Geometry: Philosophical Aspects

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Geometry: Philosophical Aspects

The theological and religious importance of geometry needs to be addressed in conjunction with the much wider question of the relationship between those religious aspirations that strive to lay hold upon abstract eternal truths embedded and embodied in God and others that emphasize the importance of contingent, temporal, and ephemeral features of existence. To some extent this antithesis reflects the differences between the ancient Greek philosophers Plato (427–347 b.c.e.) and Aristotle (384–322 b.c.e.) in their attitudes toward the status of mathematical objects.

For Plato, neither geometrical objects such as points, lines, and circles, nor arithmetical objects such as numbers could be conceived as existing in the physical world. Since two shapes could not be the same, nor two objects equal, concepts such as shape and number had to belong to a realm beyond sense and experience, the realm of forms or ideas. Aristotle rejected this notion, preferring to think of geometrical and arithmetic objects as reductive abstractions from experience that give rise to mental generalities. During the Middle Ages, the Christian philosopher Thomas Aquinas was to make much of this in terms of intellective and abstractive knowledge, and in their turn John Duns Scotus (c. 1265–1308) and William of Ockham (c. 1285–1347) were to contribute to the debate by relating the issues to questions of universal and particular knowledge. Even so, the issue of the grounding of geometrical truth did not challenge the self-evident truth of Euclidean geometry; that had to await the advent of non-Euclidean geometries and the philosophical criticisms of John Stuart Mill (1806–1873) during the nineteenth century.

Euclidean geometry

Greek mathematics culminated around 300 b.c.e. in Euclid Alexandria's Elements, whose achievement was to treat geometry axiomatically through a rigorous system of deduction. This abstraction reflected the value placed upon eternal ideas by the platonic school, and rid geometry of reliance upon particular instances of such things as circles and lines. Euclid's achievement was to classify rather than discover the theorems he systematized. He was able to see that the entire edifice of geometry could be captured in a deductive system based upon five foundational assumptions or postulates. Granted those assumptions, no reference to the physical world was required, and the truths of the theorems he was able to deduce became tautological. Albert Einstein (1879–1955), writing long after the monopoly of Euclidean geometry had been broken, reiterated essentially the same point about the relationship between geometry and experience in his essay of that name where he observed that only one assumption is required in addition to a geometrical system: the further postulate that it is a model for the real world.

Whether the configuration and behavior of the physical world conforms to a deductive geometrical system is nonetheless an open question. If it does, there is a remarkable harmony between an abstract construction of the human mind and the workings of the world—part of what contemporary physicist Stephen Weinberg has called "the unreasonable effectiveness of mathematics," but the effectiveness of geometry may say more about the limitation and consistency of human thought and action than it does about the behavior of the world. As the Italian philosopher Giambattista Vico (1668–1744) put it in the Nuovo Scienza (New Science, 1725), the dilemma arises from the fundamental question of the relationship between the "found" and the "made" (verum et factum ).

Euclidean geometry dominated mathematics for the subsequent two thousand years. Problems posed in antiquity that provided the stimuli for the development of enormous areas of mathematics—construction of a square with area exactly that of a given circle, the doubling of the volume of the altar at Delos, or the trisection of an angle (all to be solved using only straightedge and compasses)—remained unresolved until modern times, when all three were proved to be impossible. The influence of Euclidean geometry permeated education, architecture, science, and literature. The mediaeval trivium and quadrivium made geometry an essential ingredient of education. The Italian poet Dante Alighieri (1265–1321) designed the structure of Hell, Purgatory, and Heaven around it in his epic poem The Divine Comedy. First Ptolemy (c. 85–165 c.e.) and then Nicolaus Copernicus (1483–1543), Johannes Kepler (1571–1630), and Isaac Newton (1642–1727) devised their world systems upon it. Prohibition of images in Islam encouraged the intricate geometrical patterns used to decorate mosques.

Perhaps the most significant change in the study of geometry occurred with René Descartes's (1596–1650) invention of algebraic or analytic geometry as described in the Discourse on Method (1637) by envisaging geometrical figures superimposed on a grid, thereby making their properties susceptible to algebraic analysis. It is impossible to exaggerate the importance of this change, for it allowed geometry to be integrated into the calculus as discovered by Newton and Gottfried Wilhelm Leibniz (1646–1716), and so into the emerging theories of the physical world. Immanuel Kant (1724–1804) took it as an obvious a priori truth in the Critique of Pure Reason in 1781 that the sum of the angles of a triangle is 180 degrees, and Euclid continues to be taught throughout the world as a quintessential example of a deductive system to assess the potential of young mathematicians. Carl Friedrich Gauss (1777–1855), prompted by his study of curvature, was experimenting with alternative geometries, and the nineteenth century saw them proliferate through the work of Nikolai Ivanovitch Lobachevsky (1793–1856), Gauss's pupil Georg Friedrich Bernhard Riemann (1826–1866), and others.

Non-Euclidean geometries

From Euclid's own time there had been persistent attempts to deduce the "fifth postulate"—often cited as "through any point not in a given line one and only one line can be drawn parallel to the given line"—from the other four. These attempts continued until beyond Gauss's time, but he gradually became convinced that they were futile, that the fifth postulate was independent of the others, and therefore that it could be modified to produce non-Euclidean geometries. Lobachevsky was similarly obsessed with proving the fifth postulate, but, unlike Gauss, between 1826 and 1829 he worked on and eventually published his discovery of non-Euclidean geometry, thus dealing a blow to the Kantian system similar, as Carl Boyer puts it, to the impact on Pythagoreanism of the discovery of incommensurables. The "Copernicus of geometry," Lobachevsky was the first to generate an entirely consistent and coherent geometry that rejected Euclid's fifth postulate (although Janos Bólyay [1802–1860] also developed one almost simultaneously), but even he was so bemused by its counter-intuitive properties that he called it "imaginary" geometry. Non-Euclidean geometry nonetheless remained an obscure mathematical curiosity until taken up and generalized by Riemann. He realized that geometry need not be based upon quasi-Euclidean postulates at all, but could be regarded as a set of n-tuples (co-ordinates) combined according to certain rules. These rules define a metric and give rise to different kinds of Riemannian "space" governed by tensors. These spaces were to prove fundamental to the revolution in physics brought about by Einstein.

The realization that there were non-Euclidean geometries shook the foundations of mathematics and contributed to the demise of absolute foundationalism in philosophy, even though the discovery of different and mutually exclusive axiomatic geometries gave new momentum to the study of deductive systems based upon "foundations." Whereas for Kant and his predecessors there had seemed to be no element of choice in the determination of the geometry of the world because there was only one geometry, and that Euclid's, after Gauss and Lobachevsky it became necessary to add the postulate that Einstein was to remark on, that a particular geometry should also be chosen as the geometry of "real world." As he was to show in his theory of General Relativity, the "natural" geometry of the universe is not Euclidean at all, but Riemannian.

Implications for theology

In theological terms, the ubiquity and power of geometry have often been regarded as evidence for the work of God, whose mind has come on such a basis to be thought of as a perfect deductive system. But there are serious difficulties with such a view. One concerns the parallelism between deduction and determinism; another concerns the problem of the found and the made.

Deductive systems self-consciously avoid the introduction of any new material whatever; proof involves rendering explicit what is already implicit by applying the rules of deductive logic. A universe governed by physical laws equivalent to such deductive systems would be deterministic; there would quite literally be "no new thing under the sun" (Eccles. 1:9). At best, as occurs when implicit truths are rendered explicit by the articulation and proof of a new theorem in geometry, people would find themselves surprised by the unforeseen; but any freedom, either for God or for human beings, would be illusory, the outworking of an implicit and inevitable necessity.

Since Kant, people have been forced to take seriously the notion that what they regard as the intelligibility of the world is in reality the inner coherence of their modes of thought: in Vico's language, the intelligibility of the made, not the found. Objections to the employment of geometry as a model for the world reiterate this view in the context of doubts about the fine structure of the world and the limits of observation. Such doubts have been reinforced by the development of quantum mechanics and relativity, which suggest that human intuitions about the world are mistaken (although even these suggestions are still to be construed within a conceptual system, and not as grounded in a noumenal world).

Nonetheless, the expansion of elementary geometry into analytic geometry, topology, and linear algebra, preserves the sense of the "unreasonable effectiveness of mathematics," and suggests that, all doubts to the contrary notwithstanding, there may remain some sense in Newton's designation of the universe as the divine sensorium, albeit construed as a creation embodying the structure of a divine geometry, and intelligible only to a divine mind.

John Stuart Mill (1806–1873) was one of the first to challenge Kant's view that the a priori truths of geometry are necessary consequences of the possibility conditions of rational thought, in other words, that to be rational people have to think the world, amongst other things, in Euclidean terms. Mill did not know of non-Euclidean geometry, but attributed the apparent inescapability of Euclidean geometry to paucity of imagination, and its domination to the kinds of experiences to which human beings are susceptible. Mill seems to have been vindicated by the predilection of physical theory—in both quantum mechanics and relativity—for non-Euclidean geometries that defy everyday human intuitions.

Towards the end of the nineteenth century fundamental changes in philosophy of mathematics occurred, most notably the articulation of logicism by German mathematician and philosopher Gottlob Frege (1848–1925). Frege attempted to reduce arithmetic to logical categories by employing the theory of sets and the non-Euclidean geometries of Riemann that discard the intuitive notions of line and plane familiar from Euclid in favor of abstract n-tuples governed by arbitrary rules.

Attempts to reduce mathematics to logic were associated with Frege's attack on psychologism, itself a descendant of Kant's view that the nature of intelligible reality is governed not by properties of an objective world but by the rules of thought. Often called a modern platonism, Frege's work struggled to ground mathematics in an inviolable world independent of experience. Unfortunately, by adopting the theory of sets, Frege fell foul of Bertrand Russell's (1872–1970) celebrated paradox that the "set of all sets that are not members of themselves" both is and is not a member of itself, thus demonstrating that there is an apparent antinomy in the theory of sets.

Quite apart from its relevance to ontology and epistemology, and thus to theology, geometry has played a major role in the more everyday development of religion. Closely associated with the educated and priestly classes, with astronomy and astrology, with numerology and mysticism, geometry has repeatedly had an impact on the way people have viewed the order and mystery of the world. The Pythagoreans regarded number as the basis of all knowledge and truth, many religions and cults have seen mystical significance in the properties of geometrical shapes, especially the golden rectangle/ratio and the pentangle, widely employed in magic.

The fundamental religious importance of geometry nevertheless emerges from questions of the relationship between divine creative purpose, the structure and operation of the natural world, and the conceptual capacities of the human mind. If, as Einstein suggested, "the only unintelligible thing about the world is that it is intelligible," there seems to be an intrinsic harmony between all three, and geometry seems able, at least in the limit, to embody the structure of the world as found. If, as others suggest, following Kant, "the only intelligible thing about the world is that it is unintelligible," then geometry is a fabrication designed to render the world intelligible at the expense of misrepresenting its intrinsic structure, and geometry can do no more than embody a structure of the world as made in the image of the human mind.

See also Kant, Immanuel; Mathematics; Newton, Isaac; Physics