Every age has its own spiritual outlook, its own picture of the world mirroring the spirit of the time.  Why is it so important for European culture to measure its time alongside that of the antique world?  The wellsprings of all manifestations of European culture doubtless lie in antiquity.  But for more than two millenia something else in the age has forced people to turn again and again to antiquity’s spirit.  This “something” is a worldview—the very consciousness of a person of the antique age—reflecting a sense of harmony in the universe and a human being as part of this harmony within the world.

A “joyous, childish sense of the world” is frequently spoken of as an attribute of the antique consciousness.  A. F. Losev wrote, “…people of antiquity saw any truth and beauty as existing within the limits of the cosmos—a cosmos that was material, physical, palpable, spiritually animated and eternally mobile…  At the base of such a cosmos were proposed ideas, and this was true not only for idealists, but for the materialist Democritus, who also called his atoms ideas.  And in itself this Greek word, ‘idea’ (from the same root as the Latin ‘video’ and the Russian ‘vidyet,’ to see) meant ‘what is visible’—whether to the eyes or the mind.  …This physical cosmos was called God.  …The individual gods spoken of in mythology were only principles of individual aspects of what was always the same palpably material and only possible reality of the cosmos.”

Humanity was to lose this sense of the unity of the cosmos for many, many centuries, and only now, in a few areas of knowledge, has the reverse process begun—the restoration of a sense of the world’s wholeness.  This reverse process in many ways rests on what the ancients achieved.

 

Time in ancient Greece

 

Time is the phenomenon that yields to comprehension with the greatest difficulty.  In a burdensome and tortuous process, contemporary consciousness has caused society to reject the understanding of time that prevailed in the sciences in Europe in past centuries.  Time in the contemporary consciousness is understood very differently from time in the consciousness of past centuries, in a manner closely approaching some of the ancients’ concepts.  In the present day, parallels between the cosmological models of early epochs and modern scientific hypotheses constitute an object of special study.

Two among the many who have written on what the people of antiquity thought about time are Oswald Spengler and A. F. Losev.  Spengler makes an analysis of antique Greek culture in his famous Decline of the West.  In part, his study concerns two possible forms of humans’ relation to nature and history as phenomena having a temporal scale.  It makes a great difference, says Spengler, whether a person lives with a constant impression of his life as an element in a much wider narrative extending over hundreds and thousands of years or whether he senses it as a specific, self-contained, isolated thing.  For the latter type, notes Spengler, there likely is no such thing as world history.  And imagine what would happen if an entire nation were based upon this idea.  How would it perceive reality?  The world?  Life?  Spengler offers an analysis of how the Hellenic individual would view himself.  In the earthly consciousness of the Hellenes, “not merely the personal but the past overall was transmuted into a kind of…immobile, mythically formed underlayer for the constantly flowing present, and thus the story of Alexander the Great began—even before his death—to merge in the antique consciousness with the legend of Dionysus, and Caesar, at least, found it reasonable to claim himself a descendant of Venus…”  Antique culture possesses no memory, says Spengler.  All was filled with that “‘pure present’—so often praised by Goethe—in all manifestations of antique life…  What the Greeks called cosmos was an image of a world not becoming, but materially being.  It follows that the Greeks themselves were a people that had never become, but always were.”  Spengler puts forth a series of arguments to illustrate the latter assertion’s correctness.

History is known to be an interpretive art.  It is immediately evident that the conclusions in Losev’s analysis of time in antique culture strongly differ from those of Spengler.  Losev makes mention of Judaism, Persian religion, the teachings of Buddha, Chinese Taoism and Confucianism, Egyptian religion and Greek myths, observing that each noted system has its own relation to time (or rather space and time).  Only the Greek religion, writes Losev, gave a genuine sense of time as the present.  Here the eternal and the temporal fuse into a single, complete present—moreover, the one is not sacrificed for the other, but remains free and untouched.  It would be simply ridiculous, writes Losev, to understand time, in various religions and mythologies, in the style of the new European physics—that is, time as homogenous and endless, empty and dark.  One can rest assured that not only in various religions, but now, too, no one has ever experienced time in such a way.

As Losev observes, each of us has experienced certain kinds of pits and ruptures in life.  Past apocalyptic anticipations may be attribute namely to temporal condensation near an age’s end.  Time, as space, has its folds and tears.  Three seconds can be experienced as a full year, and a year as three seconds.  Religious ecstasy is characterized namely by the stoppage or coagulation of time, by the compression of past and future times into a single, indivisible point of present.  The cosmos as a whole is diverse in its temporal structure.  Let us compare, for example, time for a human and time in the life of some kind of insect.  A multitude of different times exist, they are compressible and expandable and they possess their own structural configurations.  In The Ancient Cosmos, Losev notes that the Neo-Platonians conceived of the symmetrical and concentric arrangement, around a single center—Earth—of a diverse concentration of time and space.  The latter phrase comes quite close to the ideas of modern physics…

The field of science contains concrete manifestations of all of the above.  Antique numeric thinking examines things as they are, according their magnitudes outside time and simply in the present.  This led to Euclidean geometry, mathematical statistics and a comprehensive creative system based on theories about conic sections.  As for us, we examine things in the plan of their coming into being and interaction as functions.  This has led to dynamics, analytical geometry and, from this, differential calculus.  In Greek mathematics, time is not encountered at all, and Greek physics—in being static rather than dynamic—neither knows a clock’s measure nor senses its absence.

Above are presented two points of view on the antique person’s sense of the world—from Losev and Spengler.  Both reflect one or the other extreme of the question being examined.  On the subject of varying opinions, Spengler writes, with brilliant humor, “The idealogue is horrified when anyone takes Hellenic financial problems seriously and, shall we say, in place telling us about the deep meanings of the Delphic oracle, picks the theme of the wide-ranging financial operations the oracle priests conducted with their accumulated riches.  The politician chuckles wisely over the one who squanders inspiration on the sacred formulas and vestments of Attic youths, rather than writing a book about class struggle in antiquity…”

The latter citation would make a good epigraph for any argument.

 

Can the world be comprehended through logical reasoning?

 

The modern state of our knowledge of the world has its basis in the culture of the so-called “classical” period, that is, antiquity.  In the course of the development of the history of learnedness, there is one thing in particular we owe to the Greeks.  We’ll turn for elucidation of this question to the history of the development of logical proof.

Zeno was Parmenides’ closest pupil.  He wrote only one book, presenting a series of assignments, or “aphorisms,” the goal of which lay in defending the teachings of Parmenides.  Zeno took apart his opponents’ theses and proved these theses lead to contradictions in logic.  We are informed that Zeno’s book consisted of forty such aphorisms, of which the best-known four, examined by Aristotle in the sixth chapter of his Physics, were “Dichotomy,” “Achilles,” “Arrow” and “Stadium.”  Their contents are widely known.  The aphorisms still draw mathematicians’ interest.  Aristotle proposed that in every aphorism Zeno allowed some sort of logical “lapsus,” but in reality the matter is not so simple.  The aphorisms “Dichotomy,” “Achilles” and “Arrow” are logically flawless and could not be solved by means of antique mathematics.  The results Zeno received are the same paradoxes at the heart of such concepts as continuum.  These paradoxes were explored in the course of working out the theory of manifolds.  Some see the aphorism “Stadium” as a forerunner to the principle of the relativity of motion.  Zeno’s reasoning is history’s first example of proof by pure logic.

The paradoxes of Zeno use pure logic to prove time and distance can be neither uninterrupted nor interrupted.  If time is uninterrupted, then a runner can never reach his goal.  If he is halfway there, he will need time to cover half of the remaining distance, and so on ad infinitum.  If time is uninterrupted, then an arrow cannot move, because it is located either at one point or the other and between them nothing exists.

We will require a second example, related to Euclid’s fifth postulate, to clarify the essence of the antique approach to comprehension of the world.  Euclid’s fundamental work, Elements, was written during the age of Hellenism, at the very flowering of Alexandrian science’s development.  In this work, the primary achievements of Greek mathematics were presented in a deductive-axiomatic form that subsequently became the pattern and ideal of scientific rigor.  This form was later used not only by mathematicians.  Spinoza wrote his Ethics with Euclid’s Elements before him.  The apotheosis of Euclidean geometry—in not only the substance but the character of its exposition—was Newton’s Mathematical Beginnings of Natural Philosophy.  What exactly did Euclid do?  Although the construction of theorems by means of logic was already a known art—Aristotle’s logic is the verbal equivalent of a geometrical method of proof—before Euclid, this process had not been unified around deduction from a set of axioms.  This was a substantial enough contribution that in one form or another to this day Euclid’s doctrine remains the foundation for instruction in geometry.  What is its essence?  Greek geometry (as personified by Euclid) began with fundamental propositions called axioms or postulates.  These propositions are considered the simplest, most incontestable laws of logic and geometry.  A few of them have the basic character of formal logic, such as the axiom stating that two volumes equal to the same third volume are equal to each other.  Others describe spatial relationships:  For example, the parallel axiom asserts that through every point P of a plane not lying on the same plane as line l will pass one and only one line that does not intersect l; this line is called the parallel of l.

This is Euclid’s famous fifth postulate—famous for lacking the self-evidence of the purely logical postulates of mathematics.  Whole generations of scholars have tried in every way possible to prove it cannot be violated.  In the 18th century, the Italian mathematician Saccari put great efforts into investigating various effects that would result from rejecting the parallel axiom, hoping sooner or later to arrive at some kind of contradiction in logic.  All his efforts proved futile.  The more he tried to find a contradiction among the effects of rejecting the axiom, the more the sum of the facts stemming from the rejection began to acquire their own meaning.  Moreover, all in all, the sum of facts gradually acquired the character of a geometry frighteningly odd, it seemed, by comparison with Euclidean geometry.  Nevertheless, the new geometry contained no internal contradictions.  At last, in the beginning of the 19th century, a group of scholars—the Hungarian mathematician Janos Bolyai, the Russian mathematician Lobachevskii and the German mathematician Gauss—came to the bold conclusion that rejecting the parallel axiom leads to no contradictions, but only indicates a transition to a new, “non-Euclidean” geometry.  The latter formed the foundation of the modern view of the world, together with quantum theory and Einstein’s theory of relativity.  That is, the contradiction in fact indicated Euclid’s fifth postulate as unsound, as it assumes the universe’s comprehensibility purely by means of logical reasoning.  The soundness of a so-called rational way of thinking rests namely on the ability to prove with the aid of arguments.  Here we question not the mind’s ability to prove, but the universe’s tendency to be comprehended by such reasoning, or the mind’s ability to comprehend the universe.  Greek scholars developed so-called argumentation based on general principles, and this was accompanied by the development of abstract reasoning.  This entire process was founded on the belief the universe is rational in essence and all its details can be traced through pure logic, beginning with fundamental principles.

This idea, simple at first glance, is laden with profound conclusions.

First, that the universe is truly rational, if the parallel axiom’s purely logical contradiction indicates its unsoundness.  The new geometry of Lobachevskii, as it’s called, bears great importance for the modern view of the world.  Its meaning for cosmology was revealed in 1922 by the ingenious Russian mathematician A. A. Friedmann.  That year, he published an article reporting his discovery of a solution to an equation of Einstein from which it followed that the Universe is expanding as time passes.  E. Hubble in 1929 reached the same conclusion through an experiment that revealed distant nebulas as receding.  With time as a definite element, the measurements found by A. A. Friedmann yield the space of Lobachevskii.  The space yielded by velocities in Einstein’s special theory of relativity is also Lobachevskii’s space.

Second, that Zeno’s paradoxes, also achieved through pure logic, show the true tendencies of space and time, still not studied adequately…

As it turns out, faith in the ability of the human mind, which in its day helped the Greeks begin to comprehend the world by means of pure logic, has at least twice received a brilliant affirmation.

The antique person’s faith in a rational basis for the Universe led to the development of philosophy and appearance of modern science’s foundations.  The remaining fragments of antique culture, to this day a source of delight, came into being due to yet another aspect of the antique person’s relation to the cosmos—one vividly embodied in the works of Aristotle, among others.  Losev writes that detailed acquaintance with Aristotle’s works can show the attentive reader that in the philosopher’s opinion, all things in existence are nothing other than a work of art—as a person, so the whole world.  This eloquent statement says all that is needed.

The “awakening of consciousness” in antiquity, examined in detail above, is known to have occurred at a so-called “axial time,” when similar processes also occurred in the cultures of other ancient peoples.  They happened independently of each other and nearly simultaneously (with an accuracy within two or three millenia).  This is the age of the 7th-5th centuries B.C.  In Greece, this period is defined by the concept of the “pre-Socratic.”  The same approximate time is marked by the rise of Chinese philosophy, of which the basic ideas—teachings about the elements and yin and yang, the primary forces of the universe—may in many ways be seen as closely related to the concepts of Hericlitus, Parmenides and Empedocles.  In India, at the beginning of the 6th century B.C., appears Siddhartha Gautama (Buddha)—the founder of one the most profound religious-philosophic doctrines.  In Iran, Zarathustra brought about the radical purging of mythological elements from ancient Iranian beliefs and imparted an elevated character to Iranian religion.  According to the most recent studies, namely the Iranian religious-philosophic perspective exercised the greatest influence on the thinking of the early pre-Socrateans.

The concept of an “axial” time, formulated by Karl Jaspers, shows that the depth and direction of ideas articulated at that time defined the entire spiritual movement of humanity for many centuries to come.  The subsequent evolution of European philosophy and science up to the present day (as with much done by humanity in this time) is based on the foundation laid in the “axial” time.  In the matter of developing a scientific approach to knowing the world, the Greeks, too, stayed within the bounds of questions posed by the “axial” time.  Beginning with the “axial” time, the paths of Eastern (India, China) and Western thought diverge to form two approaches to the comprehension of the world.  The approach chosen in antiquity led ultimately to the development of European civilization.

Every age has its own spiritual outlook, its own picture of the world mirroring the spirit of the time.  Why is it so important for European culture to measure its time alongside that of the antique world?  The wellsprings of all manifestations of European culture doubtless lie in antiquity.  But for more than two millenia something else in the age has forced people to turn again and again to antiquity’s spirit.  This “something” is a worldview—the very consciousness of a person of the antique age—reflecting a sense of harmony in the universe and a human being as part of this harmony within the world.

A “joyous, childish sense of the world” is frequently spoken of as an attribute of the antique consciousness.  A. F. Losev wrote, “…people of antiquity saw any truth and beauty as existing within the limits of the cosmos—a cosmos that was material, physical, palpable, spiritually animated and eternally mobile…  At the base of such a cosmos were proposed ideas, and this was true not only for idealists, but for the materialist Democritus, who also called his atoms ideas.  And in itself this Greek word, ‘idea’ (from the same root as the Latin ‘video’ and the Russian ‘vidyet,’ to see) meant ‘what is visible’—whether to the eyes or the mind.  …This physical cosmos was called God.  …The individual gods spoken of in mythology were only principles of individual aspects of what was always the same palpably material and only possible reality of the cosmos.”

Humanity was to lose this sense of the unity of the cosmos for many, many centuries, and only now, in a few areas of knowledge, has the reverse process begun—the restoration of a sense of the world’s wholeness.  This reverse process in many ways rests on what the ancients achieved.

 

Time in ancient Greece

 

Time is the phenomenon that yields to comprehension with the greatest difficulty.  In a burdensome and tortuous process, contemporary consciousness has caused society to reject the understanding of time that prevailed in the sciences in Europe in past centuries.  Time in the contemporary consciousness is understood very differently from time in the consciousness of past centuries, in a manner closely approaching some of the ancients’ concepts.  In the present day, parallels between the cosmological models of early epochs and modern scientific hypotheses constitute an object of special study.

Two among the many who have written on what the people of antiquity thought about time are Oswald Spengler and A. F. Losev.  Spengler makes an analysis of antique Greek culture in his famous Decline of the West.  In part, his study concerns two possible forms of humans’ relation to nature and history as phenomena having a temporal scale.  It makes a great difference, says Spengler, whether a person lives with a constant impression of his life as an element in a much wider narrative extending over hundreds and thousands of years or whether he senses it as a specific, self-contained, isolated thing.  For the latter type, notes Spengler, there likely is no such thing as world history.  And imagine what would happen if an entire nation were based upon this idea.  How would it perceive reality?  The world?  Life?  Spengler offers an analysis of how the Hellenic individual would view himself.  In the earthly consciousness of the Hellenes, “not merely the personal but the past overall was transmuted into a kind of…immobile, mythically formed underlayer for the constantly flowing present, and thus the story of Alexander the Great began—even before his death—to merge in the antique consciousness with the legend of Dionysus, and Caesar, at least, found it reasonable to claim himself a descendant of Venus…”  Antique culture possesses no memory, says Spengler.  All was filled with that “‘pure present’—so often praised by Goethe—in all manifestations of antique life…  What the Greeks called cosmos was an image of a world not becoming, but materially being.  It follows that the Greeks themselves were a people that had never become, but always were.”  Spengler puts forth a series of arguments to illustrate the latter assertion’s correctness.

History is known to be an interpretive art.  It is immediately evident that the conclusions in Losev’s analysis of time in antique culture strongly differ from those of Spengler.  Losev makes mention of Judaism, Persian religion, the teachings of Buddha, Chinese Taoism and Confucianism, Egyptian religion and Greek myths, observing that each noted system has its own relation to time (or rather space and time).  Only the Greek religion, writes Losev, gave a genuine sense of time as the present.  Here the eternal and the temporal fuse into a single, complete present—moreover, the one is not sacrificed for the other, but remains free and untouched.  It would be simply ridiculous, writes Losev, to understand time, in various religions and mythologies, in the style of the new European physics—that is, time as homogenous and endless, empty and dark.  One can rest assured that not only in various religions, but now, too, no one has ever experienced time in such a way.

As Losev observes, each of us has experienced certain kinds of pits and ruptures in life.  Past apocalyptic anticipations may be attribute namely to temporal condensation near an age’s end.  Time, as space, has its folds and tears.  Three seconds can be experienced as a full year, and a year as three seconds.  Religious ecstasy is characterized namely by the stoppage or coagulation of time, by the compression of past and future times into a single, indivisible point of present.  The cosmos as a whole is diverse in its temporal structure.  Let us compare, for example, time for a human and time in the life of some kind of insect.  A multitude of different times exist, they are compressible and expandable and they possess their own structural configurations.  In The Ancient Cosmos, Losev notes that the Neo-Platonians conceived of the symmetrical and concentric arrangement, around a single center—Earth—of a diverse concentration of time and space.  The latter phrase comes quite close to the ideas of modern physics…

The field of science contains concrete manifestations of all of the above.  Antique numeric thinking examines things as they are, according their magnitudes outside time and simply in the present.  This led to Euclidean geometry, mathematical statistics and a comprehensive creative system based on theories about conic sections.  As for us, we examine things in the plan of their coming into being and interaction as functions.  This has led to dynamics, analytical geometry and, from this, differential calculus.  In Greek mathematics, time is not encountered at all, and Greek physics—in being static rather than dynamic—neither knows a clock’s measure nor senses its absence.

Above are presented two points of view on the antique person’s sense of the world—from Losev and Spengler.  Both reflect one or the other extreme of the question being examined.  On the subject of varying opinions, Spengler writes, with brilliant humor, “The idealogue is horrified when anyone takes Hellenic financial problems seriously and, shall we say, in place telling us about the deep meanings of the Delphic oracle, picks the theme of the wide-ranging financial operations the oracle priests conducted with their accumulated riches.  The politician chuckles wisely over the one who squanders inspiration on the sacred formulas and vestments of Attic youths, rather than writing a book about class struggle in antiquity…”

The latter citation would make a good epigraph for any argument.

 

Can the world be comprehended through logical reasoning?

 

The modern state of our knowledge of the world has its basis in the culture of the so-called “classical” period, that is, antiquity.  In the course of the development of the history of learnedness, there is one thing in particular we owe to the Greeks.  We’ll turn for elucidation of this question to the history of the development of logical proof.

Zeno was Parmenides’ closest pupil.  He wrote only one book, presenting a series of assignments, or “aphorisms,” the goal of which lay in defending the teachings of Parmenides.  Zeno took apart his opponents’ theses and proved these theses lead to contradictions in logic.  We are informed that Zeno’s book consisted of forty such aphorisms, of which the best-known four, examined by Aristotle in the sixth chapter of his Physics, were “Dichotomy,” “Achilles,” “Arrow” and “Stadium.”  Their contents are widely known.  The aphorisms still draw mathematicians’ interest.  Aristotle proposed that in every aphorism Zeno allowed some sort of logical “lapsus,” but in reality the matter is not so simple.  The aphorisms “Dichotomy,” “Achilles” and “Arrow” are logically flawless and could not be solved by means of antique mathematics.  The results Zeno received are the same paradoxes at the heart of such concepts as continuum.  These paradoxes were explored in the course of working out the theory of manifolds.  Some see the aphorism “Stadium” as a forerunner to the principle of the relativity of motion.  Zeno’s reasoning is history’s first example of proof by pure logic.

The paradoxes of Zeno use pure logic to prove time and distance can be neither uninterrupted nor interrupted.  If time is uninterrupted, then a runner can never reach his goal.  If he is halfway there, he will need time to cover half of the remaining distance, and so on ad infinitum.  If time is uninterrupted, then an arrow cannot move, because it is located either at one point or the other and between them nothing exists.

We will require a second example, related to Euclid’s fifth postulate, to clarify the essence of the antique approach to comprehension of the world.  Euclid’s fundamental work, Elements, was written during the age of Hellenism, at the very flowering of Alexandrian science’s development.  In this work, the primary achievements of Greek mathematics were presented in a deductive-axiomatic form that subsequently became the pattern and ideal of scientific rigor.  This form was later used not only by mathematicians.  Spinoza wrote his Ethics with Euclid’s Elements before him.  The apotheosis of Euclidean geometry—in not only the substance but the character of its exposition—was Newton’s Mathematical Beginnings of Natural Philosophy.  What exactly did Euclid do?  Although the construction of theorems by means of logic was already a known art—Aristotle’s logic is the verbal equivalent of a geometrical method of proof—before Euclid, this process had not been unified around deduction from a set of axioms.  This was a substantial enough contribution that in one form or another to this day Euclid’s doctrine remains the foundation for instruction in geometry.  What is its essence?  Greek geometry (as personified by Euclid) began with fundamental propositions called axioms or postulates.  These propositions are considered the simplest, most incontestable laws of logic and geometry.  A few of them have the basic character of formal logic, such as the axiom stating that two volumes equal to the same third volume are equal to each other.  Others describe spatial relationships:  For example, the parallel axiom asserts that through every point P of a plane not lying on the same plane as line l will pass one and only one line that does not intersect l; this line is called the parallel of l.

This is Euclid’s famous fifth postulate—famous for lacking the self-evidence of the purely logical postulates of mathematics.  Whole generations of scholars have tried in every way possible to prove it cannot be violated.  In the 18th century, the Italian mathematician Saccari put great efforts into investigating various effects that would result from rejecting the parallel axiom, hoping sooner or later to arrive at some kind of contradiction in logic.  All his efforts proved futile.  The more he tried to find a contradiction among the effects of rejecting the axiom, the more the sum of the facts stemming from the rejection began to acquire their own meaning.  Moreover, all in all, the sum of facts gradually acquired the character of a geometry frighteningly odd, it seemed, by comparison with Euclidean geometry.  Nevertheless, the new geometry contained no internal contradictions.  At last, in the beginning of the 19th century, a group of scholars—the Hungarian mathematician Janos Bolyai, the Russian mathematician Lobachevskii and the German mathematician Gauss—came to the bold conclusion that rejecting the parallel axiom leads to no contradictions, but only indicates a transition to a new, “non-Euclidean” geometry.  The latter formed the foundation of the modern view of the world, together with quantum theory and Einstein’s theory of relativity.  That is, the contradiction in fact indicated Euclid’s fifth postulate as unsound, as it assumes the universe’s comprehensibility purely by means of logical reasoning.  The soundness of a so-called rational way of thinking rests namely on the ability to prove with the aid of arguments.  Here we question not the mind’s ability to prove, but the universe’s tendency to be comprehended by such reasoning, or the mind’s ability to comprehend the universe.  Greek scholars developed so-called argumentation based on general principles, and this was accompanied by the development of abstract reasoning.  This entire process was founded on the belief the universe is rational in essence and all its details can be traced through pure logic, beginning with fundamental principles.

This idea, simple at first glance, is laden with profound conclusions.

First, that the universe is truly rational, if the parallel axiom’s purely logical contradiction indicates its unsoundness.  The new geometry of Lobachevskii, as it’s called, bears great importance for the modern view of the world.  Its meaning for cosmology was revealed in 1922 by the ingenious Russian mathematician A. A. Friedmann.  That year, he published an article reporting his discovery of a solution to an equation of Einstein from which it followed that the Universe is expanding as time passes.  E. Hubble in 1929 reached the same conclusion through an experiment that revealed distant nebulas as receding.  With time as a definite element, the measurements found by A. A. Friedmann yield the space of Lobachevskii.  The space yielded by velocities in Einstein’s special theory of relativity is also Lobachevskii’s space.

Second, that Zeno’s paradoxes, also achieved through pure logic, show the true tendencies of space and time, still not studied adequately…

As it turns out, faith in the ability of the human mind, which in its day helped the Greeks begin to comprehend the world by means of pure logic, has at least twice received a brilliant affirmation.

The antique person’s faith in a rational basis for the Universe led to the development of philosophy and appearance of modern science’s foundations.  The remaining fragments of antique culture, to this day a source of delight, came into being due to yet another aspect of the antique person’s relation to the cosmos—one vividly embodied in the works of Aristotle, among others.  Losev writes that detailed acquaintance with Aristotle’s works can show the attentive reader that in the philosopher’s opinion, all things in existence are nothing other than a work of art—as a person, so the whole world.  This eloquent statement says all that is needed.

The “awakening of consciousness” in antiquity, examined in detail above, is known to have occurred at a so-called “axial time,” when similar processes also occurred in the cultures of other ancient peoples.  They happened independently of each other and nearly simultaneously (with an accuracy within two or three millenia).  This is the age of the 7th-5th centuries B.C.  In Greece, this period is defined by the concept of the “pre-Socratic.”  The same approximate time is marked by the rise of Chinese philosophy, of which the basic ideas—teachings about the elements and yin and yang, the primary forces of the universe—may in many ways be seen as closely related to the concepts of Hericlitus, Parmenides and Empedocles.  In India, at the beginning of the 6th century B.C., appears Siddhartha Gautama (Buddha)—the founder of one the most profound religious-philosophic doctrines.  In Iran, Zarathustra brought about the radical purging of mythological elements from ancient Iranian beliefs and imparted an elevated character to Iranian religion.  According to the most recent studies, namely the Iranian religious-philosophic perspective exercised the greatest influence on the thinking of the early pre-Socrateans.

The concept of an “axial” time, formulated by Karl Jaspers, shows that the depth and direction of ideas articulated at that time defined the entire spiritual movement of humanity for many centuries to come.  The subsequent evolution of European philosophy and science up to the present day (as with much done by humanity in this time) is based on the foundation laid in the “axial” time.  In the matter of developing a scientific approach to knowing the world, the Greeks, too, stayed within the bounds of questions posed by the “axial” time.  Beginning with the “axial” time, the paths of Eastern (India, China) and Western thought diverge to form two approaches to the comprehension of the world.  The approach chosen in antiquity led ultimately to the development of European civilization.