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The Discipline of Reasoning and the Value of Mathematical Study

Suppose, then, that one seeks to cultivate skill in the art of reasoning — to discipline the intellect in the precise processes of thought. Suppose further that one wishes to move beyond the uncertain domain of conjecture and probability, to escape the laborious task of weighing conflicting evidence and comparing isolated instances in order to derive general laws. Let us assume that the primary aim is to understand how to manage general propositions once established, and how to deduce from them sound and necessary conclusions.

It is evident that such intellectual discipline will be most effectively attained in those fields of study where the first principles are indisputably true. In all reasoning, errors arise from one of two sources: either from beginning with false premises, in which case even flawless reasoning cannot prevent error, or from reasoning incorrectly from true premises, in which case sound data may yield false conclusions.¹

In the mathematical, or pure, sciences — geometry, arithmetic, algebra, trigonometry, and the various branches of the calculus — we are at least assured that our initial assumptions are free from error. Their axioms are self-evident and their truths necessary. Thus, the student of mathematics may direct undivided attention to the reasoning process itself, perfecting the method of logical inference without distraction from the uncertainty of the data.²

For this reason, the mathematical sciences have long been regarded as the most exact and fruitful training in logical discipline. Founded as they are upon the simplest and most incontrovertible truths concerning space and number, they afford the intellect its most rigorous exercise in systematic and orderly thought.³

When Plato inscribed above the entrance to his Academy the injunction, “Let no one ignorant of geometry enter here,”⁴ he did not intend that geometry itself should be the primary topic of discussion. The problems that occupied his disciples were the highest and most abstract that the human mind can contemplate — questions of moral, social, and metaphysical import: the nature and destiny of man, his duties, and his relation to the divine and the unseen.

What connection, then, had geometry with such inquiries? Simply this: Plato recognized that the untrained mind, unfamiliar with exact reasoning and the legitimate derivation of conclusions from premises, was unfit to engage in such elevated speculation. The type of intellectual discipline requisite for these studies could best be acquired through geometry — the one mathematical science which, in his time, had been thoroughly formulated and reduced to a coherent system.⁵

We in England have long acted upon the same principle. Our universities require those preparing for the professions of law, the ministry, and public service to acquire at least a moderate acquaintance with mathematical studies — with curves, angles, numbers, and proportions. This requirement does not arise from any supposed practical connection between these topics and the pursuits of their future careers, but from the conviction that, in mastering them, students develop habits of precision, patience, and logical rigor. These qualities, more than any specific body of knowledge, are indispensable to sound judgment and success in every sphere of intellectual and practical life.⁶

Notes

1. J. C. Fitch, Lectures on Teaching (New York: Macmillan, 1906), 291–292.
2. Ibid.
3. John Stuart Mill, A System of Logic, 8th ed. (London: Longman, 1872), bk. 2, ch. 4.
4. Proclus, A Commentary on the First Book of Euclid’s Elements, trans. Glenn R. Morrow (Princeton: Princeton University Press, 1970), 63.
5. Plato, Republic, 526–534; see also Meno, 82b–86b.
6. Fitch, Lectures on Teaching, 292.

Bibliography

• Fitch, J. C. Lectures on Teaching. New York: Macmillan, 1906.
• Mill, John Stuart. A System of Logic. 8th ed. London: Longman, 1872.
• Plato. Republic. Translated by Paul Shorey. Cambridge, MA: Harvard University Press, 1930.
• ———. Meno. Translated by G. M. A. Grube. Indianapolis: Hackett Publishing, 1981.
• Proclus. A Commentary on the First Book of Euclid’s Elements. Translated by Glenn R. Morrow. Princeton: Princeton University Press, 1970.

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