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Author of The Analyst by George Berkeley, a critique of the foundations of calculus.

Standards of mathematical proof changed between the seventeenth and nineteenth centuries because both the objects studied and the operations performed on them changed. A finite geometrical construction, an infinite expansion, and an abstract group raise different questions about justification. There was no universal seventeenth-century indifference to rigour followed by its sudden nineteenth-century invention. The more defensible thesis is that successful methods repeatedly exposed unstated assumptions; later work made some of those assumptions into explicit definitions, hypotheses and existence arguments.

Pierre de Fermat's adequality illustrates the distinction between an effective method and its justification. For $F(a)=a(L-a)$, comparing neighbouring expressions givesDividing by $e$ and suppressing the remaining term in $e$ produces $a=L/2$. There is an independent finite verification:Thus the result can be secure even when the general procedure raises a question: why is the increment first nonzero for division and then discarded? In the parabola construction of question 2, the same issue appears in replacing a strict geometrical comparison by adequality and extracting the subtangent. A historical explanation must describe this operation on its own terms rather than silently equate it with a modern derivative.^[ref]

Isaac Newton and Gottfried Wilhelm Leibniz supplied different languages for the new calculus. Isaac Newton's Introduction to the Quadrature of Curves by Isaac Newton, first published with the 1704 Opticks, presents quantities generated by motion and considers their rates of change. Its mature language of first and last ratios seeks exact relations as increments vanish; it should not be confused with simply asserting that a small error is harmless. Gottfried Wilhelm Leibniz's 1684 Nova methodus pro maximis et minimis by Gottfried Wilhelm Leibniz presents a calculus of differentials and operational rules. Their common problem was to explain why procedures using vanishing increments establish exact results, while their chosen objects and modes of explanation differed.^[ref][ref]

George Berkeley's The Analyst by George Berkeley (1734) exposes the logical pressure point. For the cube,The finite identity does not become a legal division at $h=0$. The modern repair is to keep $h\ne 0$ throughout the identity and prove that its right-hand side tends to $3x^2$. That limit is different from a quotient evaluated at zero. George Berkeley's criticism therefore concerns the grounds of inference, not whether the established rules often produce correct answers. His theological polemic also reminds us that standards of mathematical proof could be contested outside specialist calculation.^[ref]

The infinite introduced another burden. Joseph Fourier's 1822 use of trigonometric series to represent arbitrary functions greatly extended what calculations in mathematical physics could express. But deriving Fourier coefficients by orthogonality only shows what those coefficients must be if the proposed operations are legitimate. It does not prove convergence, equality with the initial data, or interchange of an integral and an infinite sum. The example of a constant expanded in a Fourier sine series makes a hidden qualification visible: the terms all vanish at the endpoints, even when the assigned data do not. The correct theorem must specify the domain and mode of representation.^[ref]

Peter Gustav Lejeune Dirichlet's 1829 memoir supplies a particularly precise example of changing proof obligations. It criticises a convergence argument of Augustin-Louis Cauchy's which used complex substitution into an arbitrary function and an insufficient comparison between signed series. Peter Gustav Lejeune Dirichlet supplies the instructive pairThe first series converges by the alternating series test; the second diverges because their difference is the harmonic series. Nevertheless $b_n/a_n=1+(-1{)}^n/\sqrt{n}\to 1$. Thus asymptotic equality of terms does not transfer convergence between conditionally convergent signed series. A comparison principle valid for positive terms needs additional justification when cancellation is involved. The example refutes a mode of inference rather than merely correcting a numerical answer.^[ref]

Augustin-Louis Cauchy's own foundations programme is part of this change. The 1821 Cours d'analyse by Augustin-Louis Cauchy makes limits, continuity and series convergence central to elementary reasoning. Yet its Sum theorem of Cauchy also shows why announcing rigour does not make every proof adequate. In modern terminology, continuous functions can have a discontinuous pointwise limit. An explicit series version isEach term is a polynomial and the partial sum is $x-x^{N+1}$. The convergence does not hold in uniform convergence: with $x_N=1-1/N$, the error $x_N^{N+1}$ tends to $e^{-1}$ rather than zero. The modern uniform limit theorem repairs the inference by requiring one bound on the error to work throughout the domain. This example diagnoses the missing control under a pointwise reading; it does not settle every historical interpretation of Augustin-Louis Cauchy's infinitesimal language.^[ref]

The issue is the order of dependence in quantifiers. Pointwise convergence permits the threshold $N$ to depend on both the error tolerance and the point. Uniform convergence requires a threshold depending only on the tolerance. If $f_N$ is continuous near $x_0$ and the convergence is uniform, choose $N$ so that $|f-f_N|<\epsilon /3$ nearby, then use the continuity of this fixed $f_N$ to make $|f_N(x)-f_N(x_0)|<\epsilon /3$. The triangle inequality proves $|f(x)-f(x_0)|<\epsilon$. Writing out this proof reveals what a vague statement that the tail is small leaves unproved.

Another nineteenth-century development concerns the foundations of the real numbers themselves. Richard Dedekind's Continuity and Irrational Numbers by Richard Dedekind (1872) asks for an arithmetical account of the continuity often borrowed from a drawn line. A Dedekind cut separates the rational numbers into lower and upper classes. For instance,defines a gap in the rational numbers corresponding to $\sqrt{2}$. The cut is determined without assuming that a rational number fills it. The extended number system then supports existence theorems that a geometrical picture alone had suggested.^[ref]

This matters for apparently elementary proofs. Suppose $f$ is continuous on $[a,b]$ with $f(a)<0<f(b)$. Repeated bisection constructs nested intervals whose endpoint values have opposite signs and whose lengths tend to zero. To conclude that there is a common limiting point, one needs the completeness of the real numbers. Continuity then forces its value to be zero. Over the rational numbers, $f(q)=q^2-2$ changes sign between $1$ and $2$ but has no rational zero. Thus even a proof that avoids pictures must explain the existence of its limiting object. Richard Dedekind's account makes that requirement an explicit part of the mathematical setting, rather than an intuition about a continuous line.^[ref]

There is a parallel change in algebra. Heinrich Weber's 1882 definition in question 6 describes arbitrary elements through laws of composition. To assess such a definition one proves implications between axioms. The passage's finite set hypothesis implies that cancellative multiplication has an identity element and inverse elements; without that hypothesis, the positive integers under addition supply a counterexample. This is a new kind of proof obligation: one must establish that a proposed abstract definition has the intended consequences, rather than rely on their familiarity in particular numerical or permutation examples.^[ref]

The principal change was towards explicit control of operations, hypotheses and existence. Effective symbolic algorithms persisted, and geometric reasoning remained valuable. What became increasingly demanding was the passage from success in examples to a general claim: an infinitesimal step needed justification, an infinite expansion needed a convergence theorem, a limiting point needed a complete number system, and an abstract definition needed its consequences proved. The examples also prevent a simple progress story: nineteenth-century mathematicians disagreed about admissible reasoning, and a programme devoted to rigour could itself become the object of a successful critique.

Hardness is no more automatically a primary quality than redness. The comparison turns on whether one means an experienced sensation, a disposition to affect a perceiver, or a physically specified property of a material. Keeping those three levels distinct supports a useful distinction without treating either tactile or visual experience as a direct copy of an object's constitution.

John Locke's list of primary qualities includes extension, figure, number, motion and solidity. A secondary quality is instead a body's power, grounded in its constitution, to produce a sensory experience. The colour experience does not literally resemble the shape or motion of the particles producing it. Crucially, Lockean solidity is not hardness. In the former account, both water and a diamond exclude another body from the space they occupy; they differ in how readily their parts move aside. In book II, chapter IV, section 4, John Locke explicitly treats ordinary judgments of hard and soft as relative to the human body's response. Thus his own distinction does not warrant simply assigning everything felt by touch to the primary class.^[ref]

A tactile example makes the distinction intelligible. The same surface may resist a fingertip yet yield to a harder indenter. Its measured hardness also depends on the load, test geometry and material state. Its resistance is nevertheless reproducible under specified conditions. Similarly, perceived color depends on the illuminating optical spectrum, the surface's reflectance and the perceiver's color vision. A change of illumination can change an appearance without changing the surface. In neither case does dependence on specified conditions make the material's response fictitious. A disposition can be an objective feature of how something behaves in a specified interaction.

One proposed boundary therefore separates properties specified independently of a particular kind of perceiver from properties defined through perceptual responses. This classifies redness understood as a visual disposition differently from a spectral reflectance function. It likewise separates felt resistance from a specified mechanical hardness test. The boundary does not coincide with vision versus touch, or with measurable versus unmeasurable. Both colour-related and mechanical responses admit quantitative investigation; both can be relational.

There are limits to a stronger metaphysical interpretation. George Berkeley argues that perceived size, shape and motion also vary with perceptual circumstances; experience does not simply present an invariant class of primary qualities while presenting only relative secondary qualities. That challenges an easy appeal to resemblance. It does not, without further argument, show that there is no perceiver-independent structure, since variability of appearances is compatible with stable relations explaining the variation.^[ref]

The defensible conclusion is consequently qualified. Experienced hardness has no privileged immunity from the difficulties raised by redness. A distinction between perceptual responses and the physical organization explaining them remains useful, but its physical side must be established by inquiry. It is not an unrevisable list certified by whichever sense first reveals a property.

A distinction between physical constitution and perceptual response is plausible and scientifically useful. A division into wholly objective primary qualities and wholly unreal secondary qualities is much harder to defend. These are different interpretations of the primary and secondary qualities distinction.

John Locke attributes extension, solidity, shape, number and motion to bodies as primary qualities. Secondary qualities are powers to produce sensations through the constitution of bodies: a body's power to produce a colour experience is distinct from the experience itself. This distinction need not make the power unreal. Nor should the further resemblance thesis, that ideas of primary qualities resemble their causes, be confused with the more modest claim that bodily constitution explains experience.^[ref]

A modern model of colour perception shows what can be retained. For a nonfluorescent surface in fixed viewing geometry, a simplified cone cell response isHere $L$ is the incident spectral irradiance, $R$ the spectral reflectance, and $S_i$ an effective sensitivity for one of three cone cell classes. Neural processing and adaptation also matter, so these responses are not a complete identification of perceived colour. Nevertheless, the model separates the surface, illumination and observer rather than assigning every experienced variation to a changing surface.

It follows that one surface can produce different responses under different illumination. Conversely, distinct spectral reflectance functions can produce the same three responses for a specified illuminant and observer, a case of metamerism. Three weighted integrals do not in general uniquely determine a whole function. Thus identifying experienced colour with one intrinsic surface property loses both the dependence on the observer and the many-to-one character of the response.

The scientific benefits are concrete. A comparison of paints should specify illumination and observer conditions. A measurement intended to characterize a material can instead estimate its spectral reflectance, preserving information that a single colour match discards. In both cases the distinction tells an investigator what to hold fixed and what to vary. It organizes experiments and explains disagreement without requiring that one observer be hallucinating.

However, perceptual variability alone does not distinguish primary qualities from secondary qualities. George Berkeley points out that apparent extension and shape also change with viewpoint and perceiver.^[ref] A circular disc yields different projected shapes at different viewing angles. The stable object is inferred through a model of that variation; it is not simply copied into experience. Primary qualities therefore have no automatic exemption from the interpretative work involved in perception.

Nor is relationality equivalent to subjectivity. In special relativity, measured length depends on an inertial frame, but relations between such measurements are tightly constrained. Conversely, a disposition to affect a specified perceiver under specified conditions can be an objective physical fact. Classifying a property as relational does not settle whether it is real, measurable or explanatory.

The best distinction is consequently functional and revisable: distinguish the experience to be explained from physical properties and relations used to explain it. That retains the scientific benefits of separating constitution, environment and response. It does not establish an immutable list of primary qualities, a simple resemblance relation between experience and object, or a proof of scientific realism.

The distinction supplies a coherent strategy for an empiricist realism, but does not by itself justify that realism. It separates an object's relatively stable physical constitution from its effects on perceivers. The strongest defensible version concerns the explanatory roles of properties, rather than a division between wholly real properties and wholly unreal sensations.

For John Locke, primary qualities include extension, shape, solidity, number and motion; secondary qualities are powers of bodies to produce experiences such as colours and tastes through their constitution. The experienced colour need not resemble the physical cause of that experience. Importantly, a power belonging to a body is not thereby imaginary: the primary and secondary qualities distinction need not deny that an unobserved object has dispositions to affect perceivers.^[ref]

This helps empiricism accommodate perceptual variation without abandoning an external world. A surface can produce different colour experiences under different illumination while keeping approximately the same spectral reflectance. A warm and a cool hand can give different sensations in the same water; this need not mean that the water has two incompatible temperatures. The observer's initial state belongs in the explanation. Controlled changes in illumination, viewing conditions and bodily state allow repeatable predictions about experience. An empiricist can therefore treat physical objects as explanatory sources of regularities in experience, rather than demand that every appearance be copied into the object.

However, this explanatory success does not establish that ideas of primary qualities resemble mind-independent properties. We cannot step outside every perceptual and instrumental relation to compare an experience directly with its alleged original. Moreover, the motion and arrangement of unobserved constituents are themselves theoretical commitments. If empiricism permits only what is immediately given in sensation, the distinction introduces precisely the inference beyond sensation that requires defence. If empiricism permits fallible explanatory inference, scientific realism becomes possible, but the justification comes partly from that additional account of inference.

George Berkeley's challenge is that perceived extension and shape vary with viewpoint and perceiver just as colour does, and that extension cannot be experienced wholly apart from sensible qualities.^[ref] This undermines the simple test “variable appearance means merely secondary”. It does not prove that stable physical properties are absent. An obliquely viewed circular disc can produce an elliptical image; geometrical optics explains the relation between the disc and that image. Variation can be evidence for a systematic relation, rather than evidence against an external cause.

Modern physics also discourages treating Locke's list as an unrevisable inventory. Special relativity makes measured length depend on the relation between an object and an inertial frame; motion was relational even in Principle of Galilean relativity. Neither fact makes the measurements arbitrary. Conversely, spectral reflectance is an objective, measurable property even though perceived colour depends on illumination and the perceiver. Objectivity, relationality, perceptual variability and explanatory fundamentality are distinct issues.

The plausible distinction is consequently between an experience and the physical properties and relations that explain it, with the explanation open to correction. An empiricist realism can use this distinction to explain why appearances change while a world persists. It cannot obtain a proof of the external world, or a privileged catalogue of intrinsic properties, simply by naming some qualities primary. The gain is an empirically disciplined model of perception; its limit is that the realist interpretation still needs an argument.