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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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Each week '''[https://discord.gg/v2gKpkq The Road to Reality Book Club]''' tackles a chapter of Sir Roger Penrose's [[Graph,_Wall,_Tome#The_Tome | Epic Tome]]. We use these meetings as an opportunity to write down the major points to be taken from our reading. Here we attempt to sum up what we believe Penrose was trying to convey and why. The hope is that these community-generated reading notes will benefit people in the future as they go on the same journey. | Each week '''[https://discord.gg/v2gKpkq The Road to Reality Book Club]''' tackles a chapter of Sir Roger Penrose's [[Graph,_Wall,_Tome#The_Tome | Epic Tome]]. We use these meetings as an opportunity to write down the major points to be taken from our reading. Here we attempt to sum up what we believe Penrose was trying to convey and why. The hope is that these community-generated reading notes will benefit people in the future as they go on the same journey. | ||
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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=== 5.3 Multiple valuedness, natural logarithms === | === 5.3 Multiple valuedness, natural logarithms === | ||
We need to be careful with the above assertion of the logarithm, mainly since $$b^z$$ and <math>log_bw</math> are ‘many valued’. Solving the equations would require a particular choice for $$b$$ to isolate the solution. With this, the ‘base of natural logarithms’ is introduced as the [https://en.wikipedia.org/wiki/E_(mathematical_constant) number e], whose definition is the power series <math>1+1 | We need to be careful with the above assertion of the logarithm, mainly since $$b^z$$ and <math>log_bw</math> are ‘many valued’. Solving the equations would require a particular choice for $$b$$ to isolate the solution. With this, the ‘base of natural logarithms’ is introduced as the [https://en.wikipedia.org/wiki/E_(mathematical_constant) number e], whose definition is the power series <math>1+\frac{1}{1!}+\frac{1}{2!}+\frac{1}{3!}+…</math>. This power series converges for all values of z which then makes for an interesting choice to solve the ambiguity problem above. Thus we can rephrase the problem above with the natural logarithm, <math>z=logw</math> if $$w=e^z$$. | ||
However, even with this natural logarithm we run into multi-valuedness ambiguity from above. Namely that $$z$$ still has many values that lead to the same solution with $$z+2πin$$, where $$n$$ is any integer we care to choose. This represents a full rotation of $$2π$$ in the complex plane with all multiples of $$n$$ achieving the same point, $$z$$. | However, even with this natural logarithm we run into multi-valuedness ambiguity from above. Namely that $$z$$ still has many values that lead to the same solution with $$z+2πin$$, where $$n$$ is any integer we care to choose. This represents a full rotation of $$2π$$ in the complex plane with all multiples of $$n$$ achieving the same point, $$z$$. | ||
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=== 6.4 The "Eulerian" notion of a function? === | === 6.4 The "Eulerian" notion of a function? === | ||
How, then do we define the notion of a ‘Eulerian’ function? This can be accomplished in two ways. The first using complex numbers and is incredibly simple. If we extend $$f(x)$$ to $$f(z)$$ in the complex plane, then all we require is for $$f(z)$$ to be once differentiable (a kind of $$C^1$$-smooth function). That’s it, magically. We will see that this can be stated with $$f(x)$$ being an [https://en.wikipedia.org/wiki/Analytic_function analytic function]. | |||
The second method involves power series manipulations, and Penrose notes that ‘the fact that complex differentiability turns out to be equivalent to power series expansions is one of the truly great pieces of complex-number magic’. | |||
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 $$p$$. 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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=== 6.6 Integration === | === 6.6 Integration === | ||
As stated in section 6p1, integration is the inverse of differentiation as stated in the fundamental theorem of calculus. Penrose provides the following visual and explanation. If we start with the differentiated curve, the area underneath the derivative curve bounded by two points on the x-axis, and the x-axis itself is equal to the difference in heights of the original curve evaluated at the two points. | |||
[[File:Fig6p8 and6p9.png|thumb|center]] | |||
Integration is noted as making the function smoother and smoother, whereas differentiation continues to make things ‘worse’ until some functions reach a discontinuity and become ‘non-differentiable’. | |||
Penrose ends the chapter noting that there are approaches which enable the process of differentiation to be continued indefinitely, even if the function is not differentiable. One example is the [https://en.wikipedia.org/wiki/Dirac_delta_function Dirac Delta Function] which is of ‘considerable importance in quantum mechanics’. This extends our notion of $$C^n$$-functions into the negative integer space ($$C^{-1},C^{-2},...$$) and will be discussed later with complex numbers. Penrose notes that this leads us further away from the ‘Eulerian’ functions, but complex numbers provide us with an irony that expresses one of their finest magical feats of all. | |||
* If we integrated then differentiate, we get the same answer back. Non-commutative the other way. | * If we integrated then differentiate, we get the same answer back. Non-commutative the other way. | ||
== Chapter 7 Complex-number calculus == | |||
=== 7.1 Complex smoothness; holomorphic functions === | |||
An outline for section 7 is presented, in which calculus with complex numbers is introduced. The material in this chapter leads up to the explanation of [https://en.wikipedia.org/wiki/Holomorphic_function holomorphic functions], which play a vital role in much of the mathematical as well as physics material later in the book. | |||
To do so, the notion of a special type of integration along a contour in the complex plane is to be defined. This integration can then be used to solve for the coefficients of a [https://en.wikipedia.org/wiki/Taylor_series Taylor series] expression which allow for us to see that any complex function which is complex-smooth in the complex plane is necessarily analytic, or holomorphic. | |||
As will be stated in 7.3, instead of directly providing the definition of holomorphic functions, Penrose chooses to demonstrate the argument with the ingredients in order to show a "wonderful example of the way that mathematicians can often obtain their results. Neither the premise ($$f(z)$$ is complex-smooth) nor the conclusion ($$f(z)$$ is analytic) contains a hint of the notion of contour integration or the multivaluedness of a complex logarithm. Yet these ingredients provide the essential clues to the true route to finding the answer". | |||
=== 7.2 Contour integration === | |||
In the real number sense, integrals are taken from a single point $$a$$ to another point $$b$$ along the real number line. Usually the horizontal axis, and there is only one way to travel along this line (moving positive and negative along the axis). However, in the complex plane points involve two dimensions, and therefore have many such routes that allow us to get from a complex point $$a$$ to $$b$$. | |||
The [https://en.wikipedia.org/wiki/Cauchy%E2%80%93Riemann_equations Cauchy-Riemann equations] (to be formally introduced later in chapter 10) allow us to narrow our focus to find a path-specific answer, where the value of the integral on this path is the same for any other path that can be formed from the first by continuous deformation in its domain. Note that the function <math>\frac{1}{z}</math> has a hole in the domain at the origin, which can prevent a continuous deformation thereby allowing for different answers for the value of the integral depending on the path taken. | |||
Continuous deformation in this sense is defined as [https://en.wikipedia.org/wiki/Homology_(mathematics)#:~:text=When%20two%20cycles%20can%20be,in%20the%20same%20homology%20class. homologous deformation], where parts of the path can cancel each other out if they are traversed in opposite directions. This contrasts with homotopic paths where you may not cancel parts. As a visualization we take the function of <math>\frac{1}{z}</math> with a homologous path: | |||
[[File:Fig 7p3 png.png|thumb|center]] | |||
The amazing result here is that a general contour from $$a$$ to $$b$$ for the function <math>\frac{1}{z}</math> has be rephrased and shown to be equal to the result for a ''closed contour'' that loops around the point of non-analyticity (the origin in this case), regardless of where the points $$a$$ and $$b$$ (or the point of non-analyticity) lie in the complex plane. Note that since $$logz$$ is multi-valued, we need to specify the actual closed contour being used (if we looped twice rather than once, then the answer is different). | |||
=== 7.3 Power series from complex smoothness === | |||
The example in section 7p2 is a particular case for the well-known [https://en.wikipedia.org/wiki/Cauchy%27s_integral_formula Cauchy Integral Formula], which allows us to know what the function is doing at the origin (or another general point $$p$$) by what it is doing at a set of points surrounding the origin or the general point $$p$$. | |||
:<math>\frac{1}{2πi}\oint\frac{f(z)}{z-p}dz=f(p)</math> | |||
A 'higher-order' version of this formula allows us to inspect $$n$$ number of derivatives with the same relationship. | |||
:<math>\frac{n!}{2πi}\oint\frac{f(z)}{(z-p)^{n+1}}dz=f^{(n)}(p)</math> | |||
If we use this to provide the definition of a derivative at a point, we can then construct a Maclaurin formula (if using the origin, otherwise the more general [https://en.wikipedia.org/wiki/Taylor_series Taylor series]) for $$f(z)$$ using the derivatives in the coefficients of the terms. | |||
:<math> \sum_{n=0} ^ {\infty} \frac {f^{(n)}(p)}{n!} (z-p)^{n} </math> | |||
This can be shown to sum to $$f(z)$$, thereby showing the function has an actual $$n$$th derivative at the origin or general point $$p$$. This concludes the argument showing that complex smoothness in a region surrounding the origin or point implies that the function is also holomorphic. Penrose notes that neither the premise ($$f(z)$$ is complex-smooth) nor the conclusion ($$f(z)$$ is analytic) contains contour integration or multivaluedness of a complex logarithm, yet these ingredients are essential for finding the route to the answer and that this is a ‘wonderful example of the way that mathematicians can often obtain their results’. | |||
=== 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 == | |||
[[Category:Graph, Wall, Tome]] | |||
[[Category:Projects]] | |||