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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! | ||
* [[The Road to Reality|Reference Material by Chapter on the Wiki]] | * [[The Road to Reality|Reference Material by Chapter on the Wiki]] | ||
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* [https://docs.google.com/document/d/1XabH2WM24GklXoAyrGkD3XgtplcWoSH3e2tde4jL_vI/edit# Chronological guide to concepts introduced in TRTR Google Doc] | * [https://docs.google.com/document/d/1XabH2WM24GklXoAyrGkD3XgtplcWoSH3e2tde4jL_vI/edit# Chronological guide to concepts introduced in TRTR Google Doc] | ||
* [https://drive.google.com/drive/folders/1IBkAl2JrCAEEjUFT12lCs8mGnoaqQzBr Book Club Resources in Google Drive] | * [https://drive.google.com/drive/folders/1IBkAl2JrCAEEjUFT12lCs8mGnoaqQzBr Book Club Resources in Google Drive] | ||
--> | |||
== Chapter 1 The Roots of Science == | == Chapter 1 The Roots of Science == | ||
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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 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?] | 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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===2.1 The Pythagorean theorem=== | ===2.1 The Pythagorean theorem=== | ||
To explore the process of pursuing mathematical truth, Penrose outlines a few proofs of the [https://en.wikipedia.org/wiki/Pythagorean_theorem Pythagorean theorem]. The theorem can be stated as such, "For any right-angled triangle, the squared length of the hypotenuse | To explore the process of pursuing mathematical truth, Penrose outlines a few proofs of the [https://en.wikipedia.org/wiki/Pythagorean_theorem Pythagorean theorem]. The theorem can be stated as such, "For any right-angled triangle, the squared length of the hypotenuse <math>c</math> is the sum of the squared lengths of the other two sides <math>a</math> and <math>b</math> or in mathematical notation <math> a^2 + b^2 = c^2. </math> | ||
There are hundreds of proofs of the Pythagorean theorem but Penrose chooses to focus on two. The first involves filling up a plane with squares of two different sizes. Then adding a second pattern on top of tiled squares connecting the centers of the larger original squares. By translating the tilted pattern to the corner of the large square and observing the areas covered by the pattern you can show that the square on the hypotenuse is equal to the sum of the squares on the other two sides. While the outlined proof appears reasonable there are some implicit assumptions made. For instance what do you mean when we say ''square''? What is a ''right angle''? | There are hundreds of proofs of the Pythagorean theorem but Penrose chooses to focus on two. The first involves filling up a plane with squares of two different sizes. Then adding a second pattern on top of tiled squares connecting the centers of the larger original squares. By translating the tilted pattern to the corner of the large square and observing the areas covered by the pattern you can show that the square on the hypotenuse is equal to the sum of the squares on the other two sides. While the outlined proof appears reasonable there are some implicit assumptions made. For instance what do you mean when we say ''square''? What is a ''right angle''? | ||
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Euclid was trying to establish the rules which govern his geometry. Some interesting ideas start to emerge such as the indefinitely extendible geometric plane and the concept of congruence. Penrose writes "In effect, the fourth postulate is asserting the isotropy and homogeneity of space, so that a figure in one place could have the ‘same’ (i.e. congruent) geometrical shape as a figure in some other place". Surprisingly Euclid's first four postulates still align well with our understanding of a two-dimensional metric space. | Euclid was trying to establish the rules which govern his geometry. Some interesting ideas start to emerge such as the indefinitely extendible geometric plane and the concept of congruence. Penrose writes "In effect, the fourth postulate is asserting the isotropy and homogeneity of space, so that a figure in one place could have the ‘same’ (i.e. congruent) geometrical shape as a figure in some other place". Surprisingly Euclid's first four postulates still align well with our understanding of a two-dimensional metric space. | ||
Euclid's fifth postulate, also known as the [https://en.wikipedia.org/wiki/Parallel_postulate parallel postulate], was more troublesome. In Penrose words "it asserts that if two straight line segments | Euclid's fifth postulate, also known as the [https://en.wikipedia.org/wiki/Parallel_postulate parallel postulate], was more troublesome. In Penrose words "it asserts that if two straight line segments <math>a</math> and <math>b</math> in a plane both intersect another straight line <math>c</math> (so that <math>c</math> is what is called a ''transversal'' of <math>a</math> and <math>b</math>) such that the sum of the interior angles on the same side of <math>c</math> is less than two right angles, then <math>a</math> and <math>b</math>, when extended far enough on that side of <math>c</math>, will intersect somewhere". One can see that the formulation of the fifth postulate is more complicated than the rest which lead to speculation of it's validity. With the fifth postulate one can go on to properly build a square and begin to explore the world of [https://en.wikipedia.org/wiki/Euclidean_geometry Euclidean geometry]. | ||
===2.3 Similar-areas proof of the Pythagorean theorem=== | ===2.3 Similar-areas proof of the Pythagorean theorem=== | ||
Penrose revisits the Pythagorean theorem by outlining another proof. Starting with a right triangle, subdivide the shape into two smaller triangles by drawing a line perpendicular to the hypotenuse through the right angle. The two smaller triangles are said to be ''similar'' to one another meaning they have the same shape but are different sizes. This is true because each of the smaller triangles has a right angle and shares an angle with the larger triangle. The third angle known because the sum of the angles in any triangle is always the same. Knowing that the sum of the area of the two small triangles equals the area of the big triangle (by construction), we can square the sides and show that the pythagorean theorem holds. | Penrose revisits the Pythagorean theorem by outlining another proof. Starting with a right triangle, subdivide the shape into two smaller triangles by drawing a line perpendicular to the hypotenuse through the right angle. The two smaller triangles are said to be ''similar'' to one another meaning they have the same shape but are different sizes. This is true because each of the smaller triangles has a right angle and shares an angle with the larger triangle. The third angle known because the sum of the angles in any triangle is always the same. Knowing that the sum of the area of the two small triangles equals the area of the big triangle (by construction), we can square the sides and show that the pythagorean theorem holds. | ||
Again Penrose asks us to revisit our assumptions and examine which of Euclid's postulates were needed. Particularly our claim that the sum of the angles in a triangle add up to the same value of 180° (or | Again Penrose asks us to revisit our assumptions and examine which of Euclid's postulates were needed. Particularly our claim that the sum of the angles in a triangle add up to the same value of 180° (or <math>\pi</math> [https://en.wikipedia.org/wiki/Radian radians]). One must use the parallel postulate to show that this is true. Penrose asks us to consider what would it mean for the parallel postulate to be false? What would that imply? Would that make any sense? With these questions in mind we begin to explore a different kind of geometry. | ||
===2.4 Hyperbolic geometry: conformal picture=== | ===2.4 Hyperbolic geometry: conformal picture=== | ||
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'''Natural Numbers:''' | '''Natural Numbers:''' | ||
‘Counting’ numbers from 1 to | ‘Counting’ numbers from 1 to <math>\infty</math>. | ||
'''Whole Numbers:''' | '''Whole Numbers:''' | ||
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Complex numbers can be visualized graphically as a plane, where the horizontal coordinate gives the real coordinate of the number and the vertical coordinate gives the imaginary part. This helps understand the behavior of [https://en.wikipedia.org/wiki/Radius_of_convergence power series]; for example, the power series | Complex numbers can be visualized graphically as a plane, where the horizontal coordinate gives the real coordinate of the number and the vertical coordinate gives the imaginary part. This helps understand the behavior of [https://en.wikipedia.org/wiki/Radius_of_convergence power series]; for example, the power series | ||
<math>1-x^2+x^4+\cdots</math> | |||
converges to the function | converges to the function <math>1/(1+x²)</math> only when <math>|x|<1</math>, despite the fact that the function doesn't seem to have "singular" behavior anywhere on the real line. This can be explained by switching to the complex number system using <math>z=x+iy</math> whereby <math>1/(1+z²)</math> can be examined to have singularities at <math>x=i,-i</math>. | ||
With this, Penrose introduces us to the idea of the [https://mathworld.wolfram.com/RadiusofConvergence.html circle of convergence] as a circle in the complex plane centered at 0 with poles/singularities of | With this, Penrose introduces us to the idea of the [https://mathworld.wolfram.com/RadiusofConvergence.html circle of convergence] as a circle in the complex plane centered at 0 with poles/singularities of <math>f(z)</math> defining the circle radius. The series is convergent for any point z inside of this circle. | ||
Finally, the [https://en.wikipedia.org/wiki/Mandelbrot_set Mandelbrot set] is defined as the set of all points | Finally, the [https://en.wikipedia.org/wiki/Mandelbrot_set Mandelbrot set] is defined as the set of all points <math>c</math> in the complex plane so that repeated applications of the transformation mapping <math>z</math> to <math>z^2+c</math>, starting with <math>z=0</math>, do not escape to infinity. | ||
== Chapter 5 Geometry of logarithms, powers, and roots == | == Chapter 5 Geometry of logarithms, powers, and roots == | ||
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Penrose asks us to view complex addition and multiplication as transformations from the complex plane to itself, rather than just as simple addition and multiplication. The visual representations of these operations are given as the parallelogram and similar-triangle laws for addition and multiplication respectively. | Penrose asks us to view complex addition and multiplication as transformations from the complex plane to itself, rather than just as simple addition and multiplication. The visual representations of these operations are given as the parallelogram and similar-triangle laws for addition and multiplication respectively. | ||
[[File:Fig 5p1.png|thumb|center]] | [[File:Fig 5p1.png|thumb|center]] | ||
Rather than just ‘adding’ and ‘multiplying’ these can be viewed as ‘translation’ and ‘rotation’ within the complex plane. For example, multiply a real number by the complex number | Rather than just ‘adding’ and ‘multiplying’ these can be viewed as ‘translation’ and ‘rotation’ within the complex plane. For example, multiply a real number by the complex number <math>i</math> rotates the point in the complex plane π/2 and viewing the parallelogram and similar-triangle laws as translation and rotation: | ||
[[File:Fig 5p2.png|thumb|center]] | [[File:Fig 5p2.png|thumb|center]] | ||
Penrose further introduces the concept of polar coordinates where | Penrose further introduces the concept of polar coordinates where <math>r</math> is the distance from the origin and <math>θ</math> is the angle made from the real axis in an anticlockwise direction. | ||
[[File:Fig 5p4.png|thumb|center]] | [[File:Fig 5p4.png|thumb|center]] | ||
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Armed with both the cartesian and polar representations of complex numbers, it is now possible to show that the multiplication of two complex numbers leads to adding their arguments and multiplying the moduli. This, for the moduli, converts multiplication into addition. | Armed with both the cartesian and polar representations of complex numbers, it is now possible to show that the multiplication of two complex numbers leads to adding their arguments and multiplying the moduli. This, for the moduli, converts multiplication into addition. | ||
This idea is fundamental in the use of logarithms. We first start with the expression | This idea is fundamental in the use of logarithms. We first start with the expression <math>b^{m+n} = b^m \times b^n</math>, which represents the idea of [https://en.wikipedia.org/wiki/Exponentiation exponentiation]. This is easy to grasp for <math>m</math> and <math>n</math> being positive integers, as each side just represents <math>m+n</math> instances of the number <math>b</math>, all multiplied together. If <math>b</math> is positive, this law is then showed to hold for exponents that are positive integers, values of 0, negative, and fractions. If <math>b</math> is negative, we require further expansion into the complex plane. | ||
We would need a definition of | We would need a definition of <math>b^p</math> for all complex numbers <math>p,q,b</math> such that <math>b^{p+q} = b^p \times b^q</math>. If we then define the [https://en.wikipedia.org/wiki/Logarithm logarithm to the base b] as the inverse of the function <math>f(z) = b^z</math> such that <math>z=log_bw</math> for <math>w=b^z</math> then we should expect <math>z=log_b(p \times q) = log_bp + log_bq</math>. This would then convert multiplication into addition and allow for exponentiation in the complex plane. | ||
=== 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 | We need to be careful with the above assertion of the logarithm, mainly since <math>b^z</math> and <math>log_bw</math> are ‘many valued’. Solving the equations would require a particular choice for <math>b</math> 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 <math>w=e^z</math>. | ||
However, even with this natural logarithm we run into multi-valuedness ambiguity from above. Namely that | However, even with this natural logarithm we run into multi-valuedness ambiguity from above. Namely that <math>z</math> still has many values that lead to the same solution with <math>z+2πin</math>, where <math>n</math> is any integer we care to choose. This represents a full rotation of <math>2π</math> in the complex plane with all multiples of <math>n</math> achieving the same point, <math>z</math>. | ||
Penrose goes further in representing | Penrose goes further in representing <math>z</math> with polar coordinates showing <math>z=logr+iθ</math>, then <math>e^z=re^{iθ}</math>. This formulation shows us that when we multiply two complex numbers, we take the product of their moduli and the sum of their arguments (using the addition to multiplication formula introduced in 5.2). | ||
Rounding out the chapter, Penrose gives us another further representation of assuming | Rounding out the chapter, Penrose gives us another further representation of assuming <math>r=1</math>, such that we recover the ‘unit circle’ in the complex plane with <math>w=e^{iθ}</math>. We can 'encapsulate the essentials of trigonometry in the much simpler properties of complex exponential functions' on this circle by showing <math>e^{iθ}=cos(θ) + isin(θ)</math>. | ||
* | * <math>e^{i\theta}</math> is helpful notation for understanding rotating | ||
* | * <math>e^{i\theta} = cos \theta + i sin \theta</math> | ||
* (Worth looking into [https://en.wikipedia.org/wiki/Taylor_series Taylor Series], which is related.) | * (Worth looking into [https://en.wikipedia.org/wiki/Taylor_series Taylor Series], which is related.) | ||
=== 5.4 Complex Powers === | === 5.4 Complex Powers === | ||
Returning to the ambiguity problem of multi-valuedness, it seems the best way to avoid issues is when a particular choice of | Returning to the ambiguity problem of multi-valuedness, it seems the best way to avoid issues is when a particular choice of <math>logw</math> has been specified. As an example, <math>w^z</math> with <math>z=\frac{1}{2}</math>. We can specify a rotation for <math>logw</math> to achieve <math>+w^\frac{1}{2}</math>, then another rotation of <math>logw</math> to achieve <math>-w^\frac{1}{2}</math>. The sign change is achieved because of the Euler formula <math>e^{πi}=-1</math>. Note the process: | ||
<math>w^z=e^{zlogw}=e^{zre^{iθ}}=e^{ze^{iθ}}</math>, then specifying rotations for theta allows us to achieve either | <math>w^z=e^{zlogw}=e^{zre^{iθ}}=e^{ze^{iθ}}</math>, then specifying rotations for theta allows us to achieve either <math>+w^\frac{1}{2}</math> or <math>-w^\frac{1}{2}</math>. | ||
Penrose notes an interesting curiosity for the quantity | Penrose notes an interesting curiosity for the quantity <math>i^i</math>. We can specify <math>logi=\frac{1}{2}πi</math> because of the general relationship <math>logw=logr+iθ</math>. If <math>w=i</math>, then its easy to see <math>logi=\frac{1}{2}πi</math> from noting that y is on the vertical axis in the complex plane (rotation of <math>\frac{π}{2}</math>). This specification, and all rotations, amazingly achieve real number values for <math>i^i</math>. | ||
We end the section with the idea of finite multiplicative groups, or [https://en.wikipedia.org/wiki/Cyclic_group cyclic groups] Z<sub>n</sub>, which contain | We end the section with the idea of finite multiplicative groups, or [https://en.wikipedia.org/wiki/Cyclic_group cyclic groups] Z<sub>n</sub>, which contain <math>n</math> quantities ([https://en.wikipedia.org/wiki/Root_of_unity#:~:text=The%20nth%20roots%20of%20unity%20are%2C%20by%20definition%2C%20the,and%20often%20denoted%20%CE%A6n. nth roots of unity if around the unit circle]) with the property that any two can be multiplied together to get another member of the group. As an example, Penrose gives us <math>w^z=e^{ze^{i(θ+2πin)}}</math> with <math>n=3</math>, leading to three elements <math>1, ω, ω^2</math> with <math>ω=e^\frac{2πi}{3}</math>. Note <math>ω^3=1</math> and <math>ω^-1=ω^2</math>. These form a cyclic group Z<sub>3</sub> and in the complex plane, represent vertices of an equilateral triangle. Multiplication by ω rotates the triangle through <math>\frac{2}{3}π</math> anticlockwise and multiplication by <math>ω^2</math> turns it through <math>\frac{2}{3}π</math> clockwise. The cyclic group is graphically shown below: | ||
[[File:Fig5p11.png|thumb|center]] | [[File:Fig5p11.png|thumb|center]] | ||
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Penrose rounds out the chapter with some examples of complex concepts in the world of particle physics. Additive quantum numbers were briefly introduced in section 3.5, and here we are introduced to multiplicative quantum numbers, which are quantified in terms of nth roots of unity. | Penrose rounds out the chapter with some examples of complex concepts in the world of particle physics. Additive quantum numbers were briefly introduced in section 3.5, and here we are introduced to multiplicative quantum numbers, which are quantified in terms of nth roots of unity. | ||
The notion of [https://en.wikipedia.org/wiki/Parity_(physics) parity] is introduced as approximately a multiplicative quantum number with n=2, and an example is the family of particles called [https://en.wikipedia.org/wiki/Boson bosons]. Penrose notes that [https://en.wikipedia.org/wiki/Fermion fermions] could also be considered a parity group but it is not the normal convention. The distinction between these two particles are that bosons are completely restored to their original states under a | The notion of [https://en.wikipedia.org/wiki/Parity_(physics) parity] is introduced as approximately a multiplicative quantum number with n=2, and an example is the family of particles called [https://en.wikipedia.org/wiki/Boson bosons]. Penrose notes that [https://en.wikipedia.org/wiki/Fermion fermions] could also be considered a parity group but it is not the normal convention. The distinction between these two particles are that bosons are completely restored to their original states under a <math>2π</math> rotation, whereas fermions require <math>4π</math> (two rotations). Thus a multiplicative quantum number of <math>-1</math> can be assigned to a fermion and <math>+1</math> to a boson. | ||
An example of a multiplicative quantum number with | An example of a multiplicative quantum number with <math>n=3</math> relates to quarks, which have values for electric charge that are not integer multiples of the electron’s charge, but in fact <math>\frac{1}{3}</math> multiples. If <math>q</math> is the value of electric charge with respect to an electron (<math>q=-1</math> for electron charge), then quarks have q=<math>\frac{2}{3}</math> or <math>-\frac{1}{3}</math> and antiquarks q=<math>\frac{1}{3}</math> or <math>-\frac{2}{3}</math>. If we take the multiplicative quantum number <math>e^{-2qπi}</math>, then we find the values <math>1,ω,ω^2</math> from section 5p4, which constitute the cyclic group Z<sub>3</sub>. | ||
== Chapter 6 Real-number calculus == | == Chapter 6 Real-number calculus == | ||
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=== 6.2 Slopes of functions === | === 6.2 Slopes of functions === | ||
Differentiation is concerned with and calculates the rates that things change, or ‘slopes’ of these things. For the curves given in section 6p1 above, two of the three do not have unique slopes at the origin and are said to be not ''differentiable'' at the origin, or not ''smooth'' there. Further, the curve of theta(x) has a jump at the origin which is to say that it is discontinuous there, whereas | Differentiation is concerned with and calculates the rates that things change, or ‘slopes’ of these things. For the curves given in section 6p1 above, two of the three do not have unique slopes at the origin and are said to be not ''differentiable'' at the origin, or not ''smooth'' there. Further, the curve of theta(x) has a jump at the origin which is to say that it is discontinuous there, whereas <math>|x|</math> and <math>x^2</math> were continuous everywhere. | ||
Taking differentiation a step further, Penrose shows us two plots which look very similar, but are represented by different functions, | Taking differentiation a step further, Penrose shows us two plots which look very similar, but are represented by different functions, <math>x^3</math> and <math>x|x|</math>. Each are differentiable and continuous, but the difference has to do with the curvature ([https://en.wikipedia.org/wiki/Second_derivative second derivative]) at the origin. <math>x|x|</math> does not have a well-defined curvature here and is said to not be ''twice differentiable''. | ||
=== 6.3 Higher derivatives; | === 6.3 Higher derivatives; <math>C^\infty</math>-smooth functions === | ||
Looking closer at the concept of two derivatives of the same function (second derivative, or curvature), Penrose shows us the functions from 6p2 and their first and second derivatives. Note that the first derivative of | Looking closer at the concept of two derivatives of the same function (second derivative, or curvature), Penrose shows us the functions from 6p2 and their first and second derivatives. Note that the first derivative of <math>f(x)</math>, written <math>f’(x)</math>, meets the x-axis at places of local minima or maxima and the second derivative of <math>f(x)</math>, written <math>f’’(x)</math>, meets the x-axis where the curvature goes to <math>0</math> and is said to be a point of inflection. | ||
[[File:Fig 6p5.png|thumb|center]] | [[File:Fig 6p5.png|thumb|center]] | ||
In general, a function can be smooth for many derivatives and the mathematical terminology for general smoothness is to say that | In general, a function can be smooth for many derivatives and the mathematical terminology for general smoothness is to say that <math>f(x)</math> is <math>C^n</math>-smooth. It can be seen that <math>x|x|</math> is <math>C^1</math>-smooth but not <math>C^2</math>-smooth due to the discontinuity at the origin in the derivative. In general <math>x^n|x|</math> is <math>C^n</math>-smooth but not <math>C^{n+1}</math>-smooth. In fact, a function can be <math>C^\infty</math>-smooth if it is smooth for every positive integer. Note that negative integers for <math>x^n</math> immediately are not smooth for <math>x^{-1}</math> (discontinuous at the origin). | ||
Penrose notes that Euler would have required | Penrose notes that Euler would have required <math>C^\infty</math>-smooth functions to be defined as functions, and then gives the function: | ||
:<math>h(x) = \begin{cases} | :<math>h(x) = \begin{cases} | ||
0, & \mbox{if } x \le 0 \\ | 0, & \mbox{if } x \le 0 \\ | ||
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\end{cases} | \end{cases} | ||
</math> | </math> | ||
as an example of a | as an example of a <math>C^\infty</math>-smooth function but one that Euler would still not be happy with since it is two functions stuck together. | ||
=== 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 | 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 <math>f(x)</math> to <math>f(z)</math> in the complex plane, then all we require is for <math>f(z)</math> to be once differentiable (a kind of <math>C^1</math>-smooth function). That’s it, magically. We will see that this can be stated with <math>f(x)</math> 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’. | 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 | For the second method, the power series of <math>f(x)</math> 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 <math>C^\infty</math>-smooth. We must take and evaluate derivatives <math>f(x)</math> to find the coefficients, thus an infinite number of derivatives (positive integers) must exist for the power series to exist. If we evaluate <math>f(x)</math> at the origin, we call this a power series expansion about the origin. About any other point <math>p</math> would be considered a power series expansion about <math>p</math>. (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 | The power series is considered analytic if it encompasses the power series about point <math>p</math>. If it is analytic at all points of its domain, we call it an analytic function or, equivalently, a <math>C^ω</math>-smooth function. Euler would be pleased with this notion of an analytic function, which is ‘smoothier’ than the set of <math>C^\infty</math>-smooth functions (<math>h(x)</math> from 6p3 is <math>C^\infty</math>-smooth but not <math>C^ω</math>-smooth). | ||
* Physics in trying to understand reality by approximating it. | * Physics in trying to understand reality by approximating it. | ||
Line 285: | Line 286: | ||
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’. | 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 | 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 <math>C^n</math>-functions into the negative integer space (<math>C^{-1},C^{-2},...</math>) 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 (<math>f(z)</math> is complex-smooth) nor the conclusion (<math>f(z)</math> 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 <math>a</math> to another point <math>b</math> 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 <math>a</math> to <math>b</math>. | |||
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 <math>a</math> to <math>b</math> 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 <math>a</math> and <math>b</math> (or the point of non-analyticity) lie in the complex plane. Note that since <math>logz</math> 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 <math>p</math>) by what it is doing at a set of points surrounding the origin or the general point <math>p</math>. | |||
:<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 <math>n</math> 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 <math>f(z)</math> 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 <math>f(z)</math>, thereby showing the function has an actual <math>n</math>th derivative at the origin or general point <math>p</math>. 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 (<math>f(z)</math> is complex-smooth) nor the conclusion (<math>f(z)</math> 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 <math>f(z)</math> holomorphic in domain <math>D</math>, can we extend the domain to a larger <math>D’</math> so that <math>f(z)</math> 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’ <math>logz</math>. 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 <math>2πi</math>. 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]] |