I like Eric Schwitzgebel. He’s one of the wilder idea guys out there in “serious” academic philosophy.

Perhaps the most enduring-to-me idea he’s come up with is that materialism implies nations are conscious, probably. Materialists just don’t think so, he continues, because they’re “morphologically prejudiced against spatially distributed group entities”. The two intuition pumps he offers for those two parameters are rabbits (which are dumb) and swarm intelligences; accepting that they’re conscious should force you to admit that nations are, too.

I don’t have a problem with the claim, but that’s just because I think words are leaky categories. See Eliezer Yudkowsky’s A human’s guide to words. Consciousness seems to me to be “more like ‘life’ than like ‘water’ “, per Luke Muehlhauser’s Open Philanthropy report Consciousness and moral patienthood. Anyway, absurdity heuristics can misfire badly on fundamental concepts, so I don’t have much of an issue ignoring their klaxon warnings.

But I digress. This post is about Eric’s “nations are probably conscious” idea. It’s this imagery in particular that set fire to my imagination:

A planet-sized alien who squints might see the United States as a single diffuse entity consuming bananas and automobiles, wiring up communications systems, touching the moon, and regulating its smoggy exhalations – an entity that can be evaluated for the presence or absence of consciousness.

https://faculty.ucr.edu/~eschwitz/SchwitzPapers/USAconscious-140721.htm

It’s unclear by what materialist standard the U.S. lacks consciousness.

Nations, it would seem, represent and self-represent. They respond (semi-)intelligently and self-protectively, in a coordinated way, to opportunities and threats. They gather, store, and manipulate information. They show skillful attunement to environmental inputs in warring and spying on each other. Their subparts (people and larger subgroups of people) are massively informationally connected and mutually dependent, including in incredibly fancy self-regulating feedback loops.

These are the kinds of capacities and structures that materialists typically regard as the heart of mentality. Nations do all these things via the behavior of their subparts, of course; but on materialist views individual people also do what they do via the behavior of their subparts.

A planet-sized alien who squints might see individual Americans as so many buzzing pieces of a diffuse body consuming bananas and automobiles, invading Iraq, exuding waste.

http://schwitzsplinters.blogspot.com/2011/10/is-united-states-conscious.html

Elaborating:

You might say: The United States is not a biological organism.  It doesn’t have a life cycle.  It doesn’t reproduce.  It’s not biologically integrated and homeostatic.  Therefore, it’s just not the right type of thing to be conscious.

But it’s not clear that nations aren’t biological organisms.  The United States is (after all) composed of cells and organs that share genetic material, to the extent it is composed of people who are composed of cells and organs and who share genetic material.  

The United States also maintains homeostasis.  Farmers grow crops to feed non-farmers, and these nutritional resources are distributed with the help of other people via a network of roads.  Groups of people organized as import companies bring in food from the outside environment.  Medical specialists help maintain the health of their compatriots.  Soldiers defend against potential threats.  Teachers educate future generations.  Home builders, textile manufacturers, telephone companies, mail carriers, rubbish haulers, bankers, police, all contribute to the stable well-being of the organism.  Politicians and bureaucrats work top-down to ensure that certain actions are coordinated, while other types of coordination emerge spontaneously from the bottom up, just as in ordinary animals.  

Viewed telescopically, the United States is a pretty awesome animal. Now some parts of the United States also are individually sophisticated and awesome, but that subtracts nothing from the awesomeness of the U.S. as a whole – no more than we should be less awed by human biology as we discover increasing evidence of our dependence on microscopic symbionts.

Nations also reproduce – not sexually but by fission.  The United States and several other countries are fission products of Great Britain.  In the 1860s, the United States almost fissioned again.  And fissioning nations retain traits of the parent that influence the fitness of future fission products – intergenerationally stable developmental resources, if you will.  As in cellular fission, there’s a process by which subparts align into different sides and then separate physically and functionally.

On Earth, at all levels, from the molecular to the neural to the societal, there’s a vast array of competitive and cooperative pressures; at all levels, there’s a wide range of actual and possible modes of reproduction, direct and indirect; and all levels show manifold forms of symbiosis, parasitism, partial integration, agonism, and antagonism.  There isn’t as radical a difference in kind as people are inclined to think between our favorite level of organization and higher and lower levels.

I am asking you to think of the United States as a planet-sized alien might, that is, to evaluate the behaviors and capacities of the United States as a concrete, spatially distributed entity with people as some or all of its parts, an entity within which individual people play roles somewhat analogous to the role that individual cells play in your body.  If you are willing to jettison contiguism and other morphological prejudices, this is not, I think, an intolerably weird perspective.  As a house for consciousness, a rabbit brain is not clearly more sophisticated.  I leave it open whether we include objects like roads and computers as part of the body of the U.S. or instead as part of its environment.

https://faculty.ucr.edu/~eschwitz/SchwitzPapers/USAconscious-140721.htm

Maybe what’s special about consciousness is brains, in particular “their complex high order / low entropy information processing, and their role in coordinating sophisticated responsiveness to environmental stimuli”? But nations like the United States have this too:

Consider, first, the sheer quantity of information transfer among members of the United States.  The human brain contains about 10¹¹ neurons exchanging information through an average of about 10³ connections per neuron, firing at peak rates of about once every several milliseconds.  The United States, in comparison, contains only about 3 x 10⁸ people.  

But those people exchange a lot of information.  How much?  We might begin by considering how much information flows from one person to another via stimulation of the retina.  The human eye contains about 10⁸ photoreceptor cells.  Most people in the United States spend most of their time in visual environments that are largely created by the actions of people (including their own past selves).  If we count even 1/300 of this visual neuronal stimulation as the relevant sort of person-to-person information exchange, then the quantity of visual connectedness among people is similar to the neuronal connectedness within the human brain (10¹⁴ connections).  Very little of the exchanged information will make it past attentional filters for further processing, but analogous considerations apply to information exchange among neurons.  

Or here’s another way to think about the issue: If at any time 1/300th of the U.S. population is viewing internet video at 1 megabit per second, that’s a transfer rate between people of 10¹² bits per second in this one minor activity alone.^[18]  Furthermore, it seems unlikely that conscious experience requires achieving the degree of informational connectedness of the entire neuronal structure of the human brain.  If mice are conscious, they manage it with under 10⁸ neurons.

https://faculty.ucr.edu/~eschwitz/SchwitzPapers/USAconscious-140721.htm

But maybe the information exchange isn’t the right kind to engender consciousness? Maybe what you need is “some organization of the information in the service of coordinated, goal-directed responsiveness” plus “some sort of sophisticated self-monitoring”? But nations have this too:

Our information exchange is not in the form of a simply-structured massive internet download.  The United States is a goal-directed entity, flexibly self-protecting and self-preserving.  The United States responds, intelligently or semi-intelligently, to opportunities and threats – not less intelligently, I think, than a small mammal.  The United States expanded west as its population grew, developing mines and farmland in traditionally Native American territory.  When Al Qaeda struck New York, the United States responded in a variety of ways, formally and informally, in many branches and levels of government and in the populace as a whole.  Saddam Hussein shook his sword and the United States invaded Iraq.  The U.S. acts in part through its army, and the army’s movements involve perceptual or quasi-perceptual responses to inputs: The army moves around the mountain, doesn’t crash into it.  Similarly, the spy networks of the CIA detected the location of Osama bin Laden, whom the U.S. then killed.  The United States monitors space for asteroids that might threaten Earth. Is there less information, less coordination, less intelligence than in a hamster?  The Pentagon monitors the actions of the Army, and its own actions.  The Census Bureau counts us.  The State Department announces the U.S. position on foreign affairs.  The Congress passes a resolution declaring that we hate tyranny and love apple pie.  This is self-representation.  Isn’t it? 

The United States is also a social entity, communicating with other entities of its type.  It wars against Germany then reconciles then wars again.  It threatens and monitors Iran.  It cooperates with other nations in threatening and monitoring Iran.  As in other linguistic entities, some of its internal states are well known and straightforwardly reportable to others (who just won the Presidential election, the approximate unemployment rate) while others are not (how many foreign spies have infiltrated the CIA, the reason Elvis Presley sells more albums than Ella Fitzgerald).

There’s something awesomely special about brains such that they give rise to consciousness; and considered from a materialist perspective, the United States seems to be awesomely special in just the same sorts of ways.

What is it about brains, as hunks of matter, that makes them special enough to give rise to consciousness?  Looking in broad strokes at the types of things materialists tend to say in answer – things like sophisticated information processing and flexible, goal-directed environmental responsiveness, things like representation, self-representation, multiply-ordered layers of self-monitoring and information-seeking self-regulation, rich functional roles, and a content-giving historical embeddedness – it seems like the United States has all those same features.  In fact, it seems to have them in a greater degree than do some beings, like rabbits, that we ordinarily regard as conscious.

Somehow Unbundling the manager stuck with me out of all the essays I read in the Breaking Smart newsletter, probably because it resonates with my experience dealing with middle managers and wondering — genuinely by the way, not sarcastically — what purpose they actually serve within big organizations. Venkat lays it out pretty well by appealing to the historical evolution of the role.

In the 1950s, the rise of a new kind of manager over the previous half-century, the middle manager, inspired a whole era of management thinking and writing.

The middle manager — somebody who manages other managers, but isn’t the owner/leader or CEO — is the sine qua non of the no-skin-in-the-game industrial age ethos. They do not bear responsibility either for the work getting done, or for the ultimate aims of the organization being attained.

That does not mean, however, that they are useless. The middle manager evolved to occupy a niche: providing a lot of middleware support and coordination required in large-scale organizations. A lot of routine logistics, information dissemination, and large-scale coordination used to depend on it.

That niche, however, began shrinking in the late 70s. Over the last 40 years, a great many middle-management functions have been refactored into other functions, automated, outsourced, or simply done away with. Since that time, a gradual process of unbundling middle-management functions has been underway.

Unaccountable middle layers suck:

Many of the pathologies of hierarchical command-and-control organizations arise not from the structure itself, but from the existence of unaccountable middle layers.

Unlike the bottom layers with direct responsibility for getting work done, middle layers are not responsible for immediate consequences. Unlike the top layers, with accountability to stakeholders and overall organizational performance, middle layers are not responsible for ultimate consequences.

This is a structural position rife with moral hazard. There is a principal-agent problem/no-skin-in-game issue in relation to both the top and bottom of the pyramid.

So what happened again to make the role of the middle manager shrink? Why the Great Unbundling? The first quote above summarizes it, but the details are juicy. Venkat claims there are four broad phases each lasting about 10 years, each concerned with a “core unbundling principle”: autonomy, efficiency, innovation, and empathy.

Autonomy:

You can read about the autonomy phase (1975-85)  in books like Tom Peters’ Thriving on Chaos. Much of what he wrote about the era of “flattening” organizations is conventional wisdom today, but back then the core idea seemed really radical: trust the individual. The result of that early era was simply fewer managers, in fewer layers, and most importantly, more autonomous individual contributors, at least at skilled levels.

Efficiency:

After autonomy, the focus shifted to efficiency (1985-95). This led to the global spread of Japanese ideas, with much of the impact being on the factory floor. The impact on managers — a focus on process discipline, monitoring and feedback, is captured in Andy Grove’s classic High Output Management (1983).

Autonomy+efficiency in an environment of deregulation and computerization meant “bad management” was solved in a specific way: elimination, automation and outsourcing. Many middle-managerial functions such as signing off on routine paperwork or disseminating information, were simply computerized or networked away. Other functions were downcycled: it was recognized that many functions weren’t really “management” work, and outsourced to lower-cost non-manager workers, often in other countries.

Matrix management, value-chain optimization, and process re-engineering marked the later stages of this era: broadly, the era of process discipline. Process discipline helped cleave line management from project/program management, shrink the former (absorbed into “staff” functions in many cases), and frame the latter in “core competency” terms.

The rise of innovation:

In the early 90s, line management functions shrank through unbundling, abandonment, automation and outsourcing, while project/program management functions grew in importance. Managers began to be held accountable for end-to-end value addition rather than just the internal silo-functioning metrics, setting the stage for the “innovation” era.

By 1995 the stage was set for the “innovation” phase. Fueled by books such as The Innovator’s Dilemma (1997), the surviving middle managers turned into risk managers. The timing was perfect, since the rise of the Internet shifted the entire economy into an innovate-or-die/software-eating-the-world gear. Middle managers became like VCs or angel investors, stewarding agendas defined by “porftolios” of “strategic” projects with varied risk profiles and “ROI” expectations.

The autonomy to efficiency to innovation phase shifts changed middle managers, and employees, for the worse:

This 30 year/three phase drive towards increasing individual autonomy, efficiency, and innovation focus utterly transformed the staid middle manager of the 1970s. Before they were paternalistic types, taking it easy in low-stress robotic functions, insulated from harsh incentives, growing little siloed fiefdoms within safe “job descriptions”. After, they were anxious, ambitious, hungry and stressed risk managers, shepherding a dozen “value-adding strategic initiatives” through brutally Darwinian internal ecosystems.

Autonomy meant that what used to be “rank insubordination” was now “effectiveness.” In an environment of competing internal projects, loyalties became  uncertain. Efficiency meant that scrutiny of costs and performance turned brutal (if not always particularly effective). Except on Wall Street, work stopped being a bacchanalian party. The focus on innovation meant the static peacetime construct of “job description” transformed into constant maneuvering against shifting internal/external competition.

Funnily enough, I can see all this in fast-forward. I currently work at a multinational firm with O(10^4) employees which used to be the only player in its space. When I joined less than 3 years ago, the firm was somewhere between ground zero and autonomy. There’s still a ton of bureaucratic layers in place, but lots of managers who’ve worked long enough with their subordinates give the latter lots of autonomy to do the job; that’s what my own manager did to me too. This laid-back approach no longer suffices as new competitors enter our space, eating into our profit margins, in fact wiping it all out. So senior management, forced by investors to “go back into the green”, did a ton of corporate restructuring, fired the old fat cats and installed new lean cats on short-term contracts with barely realistic KPIs. This was how our new head of department entered the picture last year, phase-shifting the whole department into efficiency/innovation mode by tying everyone’s KPIs directly to the bottom line. The result so far has been chaos. We’ll see.

This is tremendously exciting! Always wanted to see something like this, and Wret is smarter than I am, so I’m glad she wrote the Reddit post Visualizing mathematical subjects (No idea re: Wret’s gender so just defaulting to ‘she/her’.)

The idea was to “make a diagram with all the math-related subjects and add some colors”. Quoting Wret:

I downloaded all the metadata of articles that where published on arXiv.org in the year 2018, with at least one subject inside of mathematics. From these I created a graph where every vertex is a subject, connecting them by an edge if there is a paper published in both of the subjects at the same time. The thickness of the edges corresponds to how often this happens.

The position of the vertices is obtained via the Fruchterman-Reingold algorithm, with some minor manual tinkering to make everything look a little bit nicer. In this first picture we use Label Propagation to obtain two big clusters (corresponding to the different colours). Perhaps they show the Algebra vs Analysis divide?

In this second picture we use Edge-Betweenness clustering to get some more detail. We still have some sort of Algebra/Analysis clusters, but a third green cluster shows up in the middle. I like to think of this as the Geometry cluster, even though Algebraic/Differential Geometry do not strictly fall into this cluster they are very close.

We also see that Statistics and Computer Science are not really mathematics as they form their own cluster. (I apologise to my statistician friends.)

Apparently nobody else seems all that impressed in the comments section though…

Peter Sarnak is one of the leading analytic number theorists of his generation. He is the Eugene Higgins Professor of Mathematics at Princeton, succeeding Andrew Wiles. He’s also a member of the permanent faculty of the School of Mathematics at the Institute for Advanced Study. He sits on the Board of Adjudicators and the selection committee for the Shaw Prize. He’s a member of the National Academy of Sciences, an AMS Fellow, and a Fellow of the Royal Society. He’s won the Polya Prize, the Ostrowski Prize, the Levi Conant Prize, the Frank Nelson Cole Prize in Number Theory, and the Wolf Prize, the highest award in math prior to the Abel, honoring a lifetime of work. A very high proportion of his doctoral students are leaders in their fields, including Fields Medalist Akshay Venkatesh, Kannan Soundararajan, Jacob Tsimerman, and Harald Helfgott.

I just had to pay my respects to a giant. Let’s move on.

From Volume 1 of the 2006 ICM Proceedings, where Peter was one of the five panelists for the closing round table talk Are pure and applied mathematics drifting apart?:

Just on a sociological level let me tell you, based on my experience, how to tell the difference between a theoretical physicist and a pure mathematician (I am not sure where the applied mathematician fits here).

A mathematician will come into you office and tell you how complicated what he is doing is “My proof is highly nontrivial it is a thousand pages long”. It is a strange discipline where to convince someone of something you have to write or make use of thousands of pages of complex arguments. Probably it means that one hasn’t yet really understood the issue at hand.

A Physicist comes into your office and he is always trying to tell you how simple and short what he is doing is and moreover it is universal and explains everything. He is lying because he is hiding 50 of more pages of calculations that he declares are trivial.

This difference in presentation explains some of the difference in culture between these disciplines. The truth is somewhere in between.

The idea that for something to be good it must be long and complicated is something that has evolved in certain quarters of mathematics, and it seems strange and wrong to me. In the end, we are always looking for the simple thing, and the real truth is somewhere in between these extreme views.

Anyway the panelists’ opinions were fine and dandy. I’d like to highlight a remark by an audience member, David Levermore, who claimed to be both pure and applied. He talks about a model that’s rapidly gaining ground in the US to resist academic balkanization, that of interdisciplinary research centers focused on a single idea or application. Of course I find this devilishly attractive, because it’s surface-level single ideas that I get interested in as a child, which just so happen to cross fields. It’s nice that they “join forces” again just like old times:

I think Martin raised a very important point – institutionally what can we do?

I think the issue is not so much we drift apart, I think that criticism is valid because we do have control of this. I think the phenomenon has to do with the expansion of human knowledge and endeavour, and that all disciplines to some measure are confronted with this, in particular universities, but also all institutions, not just academic”.

One model that does exist in the US, and is thriving in some institutions, is the development of centres – centres that focus around maybe an application or an idea, that brings together people, a mathematical paradigm or an engineering paradigm, or whatever, to work together, learn from each other and stimulate each other.

Just, for example, the mathematics department at Maryland is tied to a Norbert Wiener centre in applied harmonic analysis, which involves pure mathematicians and engineers. We have a list of several institutes like that.

And I think that if we put our minds to it, we can overcome these sort of intellectual barriers that separate us artificially, because ultimately I think the picture the whole panel has painted is that this is a human endeavour and is really the right one, and I look forward to a very good future

I didn’t even know “bursts” could be mathematically modeled in a way nontrivial enough to capture the attention of top mathematicians. Of course that’s because I’m stupid. I may still be stupid, but I am no longer ignorant, thanks to John Hopcroft’s laudatio on Jon Kleinberg, the recipient of the 2006 Nevanlinna Prize, perhaps the most comprehensive and calmly competent laudatio I have seen so far:

In order to understand a stream of information, one may organize it by topic, time, or some other parameter. In many data streams a topic suddenly appears with high frequency and then dies out. The burst of activity provides a structure that can be used to identify information in the data stream.

Jon’s work [7] on bursts developed the mathematics to organize a data stream by bursts of activity. If one is watching a news stream and the word Katrina suddenly appears, even if one does not understand English, one recognizes that an event has taken place somewhere in the world.

The question is how do you automatically detect the sudden increase in frequency of a word and distinguish the increase from a statistical fluctuation? Jon developed a model in which bursts can be efficiently detected in a statistically meaningful manner.

This still sounds like BS claims. Jon is quick to dispel skepticism by small-minded fools like myself:

Jon applied the methodology to several data streams demonstrating that his methodology could robustly and efficiently identify bursts and thereby provide a technique to organize the underlying content of the data streams. The data streams consisted of his own email, the papers that appeared in the professional conferences, FOCS and STOC, and finally the U.S. State of the Union Addresses from 1790 to 2002.

The burst analysis of Jon’s email indicated bursts in traffic when conference or proposal deadlines neared.

The burst analysis of words in papers in the FOCS and STOC conferences demonstrated that the technique finds words that suddenly increased in frequency rather then finding words of high frequency over time. Most of the words indicate the emergence or sudden increase in the importance of a technical area, although some of the bursts correspond to word usage, such as the word “how” which appeared in a number of titles in the 1982 to 1988 period.

The burst analysis of the U.S. State of the Union Addresses covered a 200 year time period from 1790 to 2002 and considered each word. Adjusting the scale parameter s produced short bursts of 5–10 years or longer bursts covering several decades. The bursts corresponded to national events up through 1970 at which time the frequency of bursts increased dramatically.

This work on bursts demonstrated that one could use the temporal structure of data streams, such as email, click streams, or search engine queries, to organize the material as well as its content. Organizing data streams around the bursts which occur, provides us with another tool for organizing material in the information age.

I’ve always wondered what The Other Guys did in their laudatios for the 2006 Fields. I’ve seen Charlie Fefferman on Terry Tao; what about the rest?

(Tangent: Charlie and Terry, I will never tire of repeating this, are like brothers. Not only were they both advised by Elias Stein at Princeton for graduate school, they were both Fields Medalists, in fact two of the top-five youngest ever, both truly world-class child prodigies with near-perfect accelerated learning trajectories who got their PhDs before age 21 and became the youngest full professors at a major American research institution and later got labeled “the best mathematician in the world” by the media at some point in their careers. A closer parallel does not exist. Yes, that’s a dare; prove me wrong.)

Today I realized I had been stupid. The URL for Charlie on Terry goes like so: http://www.icm2006.org/proceedings/Vol_I/7.pdf. Obviously you could try removing everything after ‘org’! So I did, and that was how I found the old ICM page, clearly built for a browser/internet experience half a generation removed from the present, and all the works of the Medalists.

Here’s Andrei Okounkov by Giovanni Felder, who first came to my attention because he was singled out by Ron Maimon in one of his answers on Quora. He’s a mathematical physicist of the highest order – well perhaps a notch or two below the Barry Simons of the world, but certainly the most promising under-40 representative of his subfield in 2006:

Andrei Okounkov’s initial area of research is group representation theory, with particular emphasis on combinatorial and asymptotic aspects. He used this subject as a starting point to obtain spectacular results in many different areas of mathematics and mathematical physics, from complex and real algebraic geometry to statistical mechanics, dynamical systems, probability theory and topological string theory.

The research of Okounkov has its roots in very basic notions such as partitions, which form a recurrent theme in his work. Partitions are a basic combinatorial notion at the heart of the representation theory. Okounkov started his career in this field in Moscow where he worked with G. Olshanski, through whom he came in contact with A. Vershik and his school in St. Petersburg, in particular S. Kerov.

The research programme of these mathematicians, to which Okounkov made substantial contributions, has at its core the idea that partitions and other notions of representation theory should be considered as random objects with respect to natural probability measures. This idea was further developed by Okounkov, who showed that, together with insights from geometry and ideas of high energy physics, it can be applied to the most diverse areas of mathematics.

Concluding remarks on Andrei:

Andrei Okounkov is a highly creative mathematician with both an exceptional breadth and a sense of unity of mathematics, allowing him to use and develop, with perfect ease, techniques and ideas from all branches of mathematics to reach his research objectives. His results not only settle important questions and open new avenues of research in several fields of mathematics, but they have the distinctive feature of mathematics of the very best quality: they give simple complete answers to important natural questions, they reveal hidden structures and new connections between mathematical objects and they involve new ideas and techniques with wide applicability.

Moreover, in addition to obtaining several results of this quality representing significant progress in different fields, Okounkov is able to create the ground, made of visions, intuitive ideas and techniques, where new mathematics appears. A striking example for this concerns the relation to physics: many important developments in mathematics of the last few decades have been inspired by high energy physics, whose intuition is based on notions often inaccessible to mathematics. Okounkov’s way of proceeding is to develop a mathematical intuition alternative to the intuition of high energy physics, allowing him and his collaborators to go beyond the mere verification of predictions of physicists. Thus, for example, in approaching the topological vertex of string theory, instead of stacks of D-branes and low energy effective actions we find mathematically more familiar notions such as localization and asymptotics of probability measures. As a consequence, the scope of Okounkov’s research programme goes beyond the context suggested by physics: for example the Maulik–Nekrasov– Okounkov–Pandharipande conjecture is formulated (and proved in many cases) in a setting which is much more general than the Calabi–Yau case arising in string theory.

Here’s Wendelin Werner, by Courant Institute director Charlie Newman, noting that this was the first time a probabilist had won the Fields (turns out they’re even more slighted than logicians huh?):

It is my great pleasure to briefly report on some of Wendelin Werner’s research accomplishments that have led to his being awarded a Fields Medal at this International Congress of Mathematicians of 2006. There are a number of aspects of Werner’s work that add to my pleasure in this event. One is that he was trained as a probabilist, receiving his Ph.D. in 1993 under the supervision of Jean-François Le Gall in Paris with a dissertation concerning planar Brownian Motion – which, as we shall see, plays a major role in his later work as well. Until now, Probability Theory had not been represented among Fields Medals and so I am enormously pleased to be here to witness a change in that history.

I myself was originally trained, not in Probability Theory, but in Mathematical Physics. Werner’s work, together with his collaborators such as Greg Lawler, Oded Schramm and Stas Smirnov, involves applications of Probablity and Conformal Mapping Theory to fundamental issues in Statistical Physics, as we shall discuss. A second source of pleasure is my belief that this, together with other work of recent years, represents a watershed in the interaction between Mathematics and Physics generally. Namely, mathematicians such as Werner are not only providing rigorous proofs of already existing claims in the Physics literature, but beyond that are providing quite new conceptual understanding of basic phenomena – in this case, a direct geometric picture of the intrinsically random structure of physical systems at their critical points (at least in two dimensions). One simple but important example is percolation – see Figure 1.

Permit me a somewhat more personal remark as director of the Courant Institute for the past four years. We have a scientific viewpoint, as did our predecessor institute in Göttingen – namely, that an important goal should be the elimination of artificial distinctions between the Mathematical Sciences and their applications in other Sciences – I believe Wendelin Werner’s work brilliantly lives up to that philosophy.

Yet a third source of pleasure concerns the collaborative nature of much ofWerner’s work. Beautiful and productive mathematics can be the result of many different personal workstyles. But the highly interactive style, of which Werner, together with Lawler, Schramm and his other collaborators, is a leading exemplar, appeals to many of us as simultaneously good for the soul while leading to work stronger than the sum of its parts. It is a promising sign to see Fields Medals awarded for this style of work.

Here’s the star-who-didn’t-turn-up of the 2006 ICM show, the sensational Grigori Perelman, by John Lott. Lott’s exposition is pretty dry – I’m getting used to the personality-filled writings of the Gowers and Baezes and Mazurs and Vakils of the world, it seems:

Grigory Perelman has been awarded the Fields Medal for his contributions to geometry and his revolutionary insights into the analytical and geometric structure of the Ricci flow.

Perelman was born in 1966 and received his doctorate from St. Petersburg State University. He quickly became renowned for his work in Riemannian geometry and Alexandrov geometry, the latter being a form of Riemannian geometry for metric spaces. Some of Perelman’s results in Alexandrov geometry are summarized in his 1994 ICM talk [20]. We state one of his results in Riemannian geometry. In a short and striking article, Perelman proved the so-called Soul Conjecture.

In the 1990s, Perelman shifted the focus of his research to the Ricci flow and its applications to the geometrization of three-dimensional manifolds. In three preprints [21], [22], [23] posted on the arXiv in 2002–2003, Perelman presented proofs of the Poincaré conjecture and the geometrization conjecture.

Perelman’s papers have been scrutinized in various seminars around the world. At the time of this writing, the work is still being examined.

Grigory Perelman has revolutionized the fields of geometry and topology. His work on Ricci flow is a spectacular achievement in geometric analysis. Perelman’s papers show profound originality and enormous technical skill. We will certainly be exploring Perelman’s ideas for many years to come.

And here, finally, is Charlie on Terry. Charlie’s writing has character. When I read him I can’t help but like him. He’s like Ravi Vakil in that regard, and like Scott Alexander from back in the day. His style is deceptively simple, in fact to a degree I’ve almost never seen. Maybe it’s the short sentences and simple words.

Mathematics at the highest level has several flavors. On seeing it, one might say:

(A) What amazing technical power!
(B) What a grand synthesis!
(C) How could anyone not have seen this before?
(D) Where on earth did this come from?

The work of Terence Tao encompasses all of the above. One cannot hope to capture its extraordinary range in a few pages. My goal here is simply to exhibit a few contributions by Tao and his collaborators, sufficient to produce all the reactions (A)… (D). I shall discuss the Kakeya problem, nonlinear Schrödinger equations and arithmetic progressions of primes.

I have repeatedly used the phrase “tour-de-force”; I promise that I am not exaggerating.

There are additional first-rate achievements by Tao that I have not mentioned at all. For instance, he has set forth a program [22] for proving the global existence and regularity of wave maps, by using the heat flow for harmonic maps. This has an excellent chance to work, and it may well have important applications in general relativity. I should also mention Tao’s joint work with Knutson [19] on the saturation conjecture in representation theory. It is most unusual for an analyst to solve an outstanding problem in algebra.

Tao seems to be getting stronger year by year. It is hard to imagine what can top the work he has already done, but we await Tao’s future contributions with eager anticipation.

I’ve always been interested in polymath mathematicians. Unfortunately, as math balkanizes (like with any other field) the “unifiers” get fewer and further between. Hilbert and Poincare are often agreed to be among the last ones. But just as you see the upper limits of talent-related technical power trend upward over the years in sports, so should there be Hilbert- and Poincare-equivalents today. But what do they look like? One way to find these modern-era universalist types is to go through nontechnical prize laudatios.

That’s how I found Akshay Venkatesh, who won the Fields Medal last year. Here’s how Allyn Jackson starts off describing his work:

The study of numbers has always been at the heart of mathematics. While an enormous body of knowledge has accumulated over the centuries, number theory still retains endless mysteries that grow out of the simplest concepts: Relations among the integers. Because the integers comprise the bedrock out of which all of mathematics grows, number theory has connections to many other branches of the field. Number theorists draw on ideas from analysis, algebra, combinatorics, and geometry, as well as from other fields like theoretical physics or computer science.

Even in a subject requiring such breadth, Akshay Venkatesh stands out for the startlingly original way he has connected number theory problems to deep results in other areas. Far from using them as “black boxes” to crank out solutions, Venkatesh brings fresh insights to the results and highlights their unexpected connections to number theory. In this way he has made striking advances in number theory while also greatly enriching other branches of mathematics.

Because the oeuvre of Akshay Venkatesh is so diverse, a complete overview is not possible in a short space. What follows therefore is a description of three examples of the work of Venkatesh and his co-authors that exemplify the depth and wide-ranging nature of his work.

Unfortunately WordPress doesn’t allow me to copy-paste LaTeX, so I’ll skip the substance and stick to the surface-level praise:

The final example of the work of Venkatesh is not a finished result, but rather a set of bold new conjectures he and his co-authors have formulated. Seeking to explain profound connections between phenomena in topology on the one hand and number theory on the other, these conjectures show Venkatesh as a trailblazer of new directions in research. The ideas spawned much excitement in seminars he led during the 2017-2018 academic year.

This conjectural work relates to the Langlands Program, which today drives a great deal of mathematical research. The Langlands Program envisions a web of relationships among a variety of phenomena arising in different branches of mathematics, including topology, analysis, algebra, and number theory. Mathematicians are a long way from fulfilling the whole of the Langlands Program, but some special cases have been affirmed. Perhaps the best known example is the proof in the 1990s of Fermat’s Last Theorem, carried out by Andrew Wiles with crucial input from Richard Taylor.

The Taylor-Wiles method that emerged from that work has become a powerful tool for connecting geometric objects known as elliptic curves to analytic objects known as modular forms—these are exactly the kind of connections predicted by the Langlands Program. As powerful as it is, the Taylor-Wiles method, as originally developed, applied only in a restricted setting, namely that of special geometric objects called Shimura varieties. Recent results generalizing the Taylor-Wiles method to non-Shimura varieties figure prominently in the newest work of Venkatesh.

This work centers on a set of topological objects known as locally symmetric spaces. Venkatesh and his co-authors have found that the topology of these spaces harbor unexpected symmetries. These symmetries occur in the homology groups of locally symmetric spaces; the homology group of a space can be loosely thought of as measuring the holes in the space. Venkatesh has formulated a vision for explaining these symmetries by appealing to a different mathematical area known as motivic cohomology. The explanation uses the generalized Taylor-Wiles method and, as a byproduct, might yield deeper insights into that method. The work is far from complete, but early expectations are that it will provide a key step in the ascent towards a full understanding of the Langlands Program.

Allyn ends her laudatio of Akshay like so:

Most mathematicians are either problem-solvers or theory-builders. Akshay Venkatesh is both. What is more, he is a number theorist who has developed an unusually deep understanding of several areas that are very different from number theory. This breadth of knowledge allows him to situate number theory problems in new contexts that provide just the right setting to highlight the true nature of the problems. Only 36 years of age, Venkatesh will continue to be an outstanding leader in mathematics for years to come.

Akshay is so far the only person I’ve seen described anywhere as being a standout at both problem-solving and theory-building.

From Quanta:

Most mathematicians struggle to describe the full range of Venkatesh’s diverse mathematical contributions, which build bridges from number theory to distant fields such as algebraic topology and dynamical systems. He is known for moving into an area of mathematics, transforming it, and then moving on.

“People always say there is no universal mathematician nowadays because it is too difficult,” said Emmanuel Kowalski, a mathematician at the Swiss Federal Institute of Technology Zurich. But, he said, Venkatesh’s mind “can think about anything.”

His ideas are “a vast expansion of the imagination,” said Michael Harris, a mathematician at Columbia University.

[As a graduate student] at Princeton, Venkatesh’s brilliance quickly became apparent to everyone except him. Once, when his housemate, Tony Wirth, showed Venkatesh his theoretical computer science textbook, Venkatesh instantly grasped the core concepts, Wirth recalled. “You’re dealing with a great mind when he can kind of plant himself in your area for a few minutes and really get to the bottom of it.”

And when Venkatesh first visited his adviser, Peter Sarnak, to ask for reading material, Sarnak decided to try him on a book about representation theory of semisimple groups — far from typical fare for first-year graduate students. Sarnak expected Venkatesh to return soon and demand something simpler. But a month later, Sarnak recalled, “he came back to my office explaining to me what he had read, and it was clear to me that not only was he understanding it, but he had a tremendous intuition about what it was all about.”

Venkatesh “always had remarkable insights,” said Sarnak, who is now at IAS. “It was clear that it was just a matter of time before he was going to put things together in unusual ways.”

How Akshay is so broad:

Inwardly, though, Venkatesh was struggling. “I don’t feel like I had a happy time as a graduate student,” he said. He was disturbed by the fact that Sarnak had suggested one of the key ideas in his dissertation, even though that’s not uncommon in graduate studies. “I didn’t really feel like… I had added something very original,” Venkatesh said. “I don’t think I got the sense out of graduate school that I can do research.”

When he felt discouraged, Venkatesh would dive into some mathematics textbook, often on a topic far removed from his research. “I think that’s what has always kept me going, even when my own research hasn’t gone anywhere — I’ll read something and I’ll think, ‘Well, this is really wonderful,’” he said.

Venkatesh absorbed voluminous amounts of mathematics during his graduate years. It wasn’t intended as a strategic move, but it gave him the big-picture view that now allows him to discover connections other mathematicians have missed.

I used to fantasize about a super-grad student who’d read all the yellow books — all of them. Nobody in the entire world is capable of doing this, but (logarithmically) Ed Witten is reputed to have come close, and now there’s Akshay too.

Akshay is an “essences distiller”:

Emerging from graduate school in 2002 without having yet achieved this self-awareness, Venkatesh was eager to prove himself. A collaboration with Ellenberg soon gave him the opportunity. The pair took on the task of counting “number field extensions,” the simplest of which are built by adding to the rational numbers a handful of irrational numbers that satisfy some polynomial equation. These number systems “have a little bit of irrationality, but not too much,” said Ellenberg, who is now a professor at the University of Wisconsin, Madison.

There are infinitely many number field extensions, but there’s a natural way to measure how complex each one is, and the question then becomes to count how many there are below each complexity level. Venkatesh and Ellenberg figured out a new upper bound on these counts that greatly improved on the previous state of the art, which hadn’t budged for decades.

The project, Venkatesh said, was “psychologically very important” to him. “We got into this problem ourselves, and we really made progress on it.”

The pair originally tackled the problem using some new mathematical tools they’d heard about at a conference. But as they worked to improve the exposition of their proof, they eventually found that they had simplified away all the sophisticated new tools, Ellenberg said.
“It speaks to what I now recognize as being very characteristic of Akshay,” Ellenberg said. Their first, more technical version of the paper did prove their theorem. But for Venkatesh, Ellenberg said, it’s “not so much about whether you answered the question or whether you proved the theorem. It’s about: Did you understand what’s actually going on?”

Many mathematicians besides Ellenberg are struck by Venkatesh’s ability to distill ideas down to their essence. “When he’s finished, often his proof is the proof that would be in any textbook from here on,” Sarnak said.

“There have been lots of times in mathematics when someone has explained to me the right way to think about a certain piece of mathematics,” said Frank Calegari, of the University of Chicago. “But Akshay is one of the very few people who has done that with my own work.”

High praise. More “essence”, in a groundbreaking work this time:

These generalized Riemann hypotheses are also extremely hard to prove, so for decades, mathematicians have looked for ways to sidestep the hypotheses and prove some of their many consequences directly. One of the most important of these consequences is something called subconvexity, which says, roughly speaking, that the positive and negative numbers in the sequence of numerators of an L-function quickly start balancing each other out. Subconvexity estimates for L-functions, when they can be proved, yield statistical information about patterns in whole numbers — for example, one subconvexity estimate gives a description of the variety of ways any given large number can be written as the sum of three perfect squares.

Before Venkatesh turned his focus on L-functions, subconvexity estimates were commonly done on a case-by-case basis, often involving long papers full of technicalities, Kowalski said. But in 2004, Venkatesh sent the draft of a long paper to Philippe Michel, a mathematician now at the Swiss Federal Institute of Technology Lausanne who had studied subconvexity in depth. In the paper, Venkatesh used ideas from dynamical systems — the study of systems that change over time — to solve the subconvexity problem in much greater generality than had previously been accomplished. “The method was completely new,” Michel said. “It was a big shock to me.”

Venkatesh and Michel teamed up on a second paper that used this new approach to find subconvexity estimates for a huge family of L-functions. The pair of papers, along with several others Venkatesh wrote around that time, Sarnak said, “made him already one of the leading people in the world” in number theory and dynamics.

Collaborating with Venkatesh should perhaps feel intimidating, given the breadth of his knowledge and the depth of his insights — but somehow, it doesn’t. “He’s very strong technically, but talking to him doesn’t have a technical feel at all,” said Yiannis Sakellaridis, of Rutgers University, Newark. “You just talk about the essence of things.”

But to return to the “essence distiller” comment — what do I mean? The “distill” part comes from Research Debt by Chris Olah, one of the most personally meaningful papers I’ve read. The “essence” part comes from David Mumford, in Math, beauty, and brain areas:

I think one can make a case for dividing mathematicians into several tribes depending on what most strongly drives them into their esoteric world. I like to call these tribes explorers, alchemists, wrestlers and detectives. Of course, many mathematicians move between tribes and some results are not cleanly part the property of one tribe.

Explorers are people who ask — are there objects with such and such properties and if so, how many? They feel they are discovering what lies in some distant mathematical continent and, by dint of pure thought, shining a light and reporting back what lies out there.

The most beautiful things for them are the wholly new objects that they discover (the phrase ‘bright shiny objects’ has been in vogue recently) and these are especially sought by a sub-tribe that I call Gem Collectors. …

Alchemists, on the other hand, are those whose greatest excitement comes from finding connections between two areas of math that no one had previously seen as having anything to do with each other. This is like pouring the contents of one flask into another and — something amazing occurs, like an explosion! …

In Mumford’s jargon, Akshay would be an alchemist and gem collector.

For every world-class child prodigy who “made it” (think Terry Tao’s near-perfect transition, or Wolfram’s life success) there are many who don’t (Sidis, Langan). And then there are “those in the middle”, who never achieve fame despite being tremendously technically successful, simply because they “went for the wrong thing” (w.r.t to fame-optimization). I’m thinking, in particular, of Harvey Friedman.

Mathematical logic hasn’t really been fashionable since Godel. There has only been a single logician who’s won the Fields, the devastatingly sharp Paul Cohen; there hasn’t been a single logician who’s won the Abel Prize, and it took perhaps the greatest logician of the second half of last century in Saharon Shelah to win the Wolf Prize. (I am glad that Joel David Hamkins is leading an online renaissance of sorts of this field on MO through his second-to-none voluminous expository writings. But outside of JDH and MO there really isn’t much.) But this is Friedman’s obsession, and so here we are.

(A digression: I came to know of Friedman via googology, that amateur pursuit of very large finite numbers and compact elegant ways to express them. Friedman is the greatest big-game hunter in the whole world. Everyone else, and I do mean everyone else, sticks to generating large numbers from constructions explicitly designed for that purpose. Friedman wanders into the vastness of the foundations of math, far beyond anyone else, and in the course of his meanderings occasionally comes across behemoths. His “hit rate” in this regard is probably greater than everyone else combined.)

Harvey Friedman was world-class precocious. He was also a foundations-type thinker from childhood. To quote this article about him in Nautilus:

When he was just starting to read, at age 4 or 5, Friedman remembers pointing to a dictionary and asking his mother what it was. It’s used to find out what words mean, she explained.

A few days later, he returned to her with his verdict: The volume was completely worthless. For every word he’d looked up, the dictionary had taken him in circles: from “large” to “big” to “great” and so on, until he eventually arrived back at “large” again.

“She just looked at me as if I were a really strange, peculiar child,” Friedman laughs.

That was Friedman’s first brush with foundational thinking. It would continue cropping up in innocuous places: Shortly after his introduction to the dictionary, for example, he noticed that changing the order of the items listed on his parents’ grocery bill didn’t affect the total price they ended up paying. He didn’t yet know the name for this property, but it struck a chord.

More:

It didn’t take long for his parents, who both worked in the photo typesetting business and never graduated from college, to recognize his aptitude for math. His father, Friedman says, initially had hopes of his becoming an engineer—but more than happily encouraged his budding interest in mathematics. One day, Friedman’s father came home with a ninth-grade algebra textbook he’d asked to borrow from the son of a family friend who lived down the street. Go ahead and learn this, he told his own son. Friedman quickly devoured the material. He was 9 years old.

Friedman found himself on the fast track to foundational pursuits. He skipped two grades, attended college-run summer programs for gifted students, and absorbed everything he could get his hands on—ultimately leading him to Russell’s introductory text. Decades later, Friedman still remembers its final sentences. “As the above hasty survey must have made evident,” Russell had written, “there are innumerable unsolved problems in the subject, and much work needs to be done. If any student is led into a serious study of mathematical logic by this little book, it will have served the chief purpose for which it has been written.”

Bertrand Russell, it turns out, was writing for Harvey Friedman.

The book certainly served its purpose with Friedman, who eventually decided that he had to solve those “innumerable unsolved problems.” At age 16, he passed over college altogether to enter graduate school at the Massachusetts Institute of Technology, where he immediately sought out mathematician Hilary Putnam.

The following semester, he took one of Putnam’s classes, and by his third and final year, he had formulated an agenda of sorts. He would start by working on the foundations of mathematics, he told himself, and then, after spending a few years on that, would move on to other disciplines: the foundations of mechanics, of statistics, of law, of music. The foundations of everything.

It’s hard to overstate how romantic that last passage is.

Friedman had so much mental horsepower he skipped high school, undergrad, and master’s; the only professional degree he holds is a PhD in logic from MIT, earned when he was only 18 years old. Among all technical PhD earners from HYPSMC-level institutions, only Norbert Wiener was younger; he was 17 when he got his PhD from Harvard, also in mathematical logic. (Something about logic eh? Perhaps more so than nearly any other field in math – except perhaps discrete math – raw intellectual horsepower, of the type well-measured by IQ tests, can make up for lack of experience.)

Here’s an interesting anecdote about Friedman’s PhD years I found in an oral history of the MIT math department, by Ted Martin, talking about Normal Levinson (head of the math department):

He was a great mathematical analyst, was Norman Levinson. But was he pushing all the time to hire analysts instead of topologists or algebraists? No! He would have argued against that. “Get the best people you can get.” Norman had great practical horse sense: just astonishing. He was highly respected all around the institution. People all around him in MIT thought the world and all of Norman Levinson. Everybody knew who he was, and people talked about him with great respect, people from the engineering and science schools, for example, or the engineers and scientists who were in the upper administration. How did people from other departments know who he was? Through interactions, I suppose.

I remember the year we were about to give a PhD to Harvey Friedman, a young kid who was a logician. He had come to MIT as a freshman. He was bored with the freshman curriculum but very smart, so part-way through his freshman year the math department made him a graduate student. He had been a graduate student in the department for a few years, and we were about to give him a PhD. He already had serious faculty offers from Stanford and Harvard and other places, but there was a big ruckus from the graduate school when it was time for the awarding of degrees, because he hadn’t completed the language requirement or something for the PhD. To the old, conservative engineering faculty this was an atrocity of the first magnitude. I mean, you can’t do that—give a degree to somebody who didn’t meet the requirements!

So they tied things up in the graduate school policy committee for weeks, debating it. It was Norman who convinced them to proceed without the requirements, and he did it with one single line. He said, “I don’t think the quality and reputation of MIT will be determined by the people to whom it does not give degrees.” And that ended the discussion!

He taught at Stanford as an assistant professor of philosophy at age 18, making him the youngest professor in the world at a major research institution.

Friedman’s IAS page notes the dazzling breadth of his oeuvre:

He has published extensively in topics such as model theory, proof theory and intuitionism, recursion theory, set theory, and computer science. Friedman’s additional research interests include, among others, the philosophy of mathematics, software verification, interactive educational technology, computational complexity, piano recording technology, and piano performance theory and practice.

This dazzling variety was already apparent when he was younger, and perhaps more widely appreciated, as is evident in this summary-cum-laudatio of his work by age 35 for which he won the Alan Waterman Award for “the most outstanding American scientist under thirty-five years of age in all fields of science and engineering”:

Friedman’s contributions span all branches of mathematical logic (recursion theory, proof theory, model theory, set theory, and theory of computation). He is a generalist in an age of specialization, yet his theorems often require extraordinary technical virtuosity, of which only a few selected highlights are discussed.

Friedman’s ideas have yielded radically new kinds of independence results. The kinds of statements that were proved to be independent before Friedman were mostly disguised properties of formal systems (such as Gödel’s theorem on unprovability of consistency) or assertions about abstract sets (such as the continuum hypothesis or Souslin’s hypothesis).

In contrast, Friedman’s independence results are about questions of a more concrete nature involving, for example, Borel functions or the Hilbert cube.

(You can also see, even then, how he was trying his damnedest to make logic relevant again.)

So you absolutely can’t say that he’s been a failure. And yet there’s this fallaciously-binary, but all-pervasive, knee-jerk notion that if you didn’t manage to convert the fame of your world-class precocity into the fame of adult achievement, you’ve “failed”. Friedman commands tremendous respect from those who know his work, but those voices aren’t in the majority; you know you’re in trouble when even your admirer (a colleague of Friedman’s) calls you “a solo voice in the wilderness”. And this has entirely to do with how mathematical logic is no longer fashionable these days, and nothing to do with Friedman’s intellectual output, which I can’t help but think is a bit sad.

Julie Moronuki has a very special mind. I’ve always loved her essay The unreasonable effectiveness of metaphor. I am going to rephrase wholesale from it as part of my usual “active reading” style; if you want to know what she’s talking about you’ll be much better served by reading her essay instead.

Sometimes we’ll look out at the world and try to “cleave it at the joints” somehow. Sometimes there are “natural” ways to do such cleaving; for instance, we can take in noisy irregular real-world input and somehow intuit a property called “triangularity”, despite there not being a single thing in the real world that’s actually triangular. This is well-motivated by asking of a group of things: “which one doesn’t belong?” If you choose your examples well, it becomes possible to argue that any member doesn’t belong, which turns the exercise from the boring get-the-right-answer to the interesting let’s-do-mathematical-argumentation.

In some contexts, the properties that matter for “excluding outsiders” is obvious: color, size, shape. But sometimes it isn’t so obvious. Depending on context, sometimes we are called to see each member in terms of how it’s different from the rest, and sometimes we are compelled to see how they’re all the same. Julie gives the example of the mappend operator in Haskell; it’s instructive to come up with your own.

This is the point of abstraction: finding the ways things are the same and ignoring the ways in which they differ. It lets us make these things “law-abiding”, hence letting us “concretize new layers” that we can build upon (“generalize”) further.

This process, crucially, relies on analogy and metaphor – which are not quite the same thing. Analogy is the similarity between two things, usually thought of as finding the essence they share, and is the core of cognition; metaphor is the linguistic expression of this similarity, but more fundamentally it’s when we structure our understanding of one thing in terms of another.

When the former is something we can’t directly experience, this structuring is called conceptual metaphor. Some structuring is universal: all cultures talk about time in spatial terms, e.g. “bad times are behind us”. Some isn’t, like the “orientation of time”: the future “lies ahead” for us, but “lies behind” for some cultures.

Lakoff operationalizes the defintion of conceptual metaphor as “grounded inference-preserving cross-domain mapping”. “Grounded” means there is a “ground truth” or “source frame” we understand better and are mapping from. “Inference-preserving” means inferences that hold true “on the ground” are also true “at the target”.

(to be continued)

A great quote from Barry Mazur’s essay Finding meaning in error terms, written in memory of Serge Lang, on why there are still unsolved problems in number theory:

Eratosthenes, to take an example—and other ancient Greek mathematicians—might have imagined that all they needed were a few powerful insights and then everything about numbers would be as plain, say, as facts about triangles in the setting of Euclid’s Elements of Geometry. If Eratosthenes had felt this, and if he now—transported by some time machine—dropped in to visit us, I’m sure he would be quite surprised to see what has developed.

To be sure, geometry has evolved splendidly but has expanded to higher realms and more profound structures. Nevertheless, there is hardly a question that Euclid could pose with his vocabulary about triangles that we can’t answer today. And, in stark contrast, many of the basic naive queries that Euclid or his contemporaries might have had about primes, perfect numbers, and the like, would still be open.

Sometimes, but not that often, in number theory, we get a complete answer to a question we have posed, an answer that finishes the problem off. Often something else happens: we manage to find a fine, simple, good approximation to the data, or phenomena, that interests us—perhaps after some major effort—-and then we discover that yet deeper questions lie hidden in the error term, i.e., in the measure of how badly our approximation misses its mark.

…

In a general context, once we make what we hope to be a good approximation to some numerical data, we can focus our attention to the error term that has thereby been created, namely:

Error term = Exact Value – Our “good approximation”.”

Barry gives the canonical number-theoretic example – the prime-counting function:

A telling example of this, and of how in the error term lies richness, is the manner in which we study of π(X) := the number of prime numbers less than X. The function π(X) is shown below, in various ranges as step functions giving the “staircase” of numbers of primes.

As is well known, Carl Friedrich Gauss, two centuries ago, computed tables of π(X) by hand, for X up to the millions, and offered us a probabilistic “first” guess for a nice smooth approximating curve for this data; a certain beautiful curve that, experimentally, seems to be an exceptionally good fit for the staircase of primes.

The data, as we clearly see, certainly cries out to us to guess a good approximation. If you make believe that the chances that a number N is a prime is inversely proportional to the number of digits of N you might well hit upon Gauss’s guess, which produces indeed a very good fit. In a letter written in 1849 Gauss claimed that as early as 1792 or 1793 he had already observed that the density of prime numbers over intervals of numbers of a given rough magnitude X seemed to average 1/log X.

The Riemann Hypothesis is equivalent to saying that the integral ∫ X^2 dx/log x (i.e., the area under the graph of the function 1/ log x from 2 to X) is essentially square root close to π(X). …

Barry continues in this vein, but unfortunately the WordPress editor isn’t rich enough to natively support LaTeX symbols so I can’t just copypaste from his essay.