)
Does matter have intrinsic moral standing?
Do rocks, soil, rivers, and stuff in general deserve moral consideration — or only the living, the conscious, the human? The answer turns on what matter is.
Matter is morally considerable derivatively — through what it sustains.
On this view, matter doesn't have standing on its own; it has standing through what it makes possible. Soil matters because it grows food; water matters because it sustains life and mind and practice. Asking whether the rock as such has moral standing slightly misreads …
Roads not taken Matter is morally considerable insofar as it is created or conserved good. (55%) · Matter has intrinsic moral standing as part of the relational fabric. (16%) · Matter is in flux; standing is impermanent and ritual-mediated. (4%)
4 unaligned
Should we edit the human germline? Schools split: 54% / 16% / 15% — What is a nation? Schools split: 54% / 16% / 15% — What is money? Schools split: 54% / 16% / 15% — What makes someone male or female? Schools split: 54% / 16% / 15% —
Observer · 37 dilemmas, all mainstream
Could causation work backwards? Causation runs one way — the arrow of time is real and structural. 68% Is the asymmetry between memory and anticipation a real feature of time, or just of us? The asymmetry is real because time itself has a real direction. 68% Is the arrow of time a real feature of the cosmos, or only of how we describe it? The arrow is real and structural; the asymmetry isn't an artifact of description. 68% Is environmental damage ever truly permanent? Damage is real and permanent on the relevant timescales. There is no recovery; there is only limitation. 66% Can a civilization recover from collapse? Civilizational complexity is hard to build and easy to lose; recovery is at best partial. 66% Does the second law of thermodynamics mean something morally? Entropy is what time is. The moral weight, if any, is the weight of working against the current. 66% What is our place in nature? Active in a real nature — we cultivate, steward, transform. 48% Should we colonize space? Cultivating worlds beyond Earth is the next form of stewardship. 48% Is genetic engineering of food stewardship or domination? Genetic modification is cultivation by other means. 48% Is reality fundamentally digital? No — continuous divine sustaining act, the Tao that knows no joints, the One's self-disclosure. 44% Are there indivisible units of experience? No — continuous divine presence; consciousness is the unbroken witness. 44% Is memory stored or reconstructed? Held in continuous divine or ancestral remembering — neither stored discretely nor purely reconstructed. 44% What happens to "you" when you die? A soul continues into another mode of being. 37% Can prayer for someone far away affect them? Prayer reaches because God or a cosmic ordering acts on the prayed-for. 37% Are coincidences ever more than coincidence? What looks like coincidence is providence — there is no such thing as a real coincidence. 37% Are the dead morally present to the living? The dead are present through divine memory, communion of saints, or ancestor presence. 35% Is divine omniscience compatible with human freedom? The human observer is in time, but God's vantage is not — and foreknowledge is not foreordering. 33% Does meditation reveal something genuinely timeless? Meditation participates in a real eternity — divine or cosmic — that the bounded human observer ordinarily cannot reach. 33% Does prayer change God's mind? God sees from outside time; prayer doesn't change God's mind, but it is part of how providence is enacted. 33% What makes someone the same person over time? You are a soul — what persists through change is the non-bodily aspect. 29% Is the late-stage dementia patient still the person their spouse married? The soul persists; the cognitive change is the body's, not the person's. 29% If a teleporter copied and destroyed you, would you have survived? The soul accompanies the person; engineering can't transfer it. 29% Does environmental harm in another country bind me morally? Distance doesn't dilute obligation; communion of saints / divine relation spans the cosmos. 29% Should we trust expert testimony when we can't verify it? Defer to credentialed traditions; experts are the modern analog. 28% Is religious revelation a real source of knowledge? Revelation is the paradigm case of authoritative knowledge. 28% Does an LLM 'know' the things it correctly produces? An LLM has no soul to whom revelation could be addressed; the question doesn't apply. 28% Could a fetal brain organoid in a petri dish be conscious? — — Could an AI have a mind that matters? — — Do animals have moral standing comparable to humans? — — Does history have a direction or meaning? — — How is knowledge of reality produced? — — Is salvation, liberation, or fulfillment individual or communal? — — Is truth universal, tradition-bound, situated, or constructed? — — What is marriage? — — What kind of religious-theological authority does the tradition recognize? — — When does a person begin? — — Who is the moral primary — the individual, the community, the cosmos, the class, or the species? — —
Information · 4 dilemmas, all mainstream
Is anything truly lost when someone forgets? Information is conserved — the personal pattern is held even when an individual mind loses it. 39% Does deleting your data online destroy something real? Information at the cosmic level isn't destroyed; deletion only obscures access. 39% Could the dead, in principle, be brought back? The information that constitutes a person is conserved; restoration is in principle possible. 39% Is forgiveness ontologically possible? The offense persists ontologically; forgiveness is real moral work, but it doesn't erase what was. 39%
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