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Reviews for Is God a Mathematician?:

5

Jul 26, 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 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. ...more
4

Sep 16, 2015

I was pleasantly surprised by Mario Livios 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 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.
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5

Mar 27, 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.
4

Feb 06, 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 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. ...more
4

Jun 02, 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 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"... ...more
4

Jul 10, 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.
3

Mar 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 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. ...more
4

Jul 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, 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...
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0

Jul 30, 2011

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 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. ...more
4

Feb 08, 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
4

Sep 15, 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 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. ...more
3

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, 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. ...more
2

Nov 30, 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, its not a bad read. But if youre looking for a Christian math book, this isnt the one for you. 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. ...more
5

Jan 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?
5

Aug 19, 2014

First off, I'll begin by recommending this book to anyone who: is a newbie to mathematics, is a math enthusiast, enjoys digging deeper into the history of math and the people behind it, and anyone in general who is interested in math.
Note: I think those of you who have more experience with math (and its history) might not find anything new in this book -- but it's still a fun read which you could breeze through and enjoy.

Secondly, I've thoroughly enjoyed reading this book. Livio's style is First off, I'll begin by recommending this book to anyone who: is a newbie to mathematics, is a math enthusiast, enjoys digging deeper into the history of math and the people behind it, and anyone in general who is interested in math.
Note: I think those of you who have more experience with math (and its history) might not find anything new in this book -- but it's still a fun read which you could breeze through and enjoy.

Secondly, I've thoroughly enjoyed reading this book. Livio's style is somewhat academic - in a good way. The book is fueled by his infectious enthusiasm and fascination, which translates in to a fun and inspiring read (especially if you are just starting to discover the world of math and its rich history).
Note: There is no technical texts in this book, you won't find any math doodles or riddles to think about and solve by yourself.

For Hebrew Speakers : The author gave a very nice lecture about this book, take a look here.

General Overview
Some people might find the title of the book a bit misleading, as the proposed question isn't really addressed properly (or answered, for that matter). However, this book holds its ground and keeps you entertained by trying to tackle a slightly different question: Is mathematics a human discovery , or a human invention ?

Mario Livio embarks on a journey throughout history to find some kind of an answer to this question. The book explores the ideas and achievements of famous historic personalities and influencers such Plato, Archimedes, Galileo, Descartes, Newton, Bernoulli, Boole and many others. It's a very fun and satisfying recount of how ingenious mathematical concepts came to be. Readers might ponder on the fact that many theories were created with no thought or regard for our physical reality, yet years later these theories were found to be precise and accurate in representing our physical world. How can this be?

Things You Might Find Interesting
The book is divided in to several chapters, each of which deals with a different theme, with its own set of important people - some who hold a Platonistic point of view, others who hold a Formalistic point of view.

Mystics

Pythagorean theorem; platonism

Magicians

Archimedes's genius inventions and contribution [Eureka!]; Galileo's discoveries and influence; Descartes's fusion of maths; Newton's (and Leibniz's) mind-blowing calculus [as well as an interesting recount of Newton's relationship with Hooke]

Statisticians and Probabilists

Fun tidbits about the rivalry between the Bernoulli brothers; a game that gave birth to probability theory.

Geometers

How several revolutionary thinkers flipped euclidean geometry on its head, and made way to a new era in mathematics

Logicians

I have to highlight the following: you will be quite entertained when you read Kurt Gödel's attempt to attain American citizenship!

Some Minor Quibbles
[*] As is common with Livio's other works, he is oftentimes fond of straying away from the main point. This didn't bother me too much (lots of interesting tidbits are presented this way).

[*] Loads of quotes from other sources - again, this didn't bother me too much.

[*] A good amount of unnecessary images of book covers and ancient documents - might interest the history buffs, but others might find it a waste of space.
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5

Dec 20, 2009

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 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." ...more
3

Mar 04, 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 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. ...more
4

Jan 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 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. ...more
3

Apr 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 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. ...more
4

May 22, 2010

The first half is a stale history of mathematicians, lacking in actual math. The second half is more interesting, when it gets into topology and non-Euclidean geometry. Much of the philosophy in the book suffers from the too common disease of struggling to define terms instead of presenting substantive arguments. However, the last chapter has some real gems in it, so I'd say this one's worth a read!
5

Jan 08, 2016

I am a slow reader, else normally this book can be easily completed within a week's time. Very interesting read particularly for someone interested in science and history of mathematics.
4

Oct 05, 2011

It was a bit more popular than I expected it to be but it was still good.
5

July 10, 2015

I love it so much I've heard of this quite a long time ago but this is my first chance of reading it
3

Mar 24, 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.

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...
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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.” ...more
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Jul 06, 2015

Science is an attempt to read Gods mind which is evident in the physical reality as the rules and principles which hold the world together. Livios book is an elegant attempt to tell the epic story of mans 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 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.
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