Stand-Up Comedy and Emotional Resonance

I recently read an article in The New Yorker magazine [LINK] about how the stand-up comedian Hasan Minhaj significantly exaggerated or conflated stories in his recent big stand-up comedy routines. In particular, these stories were about instances of racism or Islamophobia, including being the victim of police brutality, being part of a mosque that was infiltrated by an FBI agent, and being sent a mysterious powder that led to his child's hospitalization, that either didn't happen at all or were significantly exaggerated. As someone who has liked his work in the past and who could identify to some degree with his stand-up comedy material based on experiences as the child of immigrants from India, I found these allegations quite troubling, yet I also found myself struggling to articulate exactly why I found these allegations to be so troubling. This post is my attempt, in the current zeitgeist (as this is a very new story and new details could soon arise that would make this post irrelevant or incorrect), to make sense of these things. Follow the jump to see more.

Movie Review: Oppenheimer

I should note that the last movie review on this blog was almost exactly 12 years ago, when I had watched the movie Source Code [LINK]; going back to that post let me cringe a little again at my writing style as a college student. In any case, although I have watched many movies since then but haven't felt compelled to review them for this blog, I felt a little more compelled to do so after recently watching the movie Oppenheimer in IMAX (though not 170 mm IMAX), because of the historical & scientific significance as well as the hype around its release. That movie is essentially a dramatized adaptation of the book American Prometheus by Kai Bird & Martin J. Sherwin (which I haven't yet read), covering the life of J. Robert Oppenheimer during his career as a physicist & developer of nuclear weapons for the US and particularly focusing on his involvement in the Manhattan Project & his subsequently being stripped of a security clearance.

There were a few things that I liked about the movie. I understand that Edward Teller was ostracized by the scientific community by giving testimony that would further bolster investigators turning scientific disputes & personal friction between Teller & Oppenheimer into a reason for claiming Oppenheimer to be a national security risk, but considering that Teller's further ostracism came more when he further dug into developing nuclear arsenals & using nuclear weapons in absurd ways that signaled a weird lust for nuclear explosions, I appreciated that the movie stuck with Teller's role in the Manhattan Project (without letting later views of Teller color his portrayal during the time of the Manhattan Project) and made explicit his real-life testimony praising Oppenheimer's integrity & ultimate loyalty to the US (as opposed to other countries). I also appreciated how the movie made clear that arguments against the use of nuclear weapons after the actual bombing of Japan could be seen as facile or hypocritical when compared to similar arguments before the initial test in Los Alamos. In particular, Oppenheimer initially rationalized concerns about the US having access to the destructive power of nuclear weapons by recognizing the far greater threat to humanity of Nazi Germany getting & using such weapons first, so later claims of being disgusted by their use need to be shaped with a lot more nuance than Oppenheimer actually provided. Additionally, as I have read most of the Bhagavadgītā, I could see that Oppenheimer quoting Kṛṣṇa's line (repeating the translation that Oppenheimer used) "I am become Death, the shatterer of worlds" is arguably a misunderstanding of the philosophical implication, considering that Truman essentially had to correct Oppenheimer in the same way that Kṛṣṇa had to correct Arjuna: the US (with the president, at that time Truman, as the symbolic executor), like Kṛṣṇa, was the entity with the will to destroy, while Oppenheimer/Arjuna was the human instrument and the nuclear/celestial weapons were the insentient instruments. I wonder if more people will recognize this and thus not blindly praise Oppenheimer just for quoting the Bhagavadgītā.

There has been a lot of controversy, especially in India and also among Hindus outside of South Asia, about the depiction of a Sanskrit copy of the Bhagavadgītā during a sex scene being sacrilegious. I'm not religious, and I knew of Oppenheimer's fascination with Hindu mysticism, so I initially gave the director the benefit of the doubt that it may perhaps reflect some combined mystical view of sex & spirituality by Oppenheimer in real life, especially given that reactions about these things tend to be much harsher in India than in the US. Now that I have watched that scene, I can say that the presence of a Sanskrit copy of the Bhagavadgītā added nothing to the sex scene or to the understanding of Oppenheimer's life and was probably not something that happened in real life, so it seems to be in gratuitously bad taste. Moreover, I felt like the scenes where Oppenheimer used Kṛṣṇa's aforementioned line, including but not limited to the sex scene, made it feel cheap & unnecessary; in particular, using it first in the sex scene robbed it of the gravitas that it could have had when portraying the nuclear test explosion.

Overall, perhaps because I had some familiarity with the historical events, I felt like the director tried too hard to make an ultimately simply story about the life of a complicated person seem more complicated (as a story) & visually engaging than necessary. It is perhaps damning to the movie that I felt that despite having seen trailers where the cast of the movie encouraged people to watch it in an IMAX movie theater, I felt that I could have enjoyed it equally on a small screen in an airplane. As an example, I could see that the director was in many scenes trying to visually depict the turmoil in & tortured state of Oppenheimer's mind, but the effects often felt too overwrought with crazy pictures & loud sounds. I thus would only recommend it to people who may then be inspired to read the book (as I myself have yet to do).

On another note, there was a scene with a graph on a chalkboard for one of Oppenheimer's lectures showing a single particle tunneling quantum mechanically through a flat barrier in 1 dimension, but the wavefunction was so badly drawn that it didn't seem to show exponential suppression in space in the region of the barrier. When I saw that scene, I immediately thought that if I were a TA for a class in which he was a student and he had submitted that as part of a homework assignment, I would have deducted points.

More on Climate Categorization

This post is essentially a follow-up to a recent post [LINK] about the Köppen & Trewartha categorizations of climates; that post was in turn a follow-up to a recent post [LINK] about my intuitions of various climates. This post will discuss, more systematically & in more detail, the climates that are impossible or not typically observed in the Trewartha categorization even when consistently using the third & fourth letters to specify the hottest & coldest mean monthly temperatures respectively, the pros & cons of the Trewartha categorization, and a proposal that I thought of to address some of the cons of the Trewartha categorization. Again, I am not a trained climatologist or meteorologist; I can't guarantee that this information is accurate, and I can only say that my intuitions seem through my limited understanding to align with superficial aspects of more detailed explanations. Follow the jump to see more.

Book Review: "How Not to Be Wrong" by Jordan Ellenberg

I've recently read the book How Not to Be Wrong by Jordan Ellenberg. As the author states in the introduction, it is an exposition of simple yet profound ideas in mathematics, meant for laypeople. Topics include nonlinear phenomena (in opposition to naïve linear extrapolation), probability, Bayesian reasoning, and statistical testing of hypotheses. All chapters refer to many examples in politics, economics, and everyday life to make the concepts easier for laypeople to digest.

I found the book to be fairly easy to follow. I can't say that I learned much in terms of concepts, as these are all concepts that I've come across one way or another in school, college, graduate school, or my work now, though I did appreciate the discussion of how conspiracy theorists like to add hypotheses after the fact to make a conspiracy theory harder to fully disprove, how the fact that random fluctuations in many phenomena observed over time are time-reversal invariant implies that the phenomenon of regression toward the mean is also time-reversal invariant in a probabilistic sense, and the intuitive explanations of common causes & common effects in leading to correlations between random variables that are otherwise not causally connected. Additionally, I felt like this book did a better job than the book Algorithms to Live By by Brian Christian & Tom Griffiths (which I have reviewed on this blog before [LINK]) in having some structure in the progression from one chapter to the next and in using topics from earlier chapters in later chapters even though this book, unlike that book, didn't pretend to have a unified message. My only quibbles are the claim that the impossibility of accurately running the fundamental equations describing atmospheric & oceanic dynamics for more than 2 weeks implies impossibility in forecasting through other methods (like machine learning models looking for patterns in weather effects & progression) and the fact that the chapter connecting ideas from probability, geometry, and signal processing (particularly around error correction) took me a fair bit of effort to follow (unlike the other chapters, which tells me that laypeople will likely struggle with that chapter much more). Additionally, I think readers should be aware that the author often makes reference to sports that are mostly popular in the US and to US politics and that the author at a few points espouses more liberal or progressive political views (though I think such espousal is not gratuitous but is done in a way that fits well with broader discussions of assumptions underlying mathematical, political, and legal judgments). Overall, I think the author has done a good job of fulfilling the goal of communicating these ideas to a lay audience, so I recommend this book to anyone who might be interested in these ideas.

FOLLOW-UP: My Rough Intuition of Climate, Especially in the US

The previous post in this blog [LINK] went over my rough intuition of climate, primarily in middle latitudes like those of the US. Most of the broad categories that I described were largely aligned with the Köppen climate classification system (henceforth called the Köppen categorization). However, there is a more recent categorization known as the Trewartha climate classification system (henceforth called the Trewartha categorization) that is supposed to be more representative of middle latitudes like those of the US. Essentially, tropical, desert, and semi-arid climates, as well as polar and ice cap climates, are defined in the same ways between the two categorizations. The differences lie in the definitions of subtropical, continental, and subpolar oceanic/subarctic climates. One benefit of the Trewartha categorization is that it clearly separates boreal/subpolar climates from other oceanic and continental climates, whereas the Köppen categorization uses subcategories that could be a little more confusing. However, the definitions of subtropical, oceanic, and continental climates in the Trewartha categorization seem less justifiable to me. Follow the jump to see more details. Again, I am not a trained climatologist or meteorologist; I can't guarantee that this information is accurate, and I can only say that my intuitions seem through my limited understanding to align with superficial aspects of more detailed explanations.

My Rough Intuition of Climate, Especially in the US

For a long time, I had wondered why the climates of San Francisco, Sacramento, and Los Angeles are so different from those respectively of Richmond, DC, and Atlanta. I had read a few articles on Wikipedia on occasion, so I got a sense that it has to do in part with different ocean currents; this made sense to me, as I had become very comfortable (growing up in the DC area) with the warm waters at beaches along the East Coast in the summer, and I was always surprised by the comparatively much colder waters at beaches along the West Coast whenever I'd visit California even in the summer. I knew though that this wasn't the whole story, and I was surprised to see, for example, that even in South America, South Africa, Western Europe versus East Asia, and Australia, there were very similar contrasts in climates between cities along west versus east coasts in the middle latitudes. This made me more curious about the reasons for these similarities, so I recently went down a rabbit hole of Wikipedia articles to learn more and form an intuition about why different places have different kinds of climates. Follow the jump to see my explanation. I am not a trained climatologist or meteorologist; I can't guarantee that this information is accurate, and I can only say that my intuitions seem through my limited understanding to align with superficial aspects of more detailed explanations. I'm just putting this out there in case this intuition is helpful to anyone else as a starting point to learn more (and I recognize that an incorrect initial intuition could hurt rather than help when trying to learn more).

Book Review: "Sapiens" by Yuval Noah Harari

I've recently read the book Sapiens by Yuval Noah Harari; this was highly recommended to me 7 years ago by a friend in graduate school with whom I had enthusiastic discussions about the material in the book, but I hadn't gotten a chance to read the book until now. This book is supposed to be a history of humans, going from an evolutionary perspective for the first 2 million years since the genus Homo became distinct and then getting into the developments of language, collective myths, agriculture, urbanization, and industry. Even my summary of the flow of the book contains my opinion about its progression (though I should note that the "parts" that I speak of overlap with but aren't identical to the 4 parts that formally divide the book): the beginning part of the book seems to be a serious discussion of evolution, language, and the advent of agriculture, the middle part tries to be serious but has more inconsistencies that I find problematic, and the last part seems more clearly to be more like a "pop-history" type of book with less rigorous speculation (so I read it with a lighter heart even if the author didn't intend it that way and I therefore heavily discounted it in my overall opinion of the book); furthermore, the parts about scientific development since 1500 can be understood more clearly in other books.

There were several things that I learned from the book and several ways in which the book forced me to consider a different perspective. These are as follows, in no particular order. First, I learned that the further development of language to be able to convey detailed information, gossip, abstract ideas, and fictions happened 70,000 years ago and coincided with humans becoming dominant in the food chain and in their spread across the world. Second, I had always uncritically believed in the advent of agriculture and later urbanization as a good thing, to the extent that I've recently sometimes wondered (without being particularly informed about history & sociology) whether clashes between the Mongol invaders & Hindu natives (in contrast to earlier arrivals in India of urban Muslim traders from Mesopotamia) as well as between the expectations of Western government & the reality of the House of Saud represent clashes between urbanized cultures that have developed to a great degree versus nomadic cultures that have endured much harsher conditions and only wish to plunder the cities for wealth without care for more refined aspects of urban cultures; this book forced me to consider that individuals within nomadic tribes had much more varied diets & activities within each day, that the first few millennia of the transition to agriculture may have led to a lot of suffering compared to what came immediately before, and that the domestication of wheat can be reinterpreted as a mutalistic domestication of wheat & humans. Third, I learned that empires might be defined only by the number of culturally distinct tribes under their yokes and the flexibility of their borders in expansion, not by population or area per se. Fourth, while I had some familiarity with how capitalism & European imperialism fueled each other and with the use of scientifically-inspired racism as a justification for European colonization, I didn't have a good sense for how these all tied together until I read this book, especially in the context of scientific voyages only being funded if the scientists could tag along with naval officers ordered to colonize the lands they would reach. Fifth, I appreciated the distinction between ancient & medieval empires which grew in predictable ways by absorbing naval territories versus early modern European empires which grew unpredictably across the world with long-distance seafaring. Sixth, I appreciated the explicit call to attention about how liberal humanistic political philosophies, which profess to be atheistic in themselves & multireligious in the sense of tolerance, cast freedom & political empowerment in terms of a special nature of individuals that is drawn directly from Christian notions of creation & individual souls (though the concepts of creation & individual souls aren't unique to Christianity among traditional religions); this is something that I've pondered before but have typically glossed over, so I appreciated being challenged in this way.

There were a few points that I was happy to see in the sense of agreeing with those worldviews. These include the ideas of collective myths (not only in traditional religions but in the systems of trust that underlie monetary systems & democracies), historical predictions leading to self-fulfilling or self-negating prophecies, the existence of hierarchies of some form in almost all societies larger than about 150 people (as I've wondered, for example, if a solution to the problem of inflation coming from an immediate cash payout to everyone in a universal basic income plan would be to sprinkle it randomly upon different people at different times to ensure that the economic system doesn't stray too far from its previous state and can better respond to bigger numbers of people getting such payouts later even if that creates an effective hierarchy between those who get such payouts at a given time and those who don't), and the ways that even cultures free of external pressures can develop internal contradictions that in turn can lead to continued development of the culture as a unified entity or a split of the culture into multiple descendants. On a lighter note, I also enjoyed seeing the author, in the otherwise problematic speculation about science, point out that science fiction can only rarely, if ever, attempt to describe what would truly be alien experiences to humans, and that most science fiction stories ultimately revolve around myths & social conflicts that in one form or another have been recognizable for millennia, which leads me to the conclusion that there is no reason beyond snobbery to claim that Star Trek is science fiction while Star Wars supposedly is not (because if science fiction is defined as only portraying truly alien experiences in encounters with new technology or new intelligent species that aren't just thinly-veiled allegories for known interactions among human groups, there may only be a few books, movies, and TV series that may be called "science fiction", perhaps including Black Mirror or 2001: A Space Odyssey, and those works are rare probably exactly because readers or viewers would find them less relatable).

There were a few specific stories that I liked reading. One was of the Chinese seafarer Zheng He, as it shows that Chinese seafaring technology was as advanced as European seafaring technology around 1500 but China simply didn't have the same ambition to conquer faraway lands through seafaring. The other was of how the accompaniment of Hernán Cortés by Aztec people carrying burning incense sticks near him convinced him that the "primitive" Aztecs were treating him as a deity but was actually because he had terrible body odor due to bad hygiene, as it is a funny story, it shows that the Aztecs, immediately upon encountering Europeans, figured out what other contemporary peoples of Asia & Africa had known for many centuries (leading those peoples to set up quarantine areas for European visitors at ports), namely that Europeans had bad hygiene at that time, and it shows how the self-delusion of European winners of such conflicts (in this case Cortés believing that he was being treated as a deity so the Aztecs must have been "primitive") could persist in "official" historical narratives for many centuries.

Beyond the problematic historiography (especially ignoring the way that so many consequential scientific discoveries were made in Europe individually by people who were independently wealthy while also not clearly explaining which technological discoveries were systematically funded & used by governments, though those parts could be fixed with better writing) and excessively serious-sounding speculation about science in the last few chapters & sprinkled elsewhere in the book, there were three major points of disagreement that I had with the author, in the sense that I believe that these points strike at the fundamental arguments of the book. These are as follows.

First, the author makes a big deal throughout the book about how the global unity in understanding of political, economic, and other norms that has emerged in the last 500 years is unprecedented in all prior years of human existence. My counterargument is that this argument depends too much on the specific way that previous interactions between cultures went or on the fact that certain cultures happened to not interact. It will be based primarily on [Native] American and European cultures before and around the time of their first contact in the middle of the second millennium, as the author makes a big deal about how American tribes were among the groups that were totally isolated from the continuum of groups across Africa, Asia, and Europe (with Australian tribes being among the others). As the author argues, the lack of contact before may well have been because of a combination of technological limitations along with limitations in cultural ambition. However, in a counterfactual situation where, for example, English people looking to start local democratic governments met on a truly equal footing with Iroquois people who embodied the spirit of democracy in their local confederated governments, there is no specific reason to believe that they would have been talking past each other; the author's conception of "global unity" as a phenomenon that developed in the last 500 years with no precedent seems to depend too strongly on peoples having met or being aware of each other's existence and not enough on actual similarities between each other's cultures. Additionally, the example of Hernán Cortés and the incense sticks, along with the example (not in the book) of the origin of the ethnic slur "Indian giver" from a deliberate misunderstanding of Native Americans' attempts to barter with Europeans as gifts that were then demanded to be returned, shows that the author's view of the establishment of "global unity" depended strongly on the actual course of history (in this case Europeans deliberately ignoring what Native Americans were telling them) and not on the broader cultural similarities already present. This dependence on the actual course of history makes this a hindsight-based account that the author supposedly disclaims, making the author rather hypocritical.

Second, in later parts of the book, when the author discusses the reasons for European armies so easily conquering peoples in faraway lands, the author puts a lot of stock in the idea that success was due to the European drive for exploration of the unknown, even before that commitment to exploration started bearing systematic fruit in the forms of scientific discovery or technological advancement; conversely, the author briefly mentions and otherwise glosses over the role of more effective forms of social organization & discipline in those armies. This seems to completely undermine previous chapters in the book that so clearly emphasized the ways that communication of collective myths could lead to new forms of social organization. Perhaps this seeming contradiction can be resolved by interpreting the "drive to explore the unknown" as extending to European armies systematically developing new ways, including new forms of organization of their own armies, of dealing with unknown peoples whom they wish to conquer. However, this seems like a stupid semantic difference and again seems like the author is engaging excessively in analysis from narrow hindsight, contrary to the author's own stated claims. (UPDATE: A related point is about how the author implies that the drive for European colonists to learn about unknown cultures & explore how to systematically conquer unknown peoples led them to use what they learned about these cultures to systematically deepen existing divisions or create new divisions. I could agree that Europeans were the first to do this so systematically or so tightly coupled to the seemingly more noble goal of learning things that weren't known to them. However, I cannot agree with the idea that Europeans were the first to exploit & inflame divisions or engage in proxy wars. as ancient Egyptian kingdoms were known to have done this to the Assyrian Empire. Perhaps this can be forgiven if it turns out that this book was published before we knew about how the ancient Egyptians fomented rebellions, civil wars, and proxy wars in the Assyrian Empire, but in any case, the author's seeming unwillingness to directly assert or refute the idea that European colonists' "drive to explore" specifically included exploration of how to organize themselves better & exploit other people's weaknesses more effectively to conquer those other peoples is a much bigger problem with this book.) Moreover, the author does not attempt to explain why peoples were so consistently conquered at all by Europeans from the perspective of those conquered peoples other than simply stating the claim that those peoples couldn't imagine that their knowledge could be incomplete. I think I could do a better job than the author by imagining a counterfactual situation, using the example of Hernán Cortés encountering the Aztecs: even if the Aztecs were similarly driven as the Spanish by exploration of the unknown and had expanded their empire that way before the Spanish landed in America as historically happened, the only way from the perspective of social dynamics that I can see the Aztecs successfully repelling the invasion is by using their knowledge of dealing with unknown peoples to see through Cortés's lies into his true intentions and organize accordingly, yet there is no guarantee that knowing what to do when attempting to conquer unknown peoples would lead an empire to develop knowledge of what to do at the receiving end of a conquest attempt. Finally, in the specific cases of Europeans interacting with Native Americans, the author in a few places briefly acknowledges the role of infectious disease (used by Europeans sometimes accidentally and other times, as in the case of pox blankets in the 1763 Native American siege of the British-occupied Fort Pitt, intentionally) but otherwise glosses over this in favor of explanations based on exploration of the unknown. Yet, as the examples of Cortés as well as the slur "Indian giver" point out, it is quite plausible that Europeans, seeing how easily Native Americans were wiped out by disease, used this as propaganda to better organize themselves and portray Native Americans as weak (independent of specific technology or ideals about exploring the unknown), and I think it is irresponsible for the author to ignore this obvious possibility.

Third, there is a whole chapter about the history of traditional religions, including animistic, polytheistic, monotheistic, and atheistic religions. My problem is that the author makes too broad claims about religious trends even though there are so few surviving or [relatively] recently extinguished major traditional religions; the sample sizes are so small as to make the claims unconvincing. The author acknowledges similar problems in other contexts elsewhere but not in that chapter.

Beyond these issues, I noted several more minor issues at various points in the book. Although some of these issues personally offended me, I still categorize them as minor because I think that deleting the offending passages from the book would not significantly reduce support for or otherwise qualitatively change the main arguments of the book. These are as follows, in no particular order. Even if some of these points raise questions that have no clear answer, I think the author was irresponsible in not addressing the existence of these questions and clearly stating the lack of a clear answer.

First, the author claims that when big social orders are sustained through collective myths, those collective myths require genuine belief from members of the elite too. Recent news about how Fox News executives & star hosts privately disbelieved claims that the 2020 US presidential election was rigged but knowingly pushed such claims in public just to boost TV ratings & stock prices. On the one hand, perhaps it isn't fair to pin this on the author as this news is much more recent than the publication of the book. On the other hand, I would be curious to see how the author would react to this news now; if the author reacts by claiming to be correct because the degree of true belief among the elite was "always destined to wane at some point" or for some similar reason purely in hindsight, then that tells me that the author's approach is worthless because it would be unfalsifiable.

Second, the author does such a consistently bad job with the history of India that I have to wonder if the author's research about that specific topic consisted exclusively of books written by British colonizers to portray India to their own benefit. Problems include claims that Indo-Aryans "invaded" (as that word is usually used to imply a systematic movement of an army to bring forth a violent clash, yet there is no historical evidence for such singular violent clashes between ancient Central Asian migrants and South Asian natives), the treatment of caste in the Vedas (as even people who aren't apologists for Brahmins or deniers of the history of caste will recognize a lot of subtlety in the way the Vedas used terms associated now with caste, especially as those castes didn't exist in Indian society until after Vedic times), the treatment of caste in general (conflating jati & varna to claim that the "original" 4 varnas over time split into thousands of jatis, when the reality is much more complicated & less clear), the claim that Brahmins could have "learned" from the KKK how to enforce caste divisions (as Brahmins, especially in South India, were already brutally effective in enforcing caste divisions long before the KKK existed), and the claim that India had no national consciousness before the British Empire (which undermines the author's own prior acknowledgment of the Gupta & Maurya Empires as empires by the author's own definition). Another problematic statement by the author that I am willing to forgive given when the book was published (before the political rise of right-wing Hindu nationalism in India in the last 10 years) is the rhetorical question about whether right-wing Hindu nationalists would do away with all symbols of the Mughal Empire, include those of beauty like the Taj Mahal; the author clearly implied that they wouldn't dare to do so, but that view now seems laughably quaint. Finally, a statement by the author about religion in a broader context yields problems in the context of India. In particular, the author claims that polytheistic kings didn't try to convert conquered peoples or make them destroy temples to their own deities, but historical conflicts between Saiva & Vaisnava kingdoms in South India over religion suggest otherwise; perhaps this point can also be forgiven as a rare exception to the rule.

Third, the author's definition of an empire in terms of flexibly expanding borders still leads contemporary readers to imagine borders that strictly control the flow of people across them, which I think is historically misleading given the ease with which people could pass across. Even now, people from Mexico & countries in Central America freely pass through the international border between California & Baja California as seasonal migrant workers although the US border is otherwise very strictly controlled.

Fourth, the author makes claims about the positive cultural developments by empires, but those claims seem incomplete. Additionally, the author tries to distinguish Cyrus of Persia's claim that the empire would benefit all people from the more limited ambitions of Assyrian emperors, but this distinction is not clear at all.

Fifth, the author claims that interest (in the sense of a guaranteed geometric return on an investment) requires the existence of a currency that is not useful for any other reason. I disagree in principle because livestock and crops, which have historically been used as currencies or items of barter, have the potential to multiply over time. That said, this may be a moot point if there is no evidence for societies having charged interest directly on livestock or crops as forms of currency.

Sixth, the explanations of why some peoples did not develop agriculture until much later contact by faraway urbanized peoples seems incomplete. I agree with the idea that some peoples settled in areas where plants & animals simply could not be domesticated; the capability to be domesticated by humans is rare among species. However, this does not explain why many native peoples of America and Australia never developed agriculture even though they lived on grasslands that later turned out to support agriculture very easily; in the case of Australia, the author's omission is especially troubling given that the author explains how humans who moved to Australia 45,000 years ago had no problem with destroying the forests that were already there & replacing them with grasslands.

Seventh, near the end of the book, the author distinguishes ecological destruction from resource scarcity as the more likely cause of future human suffering or extinction. This seems to undermine an earlier chapter in which the author acknowledges that problems with allocating resources to maintain a certain standard of living under certain norms, even if the resources themselves are technically abundant, are more likely to lead to conflict. I wish the author had dealt with this more carefully.

Eighth, the author writes about the seemingly inexorable trend toward globalization and the way that world war has come to seem implausible since World War II. The failure to predict both the retreat from globalization especially since 2014 as well as the Russian military invasion of Ukraine in 2022 are forgivable given when the book was published. However, I'm more troubled that the author's claims about the implausibility of world war are phrased in ways that are unfalsifiable, as the author can always claim either a different definition of "world war" or a finite time period of validity (since the last world war) that the author would not have previously clearly stated.

Overall, I think this is still an interesting book, though I wasn't incredibly impressed by it (unlike, for example, my friend from graduate school). I'd recommend it with the caveats discussed above. In any case, as this was among the first books to go on the reading list (for books unrelated to my work) that I made for myself in graduate school, I'm glad to have finally read it.

My Time at the 2023 TRB Annual Meeting

Last month, I attended the 2023 TRB Annual Meeting. I meant to write this post immediately afterwards, but I became busy with work and with preparing for international travel which I started soon after that and finished just yesterday. Thus, for the first time in the history of this blog, there was a month with no new post. I'll try my best going forward to continue posting at least once each month, but sometimes, these issues can come up.

The conference, which was held in DC, was a lot of fun. The graduate student researcher working with me was able to present our work as a poster. Additionally, it was my first experience attending an external conference (i.e. one not hosted by UC Davis) since changing fields to get into transportation policy research, so it was an excellent opportunity for professional networking. Because I had previously attended the APS March Meeting for 3 years (2017-2019), I was prepared for a conference of a similar size and scope; in particular, I intentionally avoided overloading my schedule with presentation sessions, made time to take breaks, and made time to meet people informally. Additionally, as transportation is a much more policy-oriented field than physics and is not confined to academia, I made sure to attend TRB committee meetings, different organizations' receptions in the evenings, and other events technically outside of the conference itself. I certainly professionally got what I wanted out of it, and I look forward to attending again in the future.

Fundamental Theorem of Calculus for Functionals

I happened to think more about the idea of recovering a functional by somehow integrating its functional derivative. In the process, I realized that certain ideas that I would have to consider make this post a natural follow-up to a recent post [LINK] about mapping scalars to functions. This will become clear later in this post.

For a single variable, a function \( f(x) \) has an antiderivative \( F(x) \) such that \( f(x) = \frac{\mathrm{d}F}{\mathrm{d}x} \). One statement of the fundamental theorem of calculus is that this implies that \[ \int_{a}^{b} f(x)~\mathrm{d}x = F(b) - F(a) \] for these functions. In turn, this means \( F(x) \) can be extracted directly from \( f(x) \) through \[ F(x) = \int_{x_{0}}^{x} f(x')~\mathrm{d}x' \] in which \( x_{0} \) is chosen such that \( F(x_{0}) = 0 \).

For multiple variables, a conservative vector field \( \mathbf{f}(\mathbf{x}) \) in which \( \mathbf{f} \) must have the same number of components as \( \mathbf{x} \) can be said to have a scalar antiderivative \( F(\mathbf{x}) \) in the sense that \( \mathbf{f} \) is the gradient of \( F \), meaning \( \mathbf{f}(\mathbf{x}) = \nabla F(\mathbf{x}) \); more precisely, \( f_{i}(x_{1}, x_{2}, \ldots, x_{N}) = \frac{\partial F}{\partial x_{i}} \) for all \( i \in \{1, 2, \ldots, N \} \). (Note that if \( \mathbf{f} \) is not conservative, then it by definition cannot be written as the gradient of a scalar function! This is an important point to which I will return later in this post.) In such a case, a line integral (which, as I will emphasize again later in this post, is distinct from a functional path integral) from vector point \( \mathbf{a} \) to vector point \( \mathbf{b} \) of \( \mathbf{f} \) can be computed as \( \int \mathbf{f}(\mathbf{x}) \cdot \mathrm{d}\mathbf{x} = F(\mathbf{b}) - F(\mathbf{a}) \); more precisely, this equality holds along any contour, so if a contour is defined as \( \mathbf{x}(s) \) for \( s \in [0, 1] \), no matter what \( \mathbf{x}(s) \) actually is, as long as \( \mathbf{x}(0) = \mathbf{a} \) and \( \mathbf{x}(1) = \mathbf{b} \) hold, then \[ \sum_{i = 1}^{N} \int_{0}^{1} f_{i}(x_{1}(s), x_{2}(s), \ldots, x_{N}(s)) \frac{\mathrm{d}x_{i}}{\mathrm{d}s} \mathrm{d}s = F(\mathbf{b}) - F(\mathbf{a}) \] must also hold. This therefore suggests that \( F(\mathbf{x}) \) can be extracted from \( \mathbf{f}(\mathbf{x}) \) by relabeling \( \mathbf{x}(s) \to \mathbf{x}'(s) \), \( \mathbf{a} \) to a point such that \( F(\mathbf{a}) = 0 \), and \( \mathbf{b} \to \mathbf{x} \). Once again, if \( \mathbf{f}(\mathbf{x}) \) is not conservative, then it cannot be written as the gradient of a scalar field \( F \), and the integral \( \sum_{i = 1}^{N} \int_{0}^{1} f_{i}(x_{1}(s), x_{2}(s), \ldots, x_{N}(s)) \frac{\mathrm{d}x_{i}}{\mathrm{d}s} \mathrm{d}s \) will depend on the specific choice of \( \mathbf{x}(s) \), not just the endpoints \( \mathbf{a} \) and \( \mathbf{b} \).

For continuous functions, the generalization of a vector \( \mathbf{x} \), or more precisely \( x_{i} \) for \( i \in \{1, 2, \ldots, N\} \), is a function \( x(t) \) where \( t \) is a continuous dummy index or parameter analogous to the discrete index \( i \). This means the generalization of a scalar field \( F(\mathbf{x}) \) is the scalar functional \( F[x] \). What is the generalization of a vector field \( \mathbf{f}(\mathbf{x}) \)? To be precise, a vector field is a collection of functions \( f_{i}(x_{1}, x_{2}, \ldots, x_{N}) \) for all \( i \in \{1, 2, \ldots, N \} \). This suggests that its generalization should be a function of \( t \) and must somehow depend on \( x(t) \) as well. It is tempting therefore to write this as \( f(t, x(t)) \) for all \( t \). However, although this is a valid subset of the generalization, it is not the whole generalization, because vector fields of the form \( f_{i}(x_{i}) \) are collections of single-variable functions that do not fully capture all vector fields of the form \( f_{i}(x_{1}, x_{2}, \ldots, x_{N}) \) for all \( i \in \{1, 2, \ldots, N \} \). As a specific example, for \( N = 2 \), the vector field with components \( f_{1}(x_{1}, x_{2}) = (x_{1} - x_{2})^{2} \) and \( f_{2}(x_{1}, x_{2}) = (x_{1} + x_{2})^{3} \) cannot be written as just \( f_{1}(x_{1}) \) and \( f_{2}(x_{2}) \), as \( f_{1} \) depends on \( x_{2} \) and \( f_{2} \) depends on \( x_{1} \) as well. Similarly, in the generalization, one could imagine a function of the form \( f = \frac{x(t)}{x(t - t_{0})} \mathrm{exp}(-(t - t_{0})^{2}) \); in this case, it is not correct to write it as \( f(t, x(t)) \) because the dependence of \( f \) on \( x \) at a given dummy index value \( t \) comes through not only \( x(t) \) but also \( x(t - t_{0}) \) for some fixed parameter \( t_{0} \). Additionally, the function may depend not only on \( x \) per se but also on derivatives \( \frac{\mathrm{d}^{n} x}{\mathrm{d}t^{n}} \); the case of the first derivative \( \frac{\mathrm{d}x}{\mathrm{d}t} = \lim_{t_{0} \to 0} \frac{x(t) - x(t - t_{0})}{t_{0}} \) illustrates the connection to the aforementioned example. Therefore, the most generic way to write such a function is effectively as a functional \( f[x; t] \) with a dummy index \( t \). The example \( f = \frac{x(t)}{x(t - t_{0})} \mathrm{exp}(-(t - t_{0})^{2}) \) can be formalized as \( f[t, x] = \int_{-\infty}^{\infty} \frac{x(t')}{x(t' - t_{0})} \mathrm{exp}(-(t' - t_{0})^{2}) \delta(t - t')~\mathrm{d}t' \) where the dummy index \( t' \) is the integration variable while the dummy index \( t \) is free. (For \( N = 3 \), the condition of a vector field being conservative is often written as \( \nabla \times \mathbf{f}(\mathbf{x}) = 0 \). I have not used that condition in this post because the curl operator does not easily generalize to \( N \neq 3 \).)

If a functional \( f[x; t] \) is conservative, then there exists a functional \( F[x] \) (with no free dummy index) such that \( f \) is the functional derivative \( f[x; t] = \frac{\delta F}{\delta x(t)} \). Comparing the notation between scalar fields and functionals, \( \sum_{i} A_{i} \to \int A(t)~\mathrm{d}t \) and \( \mathrm{d}x_{i} \to \delta x(t) \), in which \( \delta x(t) \) is a small variation in a function \( x \) specifically at the index value \( t \) and nowhere else. This suggests a generalization of the fundamental theorem of calculus to functionals as follows. If \( a(t) \) and \( b(t) \) are fixed functions, then \( \int_{-\infty}^{\infty} \int f[x; t]~\delta x(t)~\mathrm{d}t = F[b] - F[a] \). More precisely, a path from the function \( a(t) \) to the function \( b(t) \) at every index value \( t \) can be parameterized by \( s \in [0, 1] \) by the map \( s \to x(t, s) \) which is a function of \( t \) for each \( s \) such that \( x(t, 0) = a(t) \) and \( x(t, 1) = b(t) \); this is why I linked this post to the most recent post on this blog. With this in mind, the fundamental theorem of calculus becomes \[ \int_{-\infty}^{\infty} \int_{0}^{1} f[x(s); t] \frac{\partial x}{\partial s}~\mathrm{d}s~\mathrm{d}t = F[b] - F[a] \] where, in the integrand, the argument \( x \) in \( f \) has the parameter \( s \) explicit but the dummy index \( t \) implicit; the point is that this equality holds regardless of the specific parameterization \( x(t, s) \) as long as \( x \) at the endpoints of \( s \) satisfies \( x(t, 0) = a(t) \) and \( x(t, 1) = b(t) \). This also means that \( F[x] \) can be recovered if \( b(t) = x(t) \) and \( a(t) \) is chosen such that \( F[a] = 0 \), in which case \[ F[x] = \int_{-\infty}^{\infty} \int_{0}^{1} f[x'(s); t]~\frac{\partial x'}{\partial s}~\mathrm{d}s~\mathrm{d}t \] (where \( x(t, s) \) has been renamed to \( x'(t, s) \) to avoid confusion with \( x(t) \)). If \( f[x; t] \) is not conservative, then there is no functional \( F[x] \) whose functional derivative with respect to \( x(t) \) would yield \( f[x; t] \); in that case, with \( x(t, 0) = a(t) \) and \( x(t, 1) = b(t) \), the integral \( \int_{-\infty}^{\infty} \int_{0}^{1} f[x(s); t] \frac{\partial x}{\partial s}~\mathrm{d}s~\mathrm{d}t \) does depend on the specific choice of parameterization \( x(t, s) \) with respect to \( s \) and not just on the functions \( a(t) \) and \( b(t) \) at the endpoints of \( s \).

As an example, consider from a previous post [LINK] the nonrelativistic Newtonian action \[ S[x] = \int_{-\infty}^{\infty} \left(\frac{m}{2} \left(\frac{\mathrm{d}x}{\mathrm{d}t}\right)^{2} + F_{0} x(t) \right)~\mathrm{d}t \] for a particle under the influence of a uniform force \( F_{0} \) (which may vanish). The first functional derivative is \[ f[x; t] = \frac{\delta S}{\delta x(t)} = F_{0} - m\frac{\mathrm{d}^{2} x}{\mathrm{d}t^{2}} \] and its vanishing would yield the usual equation of motion. The action itself vanishes for \( x(t) = 0 \), which will be helpful when using the fundamental theorem of calculus to recover the action from the equation of motion. In particular, one can parameterize \( x'(t, s) = sx(t) \) such that \( x'(t, 0) = 0 \) and \( x'(t, 1) = x(t) \). This gives the integral \( \int_{0}^{1} \left(F_{0} - ms\frac{\mathrm{d}^{2} x}{\mathrm{d}t^{2}}\right)x(t)~\mathrm{d}s = F_{0} x(t) - \frac{m}{2} x(t) \frac{\mathrm{d}^{2} x}{\mathrm{d}t^{2}} \). This is then integrated over all \( t \), so the first term is identical to the corresponding term in the definition of \( S[x] \), and the second term becomes the same as the corresponding term in the definition of \( S[x] \) after integrating over \( t \) by parts and setting the boundary conditions that \( x(t) \to 0 \) for \( |t| \to \infty \). (Other boundary conditions may require more care.) In any case, the parameterization \( x'(t, s) = sx(t) \) is not the only choice that could fulfill the boundary conditions; the salient point is that any parameterization fulfilling the boundary conditions would yield the correct action \( S[x] \).

I considered that example because I wondered whether any special formulas need to be considered if \( f[x; t] \) depends explicitly on first or second derivatives of \( x(t) \), as might be the case in nonrelativistic Newtonian mechanics. That example shows that no special formulas are needed because even if the Lagrangian explicitly depends on the velocity \( \frac{\mathrm{d}x}{\mathrm{d}t} \), the action \( S \) only explicitly depends as a functional on \( x(t) \), so proper application of functional differentiation and regular integration by parts will ensure proper accounting of each piece.

This post has been about the fundamental theorem of calculus saying that the 1-dimensional integral of a function in \( N \) dimensions along a contour, if that function is conservative, is equal to the difference between the two endpoints of its scalar antiderivative. This generalizes easily to infinite dimensions and continuous functions instead of finite-dimensional vectors. There is another fundamental theorem of calculus saying that the \( N \)-dimensional integral in a finite volume of the scalar divergence of an \( N \)-dimensional vector function, if that volume has a closed orientable surface, is equal to the \( N - 1 \)-dimensional integral of the inner product of that function with the normal vector (of unit 2-norm) at every point on the surface across the whole surface, meaning \[ \int_{V} \sum_{i = 1}^{N} \frac{\partial f_{i}}{\partial x_{i}}~\mathrm{d}V = \oint_{\partial V} \sum_{i = 1}^{N} f_{i}(x_{1}, x_{2}, \ldots, x_{N}) n_{i}(x_{1}, x_{2}, \ldots, x_{N})~\mathrm{d}S \] where \( \sum_{i = 1}^{N} |n_{i}(x_{1}, x_{2}, \ldots, x_{N})|^{2} = 1 \) for every \( \mathbf{x} \). From a purely formal perspective, this could generalize to something like \( \int_{V} \int_{-\infty}^{\infty} \frac{\delta f[x; t]}{\delta x(t)}~\mathrm{d}t~\mathcal{D}x = \oint_{\partial V} \int_{-\infty}^{\infty} f[x; t]n[x; t]~\mathrm{d}t~\mathcal{D}x \) having generalized \( \frac{\partial}{\partial x_{i}} \to \frac{\delta}{\delta x(t)} \), \( \prod_{i} \mathrm{d}x_{i} \to \mathcal{D}x \), and \( n_{i}(\mathbf{x}) \to n[x; t] \) where \( n[x; t] \) is normalized such that \( \int_{-\infty}^{\infty} |n[x; t]|^{2}~\mathrm{d}t = 1 \) for all \( x(t) \) on the surface. However, this formalism may be hard to further develop because the space has infinite dimensions. Even when working in a countable basis, it might not be possible to characterize an orientable surface enclosing a volume in an infinite-dimensional space; the surface is also infinite-dimensional. While the choice of basis is arbitrary, things become even less intuitive when choosing to work in an uncountable basis.

Mapping Scalars to Functions

In just over a year, I've written three posts for this blog about functionals, specifically about their application to probability theory [LINK], finding their stationary points [LINK], and the use of their stationary points in classical mechanics [LINK]. As a reminder, a functional is an object that maps a space of functions to a space of numbers. This got me thinking about what the reverse, namely an object that maps a space of numbers to a space of functions, looks like. To be clear, this is not the same as an ordinary function which, as an element in a space of functions, maps a space of numbers to a space of numbers.

As I thought about it more, I realized that this is a bit easier to understand and therefore more commonly encountered than a functional. An extremely glib way to describe such an object is a function of multiple variables. However, it may be more enlightening to describe this in further detail to avoid potentially deceptive images that may arise from that glib description.

In the discrete case, the matrix elements \( A_{ij} \) can be described as a map from integers to vectors, in which an integer \( j \) is associated with a vector whose elements indexed by an integer \( i \) are \( A_{ij} \). This is the essential idea behind seeing the columns of the matrix with elements \( A_{ij} \) as a collection of vectors. Formally, this maps \( i \to (j \to A_{ij}) \) where the map \( j \to A_{ij} \) defines a vector indexed by the free variable \( i \).

Similarly, in the continuous case, the function elements \( f(x, y) \) can be described as a map from numbers to functions, in which a number \( y \) is associated with a function whose elements indexed by a number \( x \) are \( f(x, y) \). Formally, this maps \( x \to (y \to f(x, y)) \) where the map \( y \to f(x, y) \) defines a function indexed by the free variable \( x \). These ideas are foundational to the development of more abstract notions of functions, like lambda calculus.