Of all books from Western civilization, only the Bible has received more intense scrutiny than Euclid’s Elements. (WILLIAM DUNHAM) | Part C’

11 Jan Of all books from Western civilization, only the Bible has received more intense scrutiny than Euclid’s Elements. (WILLIAM DUNHAM) | Part C’

It is the echo of Euclid, 23 centuries after he lived.

The work was studied by the great Italian scholars of the sixteenth century and by a young Cambridge student named Isaac Newton a century later.

We have a passage from Carl Sandburg’s biography of Abraham Lincoln that recounts how, when a young lawyer trying to sharpen his reasoning skills, the largely unschooled Lincoln

… bought the Elements of Euclid, a book twenty-three centuries old … [It) went into his carpetbag as he went out on the circuit. At night . . . he read Euclid by the light of a candle after others had dropped off to sleep.

It has often been noted that Lincoln’s prose was infleenced and enriched by his study of Shakespeare and the Bible . It is likewise obvious that many of his political arguments echo the logical development of a Euclidean proposition.

And Bertrand Russell (1872-1970) had his own fond memories of the Elements. In his autobiography, Russell penned this remarkable recollection:

At the age of eleven, I began Euclid, with my brother as tutor. This was one of the great events of my life, as dazzling as first love.

Euclid’s great genius was not so much in creating a new mathematics as in presenting the old mathematics in a thoroughly clear, organized, and logical fashion.

This is no small accomplishment.

It is important to recognize the Elements as more than just mathematical theorems and their proofs; after all, mathematicians as far back as Thales had been furnishing proofs of propositions. Euclid gave us a splendid axiomatic development of his subject, and this is a critical distinction.

He began the Elements with a few basics: 23 definitions, 5 postulates, and 5 common notions or general axioms. These were the foundations, the “givens,” of his system. He could use them at any time he chose. From these basics, he proved his first proposition. With this behind him, he could then blend his definitions, postulates, common notions, and this first proposition into a proof of his second. And on it went.

Consequently, Euclid did not just furnish proofs; he furnished them within this axiomatic framework. The advantages of such a development are significant. For one thing, it avoids circularity in reasoning.

Each proposition has a clear, unambiguous string of predecessors leading back to the original axioms.

Those familiar with computers could even draw a flow chart showing precisely which results went into the proof of a given theorem. This approach is far superior to “plunging in” to prove a proposition, for in such a case it is never clear which previous results can and cannot be used. The great danger from starting in the middle, as it were, is that to prove theorem A, one might need to use result B, which, it may turn out, cannot be proved without using theorem A itself. This results in a circular argument, the logical equivalent of a snake swallowing its own tail; in mathematics it surely leads to no good.

But the axiomatic approach has another benefit.

Since we can clearly pick out the predecessors of any proposition, we have an immediate sense of what happens if we should alter or eliminate one of our basic postulates. If, for instance, we have proved theorem A without ever using either postulate C or any result previously proved by means of postulate C, then we are assured that our theorem A remains valid even if postulate C is discarded. While this might seem a bit esoteric, just such an issue arose with respect to Euclid’s controversial fifth postulate and led to one of the longest and most profound debates in the history of mathematics.

Thus, the axiomatic development of the Elements was of major importance. Even though Euclid did not quite pull this off flawlessly, the high level of logical sophistication and his obvious success at weaving the pieces of his mathematics into a continuous fabric from the basic assumptions to the most sophisticated conclusions served as a model for all subsequent mathematical work.

To this day, in the arcane fields of topology or abstract algebra or functional analysis, mathematicians will first present the axioms and then proceed, step-by-step, to build up their wonderful theories.

It is the echo of Euclid, 23 centuries after he lived.

Journey Through Genius: THE GREAT THEOREMS OF MATHEMATICS
William Dunham

Image Α: https://en.wikipedia.org/wiki/Euclid%27s_Elements

Image Β: https://pixels.com/featured/euclid-jazzberry-blue.html?product=art-print

Image C: https://cdn.britannica.com/46/8446-050-BC92B998.jpg

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