Philosophy of Berkeley#

To comprehend the mind of Berkeley is not possible at all if we ignore the era in which he lived -

George Berkeley (1685–1753) was an Anglo-Irish philosopher and Anglican bishop who is best known for his radical philosophical position of immaterialism, also known as subjective idealism. His most famous dictum, “Esse est percipi” (to be is to be perceived), encapsulates the core of his thought. Specifically:

• Rejection of Material Substance: Berkeley fundamentally denied the existence of matter, or mind-independent material substance. He argued that there is no “stuff” or substratum that exists independently of a perceiving mind.
• Reality as Ideas: For Berkeley, everything that exists is either a mind (a “spirit”) or an “idea” in a mind. Physical objects, such as trees, houses, or tables, are not external material things but rather collections of ideas or sensations perceived by a mind.
• Direct Perception: We do not perceive an external material world that causes our ideas. Instead, we directly perceive ideas. When you see a red apple, the redness, the shape, the taste, the smell - these are all ideas in our mind. There is no underlying material “apple” that exists apart from these perceived qualities.

If “to be is to be perceived,” what happens to objects when no human mind is perceiving them (e.g., a tree in an uninhabited forest)? Berkeley’s answer is that God, as an infinite and omnipresent mind, perceives everything continuously. This divine perception ensures the consistent and orderly existence of the world, even when individual finite minds are not attending to it. God, according to Berkeley, is also the ultimate source of our sensory ideas. The regularities and patterns we observe in nature are a result of God’s consistent will and the orderly way in which the God impresses ideas upon our minds.

Berkeley’s ideas had a significant impact on subsequent philosophers, particularly David Hume, who pushed empiricism to its skeptical limits, and later idealists like Immanuel Kant and German Idealists. While often criticized for its counter-intuitive nature, Berkeley’s work forced philosophers to seriously consider the nature of perception, reality, and the relationship between mind and world.

Berkeley also engaged with the science of his day, arguing that scientific laws describe the regularities among our ideas, not inherent properties of a material world. He viewed causation not as one material object “making” another happen, but as a consistent sequence of ideas, ordained by God.

Berkeley on Mathematics#

Of all treatises written on the subject in the eighteenth century, Berkeley’s Analyst was the most sustained and penetrating critique of the methodology of the infinitesimal calculus. This work foreshadows the foundational research of the nineteenth century, and provides a link between the mathematical preoccupations of the seventeenth and eighteenth centuries and those of the nineteenth.

Broadly speaking, the mathematicians of the seventeenth and eighteenth centuries - and in particular Newton, Leibniz, and Euler - had been more concerned with exploiting and extending the techniques of the differential and integral calculus than with tidying its foundations. The mathematicians of the early nineteenth century, in contrast, bent their efforts towards placing the calculus on an unobjectionable footing - towards chasing away the obscurities that surrounded such notions as infinitesimal, limit, and differential. This project was pursued during the first two-thirds of the nineteenth century by such mathematicians as Gauss, Bolzano, Cauchy, Abel, Fourier, Riemann, and Weierstrass. Their investigations into the foundations of real analysis in turn inspired the later studies by Dedekind, Cantor, Frege, Peano, Peirce, Russell, and Hilbert of set-theory, logic, and the foundations of arithmetic.

Berkeley’s Analyst prefigures this entire development, and his philosophically-motivated criticisms of the Newtonian mathematics of the eighteenth century raise many issues that will loom large in the selections that follow. These issues may be loosely grouped under 4 headings.

1. His critique of infinitesimals raises not only the issue of justifying the central concepts of the calculus but also the more general issue of the legitimacy of the actual infinite in mathematics
2. Berkeley, as befits the empiricist philosopher who wrote An essay towards a new theory of vision, makes acute comments about the relationship between geometry, human visual perception, and the foundations of the calculus.
3. In numerous places Berkeley discusses the problem of the reference of mathematical expressions; declaring, for instance, in Of infinites, that “This plain to me we ought to use no sign without an idea answering to it”; one of his cardinal criticisms of Newton is that infinitesimals can have no empirical reference. This topic of the reference of mathematical expressions (a topic on which Berkeley occasionally shifted his position) was to be central to the development of algebra during the nineteenth century
4. All these mathematical issues - in analysis, geometry, and algebra - are intertwined in Berkeley’s thought with more general concerns about mathematical truth, the rigour of demonstrations, the applicability of mathematics to the empirical world, and the scope and limits of mathematical knowledge. Philosophers more often react to developments in mathematics than anticipate them, and in modern times perhaps only Descartes, Leibniz, and Kant can be said to have equalled Berkeley’s insight into the foundations of mathematics .

The connection between Berkeley’s mathematical interests and the leading strands of his philosophy was not adventitious: his education at Trinity College, Dublin was in mathematics and logic as well as in philosophy and the classics, and the foundations of mathematics was to be a lifelong preoccupation. His investigations into the philosophy of mathematics commenced with his first published work (in effect, his bachelor’s thesis), the Arithmetica et miscellanea mathematica; they were continued in his early, unpublished Philosophical commentaries and Of infinites - especially in Notebook B of the Commentaries, where remarks on optics and on the nature of the soul mingle with observations on algebra, geometry, and the infinitesimal calculus; they play a major though subsidiary role in his Principles of human knowledge, De motu, and Alciphron; and they are once more the centre of attention in his last major philosophical enterprise, the critique of Newton and the Newtonians in The analyst.

Berkeley’s writings show the futility of attempting to draw a sharp boundary between philosophy and mathematics. We have already observed that his philosophical reflections led him to deep criticisms of current mathematical practice. But the influence goes in the other direction as well, and his study of mathematics and the physical sciences affected his general metaphysical position—so much so that the two are often difficult to disentangle. His discussions of infinitesimals are interwoven with arguments about minima sensibilia and with his doctrine that, for sensible objects, esse is percipi; his discussion in the Principles of human knowledge of algebra and of geometric reasoning is bound up with his view of language and his rejection of abstract ideas; his critique of the foundations of the calculus is motivated by many of the same epistemological considerations that underlie his critique of the idea of material substance. It is these connections (rather than his powerful but already-known logical criticisms of Newton’s reasoning in the Principia) that give his philosophy of mathematics its depth and its strength and its present interest.

Despite the penetration of his writings—their wealth of implications both for mathematics and for philosophy - Berkeley had little actual influence on the development of mathematics. He was nobody’s inspiration, and indeed to this day is rarely thought of as a philosopher of mathematics at all. The nineteenth century remembered him principally as an ingenious paradoxer, the precursor of Hume who had denied the existence of material substance; his writings on mathematics have largely been forgotten. (Two exceptions are reproduced below: MacLaurin and the correspondence between William Rowan Hamilton and Augustus De Morgan.) But even if Berkeley did not himself initiate or influence the great period of nineteenth-century foundational research, he nevertheless glimpsed many of its central themes; and his writings are therefore an appropriate starting-point for these volumes