Is God a Mathematician?

by Mario Livio

Bestselling author and astrophysicist Mario Livio examines the lives and theories of history’s greatest mathematicians to ask how—if mathematics is an abstract construction of the human mind—it can so perfectly explain the physical world.

Nobel Laureate Eugene Wigner once wondered about “the unreasonable effectiveness of mathematics” in the formulation of the laws of nature. Is God a Mathematician? investigates why mathematics is as powerful as it is. From ancient times to the present, scientists and philosophers have marveled at how such a seemingly abstract discipline could so perfectly explain the natural world. More than that—mathematics has often made predictions, for example, about subatomic particles or cosmic phenomena that were unknown at the time, but later were proven to be true. Is mathematics ultimately invented or discovered? If, as Einstein insisted, mathematics is “a product of human thought that is independent of experience,” how can it so accurately describe and even predict the world around us?

Physicist and author Mario Livio brilliantly explores mathematical ideas from Pythagoras to the present day as he shows us how intriguing questions and ingenious answers have led to ever deeper insights into our world. This fascinating book will interest anyone curious about the human mind, the scientific world, and the relationship between them.

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

320 pages, Hardcover

First published January 1, 2009

Reviews

[Irena Pasvinter · Rating 5 out of 5]

I saw a Hebrew translation of "Is God a Mathematician?" years ago in a local bookshop. An intriguing title. Still, I didn't buy the book back then, but when I happened upon its audiobook version, I just had to add it to my oversized audible.com library, as well as a few other titles by the same author.

Judging by his name, I was sure the author was from Italy. Then my eldest daughter, who began studying physics in Technion in Haifa, asked me if I heard about Mario Livio. "Sure, this guy from Italy. I've been meaning to read his book," I said. To which she enlightened me that before moving to the US, Mario Livio was the member of physics faculty in Technion, where he taught for ten years. "Wow, now I must read his books even sooner!" I said.

Fast-forward a few years: my daughter listened to my audio of "Is God a Mathematician", told me it was great, finished her BA in physics, started her MA, attended Mario Livio's presentation of his recent book about Galileo in Technion, and this is when I finally felt I could delay no longer.;)


Mario Livio signing "Is God a Mathematician" at Powell's Technical, Feb 6 2009 (Image credits: Kirby Urner, CC BY 2.0 , via Wikimedia Commons.)

So what's with the intriguing title? Is God a Mathematician -- in other words, why is it that natural physical phenomena is so precisely described with the help of abstract mathematical language? Does mathematics truly exist and is discovered by mathematicians or is it invented as a purely artificial creation of human mind? Then why is it that often purely abstract and completely impractical mathematical theories later turn out to be indispensable tools for describing and predicting physical processes?

Those are the fascinating questions that Mario Livio, an astrophysicist and also an author of many a popular-science bestseller, sets out to discuss in "Is God a Mathematician?". He illustrates the development of mathematics and its crucial role in development of physics by telling the stories of Pythagoras, Archimedes, Euclid, Descartes, Galileo, Newton, Einstein, Gödel etc. It's never a dull moment, unless you passionately hate science and math.

Even though I've already read about all of them in other books dealing with the history of science, I not only enjoyed this masterfully-written review, but also received some new information.

For example, I haven't realized that Archimedes' writings still played such an important role at the time of Galileo. And I've never before heard about Archimedes palimpsest rediscovered in a remote Greek Orthodox monastery near Jerusalem in the 19th century and recently "re-deciphered" with the help of modern imaging technologies, which finally shed a bit of light on methods used by Archimedes to formulate problems and find solutions.

Also, who hasn't heard about Bernoulli's numbers, but I had no idea there was a whole family of Bernoulli scientists and that Bernoulli brothers fiercely competed with each other while in search for mathematical proofs. Something to read more about in the future.


One page of Archimedes Palimpsest, On Floating Bodies. (Image credits: Matthew Kon, Public domain, via Wikimedia Commons)


Image credits: Biblioteca Europea di Informazione e Cultura, Public domain, via Wikimedia Commons.


First page of a 1632 copy of "Dialogo...sopra I due Massimi Sistemi del Mondo Tolemaico, e Copernicano" by Galileo Galilei. Copy located in the Niels Bohr Library & Archives, American Institute of Physics, in College Park, Maryland. (Image credits: Galileo Galilei, Public domain, via Wikimedia Commons.)

And what about the big question in the title of the book? Big questions don't have clear-cut answers that everybody agrees about, but it's the the question itself and the never-ending journey towards the answer that counts.

P.S. The narrator of the audiobook version did a very good job, but still managed to add to my collection of audio-grievances such Stress-It-As-You-Guess-It gems as Med'ici and Mend'el (Gregor Mendel). There was something else, but it escapes me now. Anyway, my daughter, with whom I shared these stressful accidents, advised me not to be petty, and in this particular case I feel inclined to follow her advice.

Read in 2024.

[Ken Roebuck · Rating 5 out of 5]

The answer to the question "Is God a Mathematician" depends very much on your world view. Those of faith that believe in a transcendent creator God will surely answer with a resounding YES. But Atheists and other non believers are likely to think mathematics is nothing more than an invention of the human mind. Nevertheless, it remains that the universe appears to have been designed by a pure mathematician. As James Jean put it “mathematics appears to be almost too effective in describing and explaining not only the cosmos at large but even some of the most chaotic of human enterprises.” Indeed, the success of mathematics in explaining the world around us has been dubbed “the unreasonable effectiveness of mathematics." So is mathematics invented or discovered? In my opinion mathematics exists independent of human minds but God for whatever reason has given us mathematical minds with which we have used with great success to uncover the mysteries of the universe. Maybe this is part of what it means to be created in the image of God.

[Crina Bucur · Rating 4 out of 5]

I was pleasantly surprised by Mario Livio’s “Is God a Mathematician?” specifically his eloquence in walking the readers through the most significant moments in the history of mathematics and acquainting them with prominent figures on an extensive timeline from antiquity to modern days.

As the title suggests, the main focus of the book is represented by the existence of various paradigms describing how we should approach mathematics, among which two stand out as poles apart: formalism (claiming that math is invented by the human mind) and Platonism (regarding mathematics as an a priori universal language whose truths are merely discovered and otherwise independent from the human reasoning). [More on the schools of thought in mathematical philosophy - here]

Obviously this remains an open argument, but the author manages to raise pros and cons to all the theories in quite an objective manner. Regardless of the preferred approach, one cannot help but marvel at how such a seemingly abstract discipline can so superbly explain the natural world. Furthermore, if mathematics is invented, how come some of its concepts were found practical applications long after their invention?

Each chapter discusses important topics like geometry, logic, topology, statistics and probability theory, as well as major breakthroughs in adjacent fields – such as physics or astronomy (which I particularly enjoyed). The transitions are natural, the narrative style is easy to follow and the overall tone is objective. Unlike many popular science books that tend to get tedious or uninteresting after the first few chapters, it has a good structure and manages to keep the reader engaged and to arouse his/her curiosity on the subject. “Is God a Mathematician?” is an accessible book with technical concepts often explained in layman’s terms so I wouldn’t recommend it for its technical prowess, but rather for the food for thought it provides.

There is poetry in the queen of all sciences and this book succeeds in conveying it, aside from the inherent philosophical considerations surrounding the nature of mathematics.

[WarpDrive · Rating 5 out of 5]

Great book, highly recommended to anybody interested in the relationship between mathematics and physical reality. The author demonstrates his wide knowledge and culture, which is not limited only to mathematics and physics, but also to philosophy, cognitive sciences etc. A very comprehensive account, the only small defect being that the final conclusive part seems a bit rushed.

[Mircea Petcu · Rating 5 out of 5]

Este matematica descoperită sau inventată?
Comunitățile matematicienilor și filosofilor sunt divizate pe acest subiect.

Cred că matematica este inventată. Mi s-au părut convingătoare cercetările recente din neuroștiințe. Matematica nu este foarte diferită de limbaj. După cum spunea neurologul Jean-Pierre Changeux: "Pentru mine, metoda axiomatică este expresia facultăților cerebrale legate de folosirea creierului uman, căci ceea ce caracterizează limbajul este tocmai caracterul său generator."
Mai sunt cele două șocuri din matematică. Apariția geometriei neeuclidiene a arătat că ceea ce noi numim axiome sunt de fapt "definiții mascate" (Henri Poincare). Logicianul Kurt Godel a arătat că nu putem demonstra matematic că matematica funcționează (teorema incompletitudinii).

Mario Livio consideră că matematica este atât inventată, cât și descoperită. Invocă eficatatea (dă exemplu aplicarea teoriei nodurilor în compactarea ADN-ului) și coerența matematicii.

Pentru cei curioși de originea titlului: "Cu câtiva ani în urmă am ținut o conferință la Universitatea Cornell. Pe una dintre imaginile proiectate de mine se putea citi: "Este Dumnezeu matematician?" Îndată ce a apărut imaginea, l-am auzit pe un student din primul rând rostind nervos: "Oh, Doamne, sper că nu!"

Recomand

[Carduelis · Rating 4 out of 5]

Kitabın ana sorusu: "Nasıl oluyor da insan zihninin tecrübeden bağımsız, soyut bir ürünü olan matematik, gerçek dünyadaki cisimlere inanılmaz derecede uygunluk gösterebiliyor?"
Livio sorunun cevabını matematiğin tarihsel gelişimini kronolojik olarak izleyerek matematiksel kavramların ortaya çıkışı ile fiziksel gerçekliğin açıklanışı arasındaki ilişkinin nasıl oluştuğunu göstererek veriyor.

Doğa ile matematik arasında beklenmedik bağlar kuran bilim insanlarını, karmaşık olaylardan berrak yasalar çıkaran "büyücüler" olarak tanımlanıyor. Kitapta ele alınan başlıklar baya geniş: Pisagor'un sayı ve geometri ilişkisi, Platon'un ideaları, Rene Descartes'in kartezyen geometri ve cebirsel temelleri, Newton'un kütleçekim yasası, Einstein'in görelilik teoresi, olasılık ve istatistik teorileri, K.Gödel'in eksiklik kuramı, Gauss, J.Kepler, Bernoulli ailesi, B.Russell, kısada olsa Gordion düğümü ve Sicim teorisi...

Özellikle büyücüler diye isimlendirdiği dört isim üzerinde daha belirgin durduyor; Arşimet, Galileo Galilei, Descartes, Isaac Newton. Kısa özetlersek: Arşimet matematiği soyut alandan çıkarıp fiziksel problemlere uygulayarak matematik ile doğa arasındaki bağı somutlaştırmıştır. Descartes geometriyi cebirle birleştirerek mekanı sayısallaştırır. Bu düşünce miraslarını sistematik ve işleyen bir fizik teorisine dönüştüren kişi ise Isaac Newton oluyor; Galileo ve Kepler’in bulgularını birleştirerek evrensel kütleçekim yasasını formüle ediyor ve kalküsü geliştirerek hareketi matematiksel kesinlikle ifade etmeyi başarmıştır.

Kitapta sıkça "matematik keşif mi icat mı" sorusuda yer alıyor. Sorunun cevabı Platoncu geleneğe göre, matematiksel gerçekler insan zihninden bağımsız olarak vardır; biz onları tıpkı yeni bir kıta keşfeder gibi ortaya çıkarırız. Buna karşılık çağdaş yaklaşımlar matematiği insan zihninin kurduğu sembolik bir sistem, yani bir icat olarak görüyor. Livio bu iki uç arasında daha dengeli bir orta yol sunuyor: Matematiksel kavram ve sembol sistemlerini biz inşa ederiz; ancak bu sistem kurulduktan sonra ortaya çıkan ilişkiler, zorunluluklar ve sonuçlar keşfedilir diyor. Yani matematik hem icattır hem keşif; insan zihni çerçeveyi kurar, fakat o çerçevenin içindeki yapısal doğrular ondan bağımsızdır.

Livio kitabın sonunu şöyle bağlıyor: "Kısacası insanoğlu, kasıp kavuran bir merak duygusu, asla pes etmeyen bir inat, yaratıcı bir hayal gücü ve müthiş bir kararlılıkla, etrafındaki fiziksel dünyayı tanımlamasını sağlayacak uygun matematik formları geliştirmişti."

Livio kitaba ismini veren soruya kesin bir cevap veremiyor fakat matematiğin insan ile evren arasındaki en güçlü entelektüel köprü olduğunu açıkça gösteriyor.

Herkese keyifli okumalar.

[Δx Δp ≥ ½ ħ · Rating 4 out of 5]

Sangat sederhana dan menggelitik pertanyaannya, Apakah Tuhan itu Seorang Matematikawan?

Jika Anda pernah membaca buku Biografi Angka Nol-nya Charles Seife, Anda bakal langsung faham, mengapa Tuhan seolah menganakemaskan matematika dibandingkan ilmu yang lainnya. Tuhan, anehnya, seolah menaruh minat khusus pada perkembangan matematika.

Sebagai contoh, siapa yang tak kenal Golden Ratio alias "Nisbah Emas"? nyaris semua struktur makhluk hidup memiliki perbandingan nisbah emas. mulai dari susunan tulang belulang manusia, susunan bunga dan anatomi tumbuhan, proporsi sempurna pada kerang dan hewan lain, hingga perbandingan jumlah lebah jantan betina dalam satu sarang, selalu merujuk pada nisbah emas. Pembaca Da Vinci Code pasti faham dan bisa menyebutkan contoh2 lain dari nisbah emas.

Dalam ilmu fisika, kita mengenal konstanta-konstanta alam yang begitu eksak, spesifik dan memerankan peranan penting. mulai dari konstanta gravitasi--sehingga kehidupan bisa terbentuk--hingga konstanta planck yang menjaga alam semesta tetap harmonis pada tempatnya. Dalam matematika, dikenal "bilangan transenden". sebuah bilangan yang memiliki harga eksak tak berhingga namun luar biasa penting dalam menopang keberlangsungan ilmu hitung. contoh sederhana adalah pi, π, harga eksak angka ini adalah 3,14159265358.... (Anda boleh menambahkan berapapun jumlah angka dibelakang desimal. toh tak berhingga, ilmuwan sendiri baru menemukan angka eksak π sampai 1,241,100,000,000 tempat desimal).

Memangnya apa pentingnya menghitung angka desimal π sampai sekian trilyun-trilyun angka desimal di belakang koma? bayangkan hal ini. Jika Anda menghitung keliling bumi dengan rumus yang menggunakan angka π sampai 11 angka desimal saja, maka ketelitian yang diperoleh baru sampai tingkat milimeter. Jika Anda menggunakan angka π sampai 39 desimal, maka Anda akan bisa menghitung setiap keliling alam semesta hingga ketelitian atom hidrogen. (catatan: Radius alam semesta diperkirakan sekitar 46.5 miliar tahun cahaya. Radius Hidrogen = 0.00000005 milimeter).

Bayangkan jika para ilmuwan berhasil menghitung keliling alam semesta dengan menggunakan π sampai desimal terakhir. Alam semesta seolah sudah dijelajahi sampai ke pelosoknya alias semua kekuasaanNya sudah terpetakan. Dengan alasan itu, mungkin bisa menjelaskan, mengapa Tuhan membuat angka π sampai tak berhingga besarnya, hanya untuk membuktikan, kita sebagai manusia begitu kecil dan terbatas dihadapanNya.

Ini, hanyalah sebagian kecil bukti obsesi Tuhan pada matematika.

Bahkan, dalam Islam, Tuhan punya dua nama yang langsung merepresentasikan bahwa Dia "seorang" Matematikawan sejati, Al Hisab (Maha Penghitung, merujuk pada sifat-Nya yang menciptakan sesuatu dengan perhitungan cermat) dan Al Muhsy (Maha Penghitung/Perancang, yang merujuk pada sifatNya bahwa Dia memlihara alam semseta dengan cermat; takdir, nasib dan pahala, semuanya sudah diperhitungkan dengan cermat olehNya)

tapi apakah benar Dia seorang Matematikawan?

dan mengapa harus matematika?

Pertanyaan pertama jelas agak sukar dijawab, tergantung Anda melihatnya dari segi mana. tapi untuk jawaban kedua sepertinya lebih mudah. Matematika, oleh para ilmuwan sendiri disebut sebagai The Queen of the Sciences. Tak ada satupun ilmu pengetahuan yang tak menggunakan matematika dalam studinya. mulai dari ilmu eksak seperti fisika, kimia, biologi, hingga ilmu-ilmu sosial seperti ekonomi, geografi, dan politik. semuanya perlu perhitungan matematika.

Leopold Kronecker, matematikawan kesohor, pernah berujar bahwa Tuhan hanya menciptakan angka. sisanya adalah ciptaan manusia. pendapat serupa dilontarkan oleh matematikawan besar lainnya nyaris 2000 tahun sebelumnya, Phytagoras, manusia yang tergila-gila pada angka sampai menyembah dewa angka, berujar bahwa alam semesta adalah kumpulan-kumpulan angka.

Meski ditulis oleh seorang yang berprofesi sebagai astrofisikawan--kebayang, apa pekerjaannya sehari-hari? hehe--, buku ini ditulis dengan runut, lancar, dan jernih. Anda tak harus menguasai matematika banyak-banyak untuk dapat memahami buku ini sampai akhir.

Dan Anda akan terkejut, terpesona dan takjub. Tuhan, ternyata punya sisi lain. Bukan sekedar Tuhan "agama tertentu yang haus disembah", bukan sekedar Tuhan para filsuf yang menimbulkan debat dan tafsir tak berkesudahan, atau Tuhan bangsa tertentu yang membela bangsa tertentu untuk memenangkan perang melawan bangsa lain. Anda berkenalan dengan Tuhan Yang Satu. Tuhan Yang Menciptakan Keajaiban melalui penciptaan. Tuhan Yang Gemar Matematika.

[Mazola1 · Rating 4 out of 5]

In Is God A Mathematician, Mario Livio tries to explain the "unreasonable effectiveness" of mathematics to make sense of nature. Why do so many basic truths of physics, nature and the universe obey mathematical laws? Livio also tackles the question of whether mathematics is discovered (an objective truth independent of human thought) or invented (the product of human thought and reasoning). Along the way, Livio provides a fascinating mini-history of the development of math, biographies of some of the greatest mathematicians and some of the most lovely and puzzling aspects of math. The book is clearly written, and does not require any advanced or sophisticated understanding of math. If you don't love or understand math, by the time you finish this book, you will have a better understanding and appreciation of math, and you will gain some insight into why math has fascinated and obsessed some of the best thinkers to ever live, and you will understand a little of the power of math to awe the human mind.

[Jafar · Rating 3 out of 5]

So – “the unreasonable effectiveness of mathematics.” Why is it that the laws of nature are so nicely expressed by mathematical formulas, and even more strangely, how is it possible that a theorist can manipulate his equations and predict something entirely new – like a new elementary particle – which will turn out to be real? Is nature based on mathematics? And what is mathematics anyway? Is it invented or discovered? All really fascinating questions. However, most of this book is math history. If you’ve read enough math history, you can skip to the last chapter where the main questions are discussed. Spoiler: there are no clear and easy answers.

[David · Rating 4 out of 5]

In this book, Livio addresses the question of why the principles and laws of mathematics seem so "unreasonably effective" in explaining the physical world. For instance, when Newton deduced the law of gravity, he could hardly have known that these mathematical laws would for six orders of magnitude more precision than the data he originally was trying to match. In a similar way, there are numerous instances in 20th century physics of mathematical principles, previously discovered by mathematicians and considered purely as logical curiosities, turning out to be stunningly accurate as descriptions of physical phenomenon. One notable example here is the magnetic moment of the electron, whose measured value matches mathematical calculations, based on the QED theory, to 12-digit accuracy.

Livio reviews the history of math, from Pythagoras to modern mathematicians such as Lobachevsky, who discovered hyperbolic geometry, and Kurt Godel, who showed that attempts to "prove" the axioms of mathematics consistent are doomed to failure. Indeed, there are many rather interesting mini-biographies of important mathematicians through the ages.

Finally, Livio addresses the fundamental question of what is mathematical reality -- the Platonic view that math really is there (somewhere) and we just discover it, to more radical interpretations, such as the claims by some that it is only a 'social construct'. In the end, Livio offers no pat answers, only questions.

I personally am mostly a Platonist, although I acknowledge the human element in mathematics. I thoroughly reject the views of cultural relativists in this area. The mere fact that some mathematical results have been found independently by people in different lands speaks against such notions. In my own research work, on numerous occasions myself and colleagues have "discovered" by computer mathematical formulas that had lain hidden. You can't tell me that the computer found "social constructs"...

[Jimmy Ele · Rating 4 out of 5]

Interesting for the sections on the Mathematicians such as Archimedes. Did it answer the question? No. I felt like I was just baited into reading the book. Mario Livio examines the Neoplatonic ideas of the origin of Mathematics as well as the AntiPlatonist argument. He seems to side with the AntiPlatonist argument in the end. I still really enjoyed the book and it led me to put some other books on my To Read shelf. All in all, I don't consider it time wasted to have read this book.

[Julius · Rating 5 out of 5]

¿Inventamos la matemática o la descubrimos? Esta premisa, aparentemente sencilla, es uno de los enigmas filosóficos más fascinantes de la historia de la ciencia y el eje central sobre el que Mario Livio construye una obra brillante. Con la soltura de quien domina la astrofísica y (no obstante, es uno de los astrofísicos encargados de operar el telescopio Hubble), al mismo tiempo, sabe contar una buena historia, el autor nos lleva a explorar por qué las matemáticas poseen lo que el físico Eugene Wigner llamó una "irrazonable eficacia" para describir las leyes fundamentales de la naturaleza. Resulta asombroso comprobar cómo ecuaciones abstractas, concebidas en el papel por pura lógica, terminan prediciendo la existencia de fenómenos físicos o partículas subatómicas siglos después.

Para Platón, las únicas cosas que existían de un modo real y completo eran las formas y las ideas de la matemática, porque, según sostenía, sólo en la matemática se puede obtener un conocimiento absolutamente cierto y objetivo. Así, en la mente de Platón, la matemática estaba íntimamente asociada con lo divino.

A través de un recorrido histórico impecable que abarca desde la vertiente mística de Pitágoras y la teoría de las ideas de Platón hasta las revoluciones conceptuales de Newton, Leibniz, Gauss y Einstein, el libro logra desmenuzar conceptos complejos de geometría, lógica y física teórica sin abrumar con tecnicismos. Livio no se limita a exponer el avance del conocimiento, sino que penetra en la mente de los pensadores y en el contexto intelectual de cada época, ofreciendo una visión panorámica donde la filosofía y la ciencia avanzan de la mano.

El Número Ciencial es el que Pitágoras define como la extensión y producción en acto de las razones seminales que se encuentran en la Mónada o en un grupo de Mónadas.[ 27] Así, los números no eran simples herramientas para denotar cantidades: los números debían ser descubiertos, y eran los agentes formativos que actuaban en la naturaleza.

Lo más valioso de la obra es su negativa a imponer una postura dogmática. En lugar de ofrecer una respuesta categórica a la pregunta del título, el autor expone con imparcialidad el debate entre el platonismo —que defiende un universo matemático preexistente esperando ser descubierto— y el formalismo o constructivismo, que entiende la matemática como una refinada invención de la mente humana para estructurar el caos. Si bien esto puede decepcionar a quien busque una conclusión rápida, convierte al texto en una invitación honesta a la reflexión personal.

Ojo - y esto es importante. Cuando yo pensaba que el libro iba a ser otra obra más donde se describen los pensamientos de otros científicos con cierta gracia, pero no se rasca más en la superficie, Mario Livio SÍ que da su opinión sobre si los números se inventan, tienen que ser descubiertos o es el lenguaje de Dios para comunicarse con nosotros. Eso precisamente es a lo que más valor le doy en la lectura. A esos charcos en los que el autor opina.

Si la matemática no reside en el mundo platónico, fuera del espacio y del tiempo, ni en el mundo físico, ¿significa que es únicamente un invento de los seres humanos? Por supuesto que no. De hecho, mi razonamiento en la próxima sección será que la mayoría de la matemática consiste en descubrimiento.

En definitiva, es una lectura imprescindible para apasionados de la divulgación científica y la filosofía, pero también para cualquier lector curioso que alguna vez se haya maravillado ante la precisa sintonía entre la lógica abstracta y la realidad objetiva del cosmos. Una obra sumamente amena, rigurosa y estimulante que deja una huella duradera sobre nuestra manera de entender el mundo.

Por todo ello, 5⭐.

[Stephie Williams · Rating 4 out of 5]

Mario Livio examines the difficult to figure out effectiveness of mathematics in science. He also discusses the nature of mathematics, in particularly is mathematics invented or discovered? The reason for this discussion is that it becomes important to how you view the effectiveness issue, which is the major topic of the book.

After stating the “mystery” of the effectiveness of mathematics in science in chapter one, Livio discusses the Greeks views on mathematics, especially Pythagoras and Plato, in chapter two. Chapters three and four review the work of Archimedes, Galileo, Descartes, and Newton describing how they use mathematics to describe the universe, after which in chapter five covers probability and statistics. Chapter six discusses the effect of non-Euclidean geometries on the issues. Chapter seven covers the logicians and formalists attempts to secure the foundations of mathematics. Chapter seven explores the main question of the book directly, and finally Livio wraps things up by including whether mathematics is invented or discovered. He concludes it is both. We invent things like prime numbers, then discover relationships among them.

The following are some comments I have on the notes I took while reading the book. Page numbers are in brackets [] from the SIMON & SCHUSTER hardcover edition of January 2009.

[10] In an initial discussion on the invented/discover dichotomy Livio states: “Martin Gardner, the famous author of numerous texts in recreational mathematics, also takes the side of mathematics as a discovery. To him, there is no question that numbers and mathematics have their own existence, whether humans know about them or not.” (original italics) Gardner was also a theist, so a separate existence for mathematical objects and structures comes as no surprise. Of course, just because he is a theist does not make him wrong or right on the mathematical issue.

[198-201] He presents a story about Kurt Godel’s, of incompleteness fame, adventures in gaining his United States citizenship related by Oskar Morgenstern, a collegue of both Godel and Einstein at the Institute for Advanced Study in Princeton, New Jersey. Godel according to the story figured out a way that the United States could be turn into a dictatorship under the Constitution. Morgenstern and Einstein furiously tried to get Godel not to reveal this to the judge at the citizenship hearing. Godel even is reported saying to the judge: “Oh, yes, I can prove it.” (italics are mine) However, having heard this story several times before, it is never related what Godel’s proof of his claim was. There are some today that worry that Trump will attempt to become a dictator. I, however, doubt that this will ever happen.

[@227] I thought of how one could go about proving mathematical realism. I mean, where is this realm of mathematics? A mathematical heaven of sorts? It just seems unlikely that one could prove such a place exists, like trying to prove god’s existence, which so far has been an absolute failure to my knowledge.

[228] He relates Max Tegmark’s argument for the universe being mathematical, not physical. In a final theory of everything it “cannot include any concepts such as ‘subatomic particles,’ ‘vibrating strings,’ ‘warped spacetime,’ or other . . . [physical] constructs.” This seems awful close to eliminative materialism’s jettisoning of folk psychology terms (e.g. feel, think, believe, want, etc). Tegmark faces the same struggles as the Churchland’s (proponents of eliminative materialism) to show that all that exist is the brain and its states. At least the Churchland’s can show that there are not any proven validity to folk psychology as a theory of mind.

[242] Here Livio presents his view on the invention/discovery dichotomy. “’Is mathematics created or discovered?’ is the wrong question to ask because it implies that the answer has to be one or the other and that the two possibilities are mutually exclusive. Instead, I suggest that mathematics is partly created and partly discovered. Humans commonly invent mathematical concepts and discover the relations among those concepts.” I find this reasonable, but wonder does it really reveal anything profound on the issue. I have a friend who thinks the question itself is ill-posed for other reasons, but I will not try to related them here.

[243-4] Quoting Sir Michael Atiyah, “whose views on the nature of mathematics” Livio shares, on the effectiveness of mathematics in science: “If one views the brain in its evolutionary context then the mysterious success of mathematics in the physical sciences is at least partially explained. The brain evolved in order to deal with the physical world, so it should not be too surprising that it has developed a language, mathematics, that is well suited for the purpose.” I agree that an evolutionary perspective needs to be part of the answer to this issue.

[252] After asking: “Have we then solved the mystery of the effectiveness of mathematics once in for all?” he quotes Bertrand Russell from his The Problems of Philosophy: “Thus, to sum up our discussion of the value of philosophy; philosophy is to be studied, not for the sake of any definite answers to its questions, since no definitive answers can, as a rule, be known to be true, but rather for the sake of the questions themselves; because these questions enlarge our conception of what is possible, enrich our intellectual imagination and diminish the dogmatic reassurance which closes the mind against speculation; but above all because, through the greatness of the universe which philosophy contemplates, the mind is also rendered great, and becomes capable of that union with the universe which constitutes it highest good.” While I do not confer with Russell’s mystical rewards of philosophy, I do agree with Him that the asking of questions enlarges our capacity for “intellectual imagination.” In a sense it is the journey itself that is the most important thing philosophy.*

The book was better than I supposed before I started reading it. I was under the wrong impression that Livio held the views of Tegmark, based on a missed remember PBS science show - “The Math Mystery.” I was pleased when it became obvious to me that this was not the way he saw the relationship between mathematics and science. The historical sections were good, but there was nothing too new from what I already new. Still he writes well, and he explains things in understandable ways, making it an enjoyable read.

If you are interested in the relationship between science and mathematics, this book should be of interest to you. If you are looking for a definitive answer to the “mystery” you will not find it here, but this does not distract from the honest coverage that Livio provides. He does not pontificate. I will add for the nonconversant with mathematical equation the book has a very limited amount of these.

* To see a fuller exposition of my views on philosophy see my “What Is Philosophy?” blog post @ https://aquestionersjourney.wordpress...

[Tino · Rating 4 out of 5]

Quite good! And you don’t need a phd to understand this either. 4 stars.

[Nostromo · Rating 3 out of 5]

Става дума за история и философия на математиката. Мястото на математиката, като дисциплина, в книгата реално се намира само на едно място - в заглавието. Същото се отнася и за Бог (доколкото той може да се намира някъде).

Ливио се е опитал да направи едно неутрално изследване за "необяснимата ефективност на математиката" в света, който обитаваме, а бележките и библиографията в края на книгата, говорят, че се е постарал да си напише домашното. И се е получило. Като повечето домашни - сухо, педантично, но поне информативно. Информативно, но и повърхностно, защото се плъзга по математиката, както аз по чина в четвърти клас в час по въпросния предмет. Добре, че в следващите класове се научих да внимавам какво ми говорят, иначе "Математик ли е бог?" щеше да ми се стори, като изцяло нова вселена насред остатъците от средното ми образование.

Но нека не изисквам от книгата нещо, което вероятно никога не се е опитвала да бъде, а именно забавен и увлекателен пътеводител в света на механизмите и принципите на математиката. Исторически ще се запознаете с Архимед и математическите школи преди него. С Галилей и познанието за това, което милиони години е блещукало над главите на знайни и все още неизкопани фосили. С Нютон и законите на механиката. Докато във философски план ще се опитате (и няма да успеете) да си отговорите на въпроса измисляме ли математиката или просто я откриваме. Е, няма да ви спойлна много, ако ви кажа, че не е нито едното от двете, а според автора е и двете едновременно. Но ако си падате по философски празнословия, последната девета глава ще я прочетете без да заспите, което аз признавам си не успях да направя.

И все пак Ливио слага ред в една материя, която класическото образование в България предвид наблюденията ми целенасочено се опитва да забули в мистерия, така че няма да загубите много, ако отделите време да прехвърлите по-интересните пасажи от книгата. Всъщност най-забавната част от нея е цитат от друга книга...

http://thingnothing.blogspot.com/2014...

[James (JD) Dittes]

Since the enlightenment, mathematics and the sciences have ascended heights where God alone used to dwell, growing in scope and complexity and marveling the world with miracles like fusion, antibiotics, and space travel. Livio's title, "Is God a Mathematician?" isn't so much an effort to unite math and theology as it is an effort to find out how omnipotent and omniscient math can truly be.

The first half reads as a history of science--going over the ground of Archimides, Galileo, Copernicus and others (I'm not sure why all books of science cover this territory, but I especially liked Livio's analysis of Descartes).

The part I found most fascinating was the last 100 pages, where Livio goes into modern mathematical enigmas and outlines the developments of the last 150 years. I must admit, I never studied math beyond college pre-calculus, so this book was quite a challenge for me. It was an honor, then, to peer over the shoulder of a true mathematician and try to wrap my brain around this fascinating subject.

[Kamila Friedrich · Rating 4 out of 5]

Thus, to sum up our discussion of the value of philosophy;
Philosophy is to be studied, not for the sake of any definite answers
to its questions, since no definite answers can, as a rule, be known
to be true, but rather for the sake of the questions themselves;
because these questions enlarge our conception of what is possible,
enrich our intellectual imagination and diminish the dogmatic
assurance which closes the mind against speculation; but above all
because, through the greatness of the universe which philosophy
contemplates, the mind is also rendered great and becomes capable
of that union with the universe which constitutes its highest good.

[Kyle · Rating 2 out of 5]

Lots of math but not much God...felt like more of a “history of math” book than a Christian math book as I was hoping for. Great information that debates the issue of whether math is invented or discovered (which I personally believe is a mix of both). Some dry humor made it interesting. If you enjoy math, it’s not a bad read. But if you’re looking for a Christian math book, this isn’t the one for you.

[jaroiva · Rating 3 out of 5]

Připadalo mi, jako bych se vrátila na vysokou. Ty matematické disciplíny, které mě nebavily tam, mě nebavily ani tady, a ty, které mě bavily, mi připadaly tady málo rozvedené :)
Vlastně jsem nepochopila, pro koho je kniha určena.

A taky jsem čekala, že se bude víc věnovat spojení matematiky a boha. Byl tu sice letmo zmíněn Descartův důkaz existence boha, ale to opravdu jen letmo.
Jinak jde o pěkný průřez historií matematiky.
Proto bych knížku asi překřtila na "Stručné dějiny matematiky".

[Liviu · Rating 4 out of 5]

Pi in The Sky is still the gold standard for books trying to explain why mathematics fits the real world with unreasonable effectiveness as the famous Wigner quote puts it. This book is still worth a read, but does not bring anything new, though it's entertaining and well written

[Rana Rafeh · Rating 2 out of 5]

This book is neither mathematically rigorous, nor does it delve into philosophical thought enough to answer the question: Is God a Mathematician… rather it feels like discrete cherry picked examples used to support Livio’s perspective

[Ami Iida · Rating 3 out of 5]

It's various mathematical categories.

[Sajith Kumar · Rating 3 out of 5]

Science is an attempt to read God’s mind which is evident in the physical reality as the rules and principles which hold the world together. Livio’s book is an elegant attempt to tell the epic story of man’s quest to peer into nature itself and to grasp its fundamental principles with the help of his greatest intellectual tool – mathematics. Its extraordinary ability to describe the world has been a source of wonder to philosophists ever. This feat comes in two varieties. In one category named active mode, scientists deduce mathematical laws applicable to an event after carefully observing it, while in the other, passive mode, mathematical functions which were formulated long ago in totally unrelated circumstances suddenly find application to explain new discoveries in science. Judging from the closeness with which mathematical predictions approach reality, we are tempted to think that God is a mathematician. So, the answer to the rhetorical question in the title is in the affirmative and the 250-odd pages explain why it is so. It may be mentioned in passing that another book titled ‘The Loom of God’ by Clifford Pickover (reviewed earlier in this blog) also follows a similar theme. Mario Livio is a noted author who is also an astrophysicist and the head of the Office of Public Outreach at the Hubble Telescope Science Institute.

A noteworthy feature of mathematics is its strikingly effective provenance to explain natural features and phenomena. Why should it be so? Mathematics is anyway a product of human contemplation and analysis. If this fruit of human intelligence so faithfully displays an uncanny ability to explain and predict nature, it is no wonder that a group of philosophers – a large one indeed – postulated the existence of mathematics in an idealized Platonic world, whose reflections on the physical world constituted our everyday adventures. This raises the pertinent question whether mathematics is discovered or invented. The niceties of such philosophical speculation need not detain the readers, but Livio presents a deeply speculative question in an easy to digest way. The ideas of Platonic world and discovery are compatible, in the sense that the numbers and shapes already existed in a perfect, imaginary world until man stumbled upon them in a spark of intellectual brilliance. Just like America existed before it was ‘discovered’ by Columbus, or Vikings, or even by that Turkish guy – who provided some much needed comic relief in international discourse a few months ago – mathematics existed right from the universe’s moment of being. But quite a few philosophers, and such humble beings like myself, differs from this point of view. According to this theory, mathematics is an abstract concept developed by man with the help of his extraordinary ability to detect patterns in nature. The book provides ample room for general readers to get familiar with this dichotomy that surrounds mathematics’ existence.

History of science occupies a major portion of the book, but presented in an admirable way that commands attention from readers. Freely interspersed with witty anecdotes and informative quotes from authors present and past, the text stands tall as a testimony to the immense amount of research that had gone in to the publication of it. Livio identifies Archimedes, Newton and Gauss as the three greatest mathematicians of all time, but does not restrict his pen to these three. Would any discussion on the development of science through the Renaissance era be complete without a solid reference to that mathematics professor from Padua, Italy – Galileo Galilei, no less? Galileo’s trial and the stifling overlordship of blind faith over reason is a topic you would find described umpteen number of times in any book that deals with the history of science through the turbulent 17th century. Livio’s description would feel to be delightfully elegant to new readers. Old readers also would find the narration to be very congenial. This book extends the story to other mathematicians, including Descartes, and the Bernoullis. The sibling rivalry between Jakob and his brother Johann Bernoulli is brought to light with a quote from a letter the younger Johann wrote to his friend in which he exults at defeating his elder brother in the solution to a vexing problem. Mathematicians are also human, after all!

Even though Livio considers Gauss to be one of the three greatest ever mathematicians, nothing much is said about him apart from casual references in the context of non-Euclidean geometry. But this shortfall is more than leveled by the extensive discussion on the new developments in mathematics that took place during the last two centuries. The new sprouts are so revolutionary as to merit the epithet that man had broken free from the shackles of classical learning and began to explore nature in the light of a new creative spirit. A mind boggling array of discoveries had taken place in this period, but ordinary readers find it difficult to comprehend the practical purpose of many of them. Non-Euclidean geometry is however very helpful in estimating the shortest possible distance between any two points on a spherical surface. Aircrafts usually follow these shortest routes. But such hyperbolic geometry is extended to such extreme lengths that no apparent use is evident – yet! At around this time, logic was also linked to mathematics so as to strengthen the mutual foundations. Boolean algebra originated as the systematic representation of logic as ordinary algebra was to scientific concepts. Enhancement of geometry to many more dimensions than three enabled it to stand as the structural framework of advanced theories on the origins of the cosmos in the form of string theory, which postulates ten dimensions. This also shows the effectiveness of the discipline as a faithful representative of nature. But the long chapters on logic and discussions on its consistency are hard to enjoy for average readers.

A frequent source of controversy among mathematicians is the question whether its concepts should provide practical applications for human use. Such a notion itself is anathema to many practitioners who bask at the sheer glory of pure mathematics. Archimedes and G H Hardy were two mathematicians of this school. What would have been their impression when they saw their concepts eagerly accepted by the scholars and put to uses which provide immense value to their own societies? Archimedes is credited with the invention of a screw pump, levers of varying complexities, optical instruments and defensive apparatus, while much progress in cryptography is attributed to Hardy. There were mathematicians in the other camp as well, like Gerolamo Cardano, who wouldn’t conceptualize the definition of more dimensions than three because no practical utility was existent at that time, nor conceived to be feasible in the near future.

The book is splendidly written, having a good structure in presenting ideas. It is also graced with a good number of anecdotes, pictures and illustrations. There is an immense collection of notes mentioned in the main text and a sizable bibliography is listed. A nice and comprehensive index completes the attractive side of the book. On the negative part, about a quarter of the text starting from logic and its relations to mathematics is highly abstract, making life difficult for the readers. Fortunately, no harm is done even if you were to simply bypass those chapters and dive straight to the last one.

The book is highly recommended.

[Jeff Hascall · Rating 4 out of 5]

This non-fiction read explores an issue I've never mentally wrestled with before. It's a pressing question from the corner of Mathematics St. and Philosophy Blvd. "Where does math come from?" Is it an integral part of a system at the heart of the universe that we are constantly discovering or is it merely the formal method we've created and placed on top of our perception of the universe to explain what we see? Or, in Livio's words, how do we wrestle with the "unreasonable effectiveness" of mathematics? We may side with the Platonists, admitting our perceived world is just shadows on the cave wall, or the formalists, who our many systems to be constructs designed by the human brain.

To accomplish this goal, he spends the vast majority of the book walking us back through the history of math and identifies what the greatest minds of mathematics thought about the issue. As a mathematician himself, his history is robust and insightful. He ties nearly every character he reviews to their contemporaries, explaining how their views matched up to the common thought of the day. He shows how many early mathematicians reveled in the idea that there was a geometric reality beyond our own where perfect shapes actually exist. He marvels at the places where math broke into physical reality, such as how accurate Newton's predictions of gravity continue to be (even centuries later) and what this means for the discussion. Later, he brings us to non-Euclidean geometry, which shook the mathematical world by showing that other theoretical realities existed where our accepted laws did not always apply.

If the above piques your interest, the book is well-written and will hold your attention. Livio shows great talent at bringing you into the mindset of each time period. But if your eyes glaze over at the idea of reading about math, Livio does not apologize for the depth of the material nor the intellectual wrestling it requires. As someone who could only remember the highlights of past mathematicians and was hooked on the ideas, it was educational and sparked a lot of internal thought.

It's worth noting that God is not a character in the book and His existence is not discussed or debated, despite the heavy billing on the cover of the book. It's only used as a way to rephrase the central question (or get you to pick up the book in the first place). In fact, Livio deftly avoids the topic altogether, even when discussing the persecution of the church on Copernicus and Galileo.

In total, if this book's subject matter sounds interesting, let me assure you that you will probably find it as such. He provides his short version of an answer to the question at the end, although he is sure to show how both sides of the argument continue to be discussed. As any experienced tour guide, Mario Livio leaves you better educated from the journey, and curious for more.

[Alima · Rating 4 out of 5]

При взгляде на обложку рускояхысного издания у меня сложилось мнение, что в книге описаны интересные неочевидные математические законы и отношения в Космосе. Ну в таком роде, что Солнце и Земля друг от друга на n км поэтому у нас есть жизнь. Или если бы такая то планета была на столько то градусов отклонена по орбите было бы тото. Ан нет.
В книге преводится история математических открытий. Ну а задачей книги было ответить на вопрос Энштейна: математику люди открыли или придумали? Как так что по что мы придумали точь в точь отражает физическую действительность?
Последователи Платона считают математику открытием, а формалисты изобретением.
По своим убеждениям я больше платоник, однако не люблю вешать ярлыки. Наука она гораздо более многогранна, чем мы можем себе представить. Здесь и логика и физика и генетика и лингвистика тесно сплетены с математикой.
Название книги заслуживает особого внимания. В Западной доктрине наука всегда противопоставляется вере, тогда как в Исламе вся наука исновывается на вере в Всевышнего Создателя. И научные изыскания базировались на Священном Коране и Сунне Пророка. Западные ученые же придерживаются секуляристской философии и через науку пытались доказать отсутствие Бога.
И такая литература помогает проследить путь формирования человеческой мысли. (пусть и западноцентричной, здесь нужно мыслить критически)
Оцениваю на 4/5. Обложка у рускоязычного издания не удачная. Складыватся ложное представление о книге. Думаю для широкого круга она слишком специфична.

[Vedat Guven · Rating 5 out of 5]

Yazar, Hubble Uzay Teleskopu Bilim Enstitüsü Başkanı'dır. Matematiğin gelişmesi, felsefe ve matematiğin birlikteliği, sonradan matematiğin diğer bilim dallarında da kullanılmasını anlatıyor. Matematik tarihinde Arşimed için usta, Galileo için isyankar, Descartes için şüpheci ve Newton için de dev yakıştırması yapmaktadır yazar. Mendel'in gen çalışmaları ve sosyal bilimlerde de olasılık hesaplamaları ve istatistiğin gelişmesiyle matematiği artık sayısal olmayan beşeri bilimlerde de görmeye başlıyoruz.

Matematikteki gelişmeler bazen kullanımı olmadan da vuku bulabiliyor. Örneğin topolojide gördüğümüz düğüm konusu başlangıçta atom modeli hesaplamalarında kullanılıyordu. Ancak 100 küsür sene önce önerilen atom modeli yanlış olduğunu gördük sonradan. Yani çekirdek etrafında belli yörüngelerde dolaşan elektron modeli meğersem yanlışmış. Dolayısıyla Düğüm Teorisi'nin hayatımızda uygulama alanı kalmamıştı. Yine de bazı matematikçiler Düğüm Teorisi üzerine onlarca yıl çalıştı. Sonradan DNA ile ilgili analizlerde ve daha da sonrasında sicim teorisinde fazlaca uygulama alanı buldu Düğüm Teorisi. Bu durum matematiğin ne kadar da doğanın içinde mevcut olduğunu göstermektedir.

Kitabın irdelediği soru: Matematik keşif midir? İcat mıdır? Matematik felsefesinin en önemli sorusu. Platon ağırlıklı görüş keşif olduğu yönünde. Ama yeni dönem bilim insanları arasında icat olduğu görüşü de yaygın. Ne dersiniz? Sanki biraz keşif biraz icat gibi durmuyor mu?

[Robert B · Rating 3 out of 5]

The catchy title is somewhat misleading, as Livio, an astrophysicist, does not really look at any aspect of God. Instead, Livio explores “the unreasonable effectiveness” of math, asking whether math is something “out there” in the real world that people have discovered or whether it is an invention of the human mind that just happens to apply well to reality. He answers the question by examining the work of great mathematicians, including Pythagoras, Descartes, Galileo, Newton, et al. In the end, Livio decides that the question itself is wrong, that math isn’t one or the other, isn’t discovery or invention, but is partially created and partially discovered. (He also notes that its explanatory power is limited, that there is much of the world that math does not explain.) The book is a good overview of the history of some mathematical ideas but is unfortunately rather cursory in its discussion of the main question of math’s effectiveness.

[Erika Landavazo · Rating 4 out of 5]

Last two chapters are phenomenal and will wrap everything up beautifully for you. The first chapters drag especially if you’re already familiar with the discovery of gravity,… many (practically all) books of this kind start with the story of Newton and Copernicus and that can get old. I’d say your safe to skip chapters.
Chapter 8 I’d say is where it starts to pick up and get interesting as he talks about mathematicians discovering the different patterns of knots “knot theory” which then helped develop the understanding of the basic structures of our world from dna sequencing to the string theory!

[Avery Northern · Rating 4 out of 5]

My favorite thing about this book is that I went into it with zero understanding of Platonic ideals and even less understanding of how anyone could possibly think of mathematics as invented (as opposed to discovered) and came out with considerably more understanding of both of those topics.

The only reason this gets 4 stars instead of 5 is that I felt that some of the historical context was kinda unnecessary to my understanding of the concepts being discussed. Learning about mathematical history is always fun, but in this case I felt it distracted a bit from the main topics rather than adding to them.

[Vegeta · Rating 4 out of 5]

I'm amazed how precisely nature dances to the tune so carelessly, and at how the experimenters and the theorists can measure and calculate her dance to a part in a trillion.
Definitely a must read, the way in which Mario described and explained every topic and history about mathematics, plus some fascinating insights of his own is really praiseworthy.