Carl B. Boyer's "A History of Mathematics" presents mathematics as a discipline that progresses steadily upward and forward when viewed through the lens of maths and logic, rather than through the cyclical patterns of human history.
The book challenges our perspective on technological acceleration and the singularity argument, which wrongly assumes human experience is merely a back-and-forth of problem solving and finding, when in truth, it's about living the human experience itself - standing next to that which is true and beautiful.

Here are some quotes:

"the squaring of the circle, the duplication of the cube, the trisection of the angle, the ratio of incommensurable magnitudes, the paradox of motion, and the validity of infinitesimal methods." These problems can be associated, though not exclusively, with ancient thinkers like Hippocrates, Archytas, Hippias, Zeno, and Democritus.

The ancient schools with their astonishing achievements in geometric, arithmetic, and trigonometric procedures, as well as astronomical observations, have inspired considerable speculation concerning transmissions and influence. As Boyer notes, "Until now there is inadequate documentation to support any of the related major conjectures. There is however, a great deal to be learned from recent translations of these and prior texts."

The 17th century, which Boyer calls "the century of genius," produced remarkable mathematicians like Torricelli. However, as the author observes, "The discoveries of great mathematicians do not automatically become part of mathematical tradition. They may be lost to the world unless other scholars understand them and take enough interest to look at them from various points of view, clarify and generalize them and point out their implications."

Boyer explores Daniel Bernoulli's distinction between mathematical expectations and moral expectations, noting his assumption that "a small increase in a person's material means cause an increase in satisfaction that is inverse proportion to the means." This leads to the equation dM=kdPPdM=kPdP​, where M is the moral fortune, P is the physical fortune, and k is the constant of proportionality, concluding that "as the physical fortune increases geometrically, the more fortunes increase arithmetically."

The book highlights Jacobi's advice to "always invert" as "the secret success in mathematics," noting how this principle helped him achieve simplicity through inversions of functional relationships in elliptic integrals. Boyer divides the history of logic into three categories: Greek logic, scholastic logic, and mathematical logic, explaining how each stage progressively transformed logical formulas from ordinary language to specialized artificial languages.

Boyer traces the revolution in geometry to when Gauss, Lobachevsky, and Bolyai "freed themselves from preconceptions of space," and similarly notes how "the thorough-going arithmetization of analysis became possible only when, as Weierstrass foresaw, mathematicians understood that the real numbers are to be viewed as intellectual structures, rather than as intuitively given magnitudes inherited from Euclid."

In a historical irony, Boyer points out how "while Bourbaki and many other pure mathematicians pursued the goal of substituting ideas for calculations, engineers and applied mathematicians developed computers that would revive interest in numerical and arithmetic techniques, and sharply affect the composition of many departments of mathematics." He also notes that in the first half of the 20th century, "the history of computing machines involved more statisticians, physicists, and electrical engineers than mathematicians."

Carl B. Boyer's "A History of Mathematics" presents mathematics as a discipline that progresses steadily upward and forward when viewed through the lens of maths and logic, rather than through the cyclical patterns of human history.
The book challenges our perspective on technological acceleration and the singularity argument, which wrongly assumes human experience is merely a back-and-forth of problem solving and finding, when in truth, it's about living the human experience itself - standing next to that which is true and beautiful.

Here are some quotes:

"the squaring of the circle, the duplication of the cube, the trisection of the angle, the ratio of incommensurable magnitudes, the paradox of motion, and the validity of infinitesimal methods." These problems can be associated, though not exclusively, with ancient thinkers like Hippocrates, Archytas, Hippias, Zeno, and Democritus.

The ancient schools with their astonishing achievements in geometric, arithmetic, and trigonometric procedures, as well as astronomical observations, have inspired considerable speculation concerning transmissions and influence. As Boyer notes, "Until now there is inadequate documentation to support any of the related major conjectures. There is however, a great deal to be learned from recent translations of these and prior texts."

The 17th century, which Boyer calls "the century of genius," produced remarkable mathematicians like Torricelli. However, as the author observes, "The discoveries of great mathematicians do not automatically become part of mathematical tradition. They may be lost to the world unless other scholars understand them and take enough interest to look at them from various points of view, clarify and generalize them and point out their implications."

Boyer explores Daniel Bernoulli's distinction between mathematical expectations and moral expectations, noting his assumption that "a small increase in a person's material means cause an increase in satisfaction that is inverse proportion to the means." This leads to the equation dM=kdPPdM=kPdP​, where M is the moral fortune, P is the physical fortune, and k is the constant of proportionality, concluding that "as the physical fortune increases geometrically, the more fortunes increase arithmetically."

The book highlights Jacobi's advice to "always invert" as "the secret success in mathematics," noting how this principle helped him achieve simplicity through inversions of functional relationships in elliptic integrals. Boyer divides the history of logic into three categories: Greek logic, scholastic logic, and mathematical logic, explaining how each stage progressively transformed logical formulas from ordinary language to specialized artificial languages.

Boyer traces the revolution in geometry to when Gauss, Lobachevsky, and Bolyai "freed themselves from preconceptions of space," and similarly notes how "the thorough-going arithmetization of analysis became possible only when, as Weierstrass foresaw, mathematicians understood that the real numbers are to be viewed as intellectual structures, rather than as intuitively given magnitudes inherited from Euclid."

In a historical irony, Boyer points out how "while Bourbaki and many other pure mathematicians pursued the goal of substituting ideas for calculations, engineers and applied mathematicians developed computers that would revive interest in numerical and arithmetic techniques, and sharply affect the composition of many departments of mathematics." He also notes that in the first half of the 20th century, "the history of computing machines involved more statisticians, physicists, and electrical engineers than mathematicians."