What are the benefits of a rational study of mathematics?

Study of Euclid’s Elements had a tremendous influence on Galileo Galilei, Sir Isaac Newton, and Abraham Lincoln. Lincoln’s study of geometry is mentioned at the National Park Service’s website, on the page devoted to the Lincoln Memorial. In discussing his early years, they say:

Lincoln’s mother provided his early education. She was a rarity on the frontier in that she could read and shared the skill with him at an early age. The vast majority of his education was acquired by reading. He seldom was without a book and spent long hours studying Shakespeare, Byron, and even Euclid’s geometry. Despite having little formal education he triumphed with determination. Lincoln ultimately developed a talent for expression that could have led to a very different career. His “Gettysburg Address” is considered one of the most succinct and eloquently written speeches delivered by an American politician.

But there is a great deal more detail on how mathematics, specifically Euclid’s Elements, sharpened Lincoln’s reasoning skills in “An ‘Old-Fashioned’ Nationalism: Lincoln, Jefferson, and the Classical Tradition” by Drew R. McCoy. Here is an excerpt of this very interesting article:

 As one historian of mathematics has observed, “no work, except the Bible, has been more widely used, edited, and studied, and probably no work has exercised a greater influence on scientific thinking.”… Specifically, Euclid’s geometry had become, by Jefferson’s time, a testament to the power of human reason to deduce truth. On the basis of some formal definitions of terms and five postulates and five axioms whose truth was self-evident—such as, “things that are equal to the same thing are also equal to one another,” or “the whole is greater than the part”—Euclidean geometry “deduced an elaborate system of propositions that seemed both to accurately describe physical reality and to compose a flawlessly logical system.” In this sense Euclid did more than teach the principles and methods of correct reasoning in geometry; he could inspire readers of his Elements to apply reason to philosophy, economics, political theory, art, and religion, and in so doing, to arrive at truths that were as valid as mathematical truth. … As mental exercise, Lincoln’s long hours with Euclid doubtless made him a better “close reasoner,” to use Herndon’s term, and hence a more effective lawyer, which was surely his conscious purpose. But they also helped prepare him, in ways he could not have known, for the unexpected resumption of his political career after 1854. Lincoln had always been noted for his ability to reduce his thought on any given subject to the simplest and plainest terms possible; and during these critical years for the republic, his mastery of that skill allowed him to argue the case against both proslavery and popular sovereignty with something close to “Euclidean coherence.” Throughout his protracted debate with Douglas between 1858 and 1860, Lincoln “appealed repeatedly to the nature of proof in Euclid” as the appropriate standard for evaluating the arguments of the two combatants. And in a larger sense, the distinctive qualities of Lincoln’s mature political thought, including its content as well as its form and precision, appears to have owed a great deal to his immersion in Euclid. Lincoln’s ability to whittle down a complex issue to one key principle, or central axiom, directly informed his political message during the second half of the 1850s, when we might say he took a vexingly complex issue, slavery, and whittled it down to a simple issue: the humanity of the slaves. Amid the acrimonious wrangling over the complex details of the politics of slavery in the territories, Lincoln’s simple message became unmistakable: If the Negro is a man, then slavery is wrong, and must be disapproved of, and discouraged by all possible legal and constitutional means. In 1859, drawing an explicit connection between Euclid and the Declaration of Independence, Lincoln identified “the principles of Jefferson”—including, of course, the eighteenth-century Euclidean truth that “all men are created equal”—as “the definitions and axioms of free society.”