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The Road to Reality Study Notes: Difference between revisions
→6.4 The "Eulerian" notion of a function?: Slight grammatical adjustment in ultimate paragraph of section 6p4
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Chapters 1-16 focus on mathematical concepts while the later chapters use this background to describe the physical world. | Chapters 1-16 focus on mathematical concepts while the later chapters use this background to describe the physical world. | ||
* [https://discord.gg/3xgrNwJ The Portal Book Club] - We have a weekly group that meets to talk about this book. Come join us in Discord! | * [https://discord.gg/3xgrNwJ The Portal Book Club] - We have a weekly group that meets to talk about this book. Come join us in Discord! | ||
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* The Physical | * The Physical | ||
* The Mental | * The Mental | ||
as | 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 === | === 1.5 The Good, the True, and the Beautiful === | ||
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For the second method, the power series of $$f(x)$$ is introduced, <math>f(x) = a_0 + a_1x + a_2x^2 + a_3x^3 + …</math> For this series to exist then it must be $$C^\infty$$-smooth. We must take and evaluate derivatives $$f(x)$$ to find the coefficients, thus an infinite number of derivatives (positive integers) must exist for the power series to exist. If we evaluate $$f(x)$$ at the origin, we call this a power series expansion about the origin. About any other point $$p$$ would be considered a power series expansion about $$p$$. (Maclaurin Series about origin, see also [https://en.wikipedia.org/wiki/Taylor_series Taylor Series] for the general case) | For the second method, the power series of $$f(x)$$ is introduced, <math>f(x) = a_0 + a_1x + a_2x^2 + a_3x^3 + …</math> For this series to exist then it must be $$C^\infty$$-smooth. We must take and evaluate derivatives $$f(x)$$ to find the coefficients, thus an infinite number of derivatives (positive integers) must exist for the power series to exist. If we evaluate $$f(x)$$ at the origin, we call this a power series expansion about the origin. About any other point $$p$$ would be considered a power series expansion about $$p$$. (Maclaurin Series about origin, see also [https://en.wikipedia.org/wiki/Taylor_series Taylor Series] for the general case) | ||
The power series is considered analytic if it encompasses the power series about point $$p$$ | The power series is considered analytic if it encompasses the power series about point $$p$$. If it is analytic at all points of its domain, we call it an analytic function or, equivalently, a $$C^ω$$-smooth function. Euler would be pleased with this notion of an analytic function, which is ‘smoothier’ than the set of $$C^\infty$$-smooth functions ($$h(x)$$ from 6p3 is $$C^\infty$$-smooth but not $$C^ω$$-smooth). | ||
* Physics in trying to understand reality by approximating it. | * Physics in trying to understand reality by approximating it. | ||