On a personal note, immersing myself in this book—its exploration of infinity and a somewhat outdated cosmology—creates a welcome distance from the constant stream of everyday impressions, such as the political madness of the Russia–Ukraine conflict. Thinking back to what I wrote yesterday, I’m reminded that the deepest, and perhaps truly important, events are not the loudest ones. Books that are so absorbing that the loudest or most obvious happenings around us temporarily lose their volume or their apparent significance have a special value.
I also find that many Russian scholars from the Soviet era wrote exceptionally clear books—even in English translation. One that immediately comes to mind is A. Khinchin’s remarkably lucid introduction to mathematical analysis.
This book consists of four chapters. I will collect notes for each chapter here, and once I have finished all four, I will post a full review.
“One of the most vital questions that ancient Greek philosophers contended with was the structure of the world in the small . Daily experience showed that a loaf of bread could be shared by two, three, or at most ten people, and it could be broken into some ten thousand crumbs. Could the loaf of bread be further subdivided? And is there a limit to divisibility of material objects? Experience alone could not supply an answer to this question, and so the question of the limits of divisibility of objects shifted from the realm of experience to that of speculation.”
"The fifth reason - the reason Aristotle regarded as the weightiest - was that there are no bounds to thought. Specifically, there are no bounds on numbers, or on mathematical magnitudes, or on what is beyond heaven. And if what is beyond heaven is infinite, then there are many worlds."
"The first arises as a result of successive and unlimited addition of new objects, and the second is the result of delving indefinitely deeper into the structure of an object under investigation."
"Unlike Plato, who thought that the world was the work of a Demiurge (creator), Aristotle claimed that it was not created and was eternal."
"To avoid the use of the infinite, Euclid's predecessor, the Greek mathematician Eudoxus formulated an axiom which, in effect, denied the existence of infinitely small and infinitely large magnitudes."
"Cusa developed the doctrine of the maximum, that is something that cannot be ex ceeded."
"But times were changing. To solve practical problems scientists found it necessary to apply methods forbidden by Aristotelian science and to use indivisible and infinitely small magnitudes."
"Toward the end of the 17th century Newton and Leibniz independently systematized the methods of solution of a tremendous variety of problems, methods based on the use of infinitely small and infinitely large magnitudes."
“In astronomy, physics and mathematics, the end of the 17th century witnessed the triumph of ideas connected, in one way or another, with the use of the infinite”
“Einstein wrote: Newton found that observed geometric magnitudes (distances between material points) and their changes in time do not, in a physical sense, fully characterize motion . . . Thus, in addition to masses and to distances between points that change in time there exists something else that determines the occurring events; this "something" he took to be the relation to absolute space.”
“The students and followers of Newton and Leibniz used the vague concepts of the infinitely small and infinitely large, full of mystery, to sol ve the most complex problems of astronomy, physics and mechanics. They proceeded recklessly. They unceremoniously added infinitely many terms without pausing to ask whether or not the rules of operation applicable to finite sums carried over to infinite sums.”
“But at the end of the 18th century came the first signs of trouble. Cases began to accumulate where incorrect application of infinitely small and infinitely large magnitudes led to paradoxes. As a result, in the beginning of the 19th century these magnitudes were banished from mathematics and replaced by the idea of limit. In this the works of Abel , Cauchy, and Gauss, the "prince ofmathematicians," played a collective role. The following excerpt from Gauss' letter to Schumacher, written in 1 83 1 , is typical of his view of the infinite:
"I object to the use of an infinite magnitude as something completed; this is never admissible in mathematics. One must not interpret infinity literally when, strictly speaking, one has in mind a limit approached with arbitrary closeness by ratios as other things increase without bounds."
Another area where complications arose was cosmology. The natural assumptions about the uniform distribution of stars in infinite space led unexpectedly to a paradox. It turned out that their collective brightness would be thesame as if a Sun glittered at every point in the sky.
It was an image an Indian poet had in mind many centuries earlier when he exclaimed:
The sky above would shine
With boundless and awesome force
If a thousand Suns at once flashed in it."
“Riemann dealt with the problem of curvature of space in his inaugural lecture of 1854. At that time, the (laudable) custom was that a beginning instructor was expected to present a lecture before the members of the faculty who would then be in a position to judge his teaching abilities. Riemann offered a few topics for such a lecture and Gauss selected the one that interested him most - "On the hypotheses which lie at the foundations of geometry." It is safe to assume that the listeners were not greatly impressed by Riemann's pedagogical talent. The only listener who completely understood Riemann's lecture was Gauss.”
“Whereas, as noted earlier, in Newton 's physics space was completely independent of the matter in it, the new theory ruled out the very existence of empty, that is field-free, space. Also, it turned out that space and time cannot exist independently, but only in a state of inseparable connection with one another, and only as a structural property of a field.”
"After the appearance of dynamic models of the Universe there arose a number of questions now pondered by theoretical physicists : When did the scattering of the galaxies begin and what preceded it?Will the scattering persist forever or will there be a period of contraction? and so on. Today most scientists agree that many billions of years ago the Universe was in a superdense state. It appears that the density of this clot was greater than the density of matter in atomic nuclei - that is, greater than ... g/cm³.
Some computations give densities of the order of ... g/cm^3 . One thing is clear, namely that matter was in a state completely unknown to today 's physicists, and that neither gravitational laws nor quantum theory were applicable to it. Of all these theories (including the "big bang") the eminent physicist L.N. Brillouin said that "all this is too beautiful to be true and too unlikely to be believed." Nevertheless, modern science has no other theory that explains the multitude of puzzling phenomena in the Universe."
“Another question of interest to scientists is, of course, the future of theUniverse. Here there are two possible scenarios: either the Universe will continue to expand indefinitely and at some very distant time the galaxies will have drifted so far apart that the inhabitants of one won'tt be able to see any of the others, or at some moment the scattering of the galaxies will be arrested and they will begin to come closer to one another. Computations show that which of these tendencies will prevail depends on the average density of matter in the Universe. Computations show that which of these tendencies will prevail depends on the average density of matter in the Universe. If this density is less than ... , then expansion will continue forever. Otherwise, at some moment, the Universe will begin to contract. Some scientists believe that the contraction will continue until the Universe will again achieve the state of a superdense clot and the same chain of events will again unfold. One cannot help but remember the words of Heraclitus about the fire that regularly blazes up and is regularly extinguished."
"And what, in fact, is the density of matter in the Universe? What makes it difficult to answer this question is the possibility that a great part of matter is in a state that makes observations impossible."
"At the moment there are no conceivable methods of disproving the claim that our Universe is closed and
that its size does not exceed 10^25 km."
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grok:
"Chapter One of In Search for Infinity traces humanity’s long, uneasy confrontation with the idea of the infinite—from ancient philosophy through 20th-century cosmology and physics. It shows how the infinite repeatedly appeared as both an indispensable tool and a source of paradoxes, forcing science to alternately embrace, banish, and finally rigorize it.
Key historical stages
- Ancient Greece and the birth of the problem
- Questions about the cosmos (“Is the universe bounded? Did it have a beginning?”) and about matter (“Is divisibility unlimited?”) pushed thought beyond experience into speculation.
- Aristotle rejected actual infinity but accepted potential infinity. He argued that the world is eternal (not created), and that thought itself has no bounds—numbers, magnitudes, and what lies “beyond heaven” can grow without limit.
- Classical mathematics avoids the infinite
- Euclid and Eudoxus deliberately constructed geometry without infinitely large or infinitely small quantities.
- Medieval thinkers like Nicholas of Cusa introduced the idea of a maximum that cannot be exceeded, but still kept infinity theological rather than scientific.
- 17th century: The reckless triumph of the infinite
- Practical needs (ballistics, astronomy, mechanics) forced scientists to use infinitely small and infinitely large magnitudes despite philosophical prohibitions.
- Newton and Leibniz independently created the calculus, systematizing these “vague and mysterious” concepts. Their followers applied infinitesimal methods with enormous success and little rigor—adding infinite series, discarding infinite terms, etc., “recklessly” but fruitfully.
- Late 18th–early 19th century: Crisis and purification
- Paradoxes began to pile up (e.g., divergent series giving nonsense results).
- Mathematicians (Abel, Cauchy, Gauss) banished actual infinitesimals, replacing them with the rigorous concept of limit.
- Cosmology and physics: New infinities, new paradoxes
- Assuming an infinite, uniform distribution of stars led to the famous “photometric paradox” (the night sky should be infinitely bright—an image anticipated centuries earlier by an Indian poet’s vision of a thousand suns).
- Newton’s absolute space and time were challenged by relativity: Einstein showed that space-time is not an independent container but a structural property of gravitational fields; empty space cannot exist.
- 19th–20th century geometry and cosmology
- Riemann’s 1854 lecture laid the foundations for curved spaces.
- Dynamic (expanding) cosmological models raised questions about the origin and fate of the universe.
- The Big Bang picture emerged: billions of years ago the universe was an ultra-dense clot (density perhaps 10⁹³ g/cm³ or higher), a state where known physics breaks down.
- The future depends on average density: below a critical value → eternal expansion; above → eventual contraction and possible cyclic rebirth (echoing Heraclitus’ eternal fire).
- Measuring cosmic density is hampered by possible “dark” (unobservable) matter. The universe could even be closed and no larger than ~10²⁵ km without contradicting observations.
Core message of the chapter
The concept of infinity has never been a peaceful philosophical luxury; it has been forced upon science by the stubborn refusal of nature to fit into finite boxes. Each time scientists tried to eliminate infinity, new phenomena demanded its return in a more sophisticated form. By the mid-20th century, infinity—tamed by limits, curvature, and field theory—had become indispensable for describing atoms, stars, and the universe itself, even as it continued to generate deep unsolved questions about beginnings, endings, and the ultimate structure of reality."
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