A growing controversy over the multiverse and the anthropic principle has exposed a major fault line in modern physics and cosmology. Some researchers see the multiverse and the anthropic principle as inevitable, others see them as an abdication of empirical science. The controversy spans quantum mechanics, inflationary Big Bang cosmology, string theory, supersymmetry and, more generally, the proper roles of experimentation and mathematical theory in modern science.

The “many worlds interpretation” of quantum mechanicsSince the 1930s, when physicists first developed the mathematics behind quantum mechanics, researchers have found that this theory appears to

Continue reading Universe or multiverse? The war rages on

]]>A growing controversy over the multiverse and the anthropic principle has exposed a major fault line in modern physics and cosmology. Some researchers see the multiverse and the anthropic principle as inevitable, others see them as an abdication of empirical science. The controversy spans quantum mechanics, inflationary Big Bang cosmology, string theory, supersymmetry and, more generally, the proper roles of experimentation and mathematical theory in modern science.

Since the 1930s, when physicists first developed the mathematics behind quantum mechanics, researchers have found that this theory appears to govern, to extraordinary precision, the world of atomic and subatomic phenomena. Inherent in the mathematics of quantum mechanics is the notion that the world is in a superposition of many possible states, and only “chooses” one when a measurement is made. While many physicists recommend a “shut up and calculate” approach, others continue to pursue a more “reasonable” framework to explain these perplexing results.

One such framework, first advanced by Hugh Everett in 1957, is known as the “many worlds interpretation” of quantum mechanics. It posits that the real world is continually bifurcating into a vast number of parallel universes, and the reason we see only one branch realized is simply that we reside in that branch. While many physicists, to this day, resist this view, others have accepted it and apply it in day-to-day work. As physicist Sean Carroll wrote in his 2019 book *Something Deeply Hidden*, “When it comes to understanding how to quantize the universe itself, Many-Worlds seems to be the most direct path to take.” [Carroll2019, pg. 306].

The inflationary Big Bang theory of cosmology has its roots in some paradoxes first noted in the 1960s [Guth1997, pg. 25]:

*The flatness problem*. In the very early universe after the Big Bang, the ratio of the actual mass density of the universe to the “critical density” must have been exceedingly close to one (to within one part in 10^{14}). If the ratio were very slightly lower, the universe would have dispersed too rapidly for stars and galaxies to have formed, but if it were very slightly larger, the universe would have long ago recollapsed in a big crunch.*The horizon problem*. Different regions of space appear to be essentially indistinguishable, say in the intensity of the cosmic microwave background. But how can regions on opposite sides of the universe be in such close coordination today, since no physical force, not even light rays, could have traversed the distance between them since the Big Bang?

In the early 1980s, physicist Alan Guth hypothesized that in the first tiny fraction of a second after the Big Bang, the universe underwent an enormous “inflation,” wherein the fabric of space exploded by a factor of roughly 10^{30} [Guth1997, pg. 193]. The inflation hypothesis explained the two paradoxes above, and is now widely accepted in the field, although some demur. Paul Steinhart, for example, one of the early pioneers of inflationary cosmology, more recently has expressed serous doubts [Horgan2014].

In any event, one consequence of the inflationary cosmology is that the full universe created at the Big Bang is some 10^{23} times larger than our observable universe [Guth1997, pg. 186]. More recently, Andrei Linde of Stanford University and Alex Vilenkin of Tufts University observed that Guth’s theory leads to a “chaotic eternal inflation,” which has no beginning or end [Susskind2005, pg. 81]. Either way, what we have been calling the “universe” may be just one “pocket universe” amid an enormous ensemble of such universes.

For nearly 40 years, physicists have been exploring “string theory,” namely the notion that all physical phenomena are, at the lowest level of reality, tiny vibrating strings and membranes — roughly 10^{-34}cm in size, or vastly smaller even than a proton. What’s more, these strings or membranes live in a 10- or 11-dimensional space, not the 3-dimensional space that we are accustomed to. Physicist Brian Greene, author of two widely read semi-popular expositions of string theory, *The Elegant Universe* and *The Fabric of the Cosmos*, emphasizes that string theory appears to neatly unify all known physical forces, including gravity, in one elegant package [Greene2003a; Greene2011].

The original dream of string theory was to deduce a unique set of physical laws and constants — namely those that govern our universe. Instead, researchers have found that the underlying mathematics suggests a vast “landscape” of possible universes, by one reckoning 10^{500} in number, each corresponding to a different Calabi-Yau manifold, and each governed by potentially different sets of physical laws and constants. Some researchers are deeply disappointed and perplexed by this outcome, and continue to press forward to find a unique solution. Others have accepted the landscape as an unavoidable feature of the theory, and further see it as a potential solution to some long-standing paradoxes (see below). Still others cite this outcome as evidence that the string theory framework is fundamentally flawed, and question the continuation of research in the area.

Before going further, we should note that different authors employ different terminology for various multiverse varieties and realms. Physicist Paul Davies lists five realms: the universe we can now physically observe; everything within our “horizon”; the realm created in the Big Bang but beyond our horizon; the “pocket universe” that has laws similar to our own; and the full “multiverse” [Davies2007, pg. 31-32]. Max Tegmark, in his 2014 book *Our Mathematical Universe*, includes his own notion of all logically consistent mathematical structures [Tegmark2014]. Brian Greene, in his 2011 book *The Hidden Reality*, lists seven different varieties of the multiverse, including the string theory landscape [Greene2011, pg. 309].

While some researchers militantly resist the notion of a multiverse, others see lemonade in the lemons. In their view, the multiverse could resolve some nagging unexplained paradoxes, notably the fact that the universe we reside in appears to be remarkably fine-tuned for the eventual rise of intelligent life. In particular, these researchers argue that the reason that our universe is so finely tuned for the rise of intelligent life is that it is but one universe in a vast multiverse, and if ours were not so finely tuned for life, we wouldn’t be here to discuss the topic. Here are some of these finely tuned “cosmic coincidences” (see this earlier Math Scholar blog and the 2016 Lewis-Barnes book [Lewis2016] for additional details):

*Carbon resonance and the strong force*. Although researchers can explain the creation of hydrogen, helium, lithium and beryllium in the cauldron of the first 100 seconds or so after the Big Bang, the synthesis of heavier elements, beginning with carbon, relies on a very finely tuned resonance that is just energetic enough to permit a triple-helium nuclear reaction to produce a carbon nucleus. By the way, although one can imagine living organisms based on other elements, carbon is by far is the most suitable element for the construction of complex molecules, as required for any conceivable form of living or sentient beings.*The weak force and the proton-neutron balance*. Had the weak force been somewhat weaker, the amount of hydrogen in the universe would be greatly decreased, starving stars of fuel for nuclear energy and leaving the universe a cold and lifeless place.*Neutrons and the proton-to-electron mass ratio*. The neutron’s mass is very slightly more than the combined mass of a proton, an electron and a neutrino. If neutrons were very slightly less massive, then they could not decay without energy input and the universe would be entirely protons (i.e., hydrogen), but if their mass were slightly lower, then all isolated protons would decay into neutrons, and no atoms other than hydrogen, helium, lithium and beryllium, which were synthesized in the Big Bang, could form.*Anisotropy of the cosmic microwave background*. In 1992, scientists discovered that there is a very slight anisotropy in the cosmic microwave background radiation, roughly one part in 100,000, which is just enough to permit the formation of stars and galaxies.*The cosmological constant paradox*. When one calculates, based on known principles of quantum mechanics, the “vacuum energy density” of the universe, one obtains the incredible result that empty space “weighs” 10^{93}grams per cubic centimeter. Since the actual average mass density of the universe is roughly 10^{-28}grams per cc, this figure is in error by 120 orders of magnitude. Physicists, who have fretted over this discrepancy for decades, have noted that such calculations typically involve only electromagnetism, and so perhaps when other known forces are included, all terms will cancel out to exactly zero as a consequence of some heretofore unknown principle, such as supersymmetry. But these hopes were shattered in 1998 with the discovery that the universe’s expansion is accelerating, implying that the cosmological constant must be slightly positive. This means that the positive and negative contributions to the cosmological constant somehow cancel to 120-digit accuracy, yet fail to cancel beginning at the 121-st digit. Curiously, this observation is in accord with a prediction made by Nobel Prize-winning physicist Steven Weinberg in 1987, who argued that the cosmological constant must be zero to within one part in roughly 10^{120}, or else the universe either would have dispersed too fast for stars and galaxies to have formed, or would have recollapsed upon itself long ago [Weinberg1989].*Mass of the Higgs boson*. A similar coincidence has come to light recently in the wake of the 2012 discovery of the Higgs boson at the Large Hadron Collider [Overbye2012a]. Higgs was found to have a mass of 126 billion electron volts (i.e., 126 Gev). However, a calculation of interactions with other known particles yields a mass of some 10^{19}Gev. This means that the rest mass of the Higgs boson must be almost exactly the negative of this enormous number, so that when added to 10^{19}gives 126 Gev, as a result of massive and unexplained cancelation. Supersymmetry has been proposed as a solution to this paradox, but no hint of supersymmetric particles has been seen in the latest experiments at the LHC, and it is not clear that the required cancelation would occur even if the superparticles do exist. Similar difficulties afflict a number of other particle masses and forces — some are of modest size, yet others are orders of magnitude larger. These difficulties collectively are known as the “hierarchy” and “flavor” problems.*The low-entropy state of the universe*. The overall entropy (disorder) of the universe is, in the words of Lewis and Barnes, “freakishly lower than life requires.” After all, life requires, at most, a single galaxy of highly ordered matter to create chemistry and life on some planet.

For additional details, see this earlier Math Scholar blog and the 2016 Lewis-Barnes book [Lewis2016].

As mentioned above, some researchers see the multiverse as a solution to the apparent fine-tuning of the universe. According to this line of reasoning, we should not be surprised that we find ourselves in a universe that has somehow beaten the one-in-10^{120} odds to be life-friendly (to pick just the cosmological constant paradox as an example), because it had to happen somewhere in the multiverse, and, besides, if our universe were not life-friendly, then we would not be here to discuss it. In other words, these researchers propose that the multiverse, in particular the string theory landscape, actually exists in some sense, but acknowledge that the vast majority of these universes are utterly sterile — either very short-lived or else completely devoid of atoms or other information-rich structures, much less sentient beings like us contemplating the meaning of their existence.

But the multiverse and the usage of the anthropic principle to explain fine-tuning have sharply divided the physics-cosmology community. While some see these notions as inevitable, others reject them outright, and more broadly decry the underlying theories upon which they are based, including inflation, supersymmetry, string theory and other theories not strongly supported by rigorous empirical evidence. Among other things, these writers argue that the multiverse is a flagrant violation of Occam’s razor, in that it postulates an enormous ensemble of empirically unobservable universes, just to explain our own. They further see the anthropic principle as a tautology, a fundamental retreat from the quest to understand why our universe is the way it is.

Here are some examples of these critics:

In his 2006 book *Not Even Wrong*, physicist Peter Woit concluded [Woit2006, pg. 264]:

Any further progress toward understanding the most fundamental constituents of the universe will require physicists to abandon the now ossified ideology of supersymmetry and superstring theory that has dominated the last two decades.

Also in 2006, Lee Smolin described these developments as a “crisis” for the field [Smolin2006, pg. 352]:

We physicists need to confront the crisis facing us. A scientific theory [string theory and the multiverse] that makes no predictions and therefore is not subject to experiment can never fail, but such a theory can never succeed either, as long as science stands for knowledge gained from rational argument borne out by evidence. There needs to be an honest evaluation of the wisdom of sticking to a research program that has failed after decades to find grounding in either experimental results or precise mathematical formulation. String theorists need to face the possibility that they will turn out to have been wrong and others right.

In a 2015 update, Smolin repeated his concerns, specifically condemning the multiverse and the anthropic principle [Smolin2015]:

Cosmology is in crisis. Recent experiments have given us an increasingly precise narrative of the history of our universe, but attempts to interpret the data have led to a picture of a “preposterous universe” that eludes explanation in the terms familiar to scientists. …

As a result, some cosmologists suggest that there is not one universe, but an infinite number, with a huge variety of properties: the multiverse. There are an infinite number of universes in the collection that are like our universe and an infinite number that are not. But the ratio of infinity to infinity is undefined, and can be made into anything the theorist wants. Thus the multiverse theory has difficulty making any firm predictions and threatens to take us out of the realm of science. … As attractive as the idea may seem, it is basically a sleight of hand, which converts an explanatory failure into an apparent explanatory success. The success is empty because anything that might be observed about our universe could be explained as something that must, by chance, happen somewhere in the multiverse.

In the above quotation, Smolin’s comment on the “ratio of infinity to infinity” echoes the “measure problem” of cosmology: the failure to develop a convincing measure theory of possible universes in the multiverse that would permit some meaningful computation of probabilities. For details, see this Quanta magazine article: [Wolchover2014].

Most recently, physicist Sabine Hossenfelder, in her 2018 book *Lost in Math: How Beauty Leads Physics Astray*, wrote [Hossenfelder2018],

The hidden rules [a preference for elegance and mathematical beauty] served us badly. Even though we proposed an abundance of new natural laws, they all remained unconfirmed. And while I witnessed my profession slip into crisis, I slipped into my own personal crisis. I’m not sure anymore that what we do here, in the foundations of physics, is science. And if not, then why am I wasting my time with it?

Other researchers have fought back, emphasizing that the multiverse and the anthropic principle are suggested in the mathematics of several of these underlying theories. More generally, they argue that resisting the multiverse smacks of the many similar denials of a larger-than-expected world in previous eras. Tom Siegfried, for example, wrote in his 2019 book *The Number of the Heavens* [Siegfried2019],

Denying the possibility of a multiverse ignores Descartes’s exhortation to “beware of presuming too highly of ourselves” by supposing that there are “limits to the world” we are capable of correctly imagining.

Woit has responded to Siegfried by noting that Siegfried collected arguments from string theory landscape proponents such as Carroll, Deutsch, Guth, Greene, Linde, Polchinski, Rees, Susskind, Tegmark, and Weinberg, but did not fully acknowledge criticisms from writers such as Baggott, Ellis, Hossenfelder, Smolin, Penrose and Richter, and he only briefly mentioned the lack of experimental confirmation in the latest results from the Large Hadron Collider. Thus Woit argued that Siegfried has only told one side of the story [Woit2019].

John Horgan, who for many years has written articles in *Scientific American*, added the following on Carroll’s and Siegfried’s recent books [Horgan2019]:

Science is ill-served when prominent thinkers tout ideas that can never be tested and hence are, sorry, unscientific. … Shouldn’t scientists do something more productive with their time?

Here are comments by some other leading figures in the field, pro and con:

*Paul Davies*: Davies has criticized the multiverse as a flagrant violation of Occam’s razor. What’s more, he points out that if the multiverse exists, then not only would other universes like ours exist, but also vastly more universes where advanced technological civilizations acquire the power to*simulate*universes like ours on computer. Thus our entire universe, including all “intelligent” residents, could be merely avatars in some computer simulation. In that case, why should we take the “laws of nature” seriously? [Davies2007, pg. 179-185].*George F. R. Ellis*: “All in all, the case for the multiverse is inconclusive. The basic reason is the extreme flexibility of the proposal: it is more a concept than a well-defined theory. … The challenge I pose to the multiverse proponents is: can you prove that unseeable parallel universes are vital to explain the world we do see? And is the link essential and inescapable?” [Ellis2011].*David Gross*: Gross invoked Winston Churchill in urging fellow researchers to “Never, ever, ever, ever, ever, ever, ever, ever give up” in seeking a single, compelling theory that eliminates the need for multiverse-anthropic arguments [Susskind2005, pg. 355].*Stephen Hawking*: “I will describe what I see as the framework for quantum cosmology, on the basis of M theory [one formulation of string theory]. I shall adopt the no boundary proposal, and shall argue that the Anthropic Principle is essential, if one is to pick out a solution to represent our universe, from the whole zoo of solutions allowed by M theory.” [Susskind2005, pg. 353].*Andrei Linde*: “Those who dislike anthropic principles are simply in denial. This principle is not a universal weapon, but a useful tool, which allows us to concentrate on the fundamental problems of physics by separating them from the purely environmental problems, which may have an anthropic solution. One may hate the Anthropic Principle or love it, but I bet that eventually everyone is going to use it.” [Susskind2005, pg. 353].*Juan Maldacena*: “I hope [the multiverse-anthropic argument] isn’t true.” However, when asked whether he saw any hope in the other direction, he answered, “No, I’m afraid I don’t.” [Susskind2005, pg. 350].*Joseph Polchinski*: Polchinski, one of the leading researchers in string theory, saw no alternative to the multiverse-anthropic view [Susskind2005, pg. 350].*Paul Steinhardt*: “I consider this approach [the multiverse-anthropic view] to be extremely dangerous for two reasons. First, it relies on complex assumptions about physical conditions far beyond the range of conceivable observation so it is not scientifically verifiable. Secondly, I think it leads inevitably to a depressing end to science. What is the point of exploring further the randomly chosen physical properties in our tiny corner of the multiverse if most of the multiverse is so different. I think it is far too early to be so desperate. This is a dangerous idea that I am simply unwilling to contemplate.” [Steinhardt2006].*Leonard Susskind*: “The fact that [the cosmological constant] is not absent is a cataclysm for physicists, and the only way that we know how to make any sense of it is through the reviled and despised Anthropic Principle.” [Susskind2005, pg. 22].*Gerard ‘t Hooft*: “Nobody could really explain to me what it means that string theory has 10^{100}vacuum states. Before you say such a thing you must first give a rigorous definition on what string theory is, and we haven’t got such a definition. Or was it 10^{500}vacua, or 10^{10000000000}? As long as such ‘details’ are still up in the air, I feel extremely uncomfortable with the anthropic argument. … However, some form of anthropic principle I cannot rule out.” [Susskind2005, pg. 350].*Max Tegmark*: As mentioned above, Tegmark has not only endorsed the multiverse suggested by others, but has also proposed that the multiverse ultimately consists of all logically consistent mathematical structures, which actually exist, although only a minuscule fraction contain sentient observers. [Tegmark2014].*Steven Weinberg*: “For what it is worth, I hope that [the multiverse-anthropic argument] is not the case. As a theoretical physicist, I would like to see us able to make precise predictions, not vague statements that certain constants have to be in a range that is more or less favorable to life. I hope that string theory really will provide a basis for a final theory and that this theory will turn out to have enough predictive power to be able to prescribe values for all the constants of nature including the cosmological constant. We shall see.” [Weinberg1993, pg. 229].

Many of the issues surrounding the multiverse, fine tuning and the anthropic principle were nicely summarized in two recent Quanta Magazine articles [Wolchover2013; Wolchover2014].

In June 2018, a team of prominent string theorists led by Cumrun Vafa of Harvard University released a new paper that fundamentally questions the notion of a string theory-based multiverse with 10^{500} or more variations [Obied2018]. If their results are confirmed by other physicists and experimental evidence, they could seriously challenge string theory and even our entire conception of modern physics and cosmology [Moskowitz2018]. The Vafa result and its implications have been explained in detail in a very nice Quanta article by Natalie Wolchover [Wolchover2018a]. Here is a brief summary:

The main result of the Vafa team paper implies that as the universe expands, the vacuum energy density of empty space must decrease at least as fast as the rate given by a certain formula. The rule appears to hold in most string theory-based universal models, but it violates two bedrocks of modern cosmology: the accelerating expansion of the universe due to dark energy and the hypothesized “inflation” epoch of the first second after the big bang. In particular, Vafa and his colleagues argue that “de Sitter universes,” with stable, constant and positive amounts of vacuum energy density, simply are not possible under rather general assumptions of string theory. Such universes in general, and ours in particular, would reside in the “swampland” of string theory because they are not mathematically consistent. And yet here we are…

Just as significantly, the Vafa team result draws into question the widely accepted inflation epoch of the very early universe, when the universe is thought to have expanded by a factor of roughly 10^{30} or more. The trouble is, Vafa’s result implies that the “inflaton field” of energy that drove inflation must have declined too quickly to have formed a smooth and flat universe like the one we reside in.

String theorists and other physicists are divided about the Vafa team result [Wolchover2018a]. Eva Silverstein of Stanford University, a leader in efforts to construct string theory-based models of inflation, believes the result is likely false, as does her husband Shamit Kachru, co-author of the 2003 KKLT paper that is the basis of string theory-based models of de Sitter universes [Kachru2003]. But others, such as Hirosi Ooguri of the California Institute of Technology, are inclined to believe the Vafa result, since other “swampland” conjectures have withstood challenges and are now on a very solid theoretical footing [Wolchover2018a].

One possible way out is that the accelerating expansion of the universe is not due to an ever-present positive dark energy, as currently believed, but instead is due to quintessence, a hypothesized energy source that gradually decreases over tens of billions of years. If the quintessence hypothesis is true, it could revolutionize physics and cosmology.

The quintessence hypothesis will be tested in several new experiments currently underway, and some others scheduled for the future, which will analyze more carefully whether the accelerating expansion of the universe is constant or variable [Wolchover2017]. One of these experiments is the Dark Energy Survey, currently underway, which analyzes the clumpiness of galaxies. The initial results so far, released in August 2017, find the universe is 74% dark energy and 21% dark matter, with everything else (stars, galaxies, planets and us) in the remaining 5%, all of which is pretty much consistent with our current understanding so far.

Another related experiment is the Wide Field Infared Survey Telescope (WFIRST) system, which is specifically designed to study dark energy and infrared astrophysics. The Euclid telescope, currently in development, will investigate even more accurately the relationship between distance and redshift that is at the heart of modern cosmology.

As mentioned above, the theory of cosmic inflation is also challenged by the Vafa team result. Along this line, experimental systems such as the Simons Observatory will search for signatures and other evidence of cosmic inflation. This evidence will be scrutinized carefully, though, in the wake of the widely hailed 2014 announcement of evidence for inflation that subsequently bit the dust, so to speak, in the sense that the results were later explained by dust in the Milky Way [Wolchover2015].

Needless to say, if Vafa’s results are confirmed as correct, and the dark energy explanation for the accelerating expansion of the universe is reaffirmed by experimental evidence, then this may spell doom for string theory, and may fundamentally draw into question the multiverse and the anthropic principle.

But more is at issue than just whether this or that particular theory is correct. Many researchers in the field are fundamentally asking whether it is absolutely essential to have rigorous empirical testing for a theoretical line of research to be considered a legitimate branch of modern science and worth pursuing. Should the mathematical physics world stick to a very strict standard here, or is a bit more flexibility in order?

For example, string theory researchers have worked on the mathematics behind the theory for nearly 40 years. Yet despite numerous breakthroughs and advances, they are still not able to present a clear-cut experimental test, possible to conduct with present-day technology, that could definitively settle whether or not the theory is a valid description of nature. The Vafa team result, mentioned above, is perhaps the closest to a definitive test, but both sides still agree that more theoretical development and more rigorous empirical results are required to make a valid determination. If, say, ten more years elapse and theorists have still not produced a credible basis for experimental testing, will it still be prudent to continue large-scale investigations in the string theory arena? How much “rope” should be thrown to the string theory research world?

Almost everyone agrees, in the end, that empirical testing is the deciding touchstone for modern science. As Vafa explains [Wolchover2018a],

]]>Raising the question [whether string theory is consistent with dark energy] is what we should be doing. And finding evidence for or against it — that’s how we make progress.

Both traditional creationists and intelligent design writers have invoked probability arguments in criticisms of biological evolution. They argue that certain features of biology are so fantastically improbable that they could never have been produced by a purely natural, “random” process, even assuming the billions of years of history asserted by geologists and astronomers. They often equate the hypothesis of evolution to the absurd suggestion that monkeys randomly typing at a typewriter could compose a selection from the works of Shakepeare, or that an explosion in an aerospace equipment yard could produce a working 747 airliner [Dembski1998; Foster1991; Hoyle1981;

Continue reading Do probability arguments refute evolution?

]]>Both traditional creationists and intelligent design writers have invoked probability arguments in criticisms of biological evolution. They argue that certain features of biology are so fantastically improbable that they could never have been produced by a purely natural, “random” process, even assuming the billions of years of history asserted by geologists and astronomers. They often equate the hypothesis of evolution to the absurd suggestion that monkeys randomly typing at a typewriter could compose a selection from the works of Shakepeare, or that an explosion in an aerospace equipment yard could produce a working 747 airliner [Dembski1998; Foster1991; Hoyle1981; Lennox2009].

One creationist-intelligent design argument goes like this: the human alpha-globin molecule, a component of hemoglobin that performs a key oxygen transfer function, is a protein chain based on a sequence of 141 amino acids. There are 20 different amino acids common in living systems, so the number of potential chains of length 141 is 20^{141}, which is roughly 10^{183} (i.e., a one followed by 183 zeroes). These writers argue that this figure is so enormous that even after billions of years of random molecular trials, no human alpha-globin protein molecule would ever appear “at random,” and thus the hypothesis that human alpha-globin arose by an evolutionary process is decisively refuted [Foster1991, pg. 79-83; Hoyle1981, pg. 1-20; Lennox2009, pg. 163-173].

While not generally appreciated by the public at large, it is a well-known fact in the world of scientific research that arguments based on probability and statistics are fraught with potential fallacies and errors. For these reasons, rigorous courses in probability and statistics are now required of students in virtually all fields of science, and in numerous other disciplines as well. Attorneys need to be at least moderately well-versed in probability and statistics arguments and how they can go awry in the courtroom arguments [Saini2009]. In the finance world, statistical overfitting and other errors of probability and statistics are now thought to be a leading reason behind the fact that strategies and investment funds which look great on paper often disappoint in real-world usage [Bailey2014].

For numerous other examples of how seemingly improbable “coincidences” can happen, see [Hand2014].

One major fallacy in the alpha-globin argument mentioned above, common to many others of this genre, is that it ignores the fact that a large class of alpha-globin molecules can perform the essential oxygen transfer function, so that the computation of the probability of a single instance is misleadingly remote. Indeed, most of the 141 amino acids in alpha-globin can be changed without altering the key oxygen transfer function, as can be seen by noting the great variety in alpha-globin molecules across the animal kingdom (see DNA). When one revises the calculation above, based on only 25 locations essential for the oxygen transport function (which is a generous over-estimate), one obtains 10^{33} fundamentally different chains, a enormous figure but vastly smaller than 10^{183}.

A calculation such as this can be refined further, taking into account other features of alpha-globin and its related biochemistry. Some of these calculations produce probability values even more extreme than the above. But do any of these calculations really matter?

The main problem is that all such calculations, whether done accurately or not, suffer from the fatal fallacy of presuming that a structure such as human alpha-globin arose by a single all-at-once random trial event. But generating a molecule “at random” in a single shot is decidedly *not* the scientific hypothesis in question — this is a creationist theory, not a scientific theory. Instead, available evidence from hundreds of published studies on the topic has demonstrated that alpha-globin arose as the end product of a long sequence of intermediate steps, each of which was biologically useful in an earlier context. See, for example, the survey article [Hardison2012], which cites 144 papers on the topic of hemoglobin evolution.

*In short, it does not matter how good or how bad the mathematics used in the analysis is, if the underlying model is a fundamentally invalid description of the phenomenon in question.* Any simplistic probability calculation of evolution that does not take into account the step-by-step process by which the structure came to be is almost certainly fallacious and can easily mislead [Musgrave1998].

It is also important to keep in mind that the process of natural biological evolution is *not* really a “random” process. Evolution certainly has some “random” aspects, notably mutations and genetic events during reproduction. But the all-important process of natural selection, acting under the pressure of an extremely competitive landscape, often involving thousands of other individuals of the same species and other species as well, together with numerous complicated environmental pressures such as climate change, is anything but random. This strongly directional nature of natural selection, which is the essence of evolution, by itself invalidates most of these probability calculations.

With regards to hemoglobin, in particular, it has long been noted that heme, the key oxygen-carrying component of hemoglobin, is remarkably similar to chlorophyll, the molecule behind photosynthesis. The principal difference is that heme has a central iron atom, whereas chlorophyll has a central magnesium atom; otherwise they are virtually identical. This similarity can hardly be a coincidence, and in fact researchers concluded long ago that these two biomolecules must have shared a common lineage (meaning, of course, that organisms which incorporate these biomolecules must have shared a common lineage) [Hendry1980]. Here is a diagram of the two molecules [from MasterOrganicChemistry.com].

As mentioned above, some critics have equated the notion of natural evolution to the absurd suggestion that some monkeys typing randomly at a keyboard could generate a passage of Shakespeare. But this too is a fallacious argument. A recent study exhibited results of a computer program simulating natural evolution, which “evolved” segments of English text very much akin to actual passages from Charles Dickens. In many instances, a class of college students were unable to distinguish the computer-generated text segments from real text segments taken from Dickens’ *Great Expectations*. See English-text for details.

Along this line, computer programs have been written that mimic the process of biological evolution to produce novel solutions to engineering problems, in many cases superior to the best human efforts, in an approach that has been termed “genetic algorithms” or “evolutionary computing.” As a single example, in 2017 Google researchers generated 1000 image recognition algorithms, each of which were trained using state-of-the-art deep neural networks to recognize a selected set of images. They then used an array of 250 computers, each running two algorithms, to identify an image. Only the algorithm that scored higher proceeded to the next iteration, where it was changed somewhat, mimicking mutations in natural evolution. Google researchers found that their scheme could achieve accuracies as high as 94.6% [Gershgorn2017].

Closely related are advances in artificial intelligence, in which a set of computer programs “compete” to produce a superior program. One notable example is the 2016 defeat of the world’s top Go player by a computer program named AlphaGo, developed by DeepMind, in an event that surprised observers who had not expected this for decades, if ever. Then in 2017, DeepMind announced even more remarkable results: their researchers had started from scratch, programming a computer with only the rules of Go, together with a “deep learning” algorithm, and then had the program play games against itself. Within a few days it had advanced to the point that it defeated the earlier champion-beating AlphaGo program 100 games to zero. After one month, the program’s rating was as far above the world champion as the world champion was above a typical amateur [Greenmeier2017].

As an example of the many mathematical studies of evolution, researchers showed mathematically that there was plenty of time in the geologic record for evolution to have produced the complexity in nature [Wilf2010].

Numerous examples from the natural world can be cited to demonstrate the futility of trying to argue against evolution using probability — nature can and often does produce highly improbable structures and features, by the normal process of evolution:

*Lenski’s 2012 E. coli experiment*: In January 2012, a research team led by Richard Lenski at Michigan State University demonstrated that colonies of viruses can evolve a new trait in as little as 15 days. The researchers studied a virus, known as “lambda,” which infects only the bacterium E. coli. They engineered a strain of E. coli that had almost none of the molecules that this virus normally attaches to, then released them into the virus colony. In 24 of 96 separate experimental lines, the viruses evolved a strain that enabled them to attach to E. coli, using a new molecule that they had never before been observed to utilize. All of the successful runs utilized essentially the same set of four distinct mutations. Justin Meyer, a member of the research team, noted that the chances of all four mutations arising “at random” in a given experimental line (based on a superficial probability argument) are roughly one in 10^{27}(one thousand trillion trillion) [Zimmer2012]. Note also that the chances for this to happen in 24 out of 96 experimental lines are roughly one in 10^{626}.*Synthesis of RNA nucleotides and other biomolecules*: Many scientists hypothesize that RNA (a molecule similar to DNA) was involved in the origin of life (see Origin). But as recently as a few years ago, even producing the four nucleotides (components) of RNA on the primitive Earth was thought to be a “near miracle.” Nonetheless, in May 2009 a team led by John Sutherland of the University of Manchester discovered a particular combination of chemicals, very likely to have been plentiful on the early Earth, that formed the RNA nucleotide ribocytidine phosphate. By exposing the mixture to ultraviolet light, a second nucleotide of RNA was formed [Wade2009]. Then in May 2016, one of the remaining two was synthesized, by a team in Munich, Germany [Service2016]. Sutherland’s group also recently showed that two simple compounds, which almost certainly were abundant on the early Earth, can launch a cascade of chemical reactions that can produce all three major classes of biomolecules: nucleic acids, amino acids and lipids [Wade2015a]. In short, the natural production of biomolecules once thought to be “impossible” is now fairly well understood.*Hawaiian crickets*: In the 1990s, a population of crickets in Hawaii (a species introduced to the islands over 100 years ago) became victims of dive-bombing flies that targeted male crickets who were chirping to attract mates, then implanted their larvae in them. Recently, when researchers visited a region in Kauai that previously was the home to many of these chirping crickets, it was now completely silent, and they feared the crickets were now extinct in the area. Fortunately, nighttime searches found that in fact there were lots of crickets there, but very few of the males now chirped. Further study found that in just five years, or roughly 20 generations, a rather improbable mutation had arisen that inhibited the males from chirping, and this genetic trait had now spread to almost the entire population [Zuk2013, pg. 81-82].*Tibetan high-altitude genes*: In 2010, researchers analyzing DNA found that natives of the Tibetan highlands possess 30 unique genes that permit them to live well at very high altitudes: the genes foster more efficient metabolism, prevent the overproduction of red blood cells, and generate higher levels of substances that transmit oxygen to tissue. Given that the Tibetans separated from other Han Chinese only about 3,000 years ago, this is thought to be one of the fastest documented cases of evolution in humans [Wade2010b].

Does a creationist worldview, in particular the hypothesis of independent creation of each species with no common biological ancestry, provide a reasonable alternative in terms of probability?

Here it is instructive to consider transposons or “jumping genes,” namely sections of DNA that have been “copied” from one part of an organism’s genome and “pasted” seemingly at random in other locations. The human genome, for example, has over four million individual transposons in over 800 families [Mills2007]. In most cases transposons do no harm, because they “land” in an unused section of DNA, but because they are inherited they serve as excellent markers for genetic studies. Indeed, transposons have been used to classify a large number of vertebrate species into a family tree, with a result that is virtually identical to what biologists had earlier reckoned based only physical features and biological functions [Rogers2011, pg. 25-31, 86-92]. As just one example, consider the following table, where columns labeled ABCDE denote five blocks of transposons, and x and o denote that the block is present or absent in the genome [Rogers2011, pg. 89].

Transposon blocks Species A B C D E /--------- Human o x x x x /---------- Bonobo x x x x x / \--------- Chimp x x x x x /------------ Gorilla o o x x x -----|------------ Orangutan o o o x x \------------ Gibbon o o o o o

It is clear from these data that our closest primate relatives are chimpanzees and bonobos. As another example, here is a classification of four cetaceans (ocean mammals) based on transposon data [Rogers2011, pg. 27]:

Transposon blocks Species A B C D E F G H I J K L M N O P /------ Bottlenose dolphin x x x x x x x x x x x x x x x x /\------ Narwhal whale x x x x x x x x x x x x x x x x ---|------- Sperm whale x x x x x o o o o o o o o o o o \------- Humpback whale x x o o o o o o o o o o o o o o

Other examples could be listed, encompassing an even broader range of species [Rogers2011, pg. 25-31, 86-92].

Needless to say, these data, which all but scream “descent from common ancestors,” are highly problematic for creationists and others who hold that the individual species were *separately* created without common biological ancestry. Transposons typically are several thousand DNA base pair letters long, but, since there are often some disagreements from species to species, let us be very conservative and say only 1000 base pair letters long. Then for two species to share even one transposon starting at the same spot, presumably only due to random mutations since creation, the probability (according to the creationist hypothesis) is one in 4^{1000} or roughly one in 10^{600}. For 16 such common transposons, the chances are one in 4^{16000} or roughly one in 10^{9600}. What’s more, as mentioned above, an individual species typically has at least several hundred thousand such transposons. Including even part of these in the reckoning would hugely multiply these odds.

But this is not all, because we have not yet considered the fact that in each diagram above, or in other tables of real biological transposon data, there is a clear hierarchical relationship. This is by no means assured, and in fact is quite improbable — for almost all tables of “random” data, there is no hierarchical pattern, and no way to the rearrange the rows to be in a hierarchical pattern. For example, in a computer run programmed by the present author, each column of the above cetacean table was pseudorandomly shuffled (thus maintaining the same number of x and o in each column), and the program checked whether the rows of the resulting table could be rearranged to be in a hierarchical order. There were no successes in 10,000,000 trials. As a second experiment, a 4 x 16 table of pseudorandom data (with a 50-50 chance of x or o) was generated, and then the program attempted to rearrange the rows to be in a hierarchical pattern as before. There were only three successes in 10,000,000 trials.

Like the calculations mentioned earlier, these calculations are simplified and informal; more careful reckonings can be done, and one can vary the underlying assumptions. But, again, do the fine details of the calculations really matter? One way or the other, it is clear that the creationist hypothesis of separate creation does not resolve any probability paradoxes; instead it enormously magnifies them. The only other possibility, from a strict creationist worldview, is to posit that a supreme being separately created species with hundreds of thousands of transposons already in place, essentially just as we see them today. But this merely replaces a scientific disaster (the utter failure of the creationist model to explain the vast phylogenetic patterns in intron data) with a theological disaster (why did a truth-loving supreme being fill the genomes of the entire biological kingdom with vast amounts of misleading DNA evidence, all pointing unambiguously to an evolutionary descent from common ancestors, if that is not the conclusion we are to draw?). Indeed, with regards to the discomfort some have about evolution, the creationist alternative of separate creation is arguably far worse, both scientifically and theologically.

It is undeniably true that there are some perplexing features of evolution from the point of view of probability. Even at the molecular level, structures are seen that appear to be exceedingly improbable. What is the origin of these structures? How did they evolve to the forms we see today? In spite of many published papers on these topics, researchers in the field would be the first to acknowledge that there is still much that is not yet fully understood.

However, arguments based on probability, statistics or information theory that have appeared in the creationist-intelligent design literature do not help unravel these mysteries, because most of these arguments have serious fallacies:

- They presume that a given biomolecule came into existence “at random” via an all-at-once chance assemblage of atoms. But this is
*not*the scientific hypothesis of how they formed; instead, numerous published studies, covering many biomolecules, indicate that these biomolecules were the result of a long series of intermediate steps over the eons, each useful in a previous biological context. Thus such arguments are fundamentally flawed from the beginning. - They apply faulty mathematical reasoning, such as by ignoring the fact that a very wide range of biomolecules could perform a similar function to the given biomolecule. Thus the odds they provide against the formation of the given biomolecule are greatly exaggerated.
- They ignore the fact that biological evolution is fundamentally
*not*a purely “random” process — mutations may be random, but natural selection is far from random. - They ignore reams of evidence from the natural world that evolution can and often does produce highly improbable structures and features.
- Some writers attempt to invoke advanced mathematical concepts (e.g., information theory), but derive highly questionable results and misapply these results in ways that render the conclusions invalid in an evolutionary biology context.
- The creationist hypothesis of separate creation for each species does not resolve any probability paradoxes; instead it enormously magnifies them.

It is ironic that to the extent that such probability-based arguments have any validity at all, it is precisely the *creationist* hypothesis of separate, all-at-once complete formation that is falsified.

Perhaps at some time in the distant future, a super-powerful computer will be able simulate with convincing fidelity the multi-billion-year biological history of the earth, in the same way that scientists today attempt to simulate (in a much more modest scope) the earth’s weather and climate. Then, after thousands of such simulations have been performed, with different starting conditions, we might obtain some meaningful statistics on the chances involved in the origin of life, or in the formation of some class of biological structures such as hemoglobin, or in the rise of intelligent creatures such as ourselves.

Until that time, probability calculations that appear in creationist-intelligent design literature and elsewhere should be viewed with great skepticism, to say the least. As mathematician Jason Rosenhouse writes [Rosenhouse2018],

When biologists ascribe to evolution the ability to craft information-rich genomes, they are neither speculating nor guessing. The basic components of evolutionary theory are empirical facts. Genes really do mutate, sometimes leading to new functionalities. The process of gene duplication with subsequent divergence leads to the creation of information by any reasonable definition of the terms. Selection can string small variations together into directional change. On a small scale, this has all been observed. And if small increases in information are an empirical reality on human timescales, then what abstract principle of mathematics is going to rule out much larger increases on geological scales?

Then here come the ID [intelligent design] folks, full of swagger and bravado. They say the accumulated empirical evidence must yield before their back-of-the-envelope probability calculations and abstract mathematical modeling. Evolution should be abandoned in favor of the new theory of intelligent design. This theory states, in its entirety, that an intelligent agent of unspecified motives and abilities did something at some point in natural history. Not very useful.

In a larger context, one has to question whether highly technical issues such as calculations of probabilities have any place in a discussion of religion. Why attempt to “prove” God with probability, particularly when there are very serious questions as to whether such reasoning is valid? One is reminded of a passage in the New Testament: “For if the trumpet gives an uncertain sound, who shall prepare himself for the battle?” [1 Cor. 14:8]. It makes far more sense to leave such matters to peer-reviewed scientific research.

[This is based on a similar post that appeared earlier in the SMR blog.]

]]>Until a few decades ago, number theory, namely the study of prime numbers, factorization and other features of the integers, was widely regarded as the epitome of pure mathematics, completely divorced from considerations of practical utility. This sentiment was expressed most memorably by British mathematician G.H. Hardy (best known for mentoring Ramanujan and results on the Riemann Zeta function), who wrote in his book A Mathematician’s Apology (1941),

I have never done anything “useful”. No discovery of mine has made, or is likely to make, directly or indirectly, for good

Continue reading New factorization advances: Is your bank account safe?

]]>Until a few decades ago, number theory, namely the study of prime numbers, factorization and other features of the integers, was widely regarded as the epitome of pure mathematics, completely divorced from considerations of practical utility. This sentiment was expressed most memorably by British mathematician G.H. Hardy (best known for mentoring Ramanujan and results on the Riemann Zeta function), who wrote in his book A Mathematician’s Apology (1941),

I have never done anything “useful”. No discovery of mine has made, or is likely to make, directly or indirectly, for good or ill, the least difference to the amenity of the world.

It did not turn out that way. Today number theory is a centerpiece of our digital economy. In particular, integer factorization and other number theory-related techniques are key to modern cryptography, which is used numerous times each day by a typical person, say to purchase items on an e-commerce site, to reserve theatre seats, or even to merely authenticate one’s subscription on a news or social media site. In fact, it is hard to think of any topic of mathematics that is more deeply connected to the day-to-day life of a typical person than number theory.

The RSA cryptosystem, named for the authors Ron Rivest, Adi Shamir and Leonard Adleman, is one of the most widely used cryptosystems today. It is based on the observation that given large positive integers $e, d, m$ and $n$ such that $(m^e)^d \equiv m$ mod $n$, then even if one knows $e, m$ and $n$, it is typically very difficult to compute $d$. This leads to a public key cryptosystem: a public key can be used to encrypt messages, but a secret private key is required to decrypt them.

Here is a very brief overview of how the scheme works (see this Wikipedia article for additional details):

- Choose two distinct large primes $p$ and $q$ (typically 100 digits or more in length). These are normally generated by a pseudorandom number generator and then tested for primality using one of several well-known primality tests.
- Compute $n = p q$, which is then released publicly.
- Compute $\lambda(n)$, where $\lambda$ is the Carmichael totient function. In this application, since $p$ and $q$ are primes, $\lambda(n) =$ lcm $(p-1, q-1)$, where lcm means least common multiple. The numbers $p, q$ and $\lambda(n)$ are kept private.
- Choose an integer $e$ coprime to $\lambda (n)$ in the range $1 \lt e \lt \lambda(n)$. Typically $e = 2^{16} + 1 = 65537$.
- Compute $d = e^{-1}$ mod $\lambda(n)$, i.e., the multiplicative inverse of $e$ in the ring of integers modulo $\lambda(n)$, using the extended Euclidean algorithm. The number $d$ is kept private.
- To encrypt a message or message segment represented as a large integer $m$, calculate $M = m^e$ mod $n$ using the public exponent $e$. To decrypt the resulting cypher text $M$, calculate $m = M^d$ mod $n$ using the private key $d$. Exponentiation modulo a large integer can be performed very rapidly by means of the binary algorithm for exponentiation modulo $n$.

The RSA scheme is based on the difficulty of factoring large numbers. If one could factor $n = pq$ to obtain $p$ and $q$, then one could immediately find $\lambda(n) =$ lcm $(p – 1, q – 1)$ and thus could calculate the decryption key $d = e^{-1}$ mod $\lambda(n)$.

So how hard is factoring large numbers? Shortly after the RSA scheme was first announced in 1977, the authors challenged Scientific American readers to decrypt a 129-digit cyphertext, with a posted award of US$100. Rivest had estimated, given known integer factorization schemes at the time, that factoring a 129-digit integer would require 40 quadrillion years on the fastest available computers.

In the wake of the RSA scheme announcement, several new integer factorization algorithms were found, some of which were found to be significantly faster than the existing known algorithms. In April 1994, the original RSA secret message was finally decrypted, using one of these new schemes, by means of a consortium of 1600 computers from 600 volunteers worldwide. The secret text was “The Magic Words are Squeamish Ossifrage.”

Some of the current state-of-the-art integer factorization algorithms are the following:

- Continued fraction factorization.
- Quadratic sieve factorization.
- Pollard’s rho algorithm.
- General number field sieve factorization.

See this Wikipedia article for an overview of integer factorization algorithms.

The RSA cryptosystem is now supported by RSA Security, which was founded by Rivest, Shamir and Adleman in 1982 to commercialize RSA and a number of other related cryptographic systems. From 1991 to 2007, RSA Security offered the RSA challenge problems in integer factorization, with cash prizes of various sizes for successful solution. Beyond the lure of cash prizes (which are no longer offered) these challenge problems have become benchmarks for progress in the field. As of the current date (December 2019), 20 of the 54 challenges have been factored.

The most recent achievement (December 2019) was the solution of RSA-240, namely the factorization of the 795-bit (240-digit) integer

$12462036678171878406583504460810659043482037465167880575481878888328966680118821$ $08550360395702725087475098647684384586210548655379702539305718912176843182863628$ $46948405301614416430468066875699415246993185704183030512549594371372159029236099$ $= 50943595228583991455505102358084371413264838202411147318666029652182120646974670$$ $$0620316443478873837606252372049619334517$ × $24462420883831815056781313902400289665380209257893140145204122133655847709517815$$ $$5258218897735030590669041302045908071447.$

This factorization required 900 core-years of computation, using a number field sieve algorithm. Additional details are given at this announcement.

Current e-commerce standards employing the RSA scheme, for example, are based on 1024-bit and 2048-bit integers. Given that the current state-of-the-art is factorization of a 795-bit integer, it appears that systems currently in use are still safe, although most observers in the field recommend that 2048-bit integers should quickly become the universal standard.

On the other hand, a new breakthrough in factorization research could quickly change the situation. Such breakthroughs have occurred before, as mentioned above, and there is no reason to believe that any of the current factorization methods are the best possible. The only real security here is that the problem of integer factorization is so basic, and has been studied in such great depth by so many mathematicians and computer scientists worldwide, that any still-undiscovered method must require a great deal of mathematical sophistication and difficult-to-develop software.

Along this line, blockchain technology is also founded on integer-factorization-based cryptography. Thus we must accept the disquieting possibility that a major breakthrough in factorization could jeopardize billions of dollars, euros, pounds, yen and yuan held in Bitcoin accounts and other cryptocurrencies.

In 1994, American mathematician Peter Shor published a paper describing an algorithm (now called Shor’s algorithm), that permits a quantum computer to perform large integer factorization, provided it has enough working “qubits.” In recent years, as researchers have been developing ever-more-capable quantum computers, some have dreamed of a quantum computer potentially becoming a super-efficient factorization engine, one potentially far more powerful for this purpose than any conventional computer system.

However, the development of truly practical quantum computer hardware has proven to be a daunting technical challenge. Among other things, current prototype systems typically involve cryogenic tanks cooled very close to absolute zero, namely -273.15 degrees Celsius or -459.67 degrees Fahrenheit. To date, such computers have only been able to perform a few fairly simple demonstration tasks, although the Canadian firm D-Wave claims several more sophisticated successes.

In a previous Math Scholar blog, we discussed a recent announcement by Google researchers that they had finally achieved “quantum supremacy” — i.e., that they had demonstrated an application running on a quantum computer that solves a problem faster than any present-day conventional computer. Within a few days, however, IBM released a paper arguing that the Google demonstration was flawed, because, among other things, the conventional computer run that they compared with did not take full advantage of large storage systems available on present-day supercomputers. See this Math Scholar blog for details.

The latest news here is that IBM researchers have developed a new scheme to factor numbers on a quantum computer, and have used it to factor the integer $1099551473989 = 1048589$ x $1048601$. This is a remarkable advance — the previous record for factoring integers on a quantum computer was for factoring $4088459 = 2017$ x $2027$. But compared with the 240-digit integer recently factored by conventional computers (see above), it is nothing. So quantum computer factorization still has a VERY long ways to go.

Some researchers argue that truly practical quantum computing is still far in the future. Mikhail Dyakonov, for instance, wrote:

All these problems, as well as a few others I’ve not mentioned here, raise serious doubts about the future of quantum computing. There is a tremendous gap between the rudimentary but very hard experiments that have been carried out with a few qubits and the extremely developed quantum-computing theory, which relies on manipulating thousands to millions of qubits to calculate anything useful. That gap is not likely to be closed anytime soon.

So don’t cash out your Bitcoins just yet. But stay tuned — a new breakthrough may be just around the corner.

]]>Simons’ background hardly suggested that he would one day lead one of the most successful, if not the most successful, quantitative hedge fund operation in

Continue reading Jim Simons: The man who solved the market

]]>Simons’ background hardly suggested that he would one day lead one of the most successful, if not the most successful, quantitative hedge fund operation in the world.

Born in 1938 to a Jewish family that operated a small shoe factory, Simons aspired very early to be a mathematician, ultimately receiving a B.S. degree from the Massachusetts Institute of Technology at the age of 20, and a Ph.D. from the University of California, Berkeley at the age of 23. His doctoral thesis presented a new proof of Berger’s classification of the holonomy groups of Riemannian manifolds. Later, in collaboration with Shing-Shen Chern, he discovered and proved what is now known as the Chern-Simons theorem (and associated theory), which deals with 3-dimensional quantum field theory and also has applications in string theory and quantum computing. For this and some related work, in 1976 he received the Oswald Veblen Prize in Geometry.

Not satisfied with being an accomplished academic mathematician, in 1964 he joined the Communication Research Division of the Institute for Defense Analyses (IDA), near Princeton, New Jersey, to work on cryptographic problems for the U.S. government. He did excellent work, but was fired after writing a letter to the *New York Times* expressing opposition to the Vietnam War. He then accepted the position of Chair of the Department of Mathematics at Stony Brook University, a public university on Long Island about 60 miles east of New York City, where he served for nearly 12 years.

Still not satisfied, in 1978, at the age of 40, Simons resigned from his position at Stony Brook and embarked on a quixotic stab at financial markets, founding a company named Monemetrics (later renamed Renaissance Technologies).

While at IDA, Simons had briefly investigated applying an algorithm developed by Lenny Baum and Lloyd Welch to financial markets (the Baum-Welch algorithm uses data to uncover hidden states in a Markov chain process). He had also been briefly involved with a Columbian floor tile importing business. But other than that, Simons was utterly unqualified for a financial venture — he had no training or background whatsoever in business, finance or market trading. The reaction of Simons’ mathematical colleagues was a combination of astonishment and outrage.

Simons and Lenny Baum, whom he soon hired, explored whether the Baum-Welch algorithm or other mathematical-statistical approaches could be applied to financial markets. The first few years were not promising. While some efforts succeeded, others soured, leaving their investment fund nearly depleted and Simons rather depressed. But instead of giving up, he redoubled his efforts to find actionable signals in financial data.

Simons, either directly or through spin-offs that he oversaw, subsequently hired several additional staff members, including James Ax, a prominent number theorist; Elwyn Berlekamp, a prominent expert in coding theory, game theory and computer science; Rene Carmona, an expert in stochastic differential equations; and Nick Patterson, whom Simons had known at IDA. Together these researchers developed some new techniques and software that appeared to work quite well. Although there were setbacks, their investment fund, later renamed the “Medallion Fund” (reportedly a nod to the awards that Simons, Ax and others had received) started growing, from $20 million in 1988 to $66 million in 1993.

At this point, Simons concluded that he needed some additional high-powered expertise in computer science and machine learning. So he hired IBM researchers Robert Mercer and Peter Brown, who had been developing speech recognition techniques at IBM’s Yorktown Heights Research Laboratory. Mercer and Brown further improved the fund’s algorithms, data processing facilities and trading software, resulting in significant new gains. Simons then hired additional mathematicians and computer scientists, together with an eclectic group of physicists, astronomers and signal processing experts, hardly any of whom had training in business or finance.

The Medallion fund grew from $66 million in 1993 to $2.4 billion in 2000, and to $10 billion in 2010, when Simons finally resigned as Renaissance CEO, turning operation of the fund over to Mercer and Brown (although he remains Chairman of the Board). Although the fund’s total assets are now fixed at $10 billion, it continues to generate enormous profits for those fortunate enough to be investors (the fund is now closed to outside investors). From 2011 to 2018 the fund’s annual returns averaged a whopping 74% before fees, and 39% after fees, returns that are easily and consistently the best in the industry. Renaissance also now offers three funds for outside investors, namely the Renaissance Institutional Equities Fund (RIEF), the Renaissance Institutional Diversified Equities Funds (RIDGE) and the Renaissance Institutional Diversified Alpha (RIDA).

In 1994, Simons and his wife established the Simons Foundation, a New York philanthropic organization (with an annual budget currently set at $450 million) that funds research in mathematics, computer science, physics, cosmology, biology, autism and education. Among other things, the foundation operates the Quanta Magazine, which is one of the world’s best news sources in mathematics, computer science, physics and biology. Simons has also founded Math for America, which among other things grants stipends of $18,000 per year for five years to selected outstanding high school teachers.

In addition to chronicling Simons’ success, Zuckerman also recounts the numerous setbacks, failures and other incidents that nearly ruined his operation, particularly in the early days. For example, in 1989, after Berlekamp’s programmed trading in Canadian currency improbably yielded no profits, Simons called a trader at the Chicago Board of Trade, who alerted him that fraud might be involved, and that the trading firm they were using might go under. Simons quickly closed his fund’s account with that trading firm, narrowly escaping potentially disastrous losses.

In addition, Simons’ personal life has often been marked by disappointment and tragedy. In 1974, his wife Barbara Bluestein divorced him (he subsequently married Marilyn Hawrys). Then in 1996, his 34-year-old son Paul was killed while riding a bicycle, followed in 2003 when his 24-year-old son Nicholas drowned while on a trip to Bali, Indonesia. According to those who know Simons personally, these tragedies left deep scars.

Managing his team of researchers has also been very trying. Several of the original employees, including Baum, Ax and Berlekamp, subsequently left, for various reasons, ranging from health problems and a desire to live elsewhere (e.g., the West Coast) to deep dissatisfaction with how the firm was being managed.

Controversy has beset the firm at several junctures. In 2014, Simons and Renaissance Technologies, along with several other large hedge funds, came under fire for using sophisticated “barrier options” to reduce taxes. Then in 2016, Mercer’s right-wing political activism came to light. In the 2016 election, for instance, Robert Mercer and his daughter Rebekah were major donors and promoters for the Trump campaign. Recalling his own history, Simons was determined not to fire anyone because of their political beliefs, but eventually, in the wake of the controversy and organizational morale problems that ensued, Simons convinced Mercer to step down as co-CEO, although he remains as a researcher.

In general, the tale that Zuckerman tells is one of huge challenges, numerous failures and frequent personal and organizational turmoil. Simon’s success has not come easily, not by a long shot.

So what can one say of Simon’s career? Zuckerman summarizes it as follows (from the Introduction):

Since 1988, Renaissance’s flagship Medallion hedge fund has generated average annual returns of 66 percent, racking up trading profits of more than $100 billion … No one in the investment world comes close. Warren Buffet, George Soros, Peter Lynch, Steve Cohen, and Ray Dalio all fall short. …

More than the trading successes intrigued me. Early on, Simons made a decision to dig through mountains of data, employ advanced mathematics, and develop cutting-edge computer models, while others were still relying on intuition, instinct, and old-fashioned research for their own predictions. Simons inspired a revolution that has since swept the investing world. By early 2019, hedge funds and other quantitative, or

quant, investors had emerged as the market’s largest players, controlling about 30 percent of stock trading, topping the activity of both individual investors and traditional investing firms. MBAs once scoffed at the thought of relying on a scientific and systematic approach to investing, confident they could hire coders if they were ever needed. Today, coders say the same about MBAs, if they think about them at all.Simons’s pioneering methods have been embraced in almost every industry, and reach nearly every corner of everyday life. He and his team were crunching statistics more than three decades ago — long before these tactics were embraced in Silicon Valley, the halls of government, sports stadiums, doctors’ offices, military command centers, and pretty much everywhere else forecasting is required.

Simons developed strategies to corral and manage talent, turning raw brainpower and mathematical aptitude into astonishing wealth. He made money from math, and a lot of money, at that. A few decades ago, it wasn’t remotely possible.

Lately, Simons has emerged as a modern-day Medici, subsidizing the salaries of thousands of public-school math and science teachers, developing autism treatments, and expanding our understanding of the origins of life. …

I was most fascinated by a striking paradox: Simons and his team

shouldn’t have been the onesto master the market. Simons never took a single finance class, didn’t care very much for business, and, until he turned forty, only dabbled in trading. A decade later, he still hadn’t made much headway. Heck, Simons didn’t even do applied mathematics, he didtheoreticalmath, the most impractical kind. His firm, located in a sleepy town on the North Shore of Long Island, hires mathematicians and scientists whodon’t know anythingabout investing or the ways of Wall Street. Some are even outright suspicious of capitalism. Yet, Simons and his colleagues are the ones who changed the way investors approach financial markets, leaving an industry of traders, investors, and other pros in the dust. It’s as if a group of tourists, on their first trip to South America, with a few odd-looking tools and meager provisions, discovered El Dorado and proceeded to plunder the golden city, as hardened explorers looked on in frustration.

One way or another, Simons has had quite a career! He definitely fits the profile of what Dean Keith Simonton terms a true creative genius — someone who has a very high IQ, boundless energy and a never-quenched thirst to learn more and accomplish more.

His story is a great tale, presented very vividly by Zuckerman in his book: The Man Who Solved the Market: How Jim Simons Launched the Quant Revolution.

[This originally appeared at the Mathematical Investor blog.]

]]>At this point in time, the basic facts of climate change are not disputable in the least. Careful planet-wide observations by NASA and others have confirmed that 2018 was the fourth-warmest year in recorded history. The only warmer years were 2016, 2017 and 2015, respectively, and 18 of the 19 warmest years in history have occurred since 2001. Countless observational studies and supercomputer simulations have confirmed both the fact of warming and the conclusion that this warming is principally due to human activity. These studies and computations have been scrutinized in great

Continue reading The scientific debate is over: it is time to act on climate change

]]>At this point in time, the basic facts of climate change are not disputable in the least. Careful planet-wide observations by NASA and others have confirmed that 2018 was the fourth-warmest year in recorded history. The only warmer years were 2016, 2017 and 2015, respectively, and 18 of the 19 warmest years in history have occurred since 2001. Countless observational studies and supercomputer simulations have confirmed both the fact of warming and the conclusion that this warming is principally due to human activity. These studies and computations have been scrutinized in great detail by a climate science community numbering in the thousands, representing all major nations, as summarized in the latest report by the Intergovernmental Panel on Climate Change (IPCC).

Climate change skeptics continue to raise objections, claiming that there is “scientific evidence” that this mainstream consensus on climate change is wrong. But these objections have been debunked many times. Here are some examples:

*Climate change is just part of the natural cycle.***Rejoinder**: Yes, the geologic and paleontological record confirms numerous climate changes in the past. But the changes of the past 150 years, coinciding with the industrial revolution and huge increases in carbon emissions, exceeds that of the past five million years. See also this Scientific American article.*Changes are due to sunspots or galactic cosmic rays.***Rejoinder**: Scientists who have carefully monitored the sun for at least the past 20 years have seen no significant upward trend. Besides, such effects would be seen in the high atmosphere, whereas observed climate change effects are seen in the lower atmosphere. See also this Carbon Brief article.*CO2 is a small part of the atmosphere, so it can’t have a large heating effect.***Rejoinder**: The fact that CO2 results in a greenhouse warming effect has been known since 1856, and countless experimental analyses since then have confirmed heating even with only very small concentrations.*Scientists manipulate datasets to show a warming trend.***Rejoinder**: Yes, scientists adjust data, as in all experimental studies, but these adjustments are performed based on very well-understood effects, and are open to the scrutiny of peers.*Climate models are unreliable and too sensitive to carbon dioxide.***Rejoinder**Climate models are extremely sophisticated computer programs, running on some of the world’s most powerful supercomputers, and based on massive experimental datasets. These models have been produced by numerous independent international teams, which rigorously critique each other’s work, in an ongoing process of peer review. All of the latest models consistently now predict significant long-term global warming. See for example the Community Earth System Model website.

Along this line, it is worth pointing out that some previous climate skeptics have changed their mind. For example, in a remarkable New York Times Op-Ed, former climate change skeptic Richard Muller of the University of California, Berkeley, declared not only that global warming is real, but also that “humans are almost entirely the cause.”

Muller’s Berkeley Earth group approached the problem by rigorously analyzing historic temperature reports. As he described their efforts,

We carefully studied issues raised by skeptics: biases from urban heating (we duplicated our results using rural data alone), from data selection (prior groups selected fewer than 20 percent of the available temperature stations; we used virtually 100 percent), from poor station quality (we separately analyzed good stations and poor ones) and from human intervention and data adjustment (our work is completely automated and hands-off). In our papers we demonstrate that none of these potentially troublesome effects unduly biased our conclusions.

Muller noted that their record of temperatures is long enough that they could search for the fingerprint of variability in the sun’s output reaching the earth. But Muller found no such fingerprint. Global warming is real.

So how much, if any, of this warming can truly be ascribed to human activity? Muller’s Berkeley Earth group found that the record of temperatures over the past 250 years fits the increasing emissions of CO2 better than any other statistic they tried, and the magnitude of the change is entirely consistent with the known greenhouse effect of CO2.

At this point in time, at least 97% of climate science researchers agree with the central conclusion that the Earth is warming and that human activity is the primary cause. This statistic is based on multiple in-depth surveys of thousands of recently published papers in the climate science field. Further, this consensus is supported by official statements from the American Association for the Advancement of Science, the American Chemical Society, the American Geophysical Union, the American Meteorological Society, the American Physical Society, the Geological Society of America, the U.S. National Academy of Sciences and numerous other scientific societies worldwide.

In contrast, large numbers of Americans in particular continue to deny even the most basic facts. In a 2017 Pew Research Center survey, 23% denied that there is any solid evidence that the Earth has been warming, and of those who acknowledge warming, nearly half doubted that it is due to human activities.

So why are so many skeptical of the scientific consensus? According to a separate 2017 Pew Research Center survey, only 27% agreed that “almost all” scientists are in agreement; 35% said only “More than half,” and 35% said half or fewer. But even more disturbingly, only 32% agreed that the “best available scientific evidence” influences the climate scientists’ conclusions; 48% said only “some of the time”, and 18% said “not too often or never.” These results underscore a severe level of distrust of scientists in general and climate scientists in particular by the public.

According to the latest IPCC report, impacts on natural and human systems are already occurring, and even a warming of 1.5 C, which at this point can hardly be averted, will have very serious consequences, including more extreme temperature events, more instances of heavy precipitation, more severe droughts, rising sea levels damaging cities and agricultural lands, as well as enormous stress on ecosystems worldwide.

An October 2019 Scientific American article listed some of the frightening developments, just in the previous 12 months:

- In December 2018, the World Health Organization said that fossil fuel emissions are “a major contributor to health-damaging air pollution, which every year kills over seven million people.” It added that extreme weather events, which have been linked to human-caused climate change, are “a clear and present danger to health security.”
- Also in December 2018, the Global Carbon Project reported that global CO2 emissions reached an all-time high in 2018, up more than two percent after three level years. What’s more, additional increases are likely in 2019.
- In April 2019, a NASA-funded study found the mass loss of ice discharged into the ocean from Greenland glaciers had increased by a factor of six since the 1980s. Partly as a result, mean sea level has risen nearly 14 millimeters since 1972, with 7 millimeters in the past eight years. Subsequently, in July 2019, a severe Arctic heat wave resulted in 12.5 billion tons of ice melting into the ocean on a single day, the largest single-day loss on record.
- In May 2019, a United Nations biodiversity panel reported that over one million animal and plant species are threatened with extinction in the next few decades, and, further, that rates of extinction are “accelerating.”
- In September 2019, an IPCC report concluded that warming oceans, melting ice, and rising sea levels are already affecting 10 percent of the world’s population that lives in low-lying coastal areas, and that negative impacts will greatly worsen in the coming years and decades.
- In October 2019, California staggered through its third consecutive catastrophic wildfire season, with thousands of fires incinerating tens of thousands of acres, and, once again, causing billions of dollars in damages. This is in spite of the unprecedented step of pre-emptively shutting off power to large portions of the state in an effort to prevent downed power lines from generating more fires, a step that in effect reduced the world’s premier high technology leader and fifth largest economy to third-world status. The consensus of scientists is that the dry autumn weather and winds that precipitated these fires are exacerbated by climate change, with the dismal prospect for even more wildfires and power blackouts in the future.

So where do we go from here? Sixteen-year-old Swedish activist Greta Thunberg said it best in her eloquent remarks to the 2019 U.N. Climate Summit (excerpt):

]]>You have stolen my dreams and my childhood with your empty words. And yet I’m one of the lucky ones. People are suffering. People are dying. Entire ecosystems are collapsing. We are in the beginning of a mass extinction, and all you can talk about is money and fairy tales of eternal economic growth. How dare you!

For more than 30 years, the science has been crystal clear. How dare you continue to look away and come here saying that you’re doing enough, when the politics and solutions needed are still nowhere in sight.

You say you hear us and that you understand the urgency. But no matter how sad and angry I am, I do not want to believe that. Because if you really understood the situation and still kept on failing to act, then you would be evil. And that I refuse to believe.

The popular idea of cutting our emissions in half in 10 years only gives us a 50% chance of staying below 1.5 degrees [Celsius], and the risk of setting off irreversible chain reactions beyond human control.

Fifty percent may be acceptable to you. But those numbers do not include tipping points, most feedback loops, additional warming hidden by toxic air pollution or the aspects of equity and climate justice. They also rely on my generation sucking hundreds of billions of tons of your CO2 out of the air with technologies that barely exist.

So a 50% risk is simply not acceptable to us — we who have to live with the consequences.

To have a 67% chance of staying below a 1.5 degrees global temperature rise – the best odds given by the [Intergovernmental Panel on Climate Change] – the world had 420 gigatons of CO2 left to emit back on Jan. 1st, 2018. Today that figure is already down to less than 350 gigatons.

How dare you pretend that this can be solved with just ‘business as usual’ and some technical solutions? With today’s emissions levels, that remaining CO2 budget will be entirely gone within less than 8 1/2 years.

There will not be any solutions or plans presented in line with these figures here today, because these numbers are too uncomfortable. And you are still not mature enough to tell it like it is.

You are failing us. But the young people are starting to understand your betrayal. The eyes of all future generations are upon you. And if you choose to fail us, I say: We will never forgive you.

We will not let you get away with this. Right here, right now is where we draw the line. The world is waking up. And change is coming, whether you like it or not.

For at least three decades, teams of researchers have been exploring quantum computers for real-world applications in scientific research, engineering and finance. Researchers have dreamed of the day when quantum computers would first achieve “supremacy” over classical computers, in the sense that a quantum computer solving a particular problem faster than any present-day or soon-to-be-produced classical computer system.

In a Nature article dated 23 October 2019, researchers at Google announced that they have achieved exactly this.

Google researchers employed a custom-designed quantum processor, named “Sycamore,” consisting of programmable quantum

Continue reading Quantum supremacy has been achieved; or has it?

]]>For at least three decades, teams of researchers have been exploring quantum computers for real-world applications in scientific research, engineering and finance. Researchers have dreamed of the day when quantum computers would first achieve “supremacy” over classical computers, in the sense that a quantum computer solving a particular problem faster than any present-day or soon-to-be-produced classical computer system.

In a Nature article dated 23 October 2019, researchers at Google announced that they have achieved exactly this.

Google researchers employed a custom-designed quantum processor, named “Sycamore,” consisting of programmable quantum computing units, to create quantum states on 53 “qubits,” which, as they point out, corresponds to exploring a space of dimension 2^{53}, or approximately 10^{16}. The task that the researchers chose was one that admittedly was specifically chosen to be well-suited for quantum computers: the team programmed the quantum computer to run a circuit that passes the 53 qubits through a series of random operations, in effect generating a string of pseudorandom zeroes and ones. The computer then calculated the probability distribution generated by these pseudorandom trials, by sampling the circuit task one million times and recording the results.

The Google researchers’ Sycamore quantum computer system completed the task in only 3 minutes 20 seconds. They claimed that the task would require 10,000 years, even on a present-day supercomputer with 1,000,000 processors.

For additional details, see the Google researchers’ Nature paper, this Nature summary, and this New Scientist report.

The ink was barely dry on Google’s announcement when IBM released a technical paper arguing that Google’s comparison to a classical computer is flawed. The IBM researchers argue that the classical computation the Google researchers outlined (but did not actually perform) did not take full advantage of a typical present-day supercomputer’s storage system. When a storage system is strategically employed in the computation, IBM argues that the hypothetical run would complete in only 2.55 days on the “Summit” supercomputer at Oak Ridge National Laboratory, running at 87.4 Pflop/s (i.e., 87.4 quadrillion floating-point operations per second, or 8.74 x 10^{16} floating-point operations per second). IBM further argues that their computation could produce solutions with higher fidelity than the Sycamore quantum computer.

As Ciaran Gilligan-Lee of University College London observes,

Classical computers have such a large suite of things built into them that if you don’t utilise every single thing you leave yourself open for a tweaked classical algorithm to beat your quantum one.

At the least, Gilligan-Lee argues that claims of quantum supremacy “should be taken with a grain of salt.”

Another criticism of the Google team’s claim is that the particular benchmark problem they selected, namely to produce statistics from generating large numbers of pseudorandom numbers, is a highly artificial task, one that in no way reflects present-day real-world high-performance scientific computing. The pseudorandom number task is completely different from problems such as computationally simulating the earth’s climate for several decades into the future, to better understand different strategies for combatting global warming, computationally exploring the space of inorganic materials for improved solar cells or superconductors, or dynamically optimizing an investment portfolio.

One final criticism is that Google’s quantum processor is only one of many similar projects, some of which have already achieved significant success. The Canadian company D-Wave, for instance, has been producing quantum computers for several years, focusing on systems designed specifically for “quantum annealing,” namely a particular type of minimization or maximization calculation that nonetheless has broad applicability to real-world problems.

In the 1970s and 1980s, most high-end supercomputers were based on a pipelined vector architecture, where a processor performed, say, 64 identical arithmetic or logical operations on 64 sets of arguments, in a zipper-like pipelined stream. But beginning in the early 1990s, systems with a highly parallel architecture, characterized by numerous independent processor-memory nodes connected in a network, became commercially available. The high-performance computing world raced to explore these new highly parallel systems for large-scale scientific and engineering computations.

Unfortunately, however, many researchers using these systems, in their zeal to be part of a new computing movement, began to publish performance levels and comparisons with prevailing vector computers that were not carefully done. Questionable performance reports were seen in papers from universities and government laboratories as well as private industry. The present author and some others in the high-performance scientific computing field grew concerned that, in many cases, exuberance was being transformed into exaggeration and even borderline fraud, hardly becoming of rigorous, sober scientific research.

Some of the questionable practices included (see also this paper):

- Performing calculations on a small parallel systems, but reporting results scaled to a full-sized parallel system merely by multiplication (i.e., claiming runs on a system with 4096 processors, but only performing runs on a system with 1024 processors, with performance results multiplied by four).
- Reporting performance in units of operations per second, but employing inefficient algorithms that result in artificially inflated performance levels.
- Comparing highly tuned parallel computer runs with untuned programs on a vector computer system.
- Not actually performing a claimed computation on the system being compared to, but instead employing a highly questionable “rule of thumb” conversion factor.
- Employing questionable performance plots, often with extrapolated data points not clearly acknowledged in the text of the paper.
- Omitting key details in the technical paper, such as whether the computation was performed using 64-bit floating-point arithmetic (approximately 16 decimal digit accuracy) or 32-bit floating-point arithmetic (approximately 7 decimal digit accuracy, which usually runs faster but produces much less accurate final results).

At first, the present author warned the community by means of a widely circulated humorous essay; when this stratagem failed, he published a more serious technical paper that directly quoted from a number of the questionable papers (and he endured considerable sniping from some in the community as a result).

Partly as a consequence of these exaggerated performance claims, many universities and laboratories became disappointed and disillusioned with the parallel computers that they had purchased, and the parallel computing field languished. Only about ten years later, after substantially improved highly parallel computers became available, and after significantly more realistic performance benchmarks and practices were adopted in the high-performance computing community, did highly parallel computing finally gain widespread acceptance.

For additional details, see this talk by the present author.

Spanish-American philosopher George Santayana once quipped, “Those who cannot remember the past are condemned to repeat it.”

With the rise of quantum computers, as they challenge the long-running reign of highly parallel computers, we are at a juncture quite similar to the rise of parallel computer systems in the early 1990s, as they challenged the long-running reign of vector computers. In this era, as in the previous era, it is crucial that researchers, in their enthusiasm to explore the new architectures, not set aside basic principles of rigorous scientific methodology, objectivity and reproducibility.

To that end, here are some suggestions to avoid learning lessons of the past the “hard” way:

- The quantum computing field must establish clear standards for performance analyses.
- Editors of journals and referees of papers must be vigilant in enforcing these standards.
- Government funding agencies must support these standards efforts.
- Community-developed and community-endorsed benchmarks are a must.
- Benchmarks should include problems to measure specific features of a system, as well to assess performance on full-scale real-world problems typical of those that the community agrees are truly representative of computations that will actually be performed on such systems.
- Both leaders and bench-level researchers need to be frank about shortcomings of these new systems — such hard-hitting frankness is essential for the field to move forward.

The present author is as hopeful as anyone that these quantum computing systems will be successful. But let’s explore these systems with our eyes open!

]]>Introduction In a previous Math Scholar blog, we presented Archimedes’ ingenious scheme for approximating $\pi$, based on an analysis of regular circumscribed and inscribed polygons with $3 \cdot 2^k$ sides, using modern mathematical notation and techniques.

One motivation for both the previous blog and this blog is to respond to some recent writers who reject basic mathematical theory and the accepted value of $\pi$, claiming instead that they have found $\pi$ to be a different value. For example, one author asserts that $\pi = 17 – 8 \sqrt{3} =

Continue reading Pi as the limit of n-sided circumscribed and inscribed polygons

]]>**Introduction**

In a previous Math Scholar blog, we presented Archimedes’ ingenious scheme for approximating $\pi$, based on an analysis of regular circumscribed and inscribed polygons with $3 \cdot 2^k$ sides, using modern mathematical notation and techniques.

One motivation for both the previous blog and this blog is to respond to some recent writers who reject basic mathematical theory and the accepted value of $\pi$, claiming instead that they have found $\pi$ to be a different value. For example, one author asserts that $\pi = 17 – 8 \sqrt{3} = 3.1435935394\ldots$. Another author asserts that $\pi = (14 – \sqrt{2}) / 4 = 3.1464466094\ldots$. A third author promises to reveal an “exact” value of $\pi$, differing significantly from the accepted value. For other examples, see this Math Scholar blog. Of course, $\pi$ cannot possibly be given by any algebraic expression such as these, since $\pi$ was proven transcendental by Lindemann in 1882, and his proof has been checked carefully by many thousands of mathematicians since then.

To that end, the previous blog presented a rigorous proof of Archimedes’ method, demonstrating that $\pi = 3.1415926535\ldots$ and certainly not any variant values. It employed only basic algebra, trigonometry and the Pythagorean theorem, and avoided calculus, analysis or any reasoning that depends on properties of $\pi$.

**N-sided polygons**

As mentioned above, Archimedes’ construction was based on polygons with $3 \cdot 2^k$ sides, and the presentation of this scheme in the previous blog followed this convention.

However, strictly speaking from a modern perspective, $\pi$ is most commonly defined as the circumference of a circle divided by its diameter (or as the semi-circumference of a circle of radius one), where the circumference of a circle is defined as the limit of the perimeters of circumscribed or inscribed regular polygons with $n$ sides, as $n$ increases without bound. Note that this is a somewhat stricter definition than Archimedean definition, which only deals with the special case $n = 3 \cdot 2^k$, leaving open the question of other $n$.

In the spirit of adhering to the modern convention, we present here a complete proof that $\pi$ as defined by Archimedes is the same as $\pi$ based on general $n$-sided regular polygons. As a bonus, we also present a proof that the limits of the areas of these polygons is also equal to $\pi$. These proofs are a bit more difficult than with the Archimedean case $n = 3 \cdot 2^k$ (see the previous blog), but still can be presented entirely using only basic algebra and trigonometry, with no need to resort to calculus, complex numbers or any other topic not familiar to a standard high-school mathematics curriculum. As with the previous blog, traditional degree notation is used here for angles instead of the radian measure customary in professional research work, both to make the presentation easier follow and also to avoid any concepts or techniques that might be viewed as dependent on $\pi$.

We start by establishing some well-known trigonometric identities. Readers who are familiar with these identities may skip to the next section.

**LEMMA 1 (Sine, cosine and tangent of a sum)**:

$$\sin(\alpha + \beta) = \sin(\alpha) \cos(\beta) + \cos(\alpha) \sin(\beta),$$ $$\cos(\alpha + \beta) = \cos(\alpha) \cos(\beta) – \sin(\alpha) \sin(\beta),$$ $$\tan(\alpha + \beta) = \frac{\tan(\alpha) + \tan(\beta)}{1 – \tan(\alpha)\tan(\beta)}.$$ **Proof**: These identities were proved in the previous article, relying only on simple geometry and the Pythagorean theorem. See also this Wikipedia article.

**LEMMA 2 (Multi-angle inequalities)**:

Let $\alpha$ be any angle and $n \ge 2$ be any positive integer such that $0 \lt n \alpha \lt 45^\circ$. Then $\sin(n \alpha) \lt n \sin(\alpha)$ and $\tan(n \alpha) \gt n \tan(\alpha).$

**Proof**: These identities can be proved most easily using an induction argument: Clearly they hold for $n = 2$, since $\sin(2 \alpha) = 2 \sin(\alpha) \cos(\alpha) \lt 2 \sin\alpha$, and $\tan (2 \alpha) = 2 \tan (\alpha) / (1 – \tan^2(\alpha)) > 2 \tan(\alpha)$. Then assuming the inequalities hold for $n$, note that $$\sin((n+1) \alpha) = \sin(n \alpha) \cos(\alpha) + \cos(n \alpha) \sin(\alpha) \lt n \sin(\alpha) + \sin(\alpha) = (n+1) \sin(\alpha),$$ $$\tan((n+1) \alpha) = (\tan(n \alpha) + \tan(\alpha)) / (1 – \tan(n \alpha) \tan(\alpha)) \gt n \tan(\alpha) + \tan(\alpha) = (n + 1) \tan(\alpha),$$ so that the inequalities hold for $n+1$ also.

** LEMMA 3 (Ratios of multi-angle sines and tangents)**.

Let $\alpha$ be any angle and $n \ge 2$ be any positive integer such that $0 \lt n \alpha \lt 45^\circ$. Then $$\frac{\sin((n+1) \alpha)}{\sin(n \alpha)} \lt \frac{n+1}{n,}$$ $$\frac{\tan((n+1) \alpha)}{\tan(n \alpha)} \gt \frac{n+1}{n}.$$ **Proof**: First note that for $0 \lt n \alpha \lt 45^\circ$, $\sqrt{3}/2 \lt \cos(\alpha) \lt 1$ and $0 \lt \tan(\alpha) \lt 1$ (and also for $n \alpha$ in place of $\alpha$). Then we can write $$\frac{\sin((n+1) \alpha)}{\sin(n \alpha)} = \frac{\sin(n \alpha) \cos (\alpha) + \cos(n \alpha) \sin(\alpha)}{\sin(n \alpha)} = \cos (\alpha)\left(1 + \frac{\tan(\alpha)}{\tan(n \alpha)}\right) \lt 1 + \frac{\tan(\alpha)}{\tan(n \alpha)} \lt 1 + \frac{1}{n},$$ where we applied Lemma 2 at the final step. The second requires a bit more work: $$\frac{\tan((n+1)\alpha)}{\tan(n\alpha)} = \frac{\tan(n \alpha) + \tan(\alpha)}{\tan(n \alpha) (1 – \tan (n \alpha) \tan(\alpha))} = \frac{1 + \tan{\alpha}/\tan(n \alpha)}{1 – \tan(n \alpha) \tan(\alpha)} = \left(1 + \frac{\tan(\alpha)}{\tan(n \alpha)}\right) \cdot \left(1 + \frac{\tan(n \alpha) \tan(\alpha)}{1 – \tan(n \alpha) \tan(\alpha)}\right) $$ $$\gt \left(1 + \frac{\tan(\alpha)}{\tan(n \alpha)}\right) \cdot \left(1 + \tan(n \alpha) \tan(\alpha)\right) = 1 + \frac{\tan(\alpha)}{\tan(n \alpha)} + \tan(n \alpha) \tan(\alpha) + \tan^2(\alpha)$$ $$\gt 1 + \frac{\tan(\alpha)}{\tan(n \alpha)} \left(1 + \tan^2(n \alpha)\right) = 1 + \frac{\tan(\alpha)}{\tan(n \alpha) \cos^2(n \alpha)} = 1 + \frac{\sin(\alpha)}{\sin(n \alpha) \cos(n \alpha) \cos(\alpha)} > 1 + \frac{\sin(\alpha)}{\sin(n \alpha)} \gt 1 + \frac{1}{n},$$ where again we employed Lemma 2 at the final step.

**AXIOM 1 (Completeness axiom):** Every set of reals that is bounded above has a least upper bound; every set of reals that is bounded below has a greatest lower bound.

**Comment**: This fundamental axiom of real numbers merely states the property that the set of real numbers, unlike say the set of rational numbers, has no “holes.” An equivalent statement of the completeness axiom is “Every Cauchy sequence of real numbers has a limit in the real numbers.” See the Wikipedia article Completeness of the real numbers and this Chapter for details.

**THEOREM 1 (Pi as the limit of n-sided circumscribed and inscribed regular polygons)**:

**Theorem 1a**: For a circle of radius one, as the index $n \ge 6$ increases, the greatest lower bound of the semi-perimeters of circumscribed regular polygons with $n$ sides is exactly equal to the least upper bound of the semi-perimeters of inscribed regular polygons with $n$ sides, which value is exactly the same as $\pi$ as defined by Archimedes as in the previous blog.

**Theorem 1b**: For a circle of radius one, as the index $n \ge 6$ increases, the greatest lower bound of the areas of circumscribed regular polygons with $n$ sides is exactly equal to the least upper bound of the areas of inscribed regular polygons with $n$ sides, which value is exactly equal to $\pi$ as defined in Theorem 1a and in the previous blog.

**Proof strategy**: We will show that (a) the sequence of circumscribed semi-perimeters $(p_n)$ is strictly decreasing; (b) the sequence of inscribed semi-perimeters $(q_n)$ is strictly increasing; (c) all $(p_n)$ are strictly greater than all $(q_n)$; and (d) the distance between $p_n$ and $q_n$ becomes arbitrarily small for large $n$. Thus the greatest lower bound of the circumscribed semi-perimeters is equal to the least upper bound of inscribed semi-perimeters, and this common limit coincides with $\pi$ as defined by Archimedes in terms of polygons with $3 \cdot 2^k$ sides. A similar argument reaches the same conclusion for the sequence of circumscribed and inscribed areas.

**Proof**: For the entirety of this proof, we will assume $n \ge 6$. Let $p_n$ be the semi-perimeter of an $n$-sided regular circumscribed polygon for a circle with radius one, and let $q_n$ be the semi-perimeter for an $n$-sided regular inscribed polygon for a circle with radius one. By examining the figure (see also the previous blog), it is clear that $p_n = n \tan (180^\circ/n)$ and $q_n = n \sin (180^\circ/n)$. Note, for instance, that $p_6 = 6 \tan (30^\circ) = 2 \sqrt{3} = 3.46419\ldots$ and $q_6 = 6 \sin(60^\circ) = 3$. Now let $\alpha = 180^\circ/ (n(n+1))$. By applying Lemma 3, we can then write $$\frac{p_{n+1}}{p_n} = \frac{(n+1) \tan(180^\circ/(n+1))}{n \tan (180^\circ/n)} = \frac{(n+1) \tan (n \alpha)}{n \tan((n+1) \alpha)} \lt 1,$$ $$\frac{q_{n+1}}{q_n} = \frac{(n+1) \sin(180^\circ/(n+1))}{n \sin (180^\circ/n)} = \frac{(n+1) \sin(n \alpha)}{n \sin ((n+1) \alpha)} \gt 1,$$ so that $(p_n)$ is a strictly decreasing sequence and $(q_n)$ is a strictly increasing sequence. Let $\beta = 180^\circ / n$. For $n \ge 6$ note that $1/2 \lt \sqrt{3}/3 \lt \cos(\beta) \lt 1$. Then we can write $$p_n – q_n = n \tan(\beta) – n \sin(\beta) = \frac{n \sin(\beta)}{\cos(\beta)} \left(1 – \cos(\beta)\right) \gt 0,$$ so that each $p_n \gt q_n$. If $n \gt m$, then $p_n \gt q_n \gt q_m$, so $p_n \gt q_m$. Thus all $p_n$ are strictly greater than all $q_n$. In particular, since $p_6 = 2 \sqrt{3} \lt 4$, this means that all $p_n \lt 4$ and thus all $q_n \lt 4$. Similarly, since $q_6 = 3$, all $q_n \ge 3$ and thus all $p_n \gt 3$. By continuing with the last-written equality above, $$p_n – q_n = \frac{n \sin(\beta)}{\cos(\beta)} \left(1 – \cos(\beta)\right) = \frac{n \sin(\beta) (1 – \cos^2 (\beta)}{\cos(\beta)(1 + \cos(\beta))} = \frac{n \sin^3(\beta)}{\cos(\beta) (1 + \cos(\beta))} \lt 2 n \sin^3(\beta) = \frac{2 q_n^3}{n^2} \lt \frac{128}{n^2},$$ since $\cos(\beta) \gt 1/2$ and all $q_n \lt 4$. Thus the difference between $p_n$ and $q_n$ decreases to arbitrarily small values for large $n$.

Recall from the above that all $p_n \gt 3$, so that the sequence $(p_n)$ of circumscribed semi-perimeters is bounded below. Thus by Axiom 1 the sequence $(p_n)$ has a greatest lower bound $L_1$. In fact, since all $p_n$ are greater than all $q_n$, any $q_n$ is a lower bound of the entire sequence $(p_n)$, so that we may write, for any $n \ge 6$, $p_n \geq L_1 \geq q_n$. Also, all $q_n \lt 4$, so that the sequence $(q_n)$ of inscribed semi-perimeters is bounded above, and thus has a least upper bound $L_2$. And, as before, since any $p_n$ is an upper bound for the entire sequence $(q_n)$, it also follows that $p_n \geq L_2 \geq q_n$. Thus both $L_1$ and $L_2$ are “squeezed” between $p_n$ and $q_n$, which, for sufficiently large $n$, are arbitrarily close to each other (according to the last displayed equation above), so that $L_1$ must equal $L_2$. Since the sequences $(p_n)$ and $(q_n)$ include the special case $n = 3 \cdot 2^k$, and the limits for these special cases have the common value $\pi$, this means that the common limits for the full sequences $(p_n)$ and $(q_n)$ are also equal to $\pi$. This completes the proof of Theorem 1a.

For Theorem 1b, let $r_n$ and $s_n$ denote the areas of the circumscribed and inscribed regular polygons for a circle of radius one. By examining the figure above (see also the previous blog), it can be seen that $r_n = n \tan (180^\circ/n)$ and $s_n = n \sin (180^\circ/n) \cos(180^\circ/n)$. Let $\beta = 180^\circ/n$. Now note that the difference between the circumscribed and inscribed areas is $$r_n – s_n = n (\tan(\beta_n) – \sin(\beta)\cos(\beta)) = n \left(\frac{\sin(\beta}{\cos(\beta)} – \sin(\beta) \cos(\beta)\right) $$ $$= \frac{n \sin(\beta) (1 – \cos^2(\beta))}{\cos(\beta)} = \frac{n \sin^3(\beta)}{\cos(\beta)} \lt 2n \sin^3(\beta) \lt \frac{128}{n^2},$$ since the final inequality was established a few lines above. As before, it follows that the greatest lower bound of the circumscribed areas $r_n$ is exactly equal to the least upper bound of the inscribed areas $s_n$. Furthermore, since the sequence $(p_n)$ of semi-perimeters of the circumscribed polygons is *exactly the same sequence* as the sequence $(r_n)$ of areas of the circumscribed polygons, we conclude that the common limit of the areas is identical to the common limit of the semi-perimeters, namely $\pi$. This completes the proof of Theorem 1b.

**Other formulas and algorithms for Pi**

We note in conclusion that Archimedes’ scheme is just one of the many known formulas and algorithms for $\pi$. See for example this collection. One such formula is the Borwein quartic algorithm: Set $a_0 = 6 – 4\sqrt{2}$ and $y_0 = \sqrt{2} – 1$. Iterate, for $k \ge 0$, $$y_{k+1} = \frac{1 – (1 – y_k^4)^{1/4}}{1 + (1 – y_k^4)^{1/4}},$$ $$a_{k+1} = a_k (1 + y_{k+1})^4 – 2^{2k+3} (1 + y_{k+1} + y_{k+1}^2).$$ Then $1/a_k$ converges quartically to $\pi$: each iteration approximately *quadruples* the number of correct digits. Just three iterations yield 171 correct digits, which are as follows: $$3.14159265358979323846264338327950288419716939937510582097494459230781640628620899862803482$$ $$534211706798214808651328230664709384460955058223172535940812848111745028410270193\ldots$$

**Other posts in the “Simple proofs” series**

The other posts in the “Simple proofs of great theorems” series are available Here.

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The question of finding rational approximations to real numbers was first explored by the Greek scholar Diophantus of Alexandra (c. 201-285 BCE), and continues to fascinate mathematicians today, in a field known as Diophantine approximations.

It is easy to see that any real number can be approximated to any desired accuracy by simply taking the sequence of approximations given by the decimal digits out to some point, divided by the appropriate power

Continue reading New paper proves 80-year-old approximation conjecture

]]>The question of finding rational approximations to real numbers was first explored by the Greek scholar Diophantus of Alexandra (c. 201-285 BCE), and continues to fascinate mathematicians today, in a field known as Diophantine approximations.

It is easy to see that any real number can be approximated to any desired accuracy by simply taking the sequence of approximations given by the decimal digits out to some point, divided by the appropriate power of ten. For example, we can approximate $\pi$ to within one part in $10^{10}$ by simply writing $$\pi \approx \frac{31415926535}{10000000000}.$$

Researchers in the field have investigated more economical representations of this sort, in the sense of approximating a positive real number to a given tolerance by a rational whose denominator is small. One commonly used technique is the continued fraction algorithm, also known as the extended Euclidean algorithm. This operates by iteratively subtracting the greatest integer from the number, then taking the reciprocal, and then repeating, appropriately collecting the results along the way. For example, the first few approximations for $\pi$ produced in this manner are: $$3, \; 22/7, \; 333/106, \; 355/113, \; 103993/33102, \; 104348/33215, \; 208341/66317, \; \cdots.$$ The last approximation listed gives $\pi$ to approximately ten-digit accuracy, the same as for the digit expansion fraction above. See also the graph, which plots the logarithm base 10 of the error as a function of the number of steps taken in the algorithm applied to $\pi$.

An important result in this arena is Dirichlet’s approximation theorem: For any real number $\alpha$ and integer $N \ge 1$, there exists a pair of integers $(p,q), 1 \leq q \leq N$, such that $|q \alpha – p| \leq 1/N$. An immediate corollary is that given any irrational $\alpha$, the inequality $$\left|\alpha – \frac{p}{q}\right| < \frac{1}{q^2}$$ is satisfied by infinitely many distinct integer pairs $(p,q)$. One might wonder whether the exponent 2 on the right-hand side can be increased, and still yield infinitely many distinct integer pairs. For irrational algebraic numbers, Roth’s theorem shows that the 2 cannot be improved: For every irrational algebraic $\alpha$ and $\epsilon > 0$, the inequality $$\left|\alpha – \frac{p}{q}\right| < \frac{1}{q^{2 + \epsilon}}$$ has only finitely many solutions in terms of relatively prime integers $(p,q)$. Conversely, if one can demonstrate, for a given irrational constant $\alpha$, that there are infinitely many distinct solution pairs $(p,q)$ satisfying the inequality with an exponent higher than 2, then, by Roth's theorem, $\alpha$ must be transcendental.

Ever since Dirichlet first published his approximation theorem in 1840, researchers have explored generalizations, such as by considering different sequences of tolerances. In particular, let $\psi(q)$ be an arbitrary function from positive integers to nonnegative real numbers. Then given $\alpha \in [0,1]$, one can ask whether there are infinitely many pairs of integers $p,q$ such that $$\left|\alpha – \frac{p}{q}\right| \le \frac{\psi(q)}{q}.$$ As it turns out, the question is particularly difficult if $\psi(q)$ is somewhat irregular — for some $\psi$ and $\alpha$ there may be no solutions whatsoever. Nonetheless, one can show that under rather general conditions on $\psi$, this inequality will have infinitely many solutions for *almost all* $\alpha \in [0,1]$, in the measure theory sense.

One key result of this sort is Khinchin’s theorem: Suppose $\psi(q)$ is a function from the positive integers to nonnegative reals with the property that $q \psi(q)$ is decreasing, and let $\lambda$ denote Lebesque measure. Let $A$ denote the set of real $\alpha \in [0,1]$ for which the inequality $$\left|\alpha – \frac{p}{q}\right| \le \frac{\psi(q)}{q}$$ has infinitely many integer solution pairs $(p,q)$ with $0 \leq p \leq q$. Then (a): $\sum_q \psi(q) < \infty$ implies $\lambda(A) = 0$, and (b): $\sum_q \psi(q) = \infty$ implies $\lambda(A) = 1$.

In 1941, Richard J. Duffin and Albert C. Schaeffer investigated Khinchin’s theorem to see if the condition $q \psi(q)$ decreasing could be relaxed. They found that it was more natural and promising to work with integer pairs $(p,q)$ that are relatively prime. Among other things, they were able to show, by applying the first Borel-Cantelli lemma, that if $\phi(q)$ is the Euler phi function, and $\psi(q)$ satisfies $$\sum_{q=1}^\infty \frac{\phi(q) \psi(q)}{q} < \infty,$$ then $\lambda(A) = 0$, where $A$ is as defined above except that the integer pair $(p,q)$ is relatively prime. This led them to conjecture, under the same assumptions, that if $$\sum_{q=1}^\infty \frac{\phi(q) \psi(q)}{q} = \infty,$$ then $\lambda(A) = 1$. This assertion (stated more precisely below) is now known as the Duffin-Schaeffer conjecture. While there have been numerous partial results, a full-fledged proof has eluded the mathematical research community for nearly 80 years.

The surprising news is that the Duffin-Schaeffer has now been proven. In particular, in a new paper, Dimitris Koukoulopoulos of the University of Montreal and James Maynard of the University of Oxford have established the following:

**Duffin-Schaeffer conjecture (now proven by Koukoulopoulos and Maynard)**: Let $\phi(q)$ denote Euler’s phi function, and suppose that $\psi(q)$ is a function from the positive integers to nonnegative reals with the property that $$\sum_{q=1}^\infty \frac{\phi(q) \psi(q)}{q} = \infty.$$ Let $\lambda$ denote Lebesque measure, and define $A$ as the set of real $\alpha \in [0,1]$ for which the inequality $$\left|\alpha – \frac{p}{q}\right| \le \frac{\psi(q)}{q}$$ has infinitely many distinct solutions with relatively prime integers $(p,q)$. Then $\lambda(A) = 1$.

Koukoulopoulos and Maynard adopted a novel approach for this problem: they recast the problem as a question about connections between points and lines in a graph. In particular, the authors created a graph out of the integer denominators, connecting the points representing denominators with a line if they share numerous prime factors. In this way, the graph’s structure encodes the overlap between irrational numbers approximated by each denominator. With this framework in place, Koukoulopoulos and Maynard were able to analyze the graph using known techniques from graph theory, which then yielded their result.

As Dimitris Koukoulopoulos explained, “The graph is a visual aid — it’s a very beautiful language in which to think about the problem.”

Reaction from the mathematical community has been swift and very complimentary. Jeffrey Vaaler of the University of Texas, Austin (who himself published some earlier results on the Duffin-Schaeffer conjecture) declared “It’s a beautiful piece of work.” He added, “They had what I’d say was a great deal of self-confidence, which was obviously justified, to go down the path they went down.”

For additional details, see this Quanta article by Kevin Hartnett, this Scientific American article by Leila Sloman, and, of course, the Koukoulopoulos-Maynard technical paper.

]]>As of the present date (August 2019), more than 4000 exoplanets have been discovered orbiting other stars, and by the time you read this even more will have been logged. Several hundred exoplanets were announced in a July 2019 paper (although these await independent confirmation). All of this is a remarkable advance, given that the first confirmed exoplanet discovery did not occur until 1992.

Most of the discoveries mentioned above are planets that are either too large or too close to their sun to possess liquid water, much

Continue reading How many habitable exoplanets are there, really?

]]>As of the present date (August 2019), more than 4000 exoplanets have been discovered orbiting other stars, and by the time you read this even more will have been logged. Several hundred exoplanets were announced in a July 2019 paper (although these await independent confirmation). All of this is a remarkable advance, given that the first confirmed exoplanet discovery did not occur until 1992.

Most of the discoveries mentioned above are planets that are either too large or too close to their sun to possess liquid water, much less complex carbon-based compounds (see this analysis), and thus there is no conceivable chance that they harbor life even vaguely analogous to that on Earth. Thus researchers have been on the lookout for exoplanets in the circumstellar habitable zone around a star, which is loosely defined as an exoplanet that has a temperature regime capable of supporting liquid water, given sufficient atmospheric pressure, based on its distance from its host star. See this Wikipedia page, which lists more than 40 such potentially habitable exoplanets.

Along this line, an August 2019 study estimated that there are between 5 billion and 10 billion exoplanets in the Milky Way that reside in the habitable zone about their respective stars.

Among other things, researchers have focused microwave antennas and other receptors at these exoplanets, on the off chance that something might be heard at one of these locations. So far, nothing…

The public is clearly excited and fascinated by such reports. After reading some of these press reports, one might think that we are on the verge of discovering Earth 2.0, complete with little green men and women (or that we already have discovered Earth 2.0, but that “elites” are hiding the fact…). But is this type of enthusiasm really warranted, either in scientific literature or in the public arena?

Unfortunately, there are many reasons to hold the champagne. To begin with, just because an exoplanet is in a “habitable zone” about its star certainly does not mean that it actually has water, much less biological organisms. Many other factors need to be considered.

For example, Harvard researcher Laura Kreidberg has noted that the recently discovered exoplanet K2-18b, which has generated considerable excitement because its atmosphere has been confirmed to contain water, has a diameter about 2.7 times the size of Earth, making it more similar to Neptune than to Earth. What’s more, the atmospheric pressure near the rocky surface of this planet is bound to be thousands of times higher than on Earth, and the resulting temperature may exceed 2800 Celsius or 5000 Fahrenheit. There is no possible way any complex carbon-based molecule such as DNA could survive under such conditions.

In fact, as a recent New Scientist article points out, most likely *none* of the current list of 4000 exoplanets is capable of hosting life. This is because life needs much more than a water-friendly temperature regime. For example, a leading scenario for the emergence of life on Earth crucially involves ultraviolet light with a certain moderate energy level to enable simple molecules to combine to form more complex compounds.

To that end, Marcos Jusino-Maldonado and Abel Méndez, of the University of Puerto Rico at Arecibo, have defined an “abiogenesis” criterion, meaning that sufficient UV light of an appropriate energy level for abiogenesis (the origin of life from nonliving molecules) would be available. When they applied their criterion to a list of 40 known exoplanets in the habitable zone, only eight of these matched their abiogenesis condition, and most of these eight are not likely to harbor life because they have a large radius (and thus are probably not rocky planets but instead are gas giants). Only the single planet named Kepler-452b, orbiting a star 1400 light years away, remained a viable candidate. Its radius is 1.63 times that of the Earth, and it marginally meets the abiogenesis and habitability criteria.

Another major problem is that most of the “habitable” planets identified so far are planets orbiting red dwarf stars. Red dwarf stars are in the fact the most abundant and longest-living stars. Some researchers have championed such stars as likely places to hunt for exoplanets harboring life.

But as an August 2019 Scientific American article points out, red dwarf stars are notorious for frequent flares with x-rays and high-energy UV radiation that almost certainly would sterilize any planet in the “habitable” zone. In other words, if an exoplanet is close enough to a red dwarf for the star’s feeble light to permit water to exist, then it is also dangerously close for lethal radiation from stellar flares. What’s more, high-energy stellar winds would very likely strip away any protective atmosphere that any such planet might possess or develop.

Bolstering this conclusion is an August 2019 study by a team of researchers led by Laura Kreidberg of Harvard and Daniel D. B. Koll of MIT. They examined the exoplanet LHS3844b using a new astronomical technique, and showed that it lacks any significant atmosphere, very likely because its host star (a red dwarf) has stripped it away. They conclude that “hot terrestrial planets orbiting small stars may not retain substantial atmosphere.”

Other studies have found even more restrictive conditions on true life-hosting exoplanets. For instance, a team of researchers led by Paul Byrne at North Carolina State University recently found that many exoplanets, even those that are not gas giants but instead have solid crusts, might well be “toffee planets,” with surface rocks that are hot enough to slowly stretch and deform like toffee candy — see this technical paper for details. Such planets most likely would not exhibit plate tectonics, as on Earth, and thus are unlikely to enjoy the benefits of plate tectonics.

Plate tectonics and the Earth’s underlying geophysical features are now thought to be crucial to life on Earth. Among other things, plate tectonics acts as a global thermostat, regulating CO2 levels in the atmosphere to yield a moderate, long-term temperature regime. In addition, one major hazard to life on Earth is streams of high-energy particles emanating from the Sun and elsewhere, which radiation is lethally hazardous to most life. But here on Earth, almost all of this cosmic radiation is deflected by Earth’s magnetic field, which is generated by the same movement of molten iron in the Earth’s core that is the dynamo behind plate tectonics [Ward2000]. This magnetic field also significantly reduces the loss of the atmosphere to outer space.

In addition to Earth being special, the Sun and Solar System are also unusual in many ways. For example, an October 2018 Scientific American article noted that in most of the recently discovered exoplanet systems, planets tend to be of the same size — if one planet, is, say, 1.5 times the radius of Earth, the other planets in the same system are likely to be of roughly this same size also. This is in stark contrast to our Solar System, which features tiny planets such as Mercury and huge planets such as Jupiter, with roughly 20 times the radius (and 8000 times the volume) of Earth. The existence of a large planet such as Jupiter is now thought to be crucial to clearing out debris from the inner planets in the Solar System’s early life, so that, as a result, Venus, Earth and Mars have been relatively undisturbed by asteroid collisions over the past 3.8 billion years or so, allowing life to form and develop, at least on Earth [Ward2000].

Additionally, our system’s position in the Milky Way is also quite favorable: at roughly 27,000 light-years from the galactic center, our Solar System strikes a good balance between being close enough to the center to have a critical concentration of heavier elements for complex chemistry, and yet not so close as to be bathed in sterilizing radiation — only about 7% of the galaxy is in a “galactic habitable zone” by these criteria [Gribbin2018]. Along this line, roughly 85% of stellar systems in the Milky Way are binary systems (with two or more stars). Exoplanets in such systems typically have very irregular orbital patterns, almost certainly destroying any hope for a stable, long-term, life-friendly temperature/radiation regime.

A 2012 study, published in the Royal Astronomical Society of Canada, after surveying numerous criteria and other studies, found that, contrary to popular opinion, the Sun is a very special star: “[I]f one picked a star at random within our galaxy, then there is a 99.99% chance that it will *not* have the same intrinsic characteristics as our Sun and (basic) Solar System.”

See this 2018 Scientific American article by John Gribbin for additional facts and discussion.

In previous blogs (see Blog A and Blog B), we discussed the nagging puzzle known as Fermi’s paradox: If the universe (or even just the Milky Way) is teeming with life, why do we not see evidence of even a single other technological civilization? After all, if such a civilization exists at all, very likely it is thousands or millions of years more advanced, and thus exploring and even communicating with habitable planets in the Milky Way would be a relatively simple and inexpensive undertaking, even for a small group of individuals.

Numerous solutions have been proposed to Fermi’s paradox, but almost all of them have devastating rejoinders. Arguments such as “ETs are under a strict global command not to disturb Earth,” or “ETs have lost interest in space research and exploration,” or “ETs are not interested in a primitive planet such as Earth,” or “ETs have moved on to more advanced communication technologies,” all collapse under the principle of diversity, a fundamental feature of evolution. In particular, it is hardly credible that in a vast, diverse ET society (and much less credible if there are numerous such societies) that not a single individual or group of individuals has ever attempted to contact Earth, using a means of communication that an emerging technological society such as ours could quickly and easily recognize. And note that once such a signal has been sent to Earth, it cannot be called back, according to known laws of physics.

Some (see this PBS show for instance) have claimed that since only 50 years or so have elapsed since radio/TV and radio telescope transmissions began on Earth, this means that only ETs within 50 light-years of Earth (if any such exist) would even know of our existence. But this is clearly groundless, because networks of lights have been visible on Earth for hundreds of years, other evidence of civilization has been visible for thousands of years, large animal species (including early hominins) have been visible for millions of years, and atmospheric signatures of life have been evident for billions of years.

Arguments that exploration and/or communication are technologically “too difficult” for an ET society immediately founder on the fact that human society is on the verge of launching such technologies today, and ET societies, as mentioned above, are almost certainly thousands or millions of years more advanced. As a single example, since we now have rapidly improving exoplanet detection and analysis facilities, as mentioned above, surely any ET society has a far superior facility that can observe Earth. Within a few decades it will be possible to launch “von Neumann probes” that land on distant planets or asteroids, construct extra copies of themselves (with the latest software beamed from the home planet), and then launch these probes to other stars, thus exploring the entire galaxy if desired [Nicholson2013]. Such probes could then beam details of their discoveries back to the home planet and, importantly, even initiate communication with promising planets. Along this line, gravitational lenses, which utilize a star’s gravitational field as an enormously magnifying telescope, could be used to view images of distant planets such as Earth and to initiate communication with these planets [Landis2016].

So why have we not seen any such probes or communications? There is no easy answer. See this Math Scholar blog for more discussion of proposed solutions and rejoinders to Fermi’s paradox.

One cogent solution to Fermi’s paradox is the following: Perhaps the reason the heavens are silent is that Earth is an extraordinarily unique home for intelligent life, according to the criteria mentioned above and perhaps even other criteria that we do not yet understand, so that the closest Earth 2.0, if it exists at all, is exceedingly distant from our Earth. If so, this means that Earth is far more singular than anyone dreamed even a few years ago, and human society has a far greater obligation not to destroy, overheat or otherwise foul our nest — our biosphere in general, and our race in particular, are of cosmic importance.

Just as significantly, we may have to rethink the Copernican principle, namely the notion that there is nothing particularly special about human society, Earth or our position in the universe, a principle that has guided scientific research for decades if not centuries. To the contrary, it is increasingly clear that the Earth *is* rather special — at the very least, there does not appear to be any equivalent to Earth, complete with an advanced technological civilization, within hundreds of light-years of Earth. If the Copernican principle is overturned, even partially, this will mark a very significant juncture in the history of science.

On the other hand, we could hear an announcement tomorrow that not only has life been detected elsewhere, but even intelligent life, with which we can communicate. That would also be an event of incalculable significance, certainly among the most important scientific discoveries of all time.

Such considerations underscore why research into exoplanets is so important and so exciting. However this turns out, we eagerly await the new experimental findings!

]]>Yet physicists have known for many years that the standard model cannot be the

Continue reading How fast is the universe expanding? New results deepen the controversy

]]>Yet physicists have known for many years that the standard model cannot be the final answer. Most notably, quantum theory on one hand and general relativity on the other are known to be mathematically incompatible. This has led to research in string theory and loop quantum gravity as potential frameworks to resolve this incompatibility. Other difficulties may exist as well.

So how can physics advance beyond the standard model? There is only so far that mathematical theories can be taken in the absence of solid experimental results. As Sabine Hossenfelder has emphasized, beautiful mathematics published in a vacuum of experimental data can actually lead physics astray.

In a previous Math Scholar article, we described several anomalies that have arisen in recent physics experiments, any of which may potentially be a spark that leads to new physics beyond the standard model. Here are three:

*The proton radius anomaly*: This stems from the fact that careful measurements of the radius of a proton’s radius when orbited by an electron yield a radius of approximately 0.877 femtometers (i.e., 0.877 x 10^{-15}meters), whereas separate measurements of the proton’s radius when it is coupled with a muon (“muonic hydrogen”) yield a radius of 0.84 femtometers. These measurements differ by significantly more than the error bars of the two sets of experiments. See this Quanta article for details.*The neutron lifetime anomaly*: This stems from the fact that “bottle” measurements of a neutron’s average lifetime yield 879.3 seconds, whereas “beam” measurements yield 888 seconds. The error bar of the bottle measurements is just 0.75 seconds, and that of the beam measurements is just 2.1 seconds, so again the two measurements appear to be further apart than can reasonably be explained as statistical error. See this Quanta article for additional details.*The Hubble constant anomaly*: The Hubble constant H_{0}is a measure of the rate of expansion of the universe. One method to determine H_{0}is based on the flat Lambda cold dark matter (Lambda-CDM) model of the universe, combined with careful measurements of the cosmic microwave background (CMB) data from the Planck satellite. The latest (2018) result from the Planck team yielded H_{0}= 67.4, plus or minus 0.5 (the units are kilometers per second per megaparsec). Another approach is to employ a more traditional astronomical technique, based on observations of Cepheid variable stars, combined with parallax measurements as a calibration. In 2016, a team of astronomers using the Wide Field Camera 3 (WFC3) of the Hubble Space Telescope obtained the value H_{0}= 73.24, plus or minus 1.74. Again, these two values differ by significantly more than the combined error bars of the two measurements.

For each of these anomalies, experimental teams on both sides have been attempting to reduce error bars and to explore the fundamental theory to see if there is any heretofore ignored possibility of error.

In the past few months (as of August 2019), several new experimental studies have been published on the Hubble constant. In March 2019, a research team working with the Hubble Space Telescope reported that based on observations of 70 long-period Cepheid variable stars in the Large Magellanic Cloud, they were able to refine their estimate to H_{0} = 74.03, plus or minus 1.42. Needless to say, this new result does not help to resolve the discrepancy — it moves in the other direction.

In July 2019, a group reported results from another experimental approach, known as the “Tip of the Red Giant Branch” (TRGB). Their approach, which is analogous to but independent from the approach taken with Cepheid variable stars, is to analyze a surge in helium burning near the end of a red giant star’s lifetime. Using this scheme, they reported H_{0} = 69.8, plus or minus 1.7. This is slightly more than the Planck team value (67.8), but not nearly enough to close the gap with the Cepheid approach.

A third group also announced results in July 2019. This project, called H_{0} Lenses in COSMOGRAIL’s Wellspring (HoLiCOW) [yes, that is the acronym], employs gravitational lensing, namely the phenomenon predicted by general relativity that light bends as it passes near an intervening star or galaxy (see graphic above). The specific approach of the HoLiCOW project is to measure light from a very distant quasar, which is lensed by a closer galaxy. When this happens, multiple time-delayed images of the galaxy appear at the edges of the intervening galaxy, when viewed by earth-bound astronomers. The HoLiCOW project’s latest result is H_{0} = 73.3, plus or minus 1.76.

Needless to say, researchers are perplexed by the latest reports: the Planck team (based on the Lambda-CDM model) reports H_{0} = 67.4 (plus or minus 0.5); the TRGB team reports 69.8 (plus or minus 1.7); the HoLiCOW team reports 73.3 (plus or minus 1.76); and the Cepheid team reports 74.03 (plus or minus 1.42). Obviously these results cannot all simultaneously be correct. For example, the HoLiCOW team’s figure (73.3) represents a 5.3 sigma discrepancy from the Planck figure (67.4). While each of these teams is hard at work scrutinizing their methods and refining their results, there is an unsettling possibility that one or more of the underlying physical theories are just plain wrong, at least on the length and time scales involved.

Key among these theories is the Lambda-CDM model of big bang cosmology. Yet physicists and cosmologists are loath to discard this model, because it explains so much so well:

- The cosmic microwave background radiation and its properties.
- The large-scale structure and distribution of galaxies.
- The present-day observed abundances of the light elements (hydrogen, deuterium, helium and lithium).
- The accelerating expansion of the universe, as observed in measurements of distant galaxies and supernovas.

As Lloyd Knox, a cosmologist at the University of California, Davis, explains,

The Lambda-CDM model has been amazingly successful. … If there’s a major overhaul of the model, it’s hard to see how it wouldn’t look like a conspiracy. Somehow this ‘wrong’ model got it all right.

Various modifications to the Lambda-CDM model have been proposed, but while some of these changes partially alleviate the Hubble constant discrepancy, others make it worse. None are taken very seriously in the community at the present time.

For additional details and discussion, see this Scientific American article (although note that the Scientific American article’s report on the HoLiCOW measurement does not agree with the HoLiCOW team’s latest technical paper) and this Quanta article.

In spite of the temptation to jump to conclusions, throwing out the standard model or big bang cosmology, considerable caution is in order. After all, as mentioned above, in most cases anomalies are eventually resolved, usually as some defect of the experimental process or as a faulty application of the theory.

A good example of an experimental defect is the 2011 announcement by Italian scientists that neutrinos emitted at CERN (near Geneva, Switzerland) had arrived at the Gran Sasso Lab (in the Italian Alps) 60 nanoseconds sooner than if they had traveled at the speed of light. If upheld, this finding would have constituted a violation of Einstein’s theory of relativity. As it turns out, the experimental team subsequently discovered that the discrepancy was due to a loose fiber optic cable that had introduced an error in the timing system.

A good example of misapplication of underlying theory is the solar neutrino anomaly, namely a discrepancy in the number of observed neutrinos emanating from the interior of the sun from what had been predicted (incorrectly, as it turned out) based on the standard model. In 1998, researchers discovered that the anomaly could be resolved if neutrinos have a very small but nonzero mass; then, by straightforward application of standard model, the flavor of neutrinos could change enroute from the sun to the earth, thus resolving the discrepancy. Takaaki Kajita and Arthur McDonald received the 2015 Nobel Prize in physics for this discovery.

In any event, sooner or later some experimental result may be found that fundamentally upsets currently accepted theoretical theories, either for a specific framework such as Lambda-CDM big bang cosmology, or even for the foundational standard model. Are any of the above-mentioned anomalies of this earth-shaking character? Only time will tell.

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