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Is God a Mathematician?
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.
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.
320 pages, Hardcover
First published January 1, 2009
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July 27, 2011
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.
November 2, 2015
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.
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.
August 31, 2013
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.
May 16, 2010
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.
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.
February 7, 2009
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.
June 2, 2009
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"...
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"...
March 10, 2017
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.
July 26, 2016
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.
July 18, 2017
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...
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...
January 22, 2014
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...
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...
Read
July 31, 2011Since 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.
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.
February 9, 2021
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.
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.
December 1, 2017
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.
July 21, 2020
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".
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".
February 9, 2009
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
May 9, 2019
It's various mathematical categories.
September 18, 2017
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.
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.
May 22, 2017
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.
April 3, 2022
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.
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.
July 30, 2021
Two of favorite topics - God and mathematics. Numbers, algebra, geometry, calculus, etc. invented by man or are they discoveries of God's work? The book tries to answer the question while discussing some major math topics from Euclidean geometry to string theory - fun stuff.
January 26, 2018
I thoroughly enjoyed this book. The catchy title is a little misleading. The book deals with how well mathematics describes nature and with the question, Is Mathematics Invented or Discovered?
May 24, 2017
Muy lindo libro, facil de leer y a la vez muy profundo. Muy recomendable para los interesados por el fundamento filosofico de la matematica
September 20, 2010
Senior astrophysicist at the Hubble Space Telescope Science Institute and author of a few other math books aimed at the general public, Mario Livio has written a short, accessible, and in many ways profound exploration of the nature of mathematics. He centers his book around two questions: 1. "Is mathematics ultimately invented or discovered?" and 2. "Why is mathematics so effective and productive in explaining the world around us that it even yields new knowledge?"
He frames his inquiry with what physicist Roger Penrose describes as the triple mystery. The idea is that there are three worlds that people experience: the world of physical reality, the world of our minds, and the abstract world of mathematics. Then the mysteries are as follows: 1. why would world of physical reality give rise to our minds that perceive the reality? 2. why would our minds give rise to abstract mathematics? and 3. why does mathematics so effectively describe the physical reality in which we exist?
Livio jumps off from there to do a very quick run through the history of mathematics from Pythagoras and Plato to Einstein and modern day theoretical physicists by way of Archimedes, Galileo, DeCartes, Newton, Gauss, Riemann, Boole, and Russell. What he concludes in the end, is a bit of a cop out, but an utterly convincing one. Mathematics is ultimately both invented and discovered. Namely, he suggests that humans have invented certain basic concepts, like Euclid's axioms, and then they discover the implications of those axioms. These discoveries are no less true because they are built on an invention of the human mind, but they are only true because of the initial acceptance of the foundational axioms that we invented.
On the question of why math so effectively describes the physical world, Livio is still a bit baffled as we all should be. There is much to suggest that mathematics comes from our observations of our physical environment and that there is a innate way our brains make sense of those observations through mathematics. However, we have seen many times where mathematicians have worked in very abstract areas with no intention of seeing applicability to the physical world, only to see direct application to the way our universe operates. This truly is inexplicable at the moment.
He ends with the following quote from Bertrand Russell's The Problems of Philosophy that I thinks speaks to the reason for studying anything:
"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."
He frames his inquiry with what physicist Roger Penrose describes as the triple mystery. The idea is that there are three worlds that people experience: the world of physical reality, the world of our minds, and the abstract world of mathematics. Then the mysteries are as follows: 1. why would world of physical reality give rise to our minds that perceive the reality? 2. why would our minds give rise to abstract mathematics? and 3. why does mathematics so effectively describe the physical reality in which we exist?
Livio jumps off from there to do a very quick run through the history of mathematics from Pythagoras and Plato to Einstein and modern day theoretical physicists by way of Archimedes, Galileo, DeCartes, Newton, Gauss, Riemann, Boole, and Russell. What he concludes in the end, is a bit of a cop out, but an utterly convincing one. Mathematics is ultimately both invented and discovered. Namely, he suggests that humans have invented certain basic concepts, like Euclid's axioms, and then they discover the implications of those axioms. These discoveries are no less true because they are built on an invention of the human mind, but they are only true because of the initial acceptance of the foundational axioms that we invented.
On the question of why math so effectively describes the physical world, Livio is still a bit baffled as we all should be. There is much to suggest that mathematics comes from our observations of our physical environment and that there is a innate way our brains make sense of those observations through mathematics. However, we have seen many times where mathematicians have worked in very abstract areas with no intention of seeing applicability to the physical world, only to see direct application to the way our universe operates. This truly is inexplicable at the moment.
He ends with the following quote from Bertrand Russell's The Problems of Philosophy that I thinks speaks to the reason for studying anything:
"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."
May 12, 2022
Is God a Mathematician?
The short answer is “no.” But like all answers that imply a binary set of definite possibilities, the short answer is more misleading than leading. This is an excellent book. No one capable of understanding it would fail to profit from reading it. So I’m not going to focus on its myriad virtues, which you can easily find in other reviews, but on it’s few but essential problems.
The principle question the book ask is this: Was math invented or discovered? This translates to “Was the universe created (or better ‘is it governed’) by mathematical principles that are ‘ideal’ in the Platonic sense?” Is there a math that is outside the universe (in some sense not perfectly understood) whose principles are the principles that makes the universe work as it does? Along the way he also asks “Is math a language?”
The book’s slightly longer and somewhat more helpful answer is “both.” Math itself is created, but what math reveals is discovered.
I say somewhat more helpful answer because this answer maintains the uninvestigated distinction between “invention” and “discovery” that, had it been investigated, would have yielded a richer answer.
Mario Livio does not adequately define his terms. He assumes we know what it means to be invented—like the microphone—or discovered—like a new planet. The new planet was always there, but someone found it for the first time. The microphone never existed before. Someone made it. This may seem as obvious to you as it does to Livio, but even within his book, unnoticed problems arise.
First problem: prime numbers, he says, were invented. Any number of cultures had numbering systems, but most of the cultures did not have a concept of prime numbers, and they got along fine with out them. Western Mathematicians uniquely decided to invent this concept. That 9 ÷ 3 = 3 was however discovered.
Second problem: No one, Livio says, would say that Shakespeare “invented” Hamlet.
But that there are numbers that are only divisible by themselves and one is true in any counting system. They exist even if unnoticed. So do they really differ in a completely different way from the answer to a problem of division? Does it really make sense, in this dualistic thinking, to say that the concept of prime number has to be invented rather than that it has to be recognized? It may be that either option can be supported. And this brings into question the very distinction between invented and discovered.
And as for Shakespeare, one person who would have said that Shakespeare invented Hamlet was Shakespeare. “I’ll give you a verse to this note that I made yesterday in despite of my invention,” says Jacques in As You Like It to show how clever he is before having to make something up. In fact any educated person would have said so. That is what the word meant in Shakespeare’s day. What Shakespeare would never have said was that he created Hamlet.
Of course we can say that words change their meaning. So this doesn’t count against Livio.
It’s true that words change their meaning. They do so when the concepts that they supported in their previous meanings are no longer supportable. The very “invent/discover” distinction which we tend to believe is too obvious to need strict definition is one that Shakespeare would not immediately wrap his mind around. The corollary for us, who profit so much from reading Livio’s book, is that we too may need to rethink our very distinction between invention and discovery.
To come at this from another angle, Is God a Mathematician is a book build on two conundrums: if math is invented how can it predict facts about the universe that were not even suspected at the time the math was invented? How can the mathematical theory of knots, useless the purpose for which it was invented, explain the structure of molecules? The conundrum will go along way toward taking care of itself when we understand that discovery and invention do not describe two sets without common elements—a fact that this book needs to maintain that within the sent of “invented” things is a member called math, in which there are things that were discovered. If we have to have it both ways, or one way at one moment and another way at another, then the problem is certainly in the question or the model that gives rise to the question and not in the thing the question is posed to explain.
The second conundrum, which the book brings up several times but is not deeply interested in is this: Is Math a language? Livio will tell us it sort of is and sort of isn’t. He doesn’t believe much depends on a rigorous answer to this question, and he does not give one. This makes sense given the structure of thought in which the book operates. When it gets to the point where something can be or not be a language at the same time, it closes the door and goes better lighted hallways.
It does seem like a troublesome question not admitting of easy answer. If math is a language, how come small children, who are so good at acquiring languages have such trouble learning math? On the other hand, it is a symbolic structure made of signs representing concepts. It works by rules of syntax and grammar.
The problem however is only an apparent one, like a knot that is just a tangle that disappears with a tug. Math is not a language. Math is something we do in language. When I do math, I do it in English. When a French person does math, they do it in French. Math appears at first glance to be a language only because we use the same representations, the same words with the same spelling to represent the same concepts “2” is two in English and deux in French and er in Mandarin, but we all spell that concept as 2 when we do math. (A side note, Livio’s short but illuminating excursions into the history of math leave out the essential observation of the indebtedness of math to Arabic numerals.) Why do children have trouble with math? For the same reason they have trouble with logic (which no one calls a language) and with diplomacy and with any of the more complicated functions we do in language. What children acquire easily is vocabulary and syntax. Whatever it is they are capable of expressing they easily learn to express from one natural language to another.
Finally then my point is that Livio’s question about the discover v. invention of math is of the same type as his question of whether math is or is not a language. If a better vocabulary for thinking about math is developed (and I’m sure it has already been developed, though I can’t point to it at the moment), then the problem itself goes away. At least I suspect that is so.
All that said, this is a terrific book. For someone who gave up on math after three semesters of calculus it makes me re-think my choice. What I do not know because I did not get into higher mathematics is a field of wonder that I would love to explore. But life only allows us so many loves. And this peek at what I cannot explore further was infinitely worth the time I spent in the doorway.
The short answer is “no.” But like all answers that imply a binary set of definite possibilities, the short answer is more misleading than leading. This is an excellent book. No one capable of understanding it would fail to profit from reading it. So I’m not going to focus on its myriad virtues, which you can easily find in other reviews, but on it’s few but essential problems.
The principle question the book ask is this: Was math invented or discovered? This translates to “Was the universe created (or better ‘is it governed’) by mathematical principles that are ‘ideal’ in the Platonic sense?” Is there a math that is outside the universe (in some sense not perfectly understood) whose principles are the principles that makes the universe work as it does? Along the way he also asks “Is math a language?”
The book’s slightly longer and somewhat more helpful answer is “both.” Math itself is created, but what math reveals is discovered.
I say somewhat more helpful answer because this answer maintains the uninvestigated distinction between “invention” and “discovery” that, had it been investigated, would have yielded a richer answer.
Mario Livio does not adequately define his terms. He assumes we know what it means to be invented—like the microphone—or discovered—like a new planet. The new planet was always there, but someone found it for the first time. The microphone never existed before. Someone made it. This may seem as obvious to you as it does to Livio, but even within his book, unnoticed problems arise.
First problem: prime numbers, he says, were invented. Any number of cultures had numbering systems, but most of the cultures did not have a concept of prime numbers, and they got along fine with out them. Western Mathematicians uniquely decided to invent this concept. That 9 ÷ 3 = 3 was however discovered.
Second problem: No one, Livio says, would say that Shakespeare “invented” Hamlet.
But that there are numbers that are only divisible by themselves and one is true in any counting system. They exist even if unnoticed. So do they really differ in a completely different way from the answer to a problem of division? Does it really make sense, in this dualistic thinking, to say that the concept of prime number has to be invented rather than that it has to be recognized? It may be that either option can be supported. And this brings into question the very distinction between invented and discovered.
And as for Shakespeare, one person who would have said that Shakespeare invented Hamlet was Shakespeare. “I’ll give you a verse to this note that I made yesterday in despite of my invention,” says Jacques in As You Like It to show how clever he is before having to make something up. In fact any educated person would have said so. That is what the word meant in Shakespeare’s day. What Shakespeare would never have said was that he created Hamlet.
Of course we can say that words change their meaning. So this doesn’t count against Livio.
It’s true that words change their meaning. They do so when the concepts that they supported in their previous meanings are no longer supportable. The very “invent/discover” distinction which we tend to believe is too obvious to need strict definition is one that Shakespeare would not immediately wrap his mind around. The corollary for us, who profit so much from reading Livio’s book, is that we too may need to rethink our very distinction between invention and discovery.
To come at this from another angle, Is God a Mathematician is a book build on two conundrums: if math is invented how can it predict facts about the universe that were not even suspected at the time the math was invented? How can the mathematical theory of knots, useless the purpose for which it was invented, explain the structure of molecules? The conundrum will go along way toward taking care of itself when we understand that discovery and invention do not describe two sets without common elements—a fact that this book needs to maintain that within the sent of “invented” things is a member called math, in which there are things that were discovered. If we have to have it both ways, or one way at one moment and another way at another, then the problem is certainly in the question or the model that gives rise to the question and not in the thing the question is posed to explain.
The second conundrum, which the book brings up several times but is not deeply interested in is this: Is Math a language? Livio will tell us it sort of is and sort of isn’t. He doesn’t believe much depends on a rigorous answer to this question, and he does not give one. This makes sense given the structure of thought in which the book operates. When it gets to the point where something can be or not be a language at the same time, it closes the door and goes better lighted hallways.
It does seem like a troublesome question not admitting of easy answer. If math is a language, how come small children, who are so good at acquiring languages have such trouble learning math? On the other hand, it is a symbolic structure made of signs representing concepts. It works by rules of syntax and grammar.
The problem however is only an apparent one, like a knot that is just a tangle that disappears with a tug. Math is not a language. Math is something we do in language. When I do math, I do it in English. When a French person does math, they do it in French. Math appears at first glance to be a language only because we use the same representations, the same words with the same spelling to represent the same concepts “2” is two in English and deux in French and er in Mandarin, but we all spell that concept as 2 when we do math. (A side note, Livio’s short but illuminating excursions into the history of math leave out the essential observation of the indebtedness of math to Arabic numerals.) Why do children have trouble with math? For the same reason they have trouble with logic (which no one calls a language) and with diplomacy and with any of the more complicated functions we do in language. What children acquire easily is vocabulary and syntax. Whatever it is they are capable of expressing they easily learn to express from one natural language to another.
Finally then my point is that Livio’s question about the discover v. invention of math is of the same type as his question of whether math is or is not a language. If a better vocabulary for thinking about math is developed (and I’m sure it has already been developed, though I can’t point to it at the moment), then the problem itself goes away. At least I suspect that is so.
All that said, this is a terrific book. For someone who gave up on math after three semesters of calculus it makes me re-think my choice. What I do not know because I did not get into higher mathematics is a field of wonder that I would love to explore. But life only allows us so many loves. And this peek at what I cannot explore further was infinitely worth the time I spent in the doorway.
December 26, 2014
La verdad es que este libro, en general todos los que voy leyendo de este autor, ha sido una gozada. En él, el autor llama "magos" a aquellos científicos que en su día se dieron cuenta de que las matemáticas tenían una relación inseparable con la realidad. Si algo no tiene funamento matemático, decimos que le falta rigor. Pero no siempre fue así.
Y entonces al autor da un repaso de los principales personajes centrándose en esta idea y explicando muchos detalles de aquellos que no se leen en general en los libros. El autor es un historiador de la ciencia y se nota mucho por su meticulosidad al intentar encontrar as fuentes iniciales de donde saca la información.
Explica la relación entre la lógica y las matemáticas (que curiosamente no siempre estuvieron unidas) y hace que uno se replantee cosas que da por válidas sólo escucharlas.
Hay trozos en los que reconozco que se necesita cierta soltura de razonamiento. No lo recomendaría para todos los públicos sino sólo aquellos que ya han leído muchas cosas sobre historia de las matemáticas o, simple y llanamente, a apasionados de las matemáticas y la historia de la ciencia.
Y entonces al autor da un repaso de los principales personajes centrándose en esta idea y explicando muchos detalles de aquellos que no se leen en general en los libros. El autor es un historiador de la ciencia y se nota mucho por su meticulosidad al intentar encontrar as fuentes iniciales de donde saca la información.
Explica la relación entre la lógica y las matemáticas (que curiosamente no siempre estuvieron unidas) y hace que uno se replantee cosas que da por válidas sólo escucharlas.
Hay trozos en los que reconozco que se necesita cierta soltura de razonamiento. No lo recomendaría para todos los públicos sino sólo aquellos que ya han leído muchas cosas sobre historia de las matemáticas o, simple y llanamente, a apasionados de las matemáticas y la historia de la ciencia.
March 5, 2009
Written by a relatively famous physicist. Purports to examine the question as to whether math is invented or discovered. Never really gets around to 'answering' that question, and the final chapter is pretty disappointing. But there's a lot of good math history in there---mini-biographies of Newton, Galileo, Descartes, Aristotle among others. (He seems to have an odd bias against Gauss, for reasons I don't understand, and Euler barely receives mention.) Nice description of the rise of non-Euclidean geometry and how it sort of shook the world. Nice case study of knot theory, how it started as a curiosity, continued as an intellectual exercise, before the physicists found it to be very useful. If he had stuck to history, even quirky history, I'd've liked it a lot better. But the philosophy stuff just seemed thrown in there so that he'd have an excuse to write a book; it never really led anywhere.
March 25, 2018
The reality is that without mathematics, modern-day cosmologists could not have progressed even one step in attempting to understand the laws of nature.
Einstein once wondered: “How is it possible that mathematics, a product of human thought that is independent of experience [the emphasis is mine], fits so excellently the objects of physical reality?”
Penrose identifies three different “worlds”: the world of our conscious perceptions, the physical world, and the Platonic world of mathematical forms.
And now, Penrose observes, come the three mysteries. First, the world of physical reality seems to obey laws that actually reside in the world of mathematical forms. This was the puzzle that left Einstein perplexed.
Even the brief description I have presented so far already provides overwhelming evidence of a universe that is either governed by mathematics or, at the very least, susceptible to analysis through mathematics.
Does mathematics have an existence that is entirely independent of the human mind? In other words, are we merely discovering mathematical verities, just as astronomers discover previously unknown galaxies? Or, is mathematics nothing but a human invention?
Atiyah therefore believes that “man has created [the emphasis is mine] mathematics by idealizing and abstracting elements of the physical world.”
If you think that understanding whether mathematics was invented or discovered is not that important, consider how loaded the difference between “invented” and “discovered” becomes in the question: Was God invented or discovered? Or even more provocatively: Did God create humans in his own image, or did humans invent God in their own image?
examine the number of days in the lunar month—28. This number is the sum of all of its divisors (the numbers that divide it with no remainder): 28= 1 + 2 + 4 + 7 + 14. Numbers with this special property are called perfect nu...
This highlight has been truncated due to consecutive passage length restrictions.
The famous British mathematician and philosopher Alfred North Whitehead (1861–1947) remarked once that “the safest generalization that can be made about the history of western philosophy is that it is all a series of footnotes to Plato.”
Platonism in its broadest sense espouses a belief in some abstract eternal and immutable realities that are entirely independent of the transient world perceived by our senses.
Alfred North Whitehead remarked: The death of Archimedes at the hands of a Roman soldier is symbolical of a world change of the first magnitude. The Romans were a great race, but they were cursed by the sterility which waits upon practicality. They were not dreamers enough to arrive at new points of view, which could give more fundamental control over the forces of nature. No Roman lost his life because he was absorbed in the contemplation of a mathematical diagram.
In Aristarchus’s universe the Earth and the planets revolved around a stationary Sun that was located at the center (remember that this model was proposed 1,800 years before Copernicus!).
Somewhat surprisingly perhaps, Archimedes himself considered as one of his most cherished accomplishments the discovery that the volume of a sphere inscribed in a cylinder (figure 15) is always 2/3 of the volume of the cylinder. He was so pleased with this result that he requested it be engraved on his tombstone.
The Scottish poet Thomas Seggett raved: Columbus gave man lands to conquer by bloodshed, Galileo new worlds harmful to none. Which is better?
To him, mathematics was simply the language of the universe. To understand the universe, he argued, one must speak this language. God is indeed a mathematician.
Woody Allen once put it: “What if everything is an illusion and nothing exists? In that case, I definitely overpaid for my carpet.”
I have read somewhere that philosophy has always been chiefly engaged with the inter-relations of God, Nature, and Man. The earliest philosophers were Greeks who occupied themselves mainly with the relations between God and Nature, and dealt with Man separately. The Christian Church was so absorbed in the relation of God to Man as entirely to neglect Nature. Finally, modern philosophers concern themselves chiefly with the relations between Man and Nature.
Newton’s famous quote “If I have seen further it is by standing on ye shoulders of Giants” is often presented as a model for the generosity and humility that scientists are expected to display about their greatest discoveries.
A few of the first explorers of the new vistas opened by differential equations were members of the legendary Bernoulli family. Between the mid-seventeenth century and the mid-nineteenth century, this family produced no fewer than eight prominent mathematicians.
In the Principia, Russell and Whitehead defended the view that mathematics was basically an elaboration of the laws of logic, with no clear demarcation between them.
Even the German mathematician of intuitionist inclinations Leopold Kronecker (1823–91) famously declared: “God created the natural numbers, all else is the work of man.”
“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.
Hamming’s third point is that our impression of the effectiveness of mathematics may, in fact, be an illusion, since there is much in the world around us that mathematics does not really explain.
In support of this perspective I could note, for instance, that the mathematician Israïl Moseevich Gelfand was once quoted as having said: “There is only one thing which is more unreasonable than the unreasonable effectiveness of mathematics in physics, and this is the unreasonable ineffectiveness [emphasis added] of mathematics in biology.”
Einstein once wondered: “How is it possible that mathematics, a product of human thought that is independent of experience [the emphasis is mine], fits so excellently the objects of physical reality?”
Penrose identifies three different “worlds”: the world of our conscious perceptions, the physical world, and the Platonic world of mathematical forms.
And now, Penrose observes, come the three mysteries. First, the world of physical reality seems to obey laws that actually reside in the world of mathematical forms. This was the puzzle that left Einstein perplexed.
Even the brief description I have presented so far already provides overwhelming evidence of a universe that is either governed by mathematics or, at the very least, susceptible to analysis through mathematics.
Does mathematics have an existence that is entirely independent of the human mind? In other words, are we merely discovering mathematical verities, just as astronomers discover previously unknown galaxies? Or, is mathematics nothing but a human invention?
Atiyah therefore believes that “man has created [the emphasis is mine] mathematics by idealizing and abstracting elements of the physical world.”
If you think that understanding whether mathematics was invented or discovered is not that important, consider how loaded the difference between “invented” and “discovered” becomes in the question: Was God invented or discovered? Or even more provocatively: Did God create humans in his own image, or did humans invent God in their own image?
examine the number of days in the lunar month—28. This number is the sum of all of its divisors (the numbers that divide it with no remainder): 28= 1 + 2 + 4 + 7 + 14. Numbers with this special property are called perfect nu...
This highlight has been truncated due to consecutive passage length restrictions.
The famous British mathematician and philosopher Alfred North Whitehead (1861–1947) remarked once that “the safest generalization that can be made about the history of western philosophy is that it is all a series of footnotes to Plato.”
Platonism in its broadest sense espouses a belief in some abstract eternal and immutable realities that are entirely independent of the transient world perceived by our senses.
Alfred North Whitehead remarked: The death of Archimedes at the hands of a Roman soldier is symbolical of a world change of the first magnitude. The Romans were a great race, but they were cursed by the sterility which waits upon practicality. They were not dreamers enough to arrive at new points of view, which could give more fundamental control over the forces of nature. No Roman lost his life because he was absorbed in the contemplation of a mathematical diagram.
In Aristarchus’s universe the Earth and the planets revolved around a stationary Sun that was located at the center (remember that this model was proposed 1,800 years before Copernicus!).
Somewhat surprisingly perhaps, Archimedes himself considered as one of his most cherished accomplishments the discovery that the volume of a sphere inscribed in a cylinder (figure 15) is always 2/3 of the volume of the cylinder. He was so pleased with this result that he requested it be engraved on his tombstone.
The Scottish poet Thomas Seggett raved: Columbus gave man lands to conquer by bloodshed, Galileo new worlds harmful to none. Which is better?
To him, mathematics was simply the language of the universe. To understand the universe, he argued, one must speak this language. God is indeed a mathematician.
Woody Allen once put it: “What if everything is an illusion and nothing exists? In that case, I definitely overpaid for my carpet.”
I have read somewhere that philosophy has always been chiefly engaged with the inter-relations of God, Nature, and Man. The earliest philosophers were Greeks who occupied themselves mainly with the relations between God and Nature, and dealt with Man separately. The Christian Church was so absorbed in the relation of God to Man as entirely to neglect Nature. Finally, modern philosophers concern themselves chiefly with the relations between Man and Nature.
Newton’s famous quote “If I have seen further it is by standing on ye shoulders of Giants” is often presented as a model for the generosity and humility that scientists are expected to display about their greatest discoveries.
A few of the first explorers of the new vistas opened by differential equations were members of the legendary Bernoulli family. Between the mid-seventeenth century and the mid-nineteenth century, this family produced no fewer than eight prominent mathematicians.
In the Principia, Russell and Whitehead defended the view that mathematics was basically an elaboration of the laws of logic, with no clear demarcation between them.
Even the German mathematician of intuitionist inclinations Leopold Kronecker (1823–91) famously declared: “God created the natural numbers, all else is the work of man.”
“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.
Hamming’s third point is that our impression of the effectiveness of mathematics may, in fact, be an illusion, since there is much in the world around us that mathematics does not really explain.
In support of this perspective I could note, for instance, that the mathematician Israïl Moseevich Gelfand was once quoted as having said: “There is only one thing which is more unreasonable than the unreasonable effectiveness of mathematics in physics, and this is the unreasonable ineffectiveness [emphasis added] of mathematics in biology.”
October 5, 2012
Всичко около нас може да бъде представено чрез математически модели. Чрез математиката можем да формулираме теории за вселената, анализираме поведението на стоковите борси, чрез нея статистиците оптимизират и прогнозират поведението на различни социални явления, невробиолозите създават модели за работата на мозъка и т.н. Може да кажем, че математиката е езика, граматиката на науката.
Но дали тя е плод на нашето въображение или тя е даденост в природата, нещо което просто откриваме? Иначе казано - ние ли измисляме начин, с който да си обясним явленията около нас, или те са детерминирани, просто езикът на природата (или на създателя) е математиката, а ние просто разкриваме тайните? Откриваме или измисляме?
http://lammothsblog.blogspot.com/2012...
Но дали тя е плод на нашето въображение или тя е даденост в природата, нещо което просто откриваме? Иначе казано - ние ли измисляме начин, с който да си обясним явленията около нас, или те са детерминирани, просто езикът на природата (или на създателя) е математиката, а ние просто разкриваме тайните? Откриваме или измисляме?
http://lammothsblog.blogspot.com/2012...
January 13, 2017
For those who are thrown off by the title, the book mainly addresses the questions of why mathematics is so effective at modeling reality and whether mathematics was invented or discovered. It's a great read that shows many persuasive examples of the applications of mathematics. It also fleshes out the philisophical discussion of whether mathematics is invented or discovered really well, bringing in many mathematicians and philosophers ideas to the table. My favorite part was when the author was reflecting on whether that was the right framing of the discussion and eventually entertained the possibility of how mathematics is both invented and discovered.
April 27, 2009
What accounts for the uncanny ability of mathematics to model the physical world? Is mathematics purely a human construct, an external reality, or something in between. These are the questions Livio sets out to address. He spends most of the book setting the stage for that, by reviewing critical developments in math and its use for modeling, from ancient Greece to the 21st century. In the end, Livio's personal answers to the questions don't matter as much as the enjoyment of the journey he takes you on.
I believe the book would be accessible to the lay person.
I believe the book would be accessible to the lay person.
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