The tome is definitely the read for anyone interested in our current understanding of reality from the ground up. It doesn’t shy away from mathematics like most popular science accounts. Rather, as mathematics is the gasoline to travel the Road, it fuels the reader up along the journey. Yet the mathematics quickly progresses from exploring what a number exactly is to a graduate level textbook equivalent and this whirlwind journey is not to be taken lightly. Rather than a Road, I actually consider The Road to Reality to be more of a Roadmap indicating a path to reality without fully providing the reader with the tools to travel it. It did work for me–a Physics PhD student–as most of the book was putting familiar things in a grander context. However, for those not on the Road as a career, supplemental work is probably needed to get the most out of the book. In the that spirit I am presenting a condensed Roadmap in chunks of 6 chapters (there are a whopping 34 chapters covering 1123 pages) with additional links to online courses (primarily sourced from the wonderful Khan academy, Coursera, and MIT OCW).
Without further ado, I present the Roadmap I of VI of Chapters 1-6 of The Road to Reality:
1 The Roots of Science
People have always been curious about the laws which govern the universe. There is a mysterious connection between mathematics and nature, a binding between abstract entities which can never practically be realized and reality. Penrose goes through some of the philosophy behind this deep apparent truth, and digresses to discuss how physicists explore it. The abstract mathematics has a subjective beauty and simplicity to it that has traditionally helped to guide scientists and mathematicians in the right direction.
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2 An ancient theorem and a modern question
Penrose reviews geometry–starting with the triangle which possesses, in Euclidean geometry and for a right angled triangle whose sides and form the hypotenuse of length . If you want to continue on the Road, you will need to learn Geometry. The great news is this rich subject is full of visualization and has more to explore than one might expect. In some geometries, for example one defined on the surface of a sphere there are no parallel lines and a triangle doesn’t necessarily have . Penrose explores one such geometry, Hyperbolic geometry, in great detail. I would recommend familiarizing yourself with the concept of a there being different types of geometry, fully exploring Euclidian geometry before moving on, and coming back to Hyperbolic geometry later on The Road
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2.1.1 Euclidian Geometry
2.1.2 Non-Euclidian Geometry
2.1.3 Hyperbolic geometry
3 Kinds of number in the physical world
Penrose reviews different types of numbers, rational and irrational, the Real number system, and begins to address the question as to which number system is the best to represent reality-is it fundamentally discrete or continuous? As fas as we can tell thus far continuous at its base but discreteness certainly appears is quantum mechanics, e.g. in the charge of an electron. Negative numbers have a place in the physical world as well, e.g with antiparticles. Thus a good understanding of these basic issues of number theory is important as we continue on our road, and even invoke philosophical as well as physical questions
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4 Magical Complex Numbers
Penrose is big on the “magic” of complex numbers, and we will see, particularly in quantum mechanics, they indeed form a fundamental part of our description of reality. In addition, there mathematical properties can sometimes be used to our advantage to greatly simplify calculations. Thus a full understanding of complex numbers is important to continue on The Road . With complex numbers we introduce a new quantity called which is defined to be the square to and adjoin this to the real number system. Thus every complex number can be expressed as where both and are real numbers. For example , , , can all be considered part of the complex number space. We can speak of the “real” and the “complex” part of a complex number, having a real part of and a complex part of . The space of complex numbers is denoted . We can also visualize complex numbers with the real part being the axis of a graph, and the complex part being the axis. Thus we could, using the rules for computing the length of the hypotenuse of a triangle and the angle between the hypotenuse and a side, rather represent a complex number by a length and an angle. Just like we can write equations and solve them for real numbers , we can write equations and solve them for complex numbers , e.g. .
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5 Geometry of logarithms, powers, and roots
Penrose reviews the geometry of logarithms, powers and roots, in the context of complex numbers. If you aren’t familiar with these concepts in terms of real numbers, explore them in that context first and learn to manipulate them to solve real number equations before moving on to the complex numbers. When you are ready proceed to do the same with complex numbers.
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6 Real-number calculus
Ok! We are ready for calculus. The good news is that this subject is so fundamental, that just learning calculus will get you farther on the road than any other subject. In addition if you gain the confidence to tackle calculus, know that you have enough gas to truly follow the rest of The Road . Don’t get discouraged, I think this is this biggest speed bump to non-professionals. If you haven’t learned calculus before, invest most of your time here at this stage and don’t come back to the book until you are ready. Calculus concerns itself with the rate things change-quantities defined locally, and areas and volumes–quantities defined globally. If you have a function, , the rate at which it changes is called the derivative of or and the area under the function is called the integral or . Graphically if we pick a point at will be the slope of the line tangent to at . Moreover if we pick two points and the integral between and is just the area under the curve between and . This is fantastic as it turns out there are a set of easy rules to learn how to differentiate and integrate many functions on your own, and if you don’t manage there are computer algorithms which can. Thus if we represent any physical process by an equation between an item, areas/volumes, and rates of changes of the item we can make predictions simply solving the equations with our calculus techniques. For example-http://mathforum.org/library/drmath/view/53394.html
But first you will have to learn the techniques–not only to solve equations on your own, but to be able to input them to a computer, understand and follow later notation and equations, and eventually to learn how to think in a manner that will help you write your own equations which can be solved with the techniques of calculus.
6.1 Resources for learning more
Epilogue Stay tuned
Stay tuned for my next installment of the Roadmap of Chapters 7-12. Comment, like or share to speed it on its way.