A Budget of Paradoxes, Volume II

The Story

The intellectual journey of these volumes begins with an exploration of the thin line between rigorous science and fervent speculation. The narrator, a man of deep mathematical learning, opens by addressing the philosophical atheists of the previous century, notably the famous encounter between Laplace and Napoleon. When the Emperor challenges the scientist on the absence of a Creator in his work on the universe, Laplace dismisses the idea as an unnecessary hypothesis. This sets the tone for the entire work: a focus on the necessity of evidence and the danger of allowing personal beliefs to override logical demonstration. The narrative quickly shifts from the high-minded halls of academic philosophy to the cluttered desks of amateur thinkers who bombard the establishment with their "paradoxes."

As the text progresses, the reader is drawn into a series of case studies detailing the "circle-squarers," the "angle-trisectors," and those who believe the moon lacks rotary motion. The narrator does not simply mock these figures; he documents them. He recounts how men like Jellinger Symons or John Davey Hailes write letters, publish pamphlets, and even issue formal challenges to the scientific community, all grounded in a fundamental misunderstanding of basic mechanical and mathematical principles. The arc of these accounts is often circular—the same errors resurface decade after decade, passed from one self-proclaimed genius to the next, proving that ignorance of a proof is rarely a barrier to the desire to proclaim a discovery.

One of the turning points in this narrative occurs when the author shifts his focus from the paradoxers themselves to the institutional reaction they provoke. He describes the exhaustion of the Astronomer Royal, who attempts to correct these speculators, only to find that some minds are impervious to geometric evidence. The author notes that "men of science" are frequently accused of being part of a guild—the "craftsmen"—who guard their secrets behind a veil of terminology. Yet, the author argues that the true difference between a mathematician and a paradoxer is not merely knowledge, but the capacity to accept that one can be wrong. He uses the analogy of the "dunce’s cap" and the "critic’s rod," detailing how his own engagement with these individuals often leads to them feeling persecuted, which in turn fuels their zeal for further argument.

Moving through the final sections, the narrative expands into the history of scientific correspondence. The author highlights the importance of collections like the Macclesfield letters, which document the exchange of information among early modern thinkers. He demonstrates that scientific progress is a collaborative, messy, and historical endeavor, contrasting this with the isolated, singular focus of the modern paradoxer. The book ends not with a final victory over error—as the author admits that these paradoxes are likely to continue indefinitely—but with a structured taxonomy of folly. By cataloging these "moths, beetles, and butterflies" of the intellect, he seeks to bring order to a chaotic field, suggesting that the study of human error is itself a valuable, if sometimes wearying, pursuit.

The People

• Augustus De Morgan: The narrator and editor, a mathematician who views the scientific community as a repository of hard-won, demonstrable truth. He acts as a patient, if sardonic, guide who treats the eccentricities of others as a subject for serious study.
• Laplace: A celebrated French mathematician who serves as the embodiment of scientific independence. His blunt dismissal of the "Creator" as an unnecessary hypothesis defines the intellectual standard against which the more mystical paradoxers are judged.
• James Smith: A quintessential "paradoxer" who spends his energy trying to square the circle. He is portrayed as a man of means, energy, and charm, who, despite being entirely wrong, becomes a leader of a group of followers because he refuses to acknowledge the validity of established mathematical proofs.
• John Davey Hailes: An amateur astronomer who insists on his own version of the universe. He provides the book with some of its most bizarre moments, including his formal challenges to the Royal Society and his insistence on the truth of biblical miracles as a substitute for orbital mechanics.
• Jellinger Symons: An inspector of schools who causes a public controversy by claiming the moon has no rotation. He serves as an example of how a person in a position of authority can become blinded by a misunderstanding of geometry, leading to a public, protracted dispute with legitimate scientists.

In Its Own Voice

When describing the inherent difficulty of convincing a person who is already committed to their own logic, the author writes:

"To see a thing so clearly as to be satisfied that all who say they do not see it are telling wilful falsehood, is the nature of man."

Regarding the persistent, misguided belief that a simple error in the measurement of a circle can somehow be overlooked for the sake of a "greater truth," the author explains:

"If a mathematician cannot find his way round the circle without a relative error four times as big as a stockbroker's commission, he must needs be dreadfully out in his attempt to predict the time of passage of the moon."

On the nature of his own involvement with the very people he criticizes, he remarks:

"A single moth, beetle, or butterfly, may be a paltry thing; but when a cabinet is arranged by genus and species, we then begin to admire the infinite variety of a system constructed on a wonderful sameness of leading characteristics."

What It's Really About

At its heart, this work is an exploration of the boundaries of rationality. It is not merely a collection of mathematical errors, but a study of the psychological mechanism that drives individuals to reject established consensus in favor of a private, often incoherent, system of belief. The central tension lies in the nature of authority and evidence. The author is deeply concerned with how people determine what is "true" in an age where scientific advancement is accelerating, yet traditional, mystical, or purely intuitive ways of thinking remain deeply rooted.

The book argues that there is a fundamental "method of knowing" that is distinct from simply holding an opinion. When the author discusses the "circle-squarers," he is addressing the frustration of a society that allows the weight of evidence to be ignored. He posits that if a proof has been accepted by the scientific community for a century, the burden of proof rests entirely on the challenger to identify the specific flaw in that proof. The tragedy of the paradoxer is that they rarely understand the history of the argument they are trying to dismantle; they attempt to reinvent the wheel, often without realizing that the wheel has been refined over generations.

Furthermore, the book touches upon the themes of social and linguistic tribalism. The author views language, logic, and scientific notation as tools that, when used properly, facilitate human understanding. However, he observes how these same tools can be subverted by those who prefer the comfort of "Big Shibboleth" or "Little Shibboleth." The inclusion of religious and historical paradoxes alongside mathematical ones suggests that the impulse to create a closed, defensive system of thought—whether it concerns the movement of the moon or the interpretation of the Book of Revelations—is a universal human trait. The author attempts to treat the scientific community as a culture, and the paradoxers as a persistent, perhaps necessary, counter-culture. By placing these figures in a "Budget," he validates their existence as a phenomenon while simultaneously demonstrating the futility of their scientific claims. Ultimately, the work is a plea for intellectual humility and a call for a deeper appreciation of the rigorous, collective process of scientific discovery.

Why Read It Today

Today, we live in an era of abundant information, yet we remain just as prone to the allure of "alternative" facts as the Victorian subjects of this book. Reading these volumes feels like peering into a well-organized cabinet of curiosities. The author’s voice is remarkably modern—wry, sharp, and deeply suspicious of those who claim "secret" knowledge or intuitive breakthroughs that bypass hard work. If you enjoy the history of science or have ever wondered why some people seem constitutionally unable to accept a settled fact, you will find this book both comforting and enlightening.

However, be prepared for the book's density. It is not a modern pop-science narrative; it is a dense, nineteenth-century collection of essays that assumes a high level of classical education. The author frequently references figures, theological debates, and mathematical concepts that will be unfamiliar to the general reader. You will encounter footnotes upon footnotes, Latin phrases, and long, winding sentences that reflect the intellectual rigor of the era. The dialect and the specific grievances of the 1860s can feel distant, and the humor—often dry and deeply buried in the irony of the author’s tone—requires patience to uncover.

Yet, there is a rewarding rhythm to the text. You will leave the book not with a series of solutions, but with a sharpened sense of how to identify the patterns of fallacious reasoning. The author’s ability to remain "wide awake" to the folly of his subjects, even while treating them with a certain degree of anthropological respect, is what stays with you. It is a masterclass in how to be a critic: firm in one’s principles, yet never losing the ability to find the human comedy in the midst of a stubborn, persistent error. For the curious reader willing to navigate the thickets of Victorian prose, this book offers a timeless lesson in the value of skepticism and the endurance of human pride.