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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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=== 1.3 Is Plato's mathematical world "real"? === | === 1.3 Is Plato's mathematical world "real"? === | ||
Penrose asks us to consider if the world of mathematics 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 | Penrose asks us to consider if the world of mathematics 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?] | For further discussion from Penrose on this topic see [https://youtu.be/ujvS2K06dg4 Is Mathematics Invented or Discovered?] | ||
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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. | ||
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=== 7.4 Analytic continuation === | === 7.4 Analytic continuation === | ||
We now know that complex smoothness throughout a region is equivalent to the existence of a power series expansion about any point in the region. A region here is defined as a open region, where the boundary is not included in the domain. | |||
For example, if there is no singularity in the function, the region can be thought of as a circle of infinite radius. Taking <math>f(z)=\frac{1}{z}</math> however forces an infinite number of circles centered at any point with boundary radii passing through the origin (noting that an open region does not contain the boundary) to construct the domain. | |||
Now we consider the question, given a function $$f(z)$$ holomorphic in domain $$D$$, can we extend the domain to a larger $$D’$$ so that $$f(z)$$ also extends holomorphically? A procedure is formed in which we use a succession of power series about a sequence of points, forming a path where the circles of convergence overlap. This then results in a function that is uniquely determined by the values in the initial region as well as the path along which it was continued. Penrose notes this [https://en.wikipedia.org/wiki/Analytic_continuation analytic continuation] as a remarkable ‘rigidity’ about holomorphic functions. | |||
An example of this rigidity and path dependence is ‘our old friend’ $$logz$$. There is no power series expansion about the origin due to the singularity there but depending on the path chosen of points around the origin (clockwise or anti-clockwise) the function extends or subtracts in value by $$2πi$$. See chapter 5 and the euler formula (<math>e^{πi}=-1</math>) for a refresher. | |||
== Chapter 8 Riemann surfaces and complex mappings == | == Chapter 8 Riemann surfaces and complex mappings == | ||
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