Infinity

George F. R. Ellis weighs in on the concept of infinity in his excellent paper, Issues in the Philosophy of Cosmology, available on astro-ph at https://arxiv.org/abs/astro-ph/0602280. He writes:

9.3.2 Existence of Infinities

The nature of existence is significantly different if there is a finite amount of matter or objects in the universe, as opposed to there being an infinite quantity in existence. Some proposals claim there may be an infinite number of universes in a multiverse and many cosmological models have spatial sections that are infinite, implying an infinite number of particles, stars, and galaxies. However, infinity is quite different from a very large number! Following David Hilbert, one can suggest these unverifiable proposals cannot be true: the word “infinity” denotes a quantity or number that can never be attained, and so will never occur in physical reality.38 He states:

Our principal result is that the infinite is nowhere to be found in reality. It neither exists in nature nor provides a legitimate basis for rational thought . . . The role that remains for the infinite to play is solely that of an idea . . . which transcends all experience and which completes the concrete as a totality . . .

This suggests “infinity” cannot be arrived at, or realized, in a concrete physical setting; on the contrary, the concept itself implies its inability to be realized!

Thesis I2: The often claimed physical existence of infinities is questionable. The claimed existence of physically realized infinities in cosmology or multiverses raises problematic issues. One can suggest they are unphysical; in any case such claims are certainly unverifiable.

This applies in principle to both small and large scales in any single universe:

The existence of a physically existing spacetime continuum represented by a real (number) manifold at the micro-level contrasts with quantum gravity claims of a discrete spacetime structure at the Planck scale, which one might suppose was a generic aspect of fully non-linear quantum gravity theories. In terms of physical reality, this promises to get rid of the uncountable infinities the real line continuum engenders in all physical variables and fields40. There is no experiment that can prove there is a physical continuum in time or space; all we can do is test space-time structure on smaller and smaller scales, but we cannot approach the Planck scale.

Infinitely large space-sections at the macro-level raise problems as indicated by Hilbert, and leads to the infinite duplication of life and all events. We may assume space extends forever in Euclidean geometry and in many cosmological models, but we can never prove that any realised 3-space in the real universe continues in this way—it is an untestable concept, and the real spatial geometry of the universe is almost certainly not Euclidean. Thus Euclidean space is an abstraction that is probably not physically real. The infinities supposed in chaotic inflationary models derive from the presumption of pre-existing infinite Euclidean space sections, and there is no reason why those should necessarily exist. In the physical universe spatial infinities can be avoided by compact spatial sections, resulting either from positive spatial curvature, or from a choice of compact topologies in universes that have zero or negative spatial curvature. Machian considerations to do with the boundary conditions for physics suggest this is highly preferable; and if one invokes string theory as a fundamental basis for physics, the “dimensional democracy” suggests the three large spatial dimensions should also be compact, since the small (“compactified”) dimensions are all taken to be so. The best current data from CBR and other observations indeed suggest k = +1, implying closed space sections for the best-fit FL model.

The existence of an eternal universe implies that an infinite time actually exists, which has its own problems: if an event happens at any time t0, one needs an explanation as to why it did not occur before that time (as there was an infinite previous time available for it to occur); and Poincaré eternal return will be possible if the universe is truly cyclic. In any case it is not possible to prove that the universe as a whole, or even the part of the universe in which we live, is past infinite; observations cannot do so, and the physics required to guarantee this would happen (if initial conditions were right) is untestable. Even attempting to prove it is future infinite is problematic (we cannot for example guarantee the properties of the vacuum into the infinite future—it might decay into a state corresponding to a negative effective cosmological constant).

It applies to the possible nature of a multiverse. Specifying the geometry of a generic universe requires an infinite amount of information because the quantities necessary to do so are fields on spacetime, in general requiring specification at each point (or equivalently, an infinite number of Fourier coefficients): they will almost always not be algorithmically compressible. All possible values of all these components in all possible combinations will have to occur in a multiverse in which “all that can happen, does happen”. There are also an infinite number of topological possibilities. This greatly aggravates all the problems regarding infinity and the ensemble. Only in highly symmetric cases, like the FL solutions, does this data reduce to a finite number of parameters, each of which would have to occur in all possible values (which themselves are usually taken to span an infinite set, namely the entire real line). Many universes in the ensemble may themselves have infinite spatial extent and contain an infinite amount of matter, with all the problems that entails. To conceive of physical creation of an infinite set of universes (most requiring an infinite amount of information for their prescription, and many of which will themselves be spatially infinite) is at least an order of magnitude more difficult than specifying an existent infinitude of finitely specifiable objects.

One should note here particularly that problems arise in the multiverse context from the continuum of values assigned by classical theories to physical quantities. Suppose for example that we identify corresponding times in the models in an ensemble and then assume that all values of the density parameter and the cosmological constant occur at each spatial point at that time. Because these values lie in the real number continuum, this is a doubly uncountably infinite set of models. Assuming genuine physical existence of such an uncountable infinitude of universes is the antithesis of Occam’s razor. But on the other hand, if the set of realised models is either finite or countably infinite, then almost all possible models are not realised. And in any case this assumption is absurdly unprovable. We can’t observationally demonstrate a single other universe exists, let alone an infinitude. The concept of infinity is used with gay abandon in some multiverse discussions, without any concern either for the philosophical problems associated with this statement, or for its completely unverifiable character. It is an extravagant claim that should be treated with extreme caution.

38An intriguing further issue is the dual question: Does the quantity zero occur in physical reality? This is related to the idea of physical existence of nothingness, as contrasted with a vacuum. A vacuum is not nothing!

40To avoid infinities entirely would require that nothing whatever is a continuum in physical reality (since any continuum interval contains an infinite number of points). Doing without that, conceptually, would mean a complete rewrite of many things. Considering how to do so in a way compatible with observation is in my view a worthwhile project.


So, given this discussion of infinities, the answer to the doubly hypothetical question, “Can God make a rock so big he can’t pick it up?” is likely a “Yes”! – D.O.

Great Courses, Great Episodes

The Great Courses offers a number of excellent courses on DVD (also streaming and audio only). Here are my favorite episodes. (Note: This is a work in progress and more entries will be added in the future.)

Course No. 153
Einstein’s Relativity and the Quantum Revolution: Modern Physics for Non-Scientists, 2nd Edition – Richard Wolfson
Lecture 8 – Uncommon Sense—Stretching Time
“Why does the simple statement of relativity—that the laws of physics are the same for all observers in uniform motion—lead directly to absurd-seeming situations that violate our commonsense notions of space and time?”
Lecture 9 – Muons and Time-Traveling Twins
“As a dramatic example of what relativity implies, you will consider a thought experiment involving a pair of twins, one of whom goes on a journey to the stars and returns to Earth younger than her sister!”
Lecture 12 – What about E=mc2 and is Everything Relative?
“Shortly after publishing his 1905 paper on special relativity, Einstein realized that his theory required a fundamental equivalence between mass and energy, which he expressed in the equation E=mc2. Among other things, this famous formula means that the energy contained in a single raisin could power a large city for an entire day.”
Lecture 16 – Into the Heart of Matter
“With this lecture, you turn from relativity to explore the universe at the smallest scales. By the early 1900s, Ernest Rutherford and colleagues showed that atoms consist of a positively charged nucleus surrounded by negatively charged electrons whirling around it. But Rutherford’s model could not explain all the observed phenomena.”
Lecture 19 – Quantum Uncertainty—Farewell to Determinism
“Quantization places severe limits on our ability to observe nature at the atomic scale because it implies that the act of observation disturbs that which is being observed. The result is Werner Heisenberg’s famous Uncertainty Principle. What exactly does this principle say, and what are the philosophical implications?”
Lecture 21 – Quantum Weirdness and Schrödinger’s Cat
“Wave-particle duality gives rise to strange phenomena, some of which are explored in Schrödinger’s famous ‘cat in the box’ example. Philosophical debate on Schrödinger’s cat still rages.”

Course No. 158
My Favorite Universe – Neil deGrasse Tyson
Lecture 8 – In Defense of the Big Bang
“We now know without doubt how the universe began, how it evolved, and how it will end. This lecture explains and defends a “theory” far too often misunderstood.”

Course No. 415
The Will to Power: The Philosophy of Friedrich Nietzsche
Robert C. Solomon & Kathleen M. Higgins

Lecture 7 – Nietzsche and Schopenhauer on Pessimism
“Schopenhauer, the severe pessimist, is a looming presence in Nietzsche’s thought. Nietzsche felt the weight of Schopenhauer’s pessimism, and struggled to counter it by embracing “cheerfulness,” creative passion, and an aesthetic viewpoint.”
Lecture 19 – The Ranking of Values – Morality and Modernity
“Why did Nietzsche refuse to think of values as being either objective or subjective? Why did he hold that values are earthly and culture- and species-specific? Why did he argue that, in the final analysis, there are only healthy and unhealthy values, and that modern values are unhealthy?”
Lecture 22 – Resentment, Revenge, and Justice
“We continue our discussion of Nietzsche’s idea of resentment, adding to it his ideas about revenge and justice. We revisit his condemnation of asceticism, the self-denial that is often a part of extreme religious practice, in light of these new ideas.”

Course No. 443
Power over People: Classical and Modern Political Theory – Dennis Dalton
Lecture 10 – Marx’s Critique of Capitalism and the Solution of Communism
“Karl Marx’s communism provided what is probably the best known ideal society. He blamed not only private property, but the entire institution of capitalism for the inequality and injustice in society. His program has never been implemented, certainly not in the Soviet Union. Marx never advocated totalitarian or despotic rule. Although his historical determinism has been discredited, his social criticism remains relevant. The democratic dilemma boils down to this: the more liberty, the less equality; and the more equality, the less liberty.”
Special Note: I will eventually be adding more of the episodes from this excellent course as I rewatch them. (I watched this series before I began keeping track of “best” episodes.)

Course No. 700
How to Listen to and Understand Great Music, 3rd Edition – Robert Greenberg
Lecture 23 – Classical-era Form—Sonata Form, Part 1
“In Lectures 23 and 24 we examine sonata-allegro form, but first, we observe the life and personality of the extraordinary Wolfgang Mozart. We discuss the many meanings and uses of the word “sonata.” The fourth movement of Mozart’s Symphony in G Minor, K. 550, is analyzed and discussed in depth as an example.”
Special Note: I will eventually be adding more of the episodes from this excellent course as I rewatch them. (I watched this series before I began keeping track of “best” episodes.)

Course No. 730
Symphonies of Beethoven – Robert Greenberg
Lecture 11 – Symphony No. 3—The “New Path”—Heroism and Self-Expression, III
“Lectures 9 through 12 focus on Symphony No. 3, the Eroica Symphony. This key work in Beethoven’s compositional revolution resulted from his crisis of going deaf. Beethoven’s struggle with his disability raised him to a new level of creativity. Symphony No. 3 parallels his heroic battle with and ultimate triumph over adversity. The symphony’s debt to Napoleon is discussed before an analysis.”
Lecture 13 – Symphony No. 4—Consolidation of the New Aesthetic, I
“Lectures 13 through 16 examine Symphony No. 4 in historical context and in its relationship to opera buffa. Symphony No. 4 is the most infrequently heard of his symphonies. We see how it represents a return to a Classical structure. Its framework is filled with iconoclastic rhythms, harmonies, and characteristic motivic developments that mark it as a product of Beethoven’s post-Eroica period.”
Lecture 23 – Symphony No. 7—The Symphony as Dance, I
Lecture 24 – Symphony No. 7—The Symphony as Dance, II
“Lectures 23 and 24 discuss Beethoven’s Symphony No. 7 with references to the historical and personal events surrounding its composition. The essence of the symphony is seen to be the power of rhythm, and originality is seen to be an important artistic goal for Beethoven.”
Lecture 31 – Symphony No. 9—The Symphony as the World, IV
“The last five lectures are devoted to Symphony No. 9, the most influential Western musical composition of the 19th century and the most influential symphony ever written. We see how this work obliterated distinctions between the instrumental symphony and dramatic vocal works such as opera. Also discussed are Beethoven’s fall from public favor in 1815, his disastrous relationship with his nephew Karl, his artistic rebirth around 1820, his late compositions, and his death in 1827.”

Course No. 753
Great Masters: Tchaikovsky-His Life and Music – Robert Greenberg
Lecture 1 – Introduction and Early Life
“Tchaikovsky was an extremely sensitive child, obsessive about music and his mother. His private life was reflected to a rare degree in his music. His mother’s death when he was 14 years old was a shattering experience for him—one that found poignant expression in his music.”
Lecture 6 – My Great Friend
“With the generous financial support of Nadezhda von Meck, Tchaikovsky lived abroad, and in 1878 resigned from the Moscow Conservatory to compose full time. His Fourth Symphony was premiered in Moscow and was quickly followed by the brilliant Violin Concerto in D Major, which became a pillar of the repertoire within a few years.”

Course No. 754
Great Masters: Stravinsky-His Life and Music – Robert Greenberg
Lecture 2 – From Student to Professional
“Rimsky-Korsakov was so impressed with Stravinsky’s Piano Sonata in F♯ minor (1904) he agreed to take Stravinsky as a private student. In 1909, Stravinsky met the impresario Serge Diaghilev, who commissioned Stravinsky to write a ballet on the folk tale The Firebird, which was followed by the ballet Petrushka, a great success. Stravinsky’s next score, The Rite of Spring, would become arguably the most influential work of its time.”

Course No. 756
Great Masters: Mahler-His Life and Music – Robert Greenberg
Lecture 7 – Symphony No. 6, and Das Lied von der Erde
“Three events shattered the Mahlers’ lives in 1907: his resignation from the Royal Vienna Opera, the death of their elder daughter, and the diagnosis of his heart disease. In 1908, Mahler threw himself into composing Das Lied von der Erde as an attempt to find solace from the grief of his daughter’s death. The work is a symphonic song cycle about loss, grief, memory, disintegration, and transfiguration.”

Course No. 758
Great Masters: Liszt-His Life and Music – Robert Greenberg
Lecture 2 – A Born Pianist
“Liszt was surrounded by music from infancy and began to reveal his musical gifts at about age five. He stunned his teachers and, at his first performance at age 11, astonished reviewers and his audience. When Liszt was 15, his father died, sending Franz into depression and apathy for three years. He was finally blasted out of his lethargy by the July Revolution of 1830.”
Lecture 7 – Rome
“By the 1850s, Liszt became the focal point of a debate concerning program music versus absolute music and expression versus structure. Twenty years before, Liszt and his fellow young Romantic musicians had a common goal: to create a new music based on individual expression. As they grew older, many became conservative, but Liszt never lost his revolutionary spirit. But brokenhearted by the death of his daughter, he turned to the Catholic Church to find solace.”

Course No. 759
Great Masters: Robert and Clara Schumann-Their Lives and Music – Robert Greenberg
Lecture 8 – Madness
“In Düsseldorf, Robert was inspired to write the Symphony No. 3 in E-flat Major, along with trios, sonatas, orchestral works, and pieces for chorus and voice and piano. Robert and Clara also met Johannes Brahms there; he became a lifelong friend and source of strength for Clara. In 1854, Robert attempted to drown himself in the Rhine and was taken to an asylum. He died there two years later. Clara managed to sustain the family through her concerts but was dealt even more pain by the early deaths of several of her children.”

Course No. 1012
Chemistry, 2nd Edition – Frank Cardulla
Lecture 5 – The SI (Metric) System of Measurement
“Next, we continue to lay a strong foundation for our understanding of chemistry by learning about one of the key tools we’ll be using: the International System of Units (SI), or the metric system. This lecture explains why this system is so useful to scientists and lays out the prefixes and units of measurement that make up the metric system.”
Lecture 10 – The Mole
“One of the most important concepts to master in an introductory chemistry course is the concept of the mole, which provides chemists with a way to ‘count’ atoms and molecules. Learn how scientists use the mole and explore the quantitative definition of this basic unit.”
Lecture 28 – The Self-Ionization of Water
“After examining how different substances may behave when dissolved in water, we learn about the self-ionization of water and use this knowledge to solve problems. The lecture ends with a brief introduction to the pH of solutions.”
Lecture 29 – Strong Acids and Bases – General Properties
“We return to the topic of pH and learn about how pH relates to two kinds of compounds: acids and bases. Through an introductory problem, we explore the relationship of various ions within these compounds.”

Course No. 1257
Mysteries of Modern Physics: Time – Sean Carroll
Lecture 10 – Playing with Entropy
“Sharpen your understanding of entropy by examining different macroscopic systems and asking, which has higher entropy and which has lower entropy? Also evaluate James Clerk Maxwell’s famous thought experiment about a demon who seemingly defies the principle that entropy always increases.”
Lecture 15 – The Perception of Time
“Turn to the way humans perceive time, which can vary greatly from clock time. In particular, focus on experiments that shed light on our time sense. For example, tests show that even though we think we perceive the present moment, we actually live 80 milliseconds in the past.”
Lecture 16 – Memory and Consciousness
“Remembering the past and projecting into the future are crucial for human consciousness, as shown by cases where these faculties are impaired. Investigate what happens in the brain when we remember, exploring different kinds of memory and the phenomena of false memories and false forgetting.”
Lecture 20 – Black Hole Entropy
“Stephen Hawking showed that black holes emit radiation and therefore have entropy. Since the entropy in the universe today is overwhelmingly in the form of black holes and there were no black holes in the early universe, entropy must have been much lower in the deep past.”
Lecture 21 – Evolution of the Universe
“Follow the history of the universe from just after the big bang to the far future, when the universe will consist of virtually empty space at maximum entropy. Learn what is well founded and what is less certain about this picture of a universe winding down.”

Course No. 1280
Physics and Our Universe: How It All Works – Richard Wolfson
Lecture 1 – The Fundamental Science

“Take a quick trip from the subatomic to the galactic realm as an introduction to physics, the science that explains physical reality at all scales. Professor Wolfson shows how physics is the fundamental science that underlies all the natural sciences. He also describes phenomena that are still beyond its explanatory power.”
Lecture 24 – The Ideal Gas
“Delve into the deep link between thermodynamics, which looks at heat on the macroscopic scale, and statistical mechanics, which views it on the molecular level. Your starting point is the ideal gas law, which approximates the behavior of many gases, showing how temperature, pressure, and volume are connected by a simple formula.”
Lecture 44 – Cracks in the Classical Picture
“Embark on the final section of the course, which covers the revolutionary theories that superseded classical physics. Why did classical physics need to be replaced? Discover that by the late 19th century, inexplicable cracks were beginning to appear in its explanatory power.”
Special Note: This entire series is outstanding! I will eventually be adding many of the episodes of this course as I rewatch them. (I watched this series before I began keeping track of “best” episodes.)

Course No. 1360
Introduction to Astrophysics – Joshua Winn
Lecture 5 – Newton’s Hardest Problem
“Continue your exploration of motion by discovering the law of gravity just as Newton might have—by analyzing Kepler’s laws with the aid of calculus (which Newton invented for the purpose). Look at a graphical method for understanding orbits, and consider the conservation laws of angular momentum and energy in light of Emmy Noether’s theory that links conservation laws and symmetry.”
Lecture 10 – Optical Telescopes
“Consider the problem of gleaning information from the severely limited number of optical photons originating from astronomical sources. Our eyes can only do it so well, and telescopes have several major advantages: increased light-gathering power, greater sensitivity of telescopic cameras and sensors such as charge-coupled devices (CCDs), and enhanced angular and spectral resolution.”
Lecture 11 – Radio and X-Ray Telescopes
“Non-visible wavelengths compose by far the largest part of the electromagnetic spectrum. Even so, many astronomers assumed there was nothing to see in these bands. The invention of radio and X-ray telescopes proved them spectacularly wrong. Examine the challenges of detecting and focusing radio and X-ray light, and the dazzling astronomical phenomena that radiate in these wavelengths.”
Lecture 12 – The Message in a Spectrum
“Starting with the spectrum of sunlight, notice that thin dark lines are present at certain wavelengths. These absorption lines reveal the composition and temperature of the Sun’s outer atmosphere, and similar lines characterize other stars. More diffuse phenomena such as nebulae produce bright emission lines against a dark spectrum. Probe the quantum and thermodynamic events implied by these clues.”
Lecture 13 – The Properties of Stars
“Take stock of the wide range of stellar luminosities, temperatures, masses, and radii using spectra and other data. In the process, construct the celebrated Hertzsprung–Russell diagram, with its main sequence of stars in the prime of life, including the Sun. Note that two out of three stars have companions. Investigate the orbital dynamics of these binary systems.”
Lecture 15 – Why Stars Shine
“Get a crash course in nuclear physics as you explore what makes stars shine. Zero in on the Sun, working out the mass it has consumed through nuclear fusion during its 4.5-billion-year history. While it’s natural to picture the Sun as a giant furnace of nuclear bombs going off non-stop, calculations show it’s more like a collection of toasters; the Sun is luminous simply because it’s so big.”
Lecture 16 – Simple Stellar Models
“Learn how stars work by delving into stellar structure, using the Sun as a model. Relying on several physical principles and sticking to order-of-magnitude calculations, determine the pressure and temperature at the center of the Sun, and the time it takes for energy generated in the interior to reach the surface, which amounts to thousands of years. Apply your conclusions to other stars.”
Lecture 17 – White Dwarfs
“Discover the fate of solar mass stars after they exhaust their nuclear fuel. The galaxies are teeming with these dim “white dwarfs” that pack the mass of the Sun into a sphere roughly the size of Earth. Venture into quantum theory to understand what keeps these exotic stars from collapsing into black holes, and learn about the Chandrasekhar limit, which determines a white dwarf’s maximum mass.”
Lecture 18 – When Stars Grow Old
“Trace stellar evolution from two points of view. First, dive into a protostar and witness events unfold as the star begins to contract and fuse hydrogen. Exhausting that, it fuses heavier elements and eventually collapses into a white dwarf—or something even denser. Next, view this story from the outside, seeing how stellar evolution looks to observers studying stars with telescopes.”
Lecture 19 – Supernovas and Neutron Stars
“Look inside a star that weighs several solar masses to chart its demise after fusing all possible nuclear fuel. Such stars end in a gigantic explosion called a supernova, blowing off outer material and producing a super-compact neutron star, a billion times denser than a white dwarf. Study the rapid spin of neutron stars and the energy they send beaming across the cosmos.”
Lecture 20 – Gravitational Waves
“Investigate the physics of gravitational waves, a phenomenon predicted by Einstein and long thought to be undetectable. It took one of the most violent events in the universe—colliding black holes—to generate gravitational waves that could be picked up by an experiment called LIGO on Earth, a billion light years away. This remarkable achievement won LIGO scientists the 2017 Nobel Prize in Physics.”

Course No. 1434
The Queen of the Sciences: A History of Mathematics – David M. Bressoud
Lecture 2 – Babylonian and Egyptian Mathematics
“Egyptian and Mesopotamian mathematics were well developed by the time of the earliest records from the 2nd millennium B.C. Both knew how to find areas and volumes. The Babylonians solved quadratic equations using geometric methods and knew the Pythagorean theorem.”
Lecture 5 – Astronomy and the Origins of Trigonometry
“The origins of trigonometry lie in astronomy, especially in finding the length of the chord that connects the endpoints of an arc of a circle. Hipparchus discovered a solution to this problem, that was later refined by Ptolemy who authored the great astronomical work the Almagest.”
Lecture 6 – Indian Mathematics – Trigonometry Blossoms
“We journey through the Gupta Empire and the great period of Indian mathematics that lasted from A.D. 320 to 1200. Along the way, we explore the significant advances that occurred in trigonometry and other mathematical fields.”
Lecture 14 – Leibniz and the Emergence of Calculus
“Independently of Newton, Gottfried Wilhelm Leibniz discovered the techniques of calculus in the 1670s, developing the notational system still used today.”
Lecture 15 – Euler – Calculus Proves Its Promise
“Leonhard Euler dominated 18th-century mathematics so thoroughly that his contemporaries believed he had solved all important problems.”
Lecture 19 – Modern Analysis – Fourier to Carleson
“By 1800, calculus was well established as a powerful tool for solving practical problems, but its logical underpinnings were shaky. We explore the creative mathematics that addressed this problem in work from Joseph Fourier in the 19th century to Lennart Carleson in the 20th.”
Lecture 21 – Sylvester and Ramanujan – Different Worlds
“This lecture explores the contrasting careers of James Joseph Sylvester, who was instrumental in developing an American mathematical tradition, and Srinivasa Ramanujan, a poor college dropout from India who produced a rich range of new mathematics during his short life.”
Lecture 22 – Fermat’s Last Theorem – The Final Triumph
“Pierre de Fermat’s enigmatic note regarding a proof that he didn’t have space to write down sparked the most celebrated search in mathematics, lasting more than 350 years. This lecture follows the route to a proof, finally achieved in the 1990s.”
Lecture 23 – Mathematics – The Ultimate Physical Reality
“Mathematics is the key to realms outside our intuition. We begin with Maxwell’s equations and continue through general relativity, quantum mechanics, and string theory to see how mathematics enables us to work with physical realities for which our experience fails us.”
Lecture 24 – Problems and Prospects for the 21st Century
“This last lecture introduces some of the most promising and important questions in the field and examines mathematical challenges from other disciplines, especially genetics.”

Course No. 1456
Discrete Mathematics – Arthur T. Benjamin
Lecture 8 – Linear Recurrences and Fibonacci Numbers
“Investigate some interesting properties of Fibonacci numbers, which are defined using the concept of linear recurrence. In the 13th century, the Italian mathematician Leonardo of Pisa, called Fibonacci, used this sequence to solve a problem of idealized reproduction in rabbits.”
Lecture 15 – Open Secrets—Public Key Cryptography
“The idea behind public key cryptography sounds impossible: The key for encoding a secret message is publicized for all to know, yet only the recipient can reverse the procedure. Learn how this approach, widely used over the Internet, relies on Euler’s theorem in number theory.”
Lecture 16 – The Birth of Graph Theory
“This lecture introduces the last major section of the course, graph theory, covering the basic definitions, notations, and theorems. The first theorem of graph theory is yet another contribution by Euler, and you see how it applies to the popular puzzle of drawing a given shape without lifting the pencil or retracing any edge.”
Lecture 18 – Social Networks and Stable Marriages
“Apply graph theory to social networks, investigating such issues as the handshake theorem, Ramsey’s theorem, and the stable marriage theorem, which proves that in any equal collection of eligible men and women, at least one pairing exists for each person so that no extramarital affairs will take place.”
Lecture 20 – Weighted Graphs and Minimum Spanning Trees
“When you call someone on a cell phone, you can think of yourself as a leaf on a giant ‘tree’—a connected graph with no cycles. Trees have a very simple yet powerful structure that make them useful for organizing all sorts of information.”
Lecture 22 – Coloring Graphs and Maps
“According to the four-color theorem, any map can be colored in such a way that no adjacent regions are assigned the same color and, at most, four colors suffice. Learn how this problem went unsolved for centuries and has only been proved recently with computer assistance.”

Course No. 1471
Great Thinkers, Great Theorems – William Dunham
Lecture 5 – Number Theory in Euclid
“In addition to being a geometer, Euclid was a pioneering number theorist, a subject he took up in books VII, VIII, and IX of the Elements. Focus on his proof that there are infinitely many prime numbers, which Professor Dunham considers one of the greatest proofs in all of mathematics.”
Lecture 6 – The Life and Work of Archimedes
“Even more distinguished than Euclid was Archimedes, whose brilliant ideas took centuries to fully absorb. Probe the life and famous death of this absent-minded thinker, who once ran unclothed through the streets, shouting ‘Eureka!’ (‘I have found it!’) on solving a problem in his bath.”
Lecture 7 – Archimedes’ Determination of Circular Area
“See Archimedes in action by following his solution to the problem of determining circular area—a question that seems trivial today but only because he solved it so simply and decisively. His unusual strategy relied on a pair of indirect proofs.”
Lecture 8 – Heron’s Formula for Triangular Area
“Heron of Alexandria (also called Hero) is known as the inventor of a proto-steam engine many centuries before the Industrial Revolution. Discover that he was also a great mathematician who devised a curious method for determining the area of a triangle from the lengths of its three sides.”
Lecture 9 – Al-Khwarizmi and Islamic Mathematics
“With the decline of classical civilization in the West, the focus of mathematical activity shifted to the Islamic world. Investigate the proofs of the mathematician whose name gives us our term ‘algorithm’: al-Khwarizmi. His great book on equation solving also led to the term ‘algebra.'”
Lecture 10 – A Horatio Algebra Story
“Visit the ruthless world of 16th-century Italian universities, where mathematicians kept their discoveries to themselves so they could win public competitions against their rivals. Meet one of the most colorful of these figures: Gerolamo Cardano, who solved several key problems. In secret, of course.”
Lecture 11 – To the Cubic and Beyond
“Trace Cardano’s path to his greatest triumph: the solution to the cubic equation, widely considered impossible at the time. His protégé, Ludovico Ferrari, then solved the quartic equation. Norwegian mathematician Niels Abel later showed that no general solutions are possible for fifth- or higher-degree equations.”
Lecture 12 – The Heroic Century
“The 17th century saw the pace of mathematical innovations accelerate, not least in the introduction of more streamlined notation. Survey the revolutionary thinkers of this period, including John Napier, Henry Briggs, René Descartes, Blaise Pascal, and Pierre de Fermat, whose famous ‘last theorem’ would not be proved until 1995.”
Lecture 13 – The Legacy of Newton
“Explore the eventful life of Isaac Newton, one of the greatest geniuses of all time. Obsessive in his search for answers to questions from optics to alchemy to theology, he made his biggest mark in mathematics and science, inventing calculus and discovering the law of universal gravitation.”
Lecture 14 – Newton’s Infinite Series
“Start with the binomial expansion, then turn to Newton’s innovation of using fractional and negative exponents to calculate roots—an example of his creative use of infinite series. Also see how infinite series allowed Newton to approximate sine values with extraordinary accuracy.”
Lecture 16 – The Legacy of Leibniz
“Probe the career of Newton’s great rival, Gottfried Wilhelm Leibniz, who came relatively late to mathematics, plunging in during a diplomatic assignment to Paris. In short order, he discovered the ‘Leibniz series’ to represent π, and within a few years he invented calculus independently of Newton.”
Lecture 17 – The Bernoullis and the Calculus Wars
“Follow the bitter dispute between Newton and Leibniz over priority in the development of calculus. Also encounter the Swiss brothers Jakob and Johann Bernoulli, enthusiastic supporters of Leibniz. Their fierce sibling rivalry extended to their competition to outdo each other in mathematical discoveries.”
Lecture 18 – Euler, the Master
“Meet history’s most prolific mathematician, Leonhard Euler, who went blind in his sixties but kept turning out brilliant papers. A sampling of his achievements: the number e, crucial in calculus; Euler’s identity, responsible for the most beautiful theorem ever; Euler’s polyhedral formula; and Euler’s path.”
Lecture 19 – Eulers Extraordinary Sum
“Euler won his spurs as a great mathematician by finding the value of a converging infinite series that had stumped the Bernoulli brothers and everyone else who tried it. Pursue Euler’s analysis through the twists and turns that led to a brilliantly simple answer.”
Lecture 20 – Euler and the Partitioning of Numbers
“Investigate Euler’s contribution to number theory by first warming up with the concept of amicable numbers—a truly rare breed of integers until Euler vastly increased the supply. Then move on to Euler’s daring proof of a partitioning property of whole numbers.”
Lecture 21 – Gauss – the Prince of Mathematicians
“Dubbed the Prince of Mathematicians by the end of his career, Carl Friedrich Gauss was already making major contributions by his teen years. Survey his many achievements in mathematics and other fields, focusing on his proof that a regular 17-sided polygon can be constructed with compass and straightedge alone.”
Lecture 22 – The 19th Century – Rigor and Liberation
“Delve into some of the important trends of 19th-century mathematics: a quest for rigor in securing the foundations of calculus; the liberation from the physical sciences, embodied by non-Euclidean geometry; and the first significant steps toward opening the field to women.”
Lecture 23 – Cantor and the Infinite
“Another turning point of 19th-century mathematics was an increasing level of abstraction, notably in the approach to the infinite taken by Georg Cantor. Explore the paradoxes of the ‘completed’ infinite, and how Cantor resolved this mystery with transfinite numbers, exemplified by the transfinite cardinal aleph-naught.”
Lecture 24 – Beyond the Infinite
“See how it’s possible to build an infinite set that’s bigger than the set of all whole numbers, which is itself infinite. Conclude the course with Cantor’s theorem that the transcendental numbers greatly outnumber the seemingly more abundant algebraic numbers—a final example of the elegance, economy, and surprise of a mathematical masterpiece.”

Course No. 1495
Introduction to Number Theory – Edward B. Burger
Lecture 12 – The RSA Encryption Scheme
“We continue our consideration of cryptography and examine how Fermat’s 350-year-old theorem about primes applies to the modern technological world, as seen in modern banking and credit card encryption.”
Lecture 22 – Writing Real Numbers as Continued Fractions
“Real numbers are often expressed as endless decimals. Here we study an algorithm for writing real numbers as an intriguing repeated fraction-within-a-fraction expansion. Along the way, we encounter new insights about the hidden structure within the real numbers.”
Lecture 24 – A Journey’s End and the Journey Ahead
“In this final lecture, we take a step back to view the entire panorama of number theory and celebrate some of the synergistic moments when seemingly unrelated ideas came together to tell a unified story of number.”

Course No. 1802
The Search for Exoplanets: What Astronomers Know – Joshua Winn
Lecture 4 – Pioneers of Planet Searching

“Chart the history of exoplanet hunting – from a famous false signal in the 1960s, through ambiguous discoveries in the 1980s, to the big breakthrough in the 1990s, when dozens of exoplanets turned up. Astronomers were stunned to find planets unlike anything in the solar system.”
Special Note: This entire series is outstanding! I will eventually be adding most of the episodes of this course as I rewatch them. (I watched this series before I began keeping track of “best” episodes.)

Course No. 1816
The Inexplicable Universe: Unsolved Mysteries – Neil deGrasse Tyson
Lecture 4 – Inexplicable Physics

“Among the many topics you’ll learn about in this lecture are the discovery of more elements on the periodic table; muon neutrinos, tao particles, and the three regimes of matter; the secrets of string theory (which offers the hope of unifying all the particles and forces of physics); and even the hypothetical experience of traveling through a black hole.”
Special Note: This entire series is outstanding! I will eventually be adding most of the episodes of this course as I rewatch them. (I watched this series before I began keeping track of “best” episodes.)

Course No. 1830
Cosmology: The History and Nature of Our Universe – Mark Whittle
Lecture 3 – Overall Cosmic Properties

“The universe is lumpy at the scale of galaxies and galaxy clusters. But at larger scales it seems to be smooth and similar in all directions. This property of homogeneity and isotropy is called the cosmological principle.”
Lecture 4 – The Stuff of the Universe
“The most familiar constituents of the universe are atomic matter and light. Neutrinos make up another component. But by far the bulk of the universe—96%—is dark energy and dark matter. The relative amounts of these constituents have changed as the universe has expanded.”
Lecture 6 – Measuring Distances
“Astronomers use a ‘distance ladder’ of overlapping techniques to determine distances in the universe. Triangulation works for nearby stars. For progressively farther objects, observers use pulsating stars, the rotation of galaxies, and a special class of supernova explosions.”
Lecture 8 – Distances, Appearances, and Horizons
“Defining distances in cosmology is tricky, since an object’s distance continually increases with cosmic expansion. There are three important distances to consider: the emission distance, when the light set out; the current distance, when the light arrives; and the distance the light has traveled.”
Lecture 10 – Cosmic Geometry – Triangles in the Sky
“Einstein’s theory of gravity suggests that space could be positively or negatively curved, so that giant billion-light-year triangles might have angles that don’t add up to 180°. This lecture discusses the success at measuring the curvature of the universe in 1998.”
Lecture 11 – Cosmic Expansion – Keeping Track of Energy
“Has the universe’s rate of expansion always been the same? You answer this question by applying Newton’s law of gravity to an expanding sphere of matter, finding that the expansion was faster in the past and slows down over time.”
Lecture 12 – Cosmic Acceleration – Falling Outward
“You investigate why the three great eras of cosmic history—radiation, matter, and dark energy—have three characteristic kinds of expansion. These are rapid deceleration, modest deceleration, and exponential acceleration. The last is propelled by dark energy, which makes the universe fall outward.”
Lecture 13 – The Cosmic Microwave Background
“By looking sufficiently far away, and hence back in time, we can witness the ‘flash’ from the big bang itself. This arrives from all directions as a feeble glow of microwave radiation called the cosmic microwave background (CMB), discovered by chance in 1964.”
Lecture 22 – The Galaxy Web – A Relic of Primordial Sound
“A simulated intergalactic trip shows you the three-dimensional distribution of galaxies in our region of the universe. On the largest scale, galaxies form a weblike pattern that matches the peaks and troughs of the primordial sound in the early universe.”
Lecture 24 – Understanding Element Abundances
“The theory of atom genesis in the interiors of stars is confirmed by the proportions of each element throughout the cosmos. The relative abundances hardly vary from place to place, so that gold isn’t rare just on earth, it’s rare everywhere.”
Lecture 27 – Physics at Ultrahigh Temperatures
“This lecture begins your investigation of the universe during its first second, which is an immense tract of time in nature. To understand what happened, you need to know how nature behaves at ultrahigh energy and density. Fortunately, the physics is much simpler than you might think.”
Lecture 29 – Back to the GUT – Matter and Forces Emerge
“You venture into the bizarre world of the opening nanosecond. There are two primary themes: the birth of matter and the birth of forces. Near one nanosecond, the universe was filled with a dense broth of the most elementary particles. As temperatures dropped, particles began to form.”
Lecture 30 – Puzzling Problems Remain
“Although the standard big bang theory was amazingly successful, it couldn’t explain several fundamental properties of the universe: Its geometry is Euclidean, it’s smooth on the largest scales, and it was born slightly lumpy on smaller scales. The theory of cosmic inflation offers a comprehensive solution.”
Lecture 31 – Inflation Provides the Solution
“This lecture shows how the early universe might enter a brief phase of exponentially accelerating expansion, or inflation, providing a mechanism to launch the standard hot big bang universe. This picture also solves the flatness, horizon, and monopole problems that plagued the standard big-bang theory.”
Lecture 33 – Inflation’s Stunning Creativity
“All the matter and energy in stars and galaxies is exactly balanced by all the negative energy stored in the gravitational fields between the galaxies. Inflation is the mechanism that takes nothing and makes a universe—not just our universe, but potentially many.”
Lecture 34 – Fine Tuning and Anthropic Arguments
“Why does the universe have the properties it does and not some different set of laws? One approach is to see the laws as inevitable if life ever evolves to ask such questions. This position is called the anthropic argument, and its validity is hotly debated.”

Course No. 1866
The Remarkable Science of Ancient Astronomy – Bradley E. Schaefer
Lecture 10 – Origins of Western Constellations
“The human propensity for pattern recognition and storytelling has led every culture to invent constellations. Trace the birth of the star groups known in the West, many of which originated in ancient Mesopotamia. At least one constellation is almost certainly more than 14,000 years old and may be humanity’s oldest surviving creative work.”

Course No. 1872
The Life and Death of Stars – Keivan G. Stassun
Lecture 10 – Eclipses of Stars—Truth in the Shadows
“Investigate the remarkable usefulness of eclipses. When our moon passes in front of a star or one star eclipses another, astronomers can gather a treasure trove of data, such as stellar diameters. Eclipses also allow astronomers to identify planets orbiting other stars.”
Lecture 13 – E = mc2—Energy for a Star’s Life
“Probe the physics of nuclear fusion, which is the process that powers stars by turning mass into energy, according to Einstein’s famous equation. Then examine two lines of evidence that show what’s happening inside the sun, proving that nuclear reactions must indeed be taking place.”
Lecture 14 – Stars in Middle Age
“Delve deeper into the lessons of the Hertzsprung-Russell diagram, introduced in Lecture 9. One of its most important features is the main sequence curve, along which most stars are found for most of their lives. Focus on the nuclear reactions occurring inside stars during this stable period.”
Lecture 19 – Stillborn Stars
“Follow the search for brown dwarfs—objects that are larger than planets but too small to ignite stellar fires. Hear about Professor Stassun’s work that identified the mass of these elusive objects, showing the crucial role of magnetism in setting the basic properties of all stars.”
Lecture 20 – The Dark Mystery of the First Stars
“Join the hunt for the first stars in the universe, focusing on the nearby “Methuselah” star. Explore evidence that the earliest stars were giants, even by stellar standards. They may even have included mammoth dark stars composed of mysterious dark matter.”
Lecture 21 – Stars as Magnets
“Because stars spin like dynamos, they generate magnetic fields—a phenomenon that explains many features of stars. See how the slowing rate of rotation of stars like the sun allows astronomers to infer their ages. Also investigate the clock-like magnetic pulses of pulsars, which were originally thought to be signals from extraterrestrials.”
Lecture 22 – Solar Storms—The Perils of Life with a Star
“The sun and stars produce more than just light and heat. Their periodic blasts of charged particles constitute space weather. Examine this phenomenon—from beautiful aurorae to damaging bursts of high-energy particles that disrupt electronics, the climate, and even life.”

Course No. 1878
Radio Astronomy: Observing the Invisible Universe – Felix J. Lockman
Lecture 5 – Radio Telescopes and How They Work
“Radio telescopes are so large because radio waves contain such a small amount of energy. For example, the signal from a standard cell phone measured one kilometer away is five million billion times stronger than the radio signals received from a bright quasar. Learn how each of these fascinating instruments is designed to meet a specific scientific goal—accounting for their wide variation in form and size.”
Lecture 7 – Tour of the Green Bank Observatory
“The Green Bank Observatory is located within the 13,000-acre National Radio Quiet Zone straddling the border of Virginia and West Virginia. Come tour this fascinating facility where astronomers discovered radiation belts around Jupiter, the black hole at the center of our galaxy, and the first known interstellar organic molecule, and began the search for extra-terrestrial life.”
Lecture 8 – Tour of the Green Bank Telescope
“At 17 million pounds, and with more than 2,000 surface panels that can be repositioned in real time, this telescope is one of the largest moveable, land-based objects ever built. The dish could contain two side-by-side football fields, but when its panels are brought into focus, the surface has errors no larger than the thickness of a business card. Welcome to this rare insider’s view.”
Lecture 9 – Hydrogen and the Structure of Galaxies
“Using the laws of physics and electromagnetic radiation, astronomers can ‘weigh’ a galaxy by studying the distribution of its rotating hydrogen. But when they do this, it soon becomes clear something is very wrong: A huge proportion of the galaxy’s mass has simply gone missing. Welcome to the topsy-turvy world of dark matter, which we now believe accounts for a whopping 90 percent of our own Milky Way.”
Lecture 10 – Pulsars: Clocks in Space
“In the mid-1960s, astronomers discovered signals with predictable periodicity but no known source. In case these signals indicated extraterrestrial life, they were initially labeled LGM, Little Green Men. But research revealed the source of the pulsing radiation to be neutron stars. Learn how a star with a diameter of only a few kilometers and a mass similar to that of our Sun can spin around hundreds of times per second.”
Lecture 11 – Pulsars and Gravity
“A pulsar’s spin begins with its birth in a supernova and can be altered by transfer of mass from a companion star. Learn how pulsars, these precise interstellar clocks, are used to confirm Einstein’s prediction of gravitational waves by observations of a double-neutron-star system, and how we pull the pulsar signal out of the noise.”
Lecture 12 – Pulsars and the 300-Foot Telescope
“Humans constantly use radio transmission these days, for everything from military communications to garage-door openers. How can scientists determine which signals come from Earth and which come from space? Learn how the 300-foot telescope, located in two radio quiet zones, was built quickly and cheaply. It ended up studying pulsars and hydrogen in distant galaxies, and made the case for dark matter.”
Lecture 16 – Radio Stars and Early Interferometers
“When radio astronomers discovered a sky full of small radio sources of unknown origin, they built telescopes using multiple antennas to try to understand them. Learn how and why interferometers were developed and how they have helped astronomers study quasars—those massively bright, star-like objects that scientists now know only occur in galaxies whose gas is falling into a supermassive black hole.”
Lecture 18 – Active Galactic Nuclei and the VLA
“The need for a new generation of radio interferometers to untangle extragalactic radio sources led to the development of the Very Large Array (VLA) in New Mexico. With its twenty-seven radio antennas in a Y-shaped configuration, it gives both high sensitivity and high angular resolution. The VLA provided a deeper and clearer look at galaxies than ever before, and the results were astonishing.”
Lecture 19 – A Telescope as Big as the Earth
“Learn how astronomers use very-long-baseline interferometry (VLBI) with telescopes thousands of miles apart to essentially create a radio telescope as big as the Earth. With VLBI, scientists not only look deep into galactic centers, study cosmic radio sources, and weigh black holes, but also more accurately tell time, study plate tectonics, and more—right here on planet Earth.”
Lecture 20 – Galaxies and Their Gas
“In visible light, scientists had described galaxies as ‘island universes’. But since the advent of radio astronomy, we’ve seen galaxies connected by streams of neutral hydrogen, interacting with and ripping the gases from each other. Now astronomers have come to understand that these strong environmental interactions are not a secondary feature—they are key to a galaxy’s basic structure and appearance.”
Lecture 21 – Interstellar Molecular Clouds
“In the late 1960s, interstellar ammonia and water vapor were detected. Soon came formaldehyde, carbon monoxide, and the discovery of giant molecular clouds where we now know stars and planets are formed. With improvements in radio astronomy technology, today’s scientists can watch the process of star formation in other systems. The initial results are stunning.”
Lecture 22 – Star Formation and ALMA
“With an array of 66 radio antennas located in the high Chilean desert above much of the earth’s atmosphere, the Atacama Large Millimeter/submillimeter Array (ALMA) is a radio telescope tuned to the higher frequencies of radio waves. Designed to examine some of the most distant and ancient galaxies ever seen, ALMA has not only revealed new stars in the making, but planetary systems as well.”
Lecture 23 – Interstellar Chemistry and Life
“Interstellar clouds favor formation of carbon-based molecules over any other kind—not at all what statistical models predicted. In fact, interstellar clouds contain a profusion of chemicals similar to those that occur naturally on Earth. If planets are formed in this rich soup of organic molecules, is it possible life does not have to start from scratch on each planet?”
Lecture 24 – The Future of Radio Astronomy
“Learn about the newest radio telescopes and the exhilarating questions they plan to address: Did life begin in space? What is dark matter? And a new question that has just arisen in the past few years: What are fast radio bursts? No matter how powerful these new telescopes are, radio astronomers will continue pushing the limits to tell us more and more about the universe that is our home.”

Course No. 1884
Experiencing Hubble: Understanding the Greatest Images of the Universe – David M. Meyer
Lecture 5 – The Cat’s Eye Nebula – A Stellar Demise
“Turning from star birth to star death, get a preview of the sun’s distant future by examining the Cat’s Eye Nebula. Such planetary nebulae (which have nothing to do with planets) are the exposed debris of dying stars and are among the most beautiful objects in the Hubble gallery.”
Lecture 7 – The Sombrero Galaxy – An Island Universe
“In the 1920s, astronomer Edwin Hubble discovered the true nature of galaxies as ‘island universes’. Some 80 years later, the telescope named in his honor has made thousands of breathtaking pictures of galaxies. Focus on one in particular—an edge-on view of the striking Sombrero galaxy.”
Lecture 8 – Hubble’s View of Galaxies Near and Far
“Hubble’s image of the nearby galaxy NGC 3370 includes many faint galaxies in the background, exemplifying the telescope’s mission to establish an accurate distance scale to galaxies near and far—along with the related expansion rate of the universe. Discover how Hubble’s success has led to the concept of dark energy.”
Lecture 10 – Abell 2218 – A Massive Gravitational Lens
“One of the consequences of Einstein’s general theory of relativity is evident in Hubble’s picture of the galaxy cluster Abell 2218. Investigate the physics of this phenomenon, called gravitational lensing, and discover how Hubble has used it to study extremely distant galaxies as well as dark matter.”

Course No. 3130
Origin of Civilization – Scott MacEachern
Lecture 36 – Great Zimbabwe and Its Successors
“Few archaeological sites have been subjected to the degree of abuse and misrepresentation sustained by Great Zimbabwe in southeastern Africa. Nevertheless, this lecture unpacks the intriguing history of this urban center and the insights it can provide into the development of the elite.”

Course No. 3900
Ancient Civilizations of North America – Edwin Barnhart
Lecture 12 – The Wider Mississippian World
“After the fall of Cahokia, witness how Mississippian civilization flourished across eastern North America with tens of thousands of pyramid-building communities and a population in the millions. Look at the ways they were connected through their commonly held belief in a three-tiered world, as reflected in their artwork. Major sites like Spiro, Moundville, and Etowah all faded out just around 100 years before European contact, obscuring our understanding.”
Lecture 13 – De Soto Versus the Mississippians
“In 1539, Hernando de Soto of Spain landed seven ships with 600 men and hundreds of animals in present-day Florida. Follow his fruitless search for another Inca or Aztec Empire, as he instead encounters hundreds of Mississippian cities through which he led a three-year reign of terror across the land-looting, raping, disfiguring, murdering, and enslaving native peoples by the thousands.”
Lecture 19 – The Chaco Phenomenon
“Chaco Canyon contains the most sophisticated architecture ever built in ancient North America—14 Great Houses, four Great Kivas, hundreds of smaller settlements, an extensive road system, and a massive trade network. But who led these great building projects? And why do we find so little evidence of human habitation in what seems to be a major center of culture? Answer these questions and more.”
Lecture 24 – The Iroquois and Algonquians before Contact
“At the time of European contact, two main groups existed in the northeast—the hunter-gatherer Algonquian and the agrarian Iroquois. Delve into how the Iroquois created the first North American democracy as a solution to their increasing internal conflicts. Today, we know much of the U.S. Constitution is modeled on the Iroquois’ “Great League of Peace” and its 117 articles of confederation, as formally acknowledged by the U.S. in 1988.”

Course No. 4215
An Introduction to Formal Logic – Steven Gimbel
Lecture 8 – Induction in Polls and Science
“Probe two activities that could not exist without induction: polling and scientific reasoning. Neither provides absolute proof in its field of analysis, but if faults such as those in Lecture 7 are avoided, the conclusions can be impressively reliable.”

Course No. 7210
The Symphony – Robert Greenberg
Lecture 24 – Dmitri Shostakovich and His Tenth Symphony

“Dmitri Shostakovich was used and abused by the Soviet powers during much of his life. Somehow, he survived—even as his Tenth Symphony made dangerously implicit criticisms of the Soviet government.”

Course No. 7261
Understanding the Fundamentals of Music – Robert Greenberg
Lecture 9 – Intervals and Tunings

“Resuming our focus on pitch, we will turn once more to Pythagoras, and his investigation into what is now known as the overtone series. This paves the way for an examination of intervals, the evolution of tuning systems, and an introduction to the major pitch collections.”

Course No. 7270
The Concerto – Robert Greenberg
Lecture 13 – Tchaikovsky
“Excoriated by colleagues and critics alike, Tchaikovsky’s concerti ultimately triumphed to become cornerstones of the repertoire. This lecture explores his Piano Concerto no. 1 in B flat Minor, op. 23; Piano Concerto no. 2 in G Major, op. 44; and Violin Concerto in D Major, op. 35, arguably his single greatest work and one of the greatest concerti of the 19th century.”
Lecture 14 – Brahms and the Symphonic Concerto
“Johannes Brahms’s compositional style is a synthesis of the clear and concise musical forms and genres of the Classical and Baroque eras, and the melodic, harmonic, and expressive palette of the Romantic era in which he lived. This lecture examines in depth his monumental Piano Concerto no. 2 in B flat Major, op. 83.”

Course No. 8122
Albert Einstein: Physicist, Philosopher, Humanitarian – Don Howard
Lecture 1 – The Precocious Young Einstein

“The aim of these lectures is to explore Einstein the whole person and the whole thinker. You begin with an overview of the course. Then you look at important events in Einstein’s life up to the beginning of his university studies in 1896.”
Special Note: This entire series is outstanding! I will eventually be adding many of the episodes of this course as I rewatch them. (I watched this series before I began keeping track of “best” episodes.)

Course No. 8374
Understanding Russia: A Cultural History – Lynne Ann Hartnett
Lecture 10 – Alexander II, Nihilists, and Assassins

“Focus is on the reign of Alexander II, who ruled Russia from 1855 to 1881. Central to this lecture are three questions: Why did this promising reign end so violently? Did Alexander II shape developments in literature and culture? How could Russia’s last great tsar inaugurate a violent confrontation between the state and its people?”
Lecture 14 – The Rise and Fall of the Romanovs
“Here is the real story behind the Romanov dynasty, from its rise to power in 1613 to its bloody end in 1917—a tale filled with adventure, intrigue, romance, and heartbreak. It was this period that saw the Decembrist revolution, the assassination of Tsar Alexander II, and the machinations of the notorious Grigori Rasputin.”
Lecture 17 – Lenin and the Soviet Cultural Invasion
“Professor Hartnett reveals how Lenin and the Communist Party aimed to win the hearts and minds of the Soviet people through a cultural battle fought on every possible front. See how this battle was won through a militarized economy, propaganda radio, the renaming of streets, and the ‘secular sainthood’ of Lenin.”
Lecture 19 – The Tyrant is a Movie Buff: Stalinism
“Stalin and his cadre aspired to transform everyday Russian life (byt) in ways that brought forth such horrors as collectivization and the gulags. But, as you’ll learn, this was also a period where the creative work and cultural influence of writers, composers, and painters were suppressed by the terrifying mandates of Socialist Realism.”
Lecture 20 – The Soviets’ Great Patriotic War
“By the time World War II ended, the Soviets would lose 27 million men, women, and children from a total population of 200 million. In this lecture, we examine Soviet life during the Great Patriotic War and investigate how culture (including poetry and film) was used in service of the war effort.”
Lecture 21 – With Khrushchev, the Cultural Thaw
“Nikita Khrushchev emerged from the power struggles after Stalin’s death with a daring denunciation of the dictator’s cult of terror and personality. As we examine Khrushchev’s liberalization of culture, we’ll also explore its limits, including the continuation of anti-Semitism from the Stalin era, embraced under the guise of ‘anti-cosmopolitanism’.”
Lecture 22 – Soviet Byt: Shared Kitchen, Stove, and Bath
“What was everyday Soviet life like during the Khrushchev and Brezhnev periods? How and where did people live? How did they spend their leisure time? Answers to these and other questions reveal the degree to which politics affected even seemingly apolitical areas of life.”
Lecture 24 – Soviet Chaos and Russian Revenge
“On December 25, 1991, the Soviet Union came to an end. We follow the road that led to this moment under the policies of perestroika (restructuring the centrally-planned economy) and glasnost (removing rigid state censorship). Then, we conclude with a look at the rise of a new popular leader: Vladimir Putin.”

Course No. 8535
America in the Gilded Age and Progressive Era – Edward T. O’Donnell
Lecture 23 – Over There: A World Safe for Democracy

“As the Progressive Era ends, follow the complex events that led the United States into World War I. Learn how an initial federal policy of neutrality changed to one of “preparedness” and then intervention, amid conflicting public sentiments and government pro-war propaganda. Also trace the after-effects of the war on U.S. foreign policy.”
Special Note: This entire series is outstanding! I will eventually be adding many of the episodes of this course as I rewatch them. (I watched this series before I began keeping track of “best” episodes.)

Course No. 8580
Turning Points in American History – Edward T. O’Donnell
Lecture 10 – 1786 Toward a Constitution – Shays’s Rebellion

“Who was Daniel Shays? What political and economic dilemmas led to this famous farmer’s rebellion of 1786? Most important: How did this event pave the way for a reconsideration of the Articles of Confederation and the creation of the U. S. Constitution? Find out here.”
Lecture 23 – 1868 Equal Protection—The 14th Amendment
“Many legal scholars and historians have argued that the 14th Amendment, which promises equal protection under the laws, is the most important addition to the Constitution after the Bill of Rights. Here, Professor O’Donnell retells the fascinating story of how this amendment was ratified in 1868—and its turbulent history in the 20th and 21st centuries.”
Special Note: This entire series is outstanding! I will eventually be adding many of the episodes of this course as I rewatch them. (I watched this series before I began keeping track of “best” episodes.)

Course No. 30110
England, the 1960s, and the Triumph of the Beatles – Michael Shelden
Lecture 8 – The Englishness of A Hard Day’s Night
“In summer 1964, the cinematic Beatles vehicle A Hard Day’s Night broke almost every rule in Hollywood at the time. Professor Shelden reveals what lies underneath the film’s surface charm and musical numbers: an overall attitude of irreverence and defiance in the face of authority, and a challenge for audiences to think for themselves.”
Lecture 12 – Hello, Goodbye: The End of the 1960s
“In their last years together, all four of the Beatles seemed headed in new directions as they grew up—and apart. Nevertheless, witness how these final years brought a range of sounds, including protest songs, mystic melodies, anthems of friendship, and an iconic double album called simply, The Beatles, but better known as the ‘White Album.'”

Course No. 60000
The Great Questions of Philosophy and Physics – Steven Gimbel
Lecture 3 – Can Physics Explain Reality?
“If the point of physics is to explain reality, then what counts as an explanation? Starting here, Professor Gimbel goes deeper to probe what makes some explanations scientific and whether physics actually explains anything. Along the way, he explores Bertrand Russell’s rejection of the notion of cause, Carl Hempel’s account of explanation, and Nancy Cartwright’s skepticism about scientific truth.”
Lecture 4 – The Reality of Einstein’s Space
“What’s left when you take all the matter and energy out of space? Either something or nothing. Newton believed the former; his rival, Leibniz, believed the latter. Assess arguments for both views, and then see how Einstein was influenced by Leibniz’s relational picture of space to invent his special theory of relativity. Einstein’s further work on relativity led him to a startlingly new conception of space.”
Lecture 5 – The Nature of Einstein’s Time
“Consider the weirdness of time: The laws of physics are time reversable, but we never see time running backwards. Theorists have proposed that the direction of time is connected to the order of the early universe and even that time is an illusion. See how Einstein deepened the mystery with his theory of relativity, which predicts time dilation and the surprising possibility of time travel.”
Lecture 8 – Quantum States: Neither True nor False?
“Enter the quantum world, where traditional philosophical logic breaks down. First, explore the roots of quantum theory and how scientists gradually uncovered its surpassing strangeness. Clear up the meaning of the Heisenberg uncertainty principle, which is a metaphysical claim, not an epistemological one. Finally, delve into John von Neumann’s revolutionary quantum logic, working out an example.”
Lecture 10 – Wanted Dead and Alive: Schrödinger’s Cat
“The most famous paradox of quantum theory is the thought experiment showing that a cat under certain experimental conditions must be both dead and alive. Explore four proposed solutions to this conundrum, known as the measurement problem: the hidden-variable view, the Copenhagen interpretation, the idea that the human mind “collapses” a quantum state, and the many-worlds interpretation.”
Lecture 11 – The Dream of Grand Unification
“After the dust settled from the quantum revolution, physics was left with two fundamental theories: the standard model of particle physics for quantum phenomena and general relativity for gravitational interactions. Follow the quest for a grand unified theory that incorporates both. Armed with Karl Popper’s demarcation criteria, see how unifying ideas such as string theory fall short.”
Lecture 12 – The Physics of God
“The laws of physics have been invoked on both sides of the debate over the existence of God. Professor Gimbel closes the course by tracing the history of this dispute, from Newton’s belief in a Creator to today’s discussion of the “fine-tuning” of nature’s constants and whether God is responsible. Such big questions in physics inevitably bring us back to the roots of physics: philosophy.”

Course No. 80060
Music Theory: The Foundation of Great Music – Sean Atkinson
Lecture 5 – The Circle of Fifths
“Begin by defining the key of a piece of music, which is simply the musical scale that is used the most in the piece. Also discover key signatures in written music, symbols at the beginning of the musical score that indicate the key of the piece. Then grasp how the major keys all relate to each other in an orderly way, when arranged schematically according to the interval of a fifth.”
Lecture 16 – Hypermeter and Larger Musical Structures
“In listening to music, we sometimes hear the meter differently than the way it’s written on the page. Learn how the concept of hypermeter helps explain this, by showing that when measures of music are grouped into phrases, we often hear a pulse for each measure in the phrase, rather than the pulses within the measure. Explore examples of hypermeter, and how we perceive music as listeners.”

Multiverse

George F. R. Ellis writes in Issues in the Philosophy of Cosmology:

9.2 Issue H: The possible existence of multiverses
If there is a large enough ensemble of numerous universes with varying properties, it may be claimed that it becomes virtually certain that some of them will just happen to get things right, so that life can exist; and this can help explain the fine-tuned nature of many parameters whose values are otherwise unconstrained by physics.  As discussed in the previous section, there are a number of ways in which, theoretically, multiverses could be realized.  They provide a way of applying probability to the universe (because they deny the uniqueness of the universe).  However, there are a number of problems with this concept.  Besides, this proposal is observationally and experimentally untestable; thus its scientific status is debatable.

My 100-year-old uncle—a lifelong teacher and voracious reader who is still intellectually active—recently sent me Max Tegmark’s book Our Mathematical Universe: My Quest for the Ultimate Nature of Reality, published by Vintage Books in 2014. I could not have had a more engaging introduction to the concept of the Multiverse. Tegmark presents four levels of multiverses that might exist. They are

Level I Multiverse: Distant regions of space with the same laws of physics that are currently but not necessarily forever unobservable.

Level II Multiverse: Distant regions of space that may have different laws of physics and are forever unobservable.

Level III Multiverse: Quantum events at any location in space and in time cause reality to split and diverge along parallel storylines.

Level IV Multiverse: Space, time, and the Level I, II, and III multiverses all exist within mathematical structures that describe all physical existence at the most fundamental level.

There seems little question that our universe is very much larger than the part that we can observe. The vast majority of our universe is so far away that light has not yet had time to reach us from those regions. Whether we choose to call the totality of these regions the universe or a Level I multiverse is a matter of semantics.

Is our universe or the Level I multiverse infinite? Most likely not. That infinity is a useful mathematical construct is indisputable. That infinite space or infinite time exists is doubtful. Both Ellis and Tegmark agree on this and present cogent arguments as to why infinity cannot be associated with physical reality. Very, very large, or very, very small, yes, but not infinitely large or infinitely small.

Does a Level II, III, and IV multiverse exist? Tegmark thinks so, but Ellis raises several objections, noted above and elsewhere. The multiverse idea remains quite controversial, but as Tegmark writes,

Even those of my colleagues who dislike the multiverse idea now tend to grudgingly acknowledge that the basic arguments for it are reasonable. The main critique has shifted from “This makes no sense and I hate it” to “I hate it.”

I will not delve into the details of the Level II, III, and IV multiverses here. Read Tegmark’s book as he adroitly takes you through the details of eternal inflation, quantum mechanics and wave functions and the genius and tragic story of Hugh Everett III, the touching tribute to John Archibald Wheeler, and more, leading into a description of each multiverse level in detail.

I’d like to end this article with a quote from Max Tegmark from Mathematical Universe. It’s about when you think you’re the first person ever to discover something, only to find that someone else has made that discovery or had that idea before.

Gradually, I’ve come to totally change my feelings about getting scooped. First of all, the main reason I’m doing science is that I delight in discovering things, and it’s every bit as exciting to rediscover something as it is to be the first to discover it—because at the time of the discovery, you don’t know which is the case. Second, since I believe that there are other more advanced civilizations out there—in parallel universes if not our own—everything we come up with here on our particular planet is a rediscovery, and that fact clearly doesn’t spoil the fun. Third, when you discover something yourself, you probably understand it more deeply and you certainly appreciate it more. From studying history, I’ve also come to realize that a large fraction of all breakthroughs in science were repeatedly rediscovered—when the right questions are floating around and the tools to tackle them are available, many people will naturally find the same answers independently.

References
Ellis, G.F.R., Issues in the Philosophy of Cosmology, Philosophy of Physics (Handbook of the Philosophy of Science), Ed. J. Butterfield and J. Earman (Elsevier, 2006), 1183-1285.
[http://arxiv.org/abs/astro-ph/0602280]

Tegmark, Max. Our mathematical universe : my quest for the ultimate nature of reality. New York: Alfred A. Knopf, 2014.

“You passed your exam in many parallel universes—but not in this one.”

The Anthropic Question

George F. R. Ellis writes in Issues in the Philosophy of Cosmology:

9.1 Issue G: The anthropic question: Fine tuning for life
One of the most profound fundamental issues in cosmology is the Anthropic question: why does the Universe have the very special nature required in order that life can exist?  The point is that a great deal of “fine tuning” is required in order that life be possible.  There are many relationships embedded in physical laws that are not explained by physics, but are required for life to be possible; in particular various fundamental constants are highly constrained in their values if life as we know it is to exist:

Ellis goes on to quote Martin Rees.

A universe hospitable to life—what we might call a biophilic universe—has to be special in many ways … Many recipes would lead to stillborn universes with no atoms, no chemistry, and no planets; or to universes too short lived or too empty to evolve beyond sterile uniformity.

Physics does not tell us anything (yet) about why the fundamental constants and other parameters have the values they do.  These parameters include, for example, the speed of light, the Planck constant, the four fundamental forces and their relative strengths, the mass ratio of the proton and the electron, the fine-structure constant, the cosmological density parameter, Ωtot, relative to the critical density, and so on.  And, why are there four fundamental forces?  Why not five?  Or three?

Also, why do we live in a universe with three spatial dimensions and one time dimension?  Others are possible—even universes with two or more time dimensions.

But it appears that only three spatial dimensions and one time dimension is conducive to life (at least life as we know it), as shown in the diagram above (Whittle 2008).

In fact, altering almost any of the parameters would lead to a sterile universe and we could not exist.  Is the universe fine-tuned for our existence?

Let’s assume for the moment it is.  Where does that lead us?

  1. As our understanding of physics advances, we will eventually understand why these parameters must have the values that they do. -or-
  2. We will eventually learn that some of these parameters could have been different, and still support the existence of life. -or-
  3. God created the universe in such a way that life could exist -or-
  4. We’re overthinking the problem.  We live in a life-supporting universe, so of course we find the parameters are specially tuned to allow life. -or-
  5. There exist many universes with different parameters and we just happen to find ourselves in one that is conducive to life. (The multiverse idea.)

#4 is the anthropic explanation, but a deeper scientific understanding will occur if we find either #1, #2, or #5 to be true.  #3 is problematic for a couple of reasons.  First of all, how was God created?  Also, deism has a long history of explaining phenomena we don’t understand (“God of the gaps”), but in time we are able to understand each phenomenon in turn as science progresses.

The anthropic explanation itself is not controversial.  What is controversial is deciding to what degree fine tuning has occurred and how to explain it.

In recent years, the multiverse idea has become more popular because, for example, if there were a billion big bangs and therefore a billion different universes created, then it should not be at all surprising that we find ourselves in  one with just the right set of parameters to allow our existence.  However, there is one big problem with the multiverse idea.  Not only do we have no physical evidence that a multiverse exists, but we may never be able to obtain evidence that a multiverse exists, due to the cosmological horizon problem1.  If physical evidence of a multiverse is not forthcoming, then in that sense it is not any better than the deistic explanation.

To decide whether or not there is only one combination of parameters that can lead to life we need to rule out all the other combinations, and that is a tall order.  Recent work in this field suggests that there is more than one combination of parameters that could create a universe that is hospitable to life (Hossenfelder 2018).

Thinking now about why our universe is here at all, it seems there are just two possibilities:

(1)  Our universe has a supernatural origin.

(2)  Our universe has a natural origin.

If our universe has a supernatural origin, then what is the origin of the supernatural entity (e.g. God)?  If, on the other hand, our universe had a natural origin (e.g. something was created out of nothing), didn’t something have to exist (laws of physics or whatever) before the universe came into existence?  If so, what created those pre-conditions?

In either case, we are facing an infinite regression.  However, we could avoid the infinite regression by stating that something has to exist outside of time, that is to say, it has no beginning and no ending.  But isn’t this just replacing one infinity with another?

Perhaps there’s another possibility.  Just as a chimpanzee cannot possibly understand quantum mechanics, could it be that human intellect is also fundamentally limited?  Are the questions in the previous two paragraphs meaningless or nonsensical in the context of some higher intelligence?

1We appear to live in a universe that is finite but very much larger than the region that is visible to us now, or ever.

References
G.F.R. Ellis, Issues in the Philosophy of Cosmology, Philosophy of Physics (Handbook of the Philosophy of Science), Ed. J. Butterfield and J. Earman (Elsevier, 2006), 1183-1285.
[http://arxiv.org/abs/astro-ph/0602280]

Sabine Hossenfelder, Lost in Math: How Beauty Leads Physics Astray (Basic Books, 2018).

M. J. Rees, Our Cosmic Habitat (Princeton and Oxford, 2003).

Mark Whittle, “Fine Tuning and Anthropic Arguments”, Lecture 34, Course No. 1830.  Cosmology: The History and Nature of Our Universe.  The Great Courses, 2008.  DVD.
[https://www.thegreatcourses.com/courses/cosmology-the-history-and-nature-of-our-universe.html]

The Beginning

We continue our series of excerpts (and discussion) from the outstanding survey paper by George F. R. Ellis, Issues in the Philosophy of Cosmology.

Thesis D1: An initial singularity may or may not have occurred.
A start to the universe may have occurred a finite time ago, but a variety of alternatives are conceivable: eternal universes, or universes where time as we know it came into existence in one or another way.  We do not know which actually happened, although quantum gravity ideas suggest a singularity might be avoided.

If we imagine, for a moment, running the clock of the universe backwards to earlier and earlier times, its size gets smaller and its density gets larger until we reach a moment—even earlier than the putative inflationary era—when classical physics at the macroscopic level no longer applies and some (as yet unknown) quantum physics must apply to everything—even gravity.  Therein lies the problem, because if you run the clock backwards just 5.39 x 10-44 second from this time, you reach the purported moment of the Big Bang—the initial singularity.  But whoa (or perhaps woe)!  How can we say anything about the Big Bang—or even if it occurred at all—since the laws of known physics completely break down 5.39 x 10-44 second (the Planck time) after the Big Bang!  See the problem?

Perhaps the universe came into existence through a process analogous to radioactive decay where an alpha particle leaves a nucleus through quantum tunneling.  Perhaps our universe “tunneled” into existence from somewhere else, and thus our beginning isn’t really the beginning.  This is just one of many possibilities.

This is a key issue in terms of the nature of the universe: a space-time singularity is a dramatic affair, where the universe (space, time, matter) has a beginning and all of physics breaks down and so the ability to understand what happens on a scientific basis comes to an end. However eternal existence is also problematic, leading for instance to the idea of Poincaré’s eternal return: everything that ever happened will recur an infinite number of times in the future and has already occurred an infinite number of times in the past.  This is typical of the problems associated with the idea of infinity.  It is not clear in the end which is philosophically preferable: a singularity or eternal existence.  That decision will depend on what criteria of desirability one uses.

While infinity is a highly useful mathematical device, one can make a strong argument that infinities do not exist in the physical universe (or even multiverse).  Quantum physics already gives us a possible clue about the infinitely small: we appear not to be able to subdivide space or time any further than the Planck length (1.616 x 10-35 meter) or the Planck time (5.39 x 10-44 second).  We would not be able to distinguish between two points less than a Planck length apart, nor two moments in time less than a Planck time apart.  While harder to envision, might not there also be an upper limit to size?  And time?

Thesis D2: Testable physics cannot explain the initial state and hence specific nature of the universe.
A choice between different contingent possibilities has somehow occurred; the fundamental issue is what underlies this choice.  Why does the universe have one specific form rather than another, when other forms consistent with physical laws seem perfectly possible?  The reasons underlying the choice between different contingent possibilities for the universe (why one occurred rather than another) cannot be explored scientifically.  It is an issue to be examined through philosophy or metaphysics.

Metaphysics is the part of philosophy that deals with existence, space, time, cause and effect, and the like.  Metaphysics begins where physics necessarily ends due to observational limitations.

Did anything exist before the Big Bang?

Was there a Big Bang?

What are the physical properties of the very early universe, when energy densities existed that are far beyond our ability to recreate in the laboratory?

What lies beyond our particle horizon?

Are there other universes?

Why does anything exist at all?

References
Ellis, G. F. R. 2006, Issues in the Philosophy of Cosmology, Philosophy of Physics (Handbook of the Philosophy of Science), Ed. J. Butterfield and J. Earman (Elsevier, 2006), 1183-1285.
[http://arxiv.org/abs/astro-ph/0602280]

Liddle, A.R. 2015, An Introduction to Modern Cosmology, 3rd ed., Wiley, ISBN: 978-1-118-50214-3.

A Small, Big, or Really Big Universe?

George F. R. Ellis writes in section 2.4.2 of his outstanding survey paper, Issues in the Philosophy of Cosmology:

Clearly we cannot obtain any observational data on what is happening beyond the particle horizon; indeed we cannot even see that far because the universe was opaque before decoupling.  Our view of the universe is limited by the visual horizon, comprised of the worldlines of furthest matter we can observe—namely, the matter that emitted the CBR at the time of last scattering.

The picture we obtain of the LSS by measuring the CBR from satellites such as COBE and WMAP is just a view of the matter comprising the visual horizon, viewed by us at the time in the far distant past when it decoupled from radiation.

Visual horizons do indeed exist, unless we live in a small universe, spatially closed with the closure scale so small that we can have seen right around the universe since decoupling.

The major consequence of the existence of visual horizons is that many present-day speculations about the super-horizon structure of the universe—e.g. the chaotic inflationary theory—are not observationally testable, because one can obtain no definite information whatever about what lies beyond the visual horizon.  This is one of the major limits to be taken into account in our attempts to test the veracity of cosmological models.

Let’s start by defining a few of the terms that Ellis uses above.

particle horizon – the distance beyond which light has not yet had time to reach us in all the time since the Big Bang

decoupling – the time after the Big Bang when the Universe had expanded and cooled enough that it was no longer a completely ionized opaque plasma; atoms could form and photons began traveling great distances without being absorbed

worldlines – the path of a photon (or any particle or object) in 4-dimensional spacetime: its location at each and every moment in time

CBRcosmic background radiation

LSS – last scattering surface

COBECosmic Background Explorer

WMAPWilkinson Microwave Anisotropy Probe

(And, Planck should be added now, too)

Now the question.  Do we live in a small, big, or really big universe?  The best answer we can give now (or, perhaps, even in the future) is that we live in a really big universe, though it is unlikely to be infinite.  Ellis himself provides a cogent argument in section 9.3.2 of the paper referenced here that infinity, while a very useful mathematical tool, does not ever exist in physical reality.  We shall investigate this topic in a future posting.

Even though general relativity shows us how matter defines the geometry of our observable universe, it tells us nothing about the topology of our universe, in other words, its global properties.  Do we live in a wrap-around universe where if we set off in one direction and traveled long enough, we’d eventually return to the same point in spacetime?  Is the topology of our universe finite or infinite?  At the moment it appears that we are not able to observe enough of the universe to discern its topology.  If that is true, we may never be able to understand what type of universe we live in.  But observational cosmologists will continue to search for the imprint of topology on our visible universe at the largest scales.

References
Ellis, G. F. R. 2006, Issues in the Philosophy of Cosmology, Philosophy of Physics (Handbook of the Philosophy of Science), Ed. J. Butterfield and J. Earman (Elsevier, 2006), 1183-1285.
[http://arxiv.org/abs/astro-ph/0602280]

Liddle, A.R. 2015, An Introduction to Modern Cosmology, 3rd ed., Wiley, ISBN: 978-1-118-50214-3.