The Foundations of Mathematics: A Contribution to the Philosophy of Geometry

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By Charlotte Girard Posted on Nov 15, 2025
In Category - Philosophy
Carus, Paul, 1852-1919 Carus, Paul, 1852-1919
English
Ever wonder if math is something we discover or something we invent? In 1908, Paul Carus tackled this head-on in a book that's less about solving equations and more about asking the big questions. He asks: Are the rules of geometry just clever human ideas, or are they fundamental truths about the universe waiting to be found? This isn't a dry textbook. It's a philosophical showdown, written when Einstein's ideas were just starting to shake the world. It's a surprisingly readable trip into the mind of a thinker trying to pin down the very ground our logic stands on.
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first to create an interest in this important subject. Since then Dr. Seiss, of Philadelphia, has endeavoured to popularize her work on the other side of the Atlantic; and brief references have been made to the subject in such books as _Moses and Geology_, by Dr. Kinns, and in _Primeval Man_; but it was felt, for many reasons, that it was desirable to make another effort to set forth, in a more complete form, the _witness of the stars to prophetic truth_, so necessary in these last days. To the late Miss Rolleston, however, belongs the honour of collecting a mass of information bearing on this subject; but, published as it was, chiefly in the form of _notes_, unarranged and unindexed, it was suited only for, but was most valuable to, the student. She it was who performed the drudgery of collecting the facts presented by Albumazer, the Arab astronomer to the Caliphs of Grenada, 850 A.D.; and the Tables drawn up by Ulugh Beigh, the Tartar prince and astronomer, about 1450 A.D., who gives the Arabian Astronomy as it had come down from the earliest times. Modern astronomers have preserved, and still have in common use, the ancient names of over a hundred of the principal stars which have been handed down; but now these names are used merely as a convenience, and without any reference to their significance. This work is an attempt to popularize this ancient information, and to use it in the interests of truth. For the ancient astronomical facts and the names, with their signification, I am, from the very nature of the case, indebted, of course, to all who have preserved, collected, and handed them down; but for their interpretation I am alone responsible. It is for the readers to judge how far my conclusions are borne out by the evidence; and how far the foundation of our hopes of coming glory are strengthened by the prophecies which have been written in the stars of heaven, as well as in the Scriptures of truth. For the illustrations I am greatly indebted to Jamieson’s _Celestial Atlas_, 1820; Flammarion’s _L’Étoiles_; Sir John W. Lubbock’s _Stars in Six Maps_, 1883; and to the late Mr. Edward J. Cooper’s _Egyptian Scenery_, 1820. For the general presentation and arrangement of the Constellations I am responsible, while for the drawings my thanks are due to my friend Miss Amy Manson. It is the possession of “that blessed hope” of Christ’s speedy return from Heaven which will give true interest in the great subject of this book. No one can dispute the antiquity of the Signs of the Zodiac, or of the Constellations. No one can question the accuracy of the ancient star‐names which have come down to us, for they are still preserved in every good celestial atlas. And we hope that no one will be able to resist the cumulative evidence that, apart from God’s grace in Christ there is no hope for sinners now: and apart from God’s glory, as it will be manifested in the return of Christ from Heaven, there is no hope for the Church, no hope for Israel, no hope for the world, no hope for a groaning creation. In spite of all the vaunted promises of a religious World, and of a worldly Church, to remove the effects of the curse by a Social Gospel of Sanitation, we are more and more shut up to the prophecy of Gen. iii. 15, which we wait and long to see fulfilled in Christ as our only hope. This is beautifully expressed by the late Dr....

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This isn't a story with characters in the usual sense. The main "character" here is the idea of geometry itself. Paul Carus, a philosopher and editor, sets out to explore the foundation of mathematical certainty. The plot is the argument: he examines different viewpoints on whether geometric truths (like the angles of a triangle always adding up to 180 degrees) are absolute, unchanging facts of reality, or if they are useful constructs built by human minds. He guides the reader through this debate, which was heating up as new, non-Euclidean geometries challenged centuries of assumed truth.

Why You Should Read It

What's fascinating is how personal this gets. Carus isn't just reporting; he's wrestling with the ideas. You feel the excitement and tension of someone trying to lock down a slippery concept. It makes you look at the world differently—that simple triangle drawn on a page suddenly holds a deep mystery. Reading it today, with our knowledge of relativity and curved spacetime, adds another layer. You see an intelligent man grappling on the edge of a scientific revolution he couldn't fully see coming.

Final Verdict

Perfect for curious readers who enjoy philosophy or the history of science, but maybe find modern academic texts a bit dense. It's for anyone who's ever stared at a math problem and wondered, "But why is this *true*?" This book won't teach you new formulas, but it will give you a profound appreciation for the ancient, human struggle to understand the rules of our universe. Think of it as a thoughtful, accessible conversation with a sharp mind from over a century ago.



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