The Italian poet Dante Alighieri (1265 –1321) designed the structure of Hell, Purgatory, and Heaven around it in his epic poem The Divine Comedy. ![]() The mediaeval trivium and quadrivium made geometry an essential ingredient of education. The influence of Euclidean geometry permeated education, architecture, science, and literature. 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. 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 ).Įuclidean geometry dominated mathematics for the subsequent two thousand years. 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. Whether the configuration and behavior of the physical world conforms to a deductive geometrical system is nonetheless an open question. 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. Granted those assumptions, no reference to the physical world was required, and the truths of the theorems he was able to deduce became tautological. 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. Euclid's achievement was to classify rather than discover the theorems he systematized. 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. in Euclid Alexandria's Elements, whose achievement was to treat geometry axiomatically through a rigorous system of deduction. Greek mathematics culminated around 300 b.c.e. ![]() 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. 1285 –1347) were to contribute to the debate by relating the issues to questions of universal and particular knowledge. ![]() 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. Aristotle rejected this notion, preferring to think of geometrical and arithmetic objects as reductive abstractions from experience that give rise to mental generalities. 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. 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.įor 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. 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.
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