When Einstein Walked with Gödel: Excursions to the Edge of Thought

by Jim Holt

From Jim Holt, the New York Times bestselling author of Why Does the World Exist? , comes an entertaining and accessible guide to the most profound scientific and mathematical ideas of recent centuries in When Einstein Walked with Gödel : Excursions to the Edge of Thought .

Does time exist? What is infinity? Why do mirrors reverse left and right but not up and down? In this scintillating collection, Holt explores the human mind, the cosmos, and the thinkers who’ve tried to encompass the latter with the former. With his trademark clarity and humor, Holt probes the mysteries of quantum mechanics, the quest for the foundations of mathematics, and the nature of logic and truth.

Along the way, he offers intimate biographical sketches of celebrated and neglected thinkers, from the physicist Emmy Noether to the computing pioneer Alan Turing and the discoverer of fractals, Benoit Mandelbrot. Holt offers a painless and playful introduction to many of our most beautiful but least understood ideas, from Einsteinian relativity to string theory, and also invites us to consider why the greatest logician of the twentieth century believed the U.S. Constitution contained a terrible contradiction―and whether the universe truly has a future.

Genres: Science, Nonfiction, Philosophy, Physics, History, Mathematics, Essays

384 pages, Paperback

First published May 15, 2018

About the author

Jim Holt is a longtime contributor to the New Yorker -- where he has written on string theory, time, infinity, numbers, truth, and bullshit, among other subjects -- and the author of Stop Me If You've Heard This: A History and Philosophy of Jokes. He is also a frequent contributor to the New York Times and the London Review of Books. He lives in Greenwich Village.

Reviews

[Michael · Rating 5 out of 5]

A delightful set of 24 standard essays and 14 brief ones (a page or so) on the history of physics and math and current views on their relations. The essays highlight issues with Einstein’s theory of relativity (special and general), quantum mechanics, group theory, infinity and the infinitesimal, Turing’s theory of computability, Goedel’s incompleteness theorems, prime numbers and the Riemann zeta conjecture, category theory, topology, fractals, and the theory of truth. There are no equations, so we have to count on him to convey major discoveries and conflicting interpretations in words alone without distorting the truth. From a modest experience with the subject (a college course on quantum mechanics long ago and reads in recent years of popularizations by real physicists), I feel comfortable with Holt’s knowledge and competence as a science journalist. He achieved his goal of covering a lot of subjects on various themes:
My ideal is the cocktail-party chat: getting across a profound idea in brisk and amusing way to an interested friend by stripping it down to its essence …The goal is to enlighten the newcomer while providing a novel twist that will please the expert. And never to bore.

Successfully conveying the human aspects for all these figures is another skill that helped spur me onward. As a tiny example, the Nobel Prize winning physicist John Wheeler wondered is anyone made a theoretical connection between Heisenberg’s uncertainty principle and Goedel’s incompleteness theorem, so he knocked on Goedel’s door at Princeton’s Institute for Advanced Study. Goedel, huddled over a space heater, promptly threw him out.

Often there is a tragic element in the lives of the figures. Many of us are aware of Alan Turing’s sad state after WW2 due to the 2014 film, “The Imitation Game”, with his contributions to code-breaking kept secret, his persecution by “chemical castration” for being gay, and eventual suicide by poison apple. Holt tries to rectify the film’s inaccurate portrayal of him a “as a humorless and timorous nerd.” Here are a few more examples of nutters among the brilliant:

The creator of the theory of infinity, Georg Cantor, was a kabbalistic mystic who died in an insane asylum. Ada Lovelace, the cult goddess of cyber feminism … was plagued by nervous crises brought on by her obsession with atoning for the incestuous excesses of her father, Lord Byron. …Kurt Goedel, the greatest of all modern logicians, starved himself to death out of the paranoic belief that there was a universal conspiracy to poison him.

Holt takes a strange excursion with novelist David Foster Wallace’s 300-page exposition (“Everything and More”) on Cantor’s 19th century accomplishment in using set theory to prove some infinities and bigger than others, in effect creating a whole tower of infinities. Holt finds he succeeded in his goal of improving on “certain recent pop books that give such shallow and reductive accounts of Cantor’s proofs …the math is distorted and its beauty obscured” and “if he was sometimes in over his head, it is because he chose to wade through the deepest waters”. In contrast to Wallace’s admiration, Wittgenstein found that:

There is nothing awesome about the theory; it does not describe a world of timeless, transcendent, scarcely conceivable entities; it is really no more than a collection (finite) tricks of reasoning. One might imagine, Wittgenstein said, the theory of infinite sets was “created by a satirist as a kind of parody of mathematics”. …As a description of Cantor’s work on infinity, it is surely unjust. As a description of Wallace’s, it might be taken as a tribute.

Basically, Holt trusts the average man to appreciate major problems in physics. So many mathematicians wax poetic about the connection between beauty of their theorems and equations and truth. A new kind of Platonism is alive and well among many of the wizards, i.e. a belief that mathematicians are not inventing their formulas but are discovering timeless truths beyond space and time. However, for Holt:

The problem with this Platonist view of mathematics … is that it makes mathematical knowledge a miracle. If the objects of mathematics exist apart from us, living in a Platonic heaven that transcends the physical world of space and time, then how does the human mind “get in touch” with them and learn about their properties and relations? Do mathematicians have ESP? The trouble with Platonism, as the philosopher Hilary Putnam has observed, “is that it seems flatly incompatible with the simple fact that we think with our brains, and not with immaterial souls.”.

Not all mathematical proofs are elegant. In the case of the “Four Color Problem”, the proof that any map of geometrical elements can be colored with four colors without any adjacent regions getting a common color took over 700 pages of arguments and a zillion computer calculations of all the permutations. Conversely, beautiful elegant equations can lead one astray. In the case of string theory, we have a brilliant mathematical system based on vibrating membranes in nine dimensions which aims to unify the major laws of physics. It has captured the imagination and life efforts of many theoretical physicists for three decades. Because of the inaccessible six extra dimensions, experimental tests of the theory have not been possible. Moreover:
In a space of more than three dimensions, there would be no stable planetary orbits. (This was proved over a century ago by Paul Ehrenfest). Nor would there be stable orbits for electrons within atoms. Therefore, there could be no chemistry, and hence no chemically-based life forms, in a world of more than three spatial dimensions. …
So it should not come as a surprise that we find ourselves living in a three-dimensional world. (Physicists call this “anthropic” reasoning). …And we can surely sympathize with the aspiration of Mr. Square [a 2-D character in “Flatland”]—not to mention assorted Theosophists, Platonists, and cubists—to rise up into the splendor of the fourth dimension and beyond. But we need not follow them. For intellectual richness and aesthetic variety, a world of three dimensions is world enough.

I also learned some more about Einstein’s long struggle to pin down what was wrong with the standard model of quantum mechanics. Instead, of simply objecting to it on the basis of core uncertainties in reality (“God does not play dice with the universe”), he made a significant contribution in revealing it to violate the fundamental principle of locality, which posits “that the world consists of separately existing physical objects and that these objects can directly affect one another only if they come into contact” or “through causal intermediaries that bridge the distance between them.” His collaborative thought experiment known as EPR in 1927 proposed an electron in a box which then gets partitioned and the two halves moved far apart; according to the standard theory, opening one box to detect the electron, collapsing the probability wave, is predicted to instantaneously to cause the other box to be empty, a “spooky action at a distance” he took as evidence of the theory’s insufficiency. Unfortunately for Einstein’s stubborn resistence, John Stewart Bell much later designed a variant of EPR involving paired photons which allowed experimental testing. So called quantum entanglement between particles was indeed proven to involve “spooky action at a distance” according to at least three replication Thus, we must accommodate a new conception of space and the possibilities of some kind of “holistic” principle.

One chapter that was especially entertaining has Holt running around and asking notable physicists how the universe and humanity will end. Recent changes in cosmology have shaken people up. For a long time the Big Bang and movement of galaxies away from each other was predicted to be slowed by gravitational attraction, perhaps leading to a Big Crunch and another cycle. I empathize with little Alvy in Woodie Allen’s “Annie Hall”, who was much distressed with an expanding universe, despite reassurances of his psychiatrist and mother that things are safe for zillions of years and Brooklyn shows no signs of expanding. Now with unknown “dark energy” causing everything to accelerate away from each other, we are back to the idea of a “heat death”, where all matter is scattered and degraded to a virtual nothing (unless dark energy runs out of steam). Most physicists don’t really worry how all will end, while others project we will overcome the problems such as adapting ourselves to energy forms in a dust cloud or by building quantum tunneling devices to move us to another universe. In my case, I appreciate Holt’s use of the anthropic principle and a Copernician assumption of humans as not so special and likely in the Bell curve of the normal distribution to predict a duration of humankind as at most a few millions of years, in line with other mammalian species.

All in all a fun read and I think accessible to most general readers.

[Barbara K · Rating 4 out of 5]

My curiosity about theoretical physics, mathematics, and the people who study these things was piqued by reading Benjamin Labatut's When We Cease to Understand the World and The MANIAC. Actually, I suppose it actually started with the title of the first book, which kind of set off a "What?" in my brain that has stayed there since.

It's worth noting that my basic understanding of those topics is underdeveloped. My formal math and science education was limited to the minimum required by my schools, and although just like everyone else I read A Brief History of Time back in 1988, I understood very little of it. So why this current interest?

I'm going to say it's because Labatut is such a fine author. His books pull together ideas and people and present them in a stimulating way. His writing crosses a porous line between fact and fiction/interpretation in a way that assigns value, for me, to both the people and the ideas.

But this review isn't about Labatut's books, it's about this collection of essays by Jim Holt. Holt covers many of the same topics (not surprising, given that Einstein and Gödel appear in the title), but from a solely fact-based perspective. There are about 20 pieces of long form journalism (Holt has written extensively for The New Yorker), and a similar number of short ones. The long pieces are, by and large, first rate, but the short pieces are unimpressive.

Although Holt starts off with physics and mathematics, he gradually slides into philosophy. That was fine with me, although it might have been an unpleasant surprise for someone picking up the book who is interested only in science. [One of my fondest undergraduate memories is of my first semester course in philosophy. I felt I was stretching my brain for the first time ever, and I was resentful of all the “crap I learned in high school” (to quote Paul Simon’s Kodachrome). Why hadn’t anyone ever challenged me to think back then?]

Back to the book again. Because this is a collection of essays written over a number of years, there isn’t exactly a through-line, although the same people and ideas do crop up repeatedly. That actually worked for me, because the focus would be slightly different each time the people/ideas appeared, and collectively the essays helped increase my general comprehension of issues and of the timeline of developments, particularly in the 20th century.

So, why not 5 stars? Partly because of the inclusion of those short pieces, but more because Holt seems to be a bit misogynistic. Few women merit a mention, which is partly justifiable because they haven’t been a large presence in theoretical physics, mathematics, or philosophy. But there is a viciousness to his takedown of Ada Byron Lovelace that seems completely unwarranted. Based on the information he presents, Lovelace was not the godmother of computer programming, just a wannabe who yearned for legitimate fame to offset her father’s rather tarnished image. Fair enough - but there is a venom in his attack that is out of proportion to his comments on any man, even the ones he doesn’t care for. It’s as if it wasn’t enough that he put a stake through her heart, he had to burn the corpse as well. Sheesh!

That said, overall I really enjoyed these essays. I always feel enriched learning about theoretical physics, mathematics, philosophy, and cosmological eschatology (which is just fun to say).

[Matthew Harris · Rating 3 out of 5]

I don't normally write a comment regarding the star rating I've attributed to a book, but in this case I feel I should qualify my rating.

For the most part the content of this book is of great quality. Had it not been for the tedious set of shorter essays towards the end and the condescending tone I feel the author occasionally exudes this would've been a 5 star rating.

But its not 4 stars either... and that is due to the utterly unfortunate choice Jim Holt made to focus almost entirely on the "excursions to the edge of thought" that occur in a mans brain. Seriously, how in the fucking world can you overlook the rest of the population and their contribution to the wealth of human knowledge?

Of the core essays from the book (focusing on quite a broad range of subjects from the arts, technology, philosophy and science) just a single one has at its core a woman. And that essay is basically a tear down of Ada Lovelace and her contribution to computing. I'm not debating the veracity of the arguments Holt presents (although, I did detect a bit more of a bite in his writing here compared to elsewhere), I just think its a shame that rather than try to redress the balance a bit, he instead picks just a single female and seeks (solely) to undermine her achievements.

Interestingly, I didn't think I'd heard of Jim Holt prior to this book. But, while researching him, I found I had in fact come across him before: https://www.independent.co.uk/life-st...

Quite a nasty pattern, eh?

[Miglė · Rating 4 out of 5]

Tai rinkinys esė apie matematiką, analitinę filosofiją, fiziką ir šių sričių mokslininkus - jų gyvenimus, nesutarimus, klaidas ir atradimus. Autorius yra žurnalistas, taigi moka pristatyti idėjas, bet nelenda labai giliai. T.y. pristato kaip koks nors dalykas veikia, kartais su kokiu pavyzdžiu, bet neaiškina jo veikimo "mechanizmo", kad ir:

Such a peer group of mathematical objects is called, in a deliberate nod to Aristotle and Kant, a category. One category may consist of abstract surfaces. These surfaces play together, in the sense that there are natural ways of going back and forth between them that respect their general form. For example, if two surfaces have the same number of holes - like a donut and a coffee mug - one surface can, mathematically, be smoothly transformed into the other

Kai kur paduoda tokio keisto konteksto, kurio niekaip nebūčiau dagalvojusi, o ir išsamesnėse pop-science knygose dažniausiai nerasi. Pavyzdžiui, autorius mano, kad, skirtingai negu tarp Prancūzijos matematikų, tarp Rusijos matematikų labai gerai prigijo beveik metafiziškai skambanti Cantor'o (Cantor'iaus?) begalinių aibių teorija, nes Rusijoje tuomet gyvavo "Vardų garbintoų" religinė bendruomenė (kaip galima suprasti, dvasinei patirčiai išgauti karotojanti dievo vardą), iš kurios pasekėjų trys buvo garsūs matematikai. Caro laikais juos persekiojo dėl erezijos, o sovietų laikais - dėl metafizinių idėjų matematikoje, niekur nelaimėdavo, vargšeliai.

Asmenybės - apie jas skaityti man paprastai mažiau įdomu negu apie atradimus, bet čia kažkaip tikrai suteikia įdomios atmosferos. Faina, kad, nors ir stengdamasis laikytis distancijos, autorius nenueina ant to, kur "na, bet mes vis tiek negalime žinoti, kokie jie buvo, tas genialumas turi ir tamsiąją pusę...", o daugiau mažiau pagrįstai dėsto, kad štai šitas buvo genialus išradėjas, bet panašu, kad blogas žmogus (von Neumann), šita - savo genialumu įsitikinusi žvaigždė, kuriai priskiriamas jos visai nepadarytas išradimas (Ada Lovelace). Jau fb rašiau, bet man tikrai patiko, kad apie tas "istorijos moteris" galima pasakoti ir kitas istorijas, ne tik vieną tą pačią.

Buvau pamiršusi, kaip pop-sci tekstai apie fiziką man grąžina norą gyventi. Nežinau, čia turbūt asmeninis dalykas, bet tikrai, širdis dainuoja. Jei kas turit rekomendacijų pop-sci knygų apie fiziką, sakykit, vdrug būsiu neskaičius!

Kita vertus, tų esė yra labai daug ir įvairių, ir kai kurie neišvengiamai išeina nuobodesni. Kai kur nueinama į labai konkretų dalyką - pvz labai ilgai užsitesęs pasakojimas apie skandalą dėl tam tikro koncepto autorystės, kur veikė Kripke ir Marcus. Kartais - priešingai, autorius nueina į tokius vidutiniškus svarstymus apie gyvenimą ir mirtį.

Vienas buvo labai juokingas - "How will the Universe End?", kur keldamas šį klausimą (ir kitą - ar besibaigiančioje Visatoje galėtų išlikti sąmonė ir kokia forma?) autorius eina pas įvairius fizikus ir reportuoja jų atsakymus. Kiek panašu, kad jis čia nusprendė pasišaipyti iš fizikų, nes labai juokingai juos aprašinėja, o atsakymai verti science-fiction knygų.

"The most plausible answer," Dyson said, "is that conscious life will take the form of interstellar dust clouds." <...> "An ever-expanding network of charged dust particles, communicating by electromagnetic forces, has all the complexity necessary of thinking an infinite number of novel thoughts."
But, I objected, can we really imagine such a wispy thing <...> being conscious?
"Well," he said, "how do you imagine a couple of kilograms of protoplasm in someone's skull being conscious?

Gerai, ir paskutinis:

Some decades ago, the Princeton physicist John Archibald Wheeler began to wonder whether Heisenberg's uncertainty principle might not have some deep connection to Gödel's incompleteness theorem <...>. Both, after all, seem to place inherent limits on what is possible to know. But such speculation can be dangerous. "Well, one day," Wheeler recounts, "I was at the Institute for Advanced Study, and I went to Gödel's office, and there was Gödel. It was winter and Gödel had an electric heater and had his legs wrapped in a blanket. I said, 'Professor Gödel, what connection do you see between your incompleteness theorem and Heisenberg's uncertainty principle?' And Gödel got angry and threw me out of his office."

[Cheryl · Rating 4 out of 5]

Contents:

Part 1: the Moving Image of Eternity
Time---the Grand Illusion?

Part 2: Numbers in the Brain, in Platonic Heaven, and in Society
Sir Francis Galton, the Father of Statistics---and Eugenics

Part 3: Mathematics, Pure and Impure
A Mathematical Romance

Part 4: Higher Dimensions, Abstract Maps
Geometrical Creatures

Part 5: Infinity, Large and Small
Georg Cantor v. David Foster Wallace

Part 6: Heroism, Tragedy, and the Computer Age
The Ada Perplex: Was Byron's Daughter the First Coder?
Alan Turing in Life, Logic, and Death

Part 7: The Cosmos Reconsidered
The String Theory Wars: Is Beauty Truth?

Part 8: Quick Studies, A Selection of Shorter Essays

Part 9: God, Sainthood, Truth, and Bullshit
Dawkins and the Deity
On Moral Sainthood
Say Anything

"These essays were written over the last two decades. I selected them for their depth, power, and sheer beauty of the ideas they convey." Copyright 2018

[Thomas · Rating 4 out of 5]

Calling this one done a bit before the end. Honestly, the latter essays get a bit tedious. There's a lot to love about this book and a few things that kind of grated at my nerves. But overall, some of the better chapters were incredible and really made me think and want to do more research. Despite some potentially serious flaws, I'm giving this one four stars for getting me excited enough to start digging out more of my math books again.

[Arcturus · Rating 5 out of 5]

Tudtátok, hogy élete utolsó 22 évét Einstein a Princetonon töltötte, és minden reggel ráérős sétát tett a házától az egyetemig? És hogy az utolsó 12 évben egy nála fiatalabb, ismeretlenebb, elegánsabb, de a maga nemében később szintén nagyot alkotó Kurt Gödel volt a séta- és beszélgetőpartnere? Mert ez utóbbi pl. újdonság volt számomra. Mennyire belehallgattam volna egy-két ilyen beszélgetésbe! Einstein nem ismeretlen, Gödel pedig minden idők egyik legnagyobb logikusa volt, a nevéhez fűződik a nemteljességi tétel, a teljességi tétel, több húrelmélettel kapcsolatos tanulmány, és számtalan tudományfilozófiai esszé és tanulmány. Az ő történetük és Gödel szomorú sorsa csak egy a sok nagy gondolkodó, elmélkedő, tudós elme története között, Jim Holt ebben a kötetben 24 hosszabb esszét és 14 rövidebb, pár oldalas kis történetet mesél el nekünk olyan csemegékről, mint a húrelmélet, a Heisenberg-féle határozatlansági elv, az igazságelmélet, a hanggörbe, a prímszámok, relativitás, kódfejtés és sok minden más.

De nem kell aggódni, nem tör tudományos magaslatokba, egy laikus számára is teljesen érthető, sőt mi több, élvezetes és érdekes kötetet tett le az asztalra, amelyet nálunk a Typotex csapata gondozott. 

Ez a kötet elsősorban ugyanis azokról az élő, hús-vér emberekről szól, akik a fenti halhatatlan elméleteket kidolgozták. Fájóan sokan vannak köztük, akiknek sorsa tragikus volt. Gödel pl. paranoiás volt, csak a feleségében és Einsteinben bízott meg, és csak a feleségétől fogadott el ételt. Amikor az asszony kórházba került 1978-ban, Gödel nem volt hajlandó enni, és végül a koplalás következményeibe halt bele. Alan Turing története talán kevésbé ismeretlen: a zseniális matematikus és számítógép-tudós, aki fontos munkát végzett az Enigma feltörésével, homoszexuális volt. Ez pedig akkoriban büntetendő volt, kényszer-hormonkezeléseken kellett részt vennie, bebörtönözték, élete nagy részében üldöztetésnek volt kitéve, dacára a mai informatikát is megalapozó tudásának és munkájának. 1954-ben, mindössze 42 évesen, valószínűleg ciánmérgezésbe halt bele, nem tudjuk, hogy gyilkosság, öngyilkosság vagy baleset történt-e. Dmitrij Jegorov és Pavel Florenszkij, akiket - egymástól függetlenül és egész más aspektusból - a végtelen elmélete izgatott, Sztálin vaskeze alatt haltak meg. Ada Lovelace is ismert gyenge idegzetéről és persze zsenialitásáról is, de ő is korán, mindössze 37 évesen, méhnyakrák következtében távozott eme árnyékvilágból. 

Megrendítő és érdekes volt olvasni ennyi kiváló, nagy tudású és nagy hatású elme tragikus sorsáról. Sokszor elfelejtjük, hogy ezek mögött az utálatos vagy épp csodálatos elméletek, képletek és szinte érthetetlen tudományos magasságokban leledző gondolatok kidolgozói mind hús-vér emberek voltak számtalan gondolattal, hibával, összetűzésekkel másokkal, sok-sok érzéssel, családdal, vagy sokszor magányosan, elfeledve, üldöztetve. Lenyűgöző, éppen nekem való kötet volt ez, benne minden olyannal, ami érdekel: a különféle csodálatos elméletek és az emberek, akik ezek mögött vannak. Néha azonban - főleg a rövidebb esszéknél - befejezetlennek, kifejtetlennek éreztem a történetet, sokszor éreztem úgy, hogy Holt elhallgatja a saját véleményét egy-egy esetről, témáról, de valahol mégis kiviláglik a szövegből. A polcom része marad ez a kötet, az biztos, és többször is elő fogom venni, hogy egy-egy emberről vagy elméletről elolvassam még egyszer Holt gondolatait. 

[Marks54 · Rating 4 out of 5]

This is a new book of essays/columns/short pieces by an accomplished writer about philosophy. The core essays make it clear that Holt is focusing on the philosophy of science, although the collection is rounded out to include several short reviews, as well as work on the philosophy of language and naming and even the philosophy of BS.

I am never quite sure what to think of writings like this. To start with, the philosophy of science is not science. If you have any doubts, go read one of Einstein’s classic papers and see what it is like. A related point is that these essays are not really serious philosophy either. Philosophy is difficult and frequently very focused. It is also fun to work through - but it does need to be worked through. Go read some of Hilary Putnam’s essays to see.

So what are these essays? They are entertaining and readable. They also help to clarify some ideas that many have heard of but few know in detail. Einstein and quantum physics for sure, including String Theory. The coverage of Godel’s work and even Heisenberg and his uncertainty principle do a great service for general readers and point out numerous works to follow-up with if one is more interested and wishes to read more. I realized a long time ago that labor markets generally work and that big time philosophy was not in my future. But I have also loved reading philosophy and reading about philosophers and their work. For most people, a guide will be needed. Jim Holt does a fine job in this capacity. The essays are also well written and funny.

Philosophers are also academics and some of the essays touch on the dynamics of academic life featuring significant philosophy. I wish Holt had included more of this. The place of women in the occupation has not been without controversy in recent years and I found the ending piece on Kripke especially interesting. There have been other examples especially in the #metoo era, but the piece Holt includes is exceptional in showing how discussions about ideas and their lineage fit into broader academic dynamics.

[Cindy · Rating 5 out of 5]

People all over the internet want recommendations for books that will blow their minds. They're usually asking for fiction, but I think nonfiction is the way to go, and this book is the place to start. Every essay will blow your mind a little bit.

I honestly only bought this thing because it was on sale for $2, and I'm so glad I did. I feel like the cover and title did it a disservice. People who pick it up to read about Einstein and Godel will be disappointed as they're discussed mainly in only one of the essays. The cover doesn't even really give any indication that it's an essay collection. Strange marketing, especially since I think only a specific group of people would enjoy reading this. I would recommend it only to people who enjoy a good dose of philosophy with their math/physics, and don't mind a dash of biography. The biography sections are really juicy and scandalous, basically front page of People magazine material, so you may love or hate that. I sense that very strict math/science people would find the musings and the questions asked in this book kind of pointless.
I see a couple of critical reviews. They for the most part say the essays are too shallow. I think the essays basically convey the intended ideas. They don't and can't go into the mathematics of it all, simply because you can fill textbooks with the math required to explain each topic. And the thing with nonfiction reviews is that a lot of people who review nonfiction read a lot of nonfiction. This means that an essay about something like Godel's Incompleteness Theorem will seem shallow to someone who has read Godel's Proof by Ernest Nagel, which is an entire book dedicated to the topic.

View this book as a large sample platter of the most interesting ideas known to man. If you want to delve deeper into any of the ideas, feel free to pick up the corresponding text. This book actually has a Further Reading section, which makes it easy. That's what's cool about reading nonfiction, it's a choose your own adventure activity. If you liked reading about the map problem, turn to "Four Colors Suffice: How the Map Problem Was Solved" by Robin Wilson, but if you preferred the topic of prime numbers, turn to "The Music of the Primes" by Marcus du Sautoy.

[Sarah · Rating 4 out of 5]

Ugh, this book. Yeah, I'm giving it 4 stars, because it's just the kind of science writing I like. However, and this is a big however, Holt really pissed me off in a couple essays and I can't let him off for that by not leaving a review that qualifies my 4 stars. I think he has a streak of misogyny in him, and that comes across several times, most obviously in his anti-Ada Lovelace essay. You can just hear him typing out the phrase "cult goddess of cyber feminism" with a disgusted sneer on his face. It's an under-researched, over-opinionated essay that can only be described as a "hot take" looking to get some folks riled up. And in an essay on Turing, he refers to Joan Clarke as "a woman he (Turing) worked with." Clarke was a well respected cryptanalyst in her own right, to the extent of being appointed as an MBE. A final callout -- he also refers to a woman as a "person of gender," as if only women have gender as compared to the "normal" man. Get a clue. Aside from that, a subsequent essay on Alan Turing that hints at him not committing suicide is just bonkers, especially since Holt's very next writing about Turing unquestionably mentions his suicide. Holt seems to like controversy for the sake of controversy, which is probably why he spent an ungodly amount of words on a philosophical kerfuffle that must only be interesting to academics. So why 4 stars? The writing is fantastic, and the first 13 essays were fascinating, even for someone who's read a ton on Godel, Mandelbrot, Einstein, et al. I thought his treatment of David Foster Wallace's work on infinity was kind and fair while also being slightly critical. All of which adds up to me thinking Holt approaches his work in two different mindsets each time he sits down to write. I know which Holt I prefer.

[D · Rating 2 out of 5]

I agree with this review. Worse, I did not manage to finish the book since the quality/interest went rapidly down hill in the second half.

The first story on the relationship between Einstein and Godel is not uninteresting but lots more detail can be found in this book which contains an account of his wife's life.

A minor quip on the hilarious Boganov scandal: Wikipedia confirms that the bogus articles were not about string theory as the author mistakenly claims, probably because there was a stray reference to an article by a string theorist in the bibliography.

By the time another section started explaining for the second time the unimpressive 'Copernican' principle, I had enough and put the book aside. All in all, a disappointment.

[Gerald · Rating 2 out of 5]

Most of this is Holt not Einstein or Godel

Only the introduction describes how the two great men walked together at Princeton. The rest of the chapters are the author's simplified explanations of his understanding of the latest thinking in relativity and mathematics. He asserts Godel said time does not exist but doesn't explain the notion well. Time need not be a factor in Einstein's equations. But that doesn't mean it isn't a consequence of being in a particular frame of reference. Otherwise rocket scientists wouldn't be able to reach their targets!

[Eric · Rating 4 out of 5]

TL;DR

Well-written, thoroughly researched and argued collection of essays wrestling with the difficult topics of mathematics. Recommended for anyone who wants an accessible but challenging look at some of humanity’s deepest problems. This book will make you think.

Disclosure

Farrar, Straus, and Giroux provided an advanced electronic copy in exchange for an honest review. Review cross-posted at my website: PrimmLife

Review


Math and philosophy have always been two sides of the same coin to me. One uses numbers and symbols to build logical arguments, and the other uses words as its tools. While the goals look different, they are actually very similar. Both disciplines seek to explain the beauty of the world they observe around them, and let’s face it, popular culture views the practitioners of both as intelligent ascetic scholars locked in their academic towers living lives wholly of the mind writing and creating papers for other scholars. But there exists certain scholars who wish to bridge the gap between those inside and outside the academic disciplines. When Einstein Walked with Gödel: Excursions to the Edge of Thought by Jim Holt from Farrar, Straus, & Giroux seeks to bring a philosopher’s gaze to the discipline of mathematics, and he largely succeeds. This collection of previously published essays paints a picture of a careful writer wrestling with the mathematician’s place in history. Mr. Holt writes with authority over a wide range of ideas, and he’s found the right balance of challenging and accessibility. When Einstein Walked with Gödel is an excellent addition to my growing list of scientific, mathematical, and philosophical publications; recommended for anyone who wants a survey of the deep ideas in those fields.

Walking with Big Ideas

From infinity to string theory to Mandelbrot to the Riemann Zeta conjecture, Mr. Holt covers the big ideas of mathematics. The essays approach these ideas from a descriptive rather than technical standpoint, and many have a historical and biographical focus that place the mathematical concept in time. To organize the book, essays are grouped into nine sections. Part eight consists of shorter, quicker essays that are unique to this book. Overall, the book covers a lot of ground but doesn’t just skim. Mr. Holt dives into the ideas to produce thorough and enlightening work.

Writing

Receiving an Advanced Reader Copy (ARC) of When Einstein Walked with Gödel made me ridiculously happy. It hits two of my favorite areas of reading, philosophy and math, but it does so in a pragmatic way. When I pick up scientific/philosophy books, I’m looking for the middle ground of being understandable without being condescending or written with too much hand-holding. Collections like this should be – as it says in the description – accessible. What does accessible mean, though? To me, an accessible scholarly work means that a deep knowledge is not necessary, but the text still requires intellectual interaction. When Einstein Walked with Gödel hits this sweet spot of shining light on tough ideas while not being out of my intellectual capabilities. Mr. Holt accomplishes this through a mix of translating academic ideas from math to practical descriptions, biography, history, and book reviews. It’s clear that Mr. Holt does exhaustive research for his articles and thinks deeply about each topic. With any scientific work, the writing determines the success of the book, and Mr. Holt’s writing is excellent, challenging, and precise. In nearly every essay, Mr. Holt slips easily into a very high diction that had me running to the dictionary; he loved slipping in $100 words or haughty phrases. These moments felt self-congratulatory and knocked me out of the essay but each occurrence was momentary.

The Problem with Collections

As this is a collection of essays and not one whole work, each section provides a new topic, a new argument. Like any collection, some work better than others. The titular essay and "Truth and Reference: A Philosophical Feud" stood out as the two best. The three essays dealing with the mathematical concept of infinity, and, in particular, "The Dangerous Idea of the Infinitesimal," shifted my perspective on the slippery concept in an entertaining way. Mr. Holt’s shorter essays mostly succeeded as more philosophical sketches than essays. Though I think of them more as intellectual snacks when compared to the full meals that are his essays, these short pieces will make you think, or, at least, say “aha.” Overall, When Einstein Walked with Gödel avoids the problem with collection by being filled with many strong pieces. While there were a few weaker essays, only one stood out as a real negative.

Poor Ada Lovelace

In the "The Ada Perplex", Jim Holt questions whether Byron’s daughter deserves the historical position that she occupies, and the well-researched essay casts doubt on her position. But I can’t help notice that the reasons used by Mr. Holt align with historical methods of erasing the contributions of women. Hell, just the cover of Joanna Russ’s How to Suppress Women’s Writing lists examples of the arguments that Mr. Holt used to show that Ada may not deserve her vaunted place. Even upon finishing the whole collection, I still don’t know what to make of the Ada Lovelace essay because her contributions are debated but not erased. While this essay attempts to walk the thin line between shedding light on a necessary debate and controversy, I don’t think it succeeds. Her collaboration with Babbage should be scrutinized, but she shouldn’t be discounted as a contributor. Mr. Holt, clearly, researched the hell out of this topic, and if he found obvious evidence for his conclusion, it wasn’t effectively argued in his essay. In the later essay "Truth and Reference: A Philosophical Feud," Mr. Holt succeeds in clearly laying out a controversy without erasing Ruth Barcan Marcus’s contributions, which shows he possesses the capability to write about a difficult topic without erasure. It could be that the second essay succeeds where the first doesn’t because of historical distance between the two events. I just wish he’d handled Ada’s case with a little more care.

Gödel

Gödel and his incompleteness theorem pop up frequently throughout this collection. Whether a byproduct of research, personal bias, or just being right there on his mind, Mr. Holt slips Kurt Gödel into many of the essays. While noticeable, the reference never feels forced. Much the same way a physicist drops Einstein or Feynman into conversation, Mr. Holt name-checks Gödel. All of these references indicate where Mr. Holt places the Austrian in the pantheon of great mathematicians. Simply because of Mr. Holt’s his intellectual rigor and repeated usage of Gödel throughout the collection, he made me curious to learn more about the mathematician and his incompleteness theorem, which is currently beyond my grasp, not that I’ve spent much time on it. However, this is one of the things I enjoyed about the book because it places Mr. Holt in an intellectual lineage. Before this collection, I knew little of Kurt Gödel other than he had an important theorem. Now, I owe Mr. Holt a debt for piquing my curiosity about this strange genius.

Conclusion

When Einstein Walked with Gödel: Excursions to the Edge of Thought exceeded my expectations all around. While some essays missed and one failed, overall the collection presents an enticing, in-depth look at big ideas, maybe some of the biggest ideas in human intellectual history. This accessible yet challenging book educated me on every page without boring me. The very best essays stuck with me long after I finished reading them, and I realized it was because Jim Holt had taken exceedingly complex ideas and distilled them down to their essences. In When Einstein Walked with Gödel: Excursions to the Edge of Thought, Jim Holt translated academic beauty into the layperson’s splendor.

8 out of 10

[Jim · Rating 4 out of 5]

Source is Libby audio and also Nashua L owns a hardbound copy.
Over time I have read about Number Theory & Quantum Physics and remember precious little. So if you asked me about Godel. I would say something like he wrote about Logic and had a proof that spoke to there are things that can't be proved in a number system using a little paradox and this was eye opening for many mathematicians at the time. Sort of like the arrow paradox.

Thought writing some little factoids I pick up may help me remember these ideas for a longer period.

Einstein theory of relativity states there is no Universal Now.
Goeddel theory of incompleteness is based on the conjecture : This statement is unprovable.
He has a first and a second theorem of incompleteness.
Zipf law and Mandelbrot section is interesting

[Algirdas · Rating 4 out of 5]

Esė rinkinys apie fizikus ir jų draugus matematikus. Daug visko įdomaus.

[ini · Rating 5 out of 5]

My first impression is very positiv about this book. It compact with all information I needed. And you will learn more about there personal Life.

The book "When Einstein walked with Gödel" by Jim Holt, published in 2018, is about the Life of scientist success and tragedy. The book begins with the most famous scientist. It tells the work, the life and the feeling of the scientists. Many have a different image of scientists and I think it's good that this book addresses it more clearly. They have a beautiful similar novel life story with the most important theories in a book, which is a positive aspect.

Overall, the reading of this book is interesting and compact with scientific information. Therefore, "When Einstein Walked with Gödel" is highly recommended for anyone interested in physics, mathematics, history and/or philosophy

[Joseph Carrabis · Rating 2 out of 5]

I wrote a review of this book. Now it’s missing. Holt would turn that fact into a chapter on any number of topics.
Not quite popsci, not quite personal philosophical explorations, not deep enough for research, too deep for most readers’ pleasure reading. Holt slides over some material and delves into others. Not sure what his selection algorithm is. Am sure he’d write a chapter about it. Or use it as a leaping place to some other topic.

[Jon Stout · Rating 4 out of 5]

Consisting of article-like chapters that describe people, personal interactions and issues in the scientific and intellectual world, this book shows remarkable knowledge of both scientific literature and history. I was surprised that the author also included philosophical topics, and was remarkably conversant with some of the most important philosophers and philosophical issues of the current times.

The title essay refers to Albert Einstein’s friendship with Kurt Gödel, when they were both at the Princeton Institute for Advanced Studies, and casts light on their individual personalities as well as their interactions regarding theory, both mathematical and physical. Various essays deal extensively with such topics such as Gödel’s Proof, quantum physics, neuroscience and computation.

There are also personal stories regarding scientific and intellectual issues, for example the story of Ida Lovelace, often described as the very first computer programmer. Holt’s portrayal describes her as being more of a popularizer than an expert formulator of algorithms, but she nonetheless, along with Charles Babbage, played an important role in the early development of computer theory.

I was impressed with Holt’s discussion of the philosophical issue of naming, that is, whether a proper name is really a definite description of something in the world, or whether it is tied to the thing named (is a “rigid designator”) apart from any contingent description. This is likely to make the eyes of non-philosophers glaze over, but Holt manages to make it interesting. He examines the “rigid designator” position, usually attributed to Saul Kripke, and considers whether it might have come from Ruth Marcus, one of Kripke’s college professors, in other words an underappreciated female antecedent. Holt gives an even-handed treatment, and remarkably seems to have been on good terms with both philosophers. This somewhat gossipy treatment is tied in with a good discussion of why the philosophical issue matters in the first place.

I enjoyed the book. My only criticism would be that Holt often does a good job of describing both sides of an issue without coming to any conclusion, or pressing the issue to the end. Some of the issues seemed frivolous, such as discussing the boundary between astrology and astronomy, but most of the articles were insightful and entertaining.

[Philipp · Rating 4 out of 5]

A collection of book reviews and essays, mostly from NYRB I think - in structure these are very similar to Freeman Dyson's collections of essays, i.e., there's a book to be reviewed, but the book's themes are used as a starting point for an essay about the themes, not a evaluation of the book itself.

The themes of the essays sometimes overlap, but Holt is much more interested in the philosophy of mathematics, i.e., what's more fun to think about, the infinity of numbers between 1 and 2, or the infinity of large numbers? There's a history of the 'solve' of the four-colour-theorem and the associated problems (it was essentially brute forced with computers, which angered many, which is what the essay is about!)

There's a needlessly mean essay about Ada Lovelace in there (the gist: she was a kind of Paris Hilton who couldn't even do basic maths, it was all Babbage), but the rest is gold.

Favorite quote, from a review of David Foster Wallace's history of infinity:

And if he was sometimes in over his head, it is because he chose to wade through the deepest waters.

Other quotes:

De Prony found his inspiration in Adam Smith’s Wealth of Nations—specifically, in Smith’s account of the division of labor in a pin factory. Hiring a hundred or so Parisian hairdressers who had been thrown out of work when their clients lost their pompadoured heads to guillotines during the Reign of Terror, de Prony set up a kind of arithmetic assembly line that would, as he put it, “manufacture logarithms as one manufactures pins.” The individual hairdressers had no special mathematical abilities: all they could do was add, subtract, and cut hair. The intelligence was in their organization.

This surprised me a lot; there has been a lot of hype about Liu Cixin's The Three-Body Problem, in which there's a very similar machine using soldiers as 'pins'. Amazing that this existed in reality!

On the eternity of the universe:

Why should we want the universe to last forever, anyway? Look—either the universe has a purpose or it doesn’t. If it doesn’t, then it is absurd. If it does have a purpose, then there are two possibilities: either this purpose is eventually achieved, or it is never achieved. If it is never achieved, then the universe is futile. But if it is eventually achieved, then any further existence of the universe is pointless. So, no matter how you slice it, an eternal universe is either (a) absurd, (b) futile, or (c) eventually pointless.

Channeling Discworld's Death in Hogfather:

As long as there are no decisive arguments for or against the existence of God, a certain number of smart people will go on believing in him, just as smart people reflexively believe in other things for which they have no knockdown philosophical arguments, like free will, or objective values, or the existence of other minds.

If all of these soundbites sound boring to you, steer clear, it's mostly like that!

[Roberto Rigolin F Lopes · Rating 5 out of 5]

We are in 2018, Jim collected a bunch of essays on uncomfortable topics here called the edge of thought. Check out this unusual question. What is likely to happen with laughter and numbers at the year one million? Thrilling. And it gets better. Remember that the intercourse between mathematics and physics has been pushing human intelligence to its limits. The essays on mathematics invite you to: digest infinities with different sizes, touch infinite small numbers and jump in places with many dimensions. In sequence, come computers creating very long proofs together with the decision problem which is equivalent to the halting problem. Followed by physics searching for a theory of everything that may help us to explore the energy of our sun so civilization can spread over the galaxy. And much more. But no bullshit. Although there is an essay on BS. What?! Surely these edgy topics will demand some brainpower. Pure delight.

[Eduardo Montiel · Rating 5 out of 5]

A rabbit hole treasure chest. A collection of 25 essays on diverse topics: Einstein’s theory of relativity (special and general), quantum mechanics, group theory, infinity and the infinitesimal, Turing’s theory of computability and the “decision problem,” Gödel’s incompleteness theorems, prime numbers and the Riemann zeta conjecture, category theory, topology, higher dimensions, fractals, statistical regression and the “bell curve,” the theory of truth, string theory, Newcomb's problem, etc.

The ideas that Holt presents all bear crucially on our most general conception of the world (metaphysics), on how we come to attain and justify our knowledge (epistemology), and even on how we conduct our lives (ethics).

I thoroughly enjoyed reading each essay. Holt is truly one of the best modern science writers. He provides a fresh and impartial perspective on each topic and is gifted at the art of providing parsimonious explanations to complex subjects (no easy task).

Some of the essays are too short and cover the topics only sparingly (the final essays). Nonetheless, you can research any of these topics and read more about them. I personally found fractals and Mandelbrot sets to be fascinating:

Other self-similar phenomena, each with its distinctive form, include clouds, coastlines, bolts of lightning, clusters of galaxies, the network of blood vessels in our bodies, and, quite possibly, the pattern of ups and downs in financial markets.

In the late 1970s, he became famous for popularizing the idea of self-similarity and for coining the word “fractal” (from the Latin fractus, meaning “broken”) to designate self-similar forms. In 1980, he discovered the “Mandelbrot set,” whose shape—it looks a bit like a warty snowman or beetle—came to represent the newly fashionable science of chaos.

The world we live in, he observes, is an “infinite sea of complexity.” Yet it contains two “islands of simplicity.” One of these, the Euclidean simplicity of smooth forms, was discovered by the ancients. The other, the fractal simplicity of self-similar roughness, was largely discovered by Mandelbrot himself.

4.5 / 5. September 2022.

Highlights

Godel / Einstein

Both Gödel and Einstein insisted that the world is independent of our minds yet rationally organized and open to human understanding.

In his March paper, on the photoelectric effect, he deduced that light came in discrete particles, which were later dubbed photons. In his April and May papers, he established once and for all the reality of atoms, giving a theoretical estimate of their size and showing how their bumping around caused Brownian motion.

First, the laws of physics are absolute: the same laws must be valid for all observers. Second, the speed of light is absolute; it, too, is the same for all observers. The second principle, though less obvious, had the same sort of logic to recommend

To make these laws absolute, he made distance and time relative.

Working from his two basic principles, Einstein proved that whether an observer deems two events to be happening “at the same time” depends on his state of motion. In other words, there is no universal now. With different observers slicing up the timescape into “past,” “present,” and “future” in different ways, it seems to follow that all moments coexist with equal reality.

The conclusion—that no logical system can capture all the truths of mathematics—is known as the first incompleteness theorem. Gödel also proved that no logical system for mathematics could, by its own devices, be shown to be free from inconsistency, a result known as the second incompleteness theorem.

If the laws of physics were to provide a truly objective description of nature, they ought to be valid for observers moving in any way relative to one another—spinning, accelerating, spiraling, whatever. It was thus that Einstein made the transition from his “special” theory of relativity of 1905 to his “general” theory, whose equations he worked out over the next decade and published in 1916. What made those equations so powerful was that they explained gravity, the force that governs the overall shape of the cosmos.

Einstein had shown that the flow of time depended on motion and gravity and that the division of events into “past” and “future” was relative. Gödel took a more radical view: he believed that time, as it was intuitively understood, did not exist at all.

Time

For little children, however, time goes quite slowly. Owing to the endless novelty of a child’s experience, a single summer can stretch out into an eternity. It has been estimated that by the age of eight, one has subjectively lived two-thirds of one’s life.

What science can tell us something about is the psychology of time’s passage. Our conscious now—what William James dubbed the “specious present”—is actually an interval of about three seconds. That is the span over which our brains knit up arriving sense data into a unified experience.

“Time is nature’s way to keep everything from happening all at once.”

Other

The human memory, unlike that of a computer, has evolved to be associative, which makes it ill-suited to arithmetic, where bits of knowledge must be kept from interfering with one another: if you’re trying to retrieve the result of multiplying 7 × 6, the reflex activation of 7 + 6 and 7 × 5 can be disastrous.

Evidently, the number sense has an even longer evolutionary history than that of laughter. So again, by the Copernican principle, we can be quite certain that numbers will be around in the Year Million.

Mathematics, after all, is supposed to be the most universal part of human civilization. All terrestrial cultures count, so all terrestrial cultures have number. If there is intelligent life elsewhere in the cosmos, we would expect the same. The one earmark of civilization that is likely to be recognized across the universe is number.

“Zeta” refers to the zeta function, a creature of higher mathematics that, as Riemann was the first to realize, holds the secret of the primes. In 1859, in a brief but exceedingly profound paper, Riemann put forward a hypothesis about the zeta function. If his hypothesis is true, then there is a hidden harmony to the primes, one that is rather beautiful. If it is false, then the music of the primes could turn out to be somewhat ugly, like that produced by an orchestra out of balance.

The zeta function, fittingly, has its origins in music. If you pluck a violin string, it vibrates to create not only the note to which it is tuned but also all possible overtones. Mathematically, this combination of sounds corresponds to the infinite sum 1 + ½ + ⅓ + ¼ +…, which is known as the harmonic series.

Riemann was able to do a marvelous thing: he produced, for the first time ever, a formula that described exactly how the infinity of primes arranged themselves in the number sequence.

Oddly enough, the bell curve—also known as the normal or Gaussian distribution (after Carl Friedrich Gauss, one of its multiple discoverers)—first arose in astronomy.

Why is the bell curve so ubiquitous? Mathematics yields the answer. It is guaranteed to arise whenever some variable (like human height) is determined by lots of little causes (genes, diet, health, and so on) operating more or less independently.

“Regression to the mean motivates almost every variety of risk-taking and forecasting. It is at the root of homilies like ‘What goes up must come down,’ ‘Pride goeth before a fall,’ and ‘From shirtsleeves to shirtsleeves in three generations.’”

regression is a matter of pure mathematics, not an empirical force.

These have been likened to the yin and yang of mathematics: geometry is space, algebra is time; geometry is like painting, algebra is like music.

Harvard linguist named George Kingsley Zipf, this law concerns the frequency with which different words occur in written texts—newspaper articles, books, and so on. The most frequently occurring word in written English is “the,” followed by “of” and then “and.” Zipf ranked all the words in a large variety of written texts in this way and then plotted their frequency of usage. The resulting curve had an odd shape.

Zipf’s law , which has been shown to hold for all languages, may seem a trifle. But the same basic principle turns out to be valid for a great variety of phenomena, including the size of islands, the populations of cities, the amount of time a book spends on the bestseller list, the number of links to a given website, and—as the Italian economist Vilfredo Pareto had discovered in the 1890s—a country’s distribution of income and wealth. All of these are examples of “power law” distributions.

The only known way to make relativity theory consistent with quantum theory is by supposing that the basic objects that make up our universe are not one-dimensional particles but two-dimensional strings and still-higher-dimensional “branes” (a term derived from “membrane”). Moreover, if the unified theory—called string theory, or sometimes M-theory—is to be mathematically coherent, these strings and branes must be vibrating in a space that has no fewer than nine dimensions.

Euler was perhaps the most prolific mathematician in history. Among the discoveries he made, while shuttling between the courts of Frederick the Great and Catherine the Great, was the formula V − E + F = 2, which was not long ago voted the second most beautiful theorem in mathematics. (The winner of the beauty contest, according to a 1988 survey published in The Mathematical Intelligencer, was eiπ = −1.)

the infinity of sets of things is greater than the infinity of things. The beauty of this principle, which has come to be known as Cantor’s theorem, is that it can be applied over and over again. Given any infinite set, you can always come up with a larger infinity by considering its “power set”—the set of all subsets that can be formed from it.

Essentially, calculus deals with curves. Its two basic operations are finding the direction of a curve at a given point (the “derivative”) and the area bounded by a curve (the “integral”). Curves are mathematically represented by “functions.” Some functions, like the sine wave, are nice and smooth; they are called continuous. But others are riddled with breaks and jumps: discontinuities.

In everyday parlance, “infinitesimal” is loosely used to refer to things that are extremely tiny by human standards, too small to be worth measuring.

The power of the calculus was matched by its versatility. It made possible the quantitative handling of all varieties of continuous change. The differential calculus showed how to represent the rate of change as a ratio of infinitesimals. The integral calculus showed how to sum up an infinite number of such changes to describe the overall evolution of the phenomenon in question.

Can you determine, in principle, whether a conjecture can be proved true or false? The decision problem calls for a mechanical set of rules for deciding whether such an inference is valid, one that is guaranteed to yield a yes-or-no answer in a finite amount of time. Such a method would be particularly useful to mathematicians, because it would allow them to resolve many of the conundrums in their field—like Fermat’s last theorem, or Goldbach’s conjecture—by brute force.

“The history of digital computing,” Dyson writes in his 2012 book, Turing’s Cathedral, “can be divided into an Old Testament whose prophets, led by Leibniz, supplied the logic, and a New Testament whose prophets, led by von Neumann, built the machines. Alan Turing arrived in between.”

“Fluid” intelligence is one’s ability to solve abstract problems, like logic puzzles. “Crystallized” intelligence is one’s store of information about the world, including learned shortcuts for making inferences about it.

The only reason we used to read big long novels before the advent of the Internet was that we were living in an information-impoverished environment. Our “pleasure cycles” are now tied to the web, the literary critic Sam Anderson claimed in a 2009 cover story in New York magazine, “In Defense of Distraction.” “It’s too late,” he declared, “to just retreat to a quieter time.”

In fact, beyond a certain minimum IQ threshold—about one standard deviation above average, or an IQ of 115—there is no correlation at all between intelligence and creativity.

In the decades after this dual revolution, most of the action was on the quantum side. In addition to gravity, there are three basic forces that govern nature: electromagnetism, the “strong” force (which holds the nucleus of an atom together), and the “weak” force (which causes radioactive decay). Eventually, physicists managed to incorporate all three into the framework of quantum mechanics, creating the “standard model” of particle physics.

The standard model tells how nature behaves on the scale of molecules, atoms, electrons, and on down, where the force of gravity is weak enough to be overlooked.

General relativity tells how nature behaves on the scale of apples, planets, galaxies, and on up, where quantum uncertainties average out and can be ignored. Between the two theories, all nature seems to be covered.

Witten’s papers are models of depth and clarity. Other physicists attack problems by doing complicated calculations; he solves them by reasoning from first principles. Witten once said that “the greatest intellectual thrill of my life” was learning that string theory could encompass both gravity and quantum mechanics.

Second, it is discriminating: an experiment done on one photon in an entangled pair affects only its partner, wherever that partner may be, leaving all other photons, near and far, untouched. The discriminating nature of entanglement again stands in contrast to gravity, where a disturbance created by the jostling of one atom will ripple out to affect every atom in the universe.

The shortest spatial span that has any meaning is the Planck length, which is about 10−35 meters (about 20 orders of magnitude smaller than a proton). The shortest possible tick of an imaginary clock (sometimes called a chronon) is the Planck time, about 10−43 seconds. (This is the time it takes light to cross a distance equal to the Planck length.)

Stigler’s law of eponymy. This law, which in its simplest form states that “no scientific discovery is named after its original discoverer,”

One of Nietzsche’s more notorious doctrines is perspectivism—the idea that we are condemned to see the world from a partial and distorted perspective, one defined by our interests and values.

[Goatboy · Rating 4 out of 5]

Very good collection of essays/articles about a wide range of topics in science, mathematics, philosophy and other disciplines. Was most enlightened by the articles on mathematicians (who apparently tend toward the slightly crazy). Thought the collection lagged a bit in the last third, but did enjoy the chapter on the philosophical feud related to naming and reference. Topics and people discussed definitely added to my reading list.

[jazlyn · Rating 4 out of 5]

paradigm shifting ... i learned so much math and so much about humanity. the perfect blend of high minded academic bs, actually touching humanity, and a lot of humor. rly rly loved it.

[Jacob Alcott · Rating 3 out of 5]

Wonderful jumping off point for someone who is interested and curious about math, physics, and where the two dip into philosophy.

[Fernando Conde · Rating 5 out of 5]

El mejor libro de divulgación que me he leído 😌😌

[Ahlam · Rating 4 out of 5]

fun collection of essays and a good way to figure out what i want to know more about!

for later: the copernican principle, gödel (just bought a book on this, i want to actually get the incompleteness theorem), turing/ recursion theorem, georg cantor/ set theory, “on bullshit”

also i KNEW ada lovelace didn’t know shit about programming… vindicated

[Alex Roman · Rating 4 out of 5]

I really enjoyed most of this book, but I feel like it became very hit or miss towards the latter third or perhaps even the latter half. Definitely some great tidbits and ideas.

Ignoring the few total misses such as the last two essays, this would have been a great book to give to myself near the beginning of my undergrad. There are lots of things it took me a long time to find out which are described on these pages. This part earns the book four stars. However in truth, about half of the length of this book should earn more of a three or in a couple cases two star designation. So I’ll put my final score around three and a third.

[B. Rule · Rating 3 out of 5]

A collection of essays on interesting topics that unfortunately is less than the sum of its parts. Holt is an engaging writer with a rock-solid knowledge of mathematical and philosophical topics, and he wrote one of the best books I've read in a long time ("Why Does the World Exist?"), but this one falls far short of that lofty exemplar even though it covers many of the same topics.

The main problem is that this collects essays that appeared in other publications over a period of many, many years (at least one clearly dates back to the mid-90's, and receives an outpouring of footnotes in a hasty attempt to bring it up to date). Further, those essays often overlap in topic. So, while they may stand well alone, in succession they become repetitive, belabored, and also sometimes inconsistent. A thinker praised in one essay may turn up as the target of a hit piece later in the volume. The reader is often introduced to a physicist or mathematician and then receives another introduction afresh later in the book, which wastes pages and feels a bit absent-minded at best. Further, a lot of these lack the depth of Holt's book-length excursions on the questions of existential import. That magazine quality occasionally reveals something of the threadbare in Holt's argumentation: in his haste to find an "angle", there are a lot more straw men set up, pompous declarations on the soundness of arguments without the legwork to show it, and general pettiness on display.

I'd still rather read Holt than the vast majority of authors out there, but I was disappointed in this. It felt more like a cash-in due to the popularity of his earlier book rather than something that urgently needed to be published. There were definitely parts I liked though. I enjoyed his essay about Dawkins, for instance. Although I know Holt to be an agnostic with some antipathy towards organized religion, I thought he did a good job of showing the weaknesses in Dawkins' case against theism (even though I'm guessing Holt agrees with Dawkins more than he differs). The best pieces are often the ones with an inside baseball exploration of professional conflicts that most of us would never hear about, but which deal with topics of supreme interest to all of us (e.g., the ultimate fate of the universe, string theory and/or the theory of everything, the nature of truth, etc.).

[Tracy Rowan · Rating 4 out of 5]

I read a lot of audiobooks about science. I don't always understand everything I hear, but the format does make that easier for me. Holt's collection of essays on science and (to a lesser degree) philosophy range from the easily comprehensible to the sort of things that would make my eyes glaze over if I was reading hard copy, but for the most part he does a great job of making a lot of complex scientific ideas much clearer and more accessible.

His discussions of physics and mathematics, which make up the bulk of the book, made a good deal of sense to me as I listened. Not that I could reproduce the formulae or equations involved. But Holt manages to give a layperson the ability to grasp some difficult concepts with the clarity of his prose.

And then there's the philosophy part which sometimes utterly eludes me because so much of it is counter-intuitive.  Still, it's almost as interesting to hear about the battles over who took credit for what, even if I don't begin to understand the What part, as it is to get the lowdown on Einstein's problems with "spooky action at a distance" which name could have been applied to gravity before science became aware of how forces work, or Gödel's paranoia that people were trying to poison him, leading him to effectively starve himself to death. Certainly some of the most interesting parts were Holt's discussion of the life and work of Alan Turing, who these days seems to be more famous as a gay martyr than as a brilliant mathematician who, in breaking the Enigma code, helped win WWII.

It's one of those books that veers from the chatty and informative to the murkily complex. Some of it is a joy to read, some went the proverbial route of in one ear and out the other. Still, I feel as if I got a great deal of both pleasure and information out of it, and I think that's all I can reasonably expect.

[Kamal · Rating 3 out of 5]

I received this book as a gift a very long time ago and just bounced around between a few of the essays/short stories that had interesting titles. I decided to finish it before the year ended, so that I could finally move on lol. I wanted the book, but I think my brain was smoother at the time, so I couldn’t really pay attention to everything in it. FYI this is pop science. You don’t need a dense academic background to enjoy it.

Some parts are really great but more are boring. If you love academic history (with a focus on STEM and Philosophy) you will probably enjoy this. I particularly loved the parts involving Kurt Gödel and Alan Turing. The author does a relatively decent job of calling out academia as being male dominated. There is even a very good essay detailing the theft of an idea from a female philosopher by a male one. The author is a bit condescending though. He also could have wrote about other groups’ accomplishments in science, but did not. At least, he did not include any in this book.

As someone who participated, albeit briefly (~2 years), in academia. It is a very Western (majority white) male dominated field. A club that can be hard to break into for anyone who doesn’t fit that mold. The book talks a few times about the stagnation of certain fields and how things haven’t moved much. I wonder if it is because there aren’t many other perspectives besides Western and Male.

In a different chapter it talks about Stigler’s law of eponymy, the idea that no scientific discovery is named after its original discoverer. Makes me wonder how many things we are crediting to the wrong people. I mean if Watson and Crick didn’t credit Rosalind Franklin for her contributions to molecular biology, who else has been wrongfully left out from history?

I’m rambling. The book is a 3.5/5.