Is God a Mathematician? Is Mathematics the language of the universe? Is it an invention or a discovery? Mario Livio tries to present arguments for and against these questions with chapters from mathematical history, concepts from the umpteen branches of mathematics, anecdotes from the lives of great Mathematicians and verbatim quotes from their journals in a 252-pager thriller. The story-telling in the book is fantastic and I found the latter half of the book particularly gripping. A note of trivia about the author - Mario Livio is the same author who wrote about the "golden ratio", a book and the ratio that got a mention in Dan Brown's Da Vinci Code.
The first half of the book presents the mathematical views of the ancient world like the Pythagorean mathematicians and the works of Euclid and Plato. He talks about the Tetraktys and Gnomons, tools developed by the Pythagorean mathematicians that proved certain theorems by inspection. Mario Livio then dwells into the 4 people whom he considers as the giants in Mathematical progress, in chronological order, giving historical insights into their lives and work. The author talks about Archimedes, who quoted “Give me a place to stand, and I will move the Earth”, for his genius in designing pulley-based contraptions, his work on displacements of liquids and his work on finding volumes of 3-d objects in an age where integral calculus was not known. He then moves to Galileo and his work on the acceleration of falling bodies, his evangelism of astronomy and the invention of the telescope. Then on the work of Rene Descartes, that led to the unification of algebra and geometry with concepts what we call as "Analytical Geometry", a breakthrough in math that made the very foundations of geometry stronger and provable. The fourth great one is of course Newton, for his work on gravity that changed not only the perspective of how we view the falling apple but also it changed the way we looks at the solar system and the universe. As a by product, Newton independently gave us calculus along with Leibniz. Mario Livio goes to exhibit the platonic view these mathematical giants had almost convincing the reader that Math had to be a discovery rather than an invention. But hold on, now there is a twist to this tale.
Till this time in history, Euclidean geometry was considered the ultimate truth and the perceived universe seemed to be behaving consistently with these postulates. Euclid's 5th postulate, the parallel postulate, was taken for granted till geometries of curved surfaces came into existence. Curved surfaces posed a new difficulty, it put Euclidean geometry in a fix as it did not behave consistently with the Euclidean postulates. These new-Euclidean surfaces and geometries gave way in the belief that after all Math is an invention, a figment of human thought. Mario Livio goes into the lesser known mathematicians of the non-Euclidean geometries and mentions names like Gauss and Riemann and skims through their work, the latter's geometry forming the building block for Einstein's theories in the future.
The author moves onto Logic and Mathematics, their relationship, discussing the works of George Boole (Boolean algebra), Bertrand Russell (Russell's paradox) and Kurt Godel (Incompleteness theorem) among others. He briefly writes about Godel's "Incompleteness theorem", another shock that rocked the Platonists. Godel showed that any formal system that is powerful enough to be of any interest is inherently either incomplete or inconsistent.
The last few chapters talk about some examples that show the difference between "active" and "passive" applications of Mathematics. Mario Livio covers Knot theory, a mathematical theory that was studied ages before it's application was found, in decoding DNA knots and strands (passive application). He gives a brief description of string theory too in this context. The theories of relativity - special and general are also discussed and their applications in adjusting clocks onboard satellites is given. The author then discusses the opinion of cognitive scientists on the same, which is mostly that Mathematics is an invention of the human intellect. Mario Livio concludes with the answer "it depends", there is math that occurs in nature whose characteristics we have conceptualized in our minds and there is math that purely occurs as thought.
Overall, an interesting read that showcases mathematics - invented or discovered. The book concludes with a paragraph from Bertrand Russell's essay The Problems 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 also is rendered great, and becomes capable of that union with the universe which constitutes its highest good.
The first half of the book presents the mathematical views of the ancient world like the Pythagorean mathematicians and the works of Euclid and Plato. He talks about the Tetraktys and Gnomons, tools developed by the Pythagorean mathematicians that proved certain theorems by inspection. Mario Livio then dwells into the 4 people whom he considers as the giants in Mathematical progress, in chronological order, giving historical insights into their lives and work. The author talks about Archimedes, who quoted “Give me a place to stand, and I will move the Earth”, for his genius in designing pulley-based contraptions, his work on displacements of liquids and his work on finding volumes of 3-d objects in an age where integral calculus was not known. He then moves to Galileo and his work on the acceleration of falling bodies, his evangelism of astronomy and the invention of the telescope. Then on the work of Rene Descartes, that led to the unification of algebra and geometry with concepts what we call as "Analytical Geometry", a breakthrough in math that made the very foundations of geometry stronger and provable. The fourth great one is of course Newton, for his work on gravity that changed not only the perspective of how we view the falling apple but also it changed the way we looks at the solar system and the universe. As a by product, Newton independently gave us calculus along with Leibniz. Mario Livio goes to exhibit the platonic view these mathematical giants had almost convincing the reader that Math had to be a discovery rather than an invention. But hold on, now there is a twist to this tale.
Till this time in history, Euclidean geometry was considered the ultimate truth and the perceived universe seemed to be behaving consistently with these postulates. Euclid's 5th postulate, the parallel postulate, was taken for granted till geometries of curved surfaces came into existence. Curved surfaces posed a new difficulty, it put Euclidean geometry in a fix as it did not behave consistently with the Euclidean postulates. These new-Euclidean surfaces and geometries gave way in the belief that after all Math is an invention, a figment of human thought. Mario Livio goes into the lesser known mathematicians of the non-Euclidean geometries and mentions names like Gauss and Riemann and skims through their work, the latter's geometry forming the building block for Einstein's theories in the future.
The author moves onto Logic and Mathematics, their relationship, discussing the works of George Boole (Boolean algebra), Bertrand Russell (Russell's paradox) and Kurt Godel (Incompleteness theorem) among others. He briefly writes about Godel's "Incompleteness theorem", another shock that rocked the Platonists. Godel showed that any formal system that is powerful enough to be of any interest is inherently either incomplete or inconsistent.
The last few chapters talk about some examples that show the difference between "active" and "passive" applications of Mathematics. Mario Livio covers Knot theory, a mathematical theory that was studied ages before it's application was found, in decoding DNA knots and strands (passive application). He gives a brief description of string theory too in this context. The theories of relativity - special and general are also discussed and their applications in adjusting clocks onboard satellites is given. The author then discusses the opinion of cognitive scientists on the same, which is mostly that Mathematics is an invention of the human intellect. Mario Livio concludes with the answer "it depends", there is math that occurs in nature whose characteristics we have conceptualized in our minds and there is math that purely occurs as thought.
Overall, an interesting read that showcases mathematics - invented or discovered. The book concludes with a paragraph from Bertrand Russell's essay The Problems 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 also is rendered great, and becomes capable of that union with the universe which constitutes its highest good.
