Editing The Road to Reality Study Notes (section)

== Chapter 1 The Roots of Science ==

=== 1.1 The quest for the forces that shape the world ===

Understanding natural processes has been a common pursuit since the dawn of humanity. 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. It became apparent that mathematics unlocked deep truths about the universe. 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. One famous example from the ancient Greeks is the association between [https://en.wikipedia.org/wiki/Platonic_solid Platonic solids] and the basic elementary states of matter.

=== 1.2 Mathematical truth ===

There was a need to define a more rigorous method for differentiating truth claims. The Greek philosopher [https://en.wikipedia.org/wiki/Thales_of_Miletus Thales of Miletus] (c. 625-547 BC) and [https://en.wikipedia.org/wiki/Pythagoras Pythagoras of Samos] (c. 572-497 BC) are considered to be the first to introduce the concept of ''mathematical proof''. Developing a rigorous mathematical framework was central to the development of science. Mathematical proof allowed for much stronger statements to be made about relationships between the arithmetic of numbers and the geometry of physical space. 

A 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. If the mathematician hasn't broken any rules then the new statement is called a ''theorem''. The most fundamental mathematical statements, from which all other proofs are built, are called ''axioms'' and their validity is taken to be self-evident. Mathematicians trust that the axioms, on which their theorems depend, are actually ''true''.  The Greek philosopher [https://en.wikipedia.org/wiki/Plato Plato] (c.429-347 BC) believed that mathematical proofs referred not to actual physical objects but to certain idealized entities. Physical manifestations of geometric objects could come close to the Platonic world of mathematical forms, but they were always approximations. To Plato the idealized mathematical world of forms was a place of absolute truth, but inaccessible from the physical world.

=== 1.3 Is Plato's mathematical world "real"? ===
Penrose asks us to consider if the world of mathematics is in any sense ''real''. He claims that objective truths are revealed through mathematics and that it is not a subjective matter of opinion. He uses [https://en.wikipedia.org/wiki/Fermat%27s_Last_Theorem Fermat's last theorem] as a point to consider what it would mean for mathematical statements to be subjective. 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". Penrose introduces a more complicated mathematical notion, the [https://en.wikipedia.org/wiki/Axiom_of_choice axiom of choice], which has been debated amongst mathematicians. 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". Finally he discusses the [https://en.wikipedia.org/wiki/Mandelbrot_set Mandelbrot set] and claims that it exists in a place outside of time and space and was only uncovered by Mandelbrot. Any mathematical notion can be thought of as existing in that place. Penrose invites the reader to reconsider their notions of reality beyond the matter and stuff that makes up the physical world. 

For further discussion from Penrose on this topic see [https://youtu.be/ujvS2K06dg4 Is Mathematics Invented or Discovered?]

=== 1.4 Three worlds and three deep mysteries ===
[[File:Three worlds penrose.png|thumb|Figure 1.3: Penrose three worlds]]
Penrose outlines his conception of three worlds:
* The Platonic Mathematical
* The Physical
* The Mental
as shown in figure 1.3. Connections between these worlds present a great number of mysteries. The main focus of the book is to explore the connection between mathematics and its use in describing the physical world. Note that only a small subset of the mathematical world is utilized in describing the physical world. The reason why mathematics can describe the physical world so accurately is unknown. Moving in the counter clockwise direction there is a mysterious connection between the physical world and that of the mind. He believes that there must be some basis of consciousness in physical reality, but it is still unknown. Finally there is the connection between the mental world and mathematics. Penrose believes that there is no mathematical notion beyond our mental construction. He highlights that this figure represents many of his prejudices and might upset some people. Maybe the connections from one world do not fully describe the other, or are incomplete.  Penrose believes that not much progress can be made with respect to the mental world until we know much more about the physical world.

=== 1.5 The Good, the True, and the Beautiful ===
The full conception of [https://en.wikipedia.org/wiki/Theory_of_forms Plato's theory of forms] was not limited to only mathematical notions. Mathematics was linked to the concept of ''Truth'' but Plato was also interested in the absolute idealized forms of ''Beauty'' and ''Good''. Beauty plays an important role in many mathematical discoveries and is often used as a guide to the truth. Questions of morality are of less relevance in this context but are critical with respect to the mental world. Moral debates are outside of the scope of this book but must be considered as science and technology progress. Penrose notes that figure 1.3 has purposely been constructed to be paradoxical in the sense that each world is entirely encompassed by the next. He writes "There may be a sense in which the three worlds are not separate at all, but merely reflect, individually, aspects of a deeper truth about the world as a whole of which we have little conception at the present time."