The reason why mathematics can describe the physical world so accurately is unknown. Note that only a small subset of the mathematical world is utilized in describing the physical world. The main focus of the book is to explore the connection between mathematics and its use in describing the physical world. Connections between these worlds present a great number of mysteries. ![]() Penrose outlines his conception of three worlds:Īs shown in figure 1.3. Penrose invites the reader to reconsider their notions of reality beyond the matter and stuff that makes up the physical world.įor further discussion from Penrose on this topic see Is Mathematics Invented or Discovered? 1.4 Three worlds and three deep mysteries Any mathematical notion can be thought of as existing in that place. Finally he discusses the Mandelbrot set and claims that it exists in a place outside of time and space and was only uncovered by Mandelbrot. He notes that "questions as to whether some particular proposal for a mathematical entity is or is not to be regarded as having objective existence can be delicate and sometimes technical". Penrose introduces a more complicated mathematical notion, the axiom of choice, which has been debated amongst mathematicians. He shows that "the issue is the objectivity of the Fermat assertion itself, not whether anyone’s particular demonstration of it (or of its negation) might happen to be convincing to the mathematical community of any particular time". ![]() He uses Fermat's last theorem as a point to consider what it would mean for mathematical statements to be subjective. He claims that objective truths are revealed through mathematics and that it is not a subjective matter of opinion. Penrose asks us to consider if the world of mathematics is in any sense real. To Plato the idealized mathematical world of forms was a place of absolute truth, but inaccessible from the physical world.ġ.3 Is Plato's mathematical world "real"? Physical manifestations of geometric objects could come close to the Platonic world of mathematical forms, but they were always approximations. The Greek philosopher Plato (c.429-347 BC) believed that mathematical proofs referred not to actual physical objects but to certain idealized entities. Mathematicians trust that the axioms, on which their theorems depend, are actually true. The most fundamental mathematical statements, from which all other proofs are built, are called axioms and their validity is taken to be self-evident. If the mathematician hasn't broken any rules then the new statement is called a theorem. Mathematical proof allowed for much stronger statements to be made about relationships between the arithmetic of numbers and the geometry of physical space.Ī mathematical proof is essentially an argument in which one starts from a mathematical statement, which is taken to be true, and using only logical rules arrives at a new mathematical statement. Developing a rigorous mathematical framework was central to the development of science. 572-497 BC) are considered to be the first to introduce the concept of mathematical proof. The Greek philosopher Thales of Miletus (c. There was a need to define a more rigorous method for differentiating truth claims. One famous example from the ancient Greeks is the association between Platonic solids and the basic elementary states of matter. Many people in ancient times allowed their imaginations to be carried away by their fascination with the subject, leading to mystical associations with mathematical objects. ![]() It became apparent that mathematics unlocked deep truths about the universe. After many millennia of chaos and frustration, it was discovered that the regular movement of celestial bodies, such as the sun and moon, could be described mathematically. Understanding natural processes has been a common pursuit since the dawn of humanity. Reference Material by Chapter on the WikiĬhapter 1 The Roots of Science 1.1 The quest for the forces that shape the world.The Portal Book Club - We have a weekly group that meets to talk about this book.The hope is that these community-generated reading notes will benefit people in the future as they go on the same journey.Ĭhapters 1-16 focus on mathematical concepts while the later chapters use this background to describe the physical world. Here we attempt to sum up what we believe Penrose was trying to convey and why. We use these meetings as an opportunity to write down the major points to be taken from our reading. Each week The Road to Reality Book Club tackles a chapter of Sir Roger Penrose's Epic Tome.
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