Quark Central

The Road to Reality

NOTE before you begin: As this is not a scholarly paper, I am being somewhat lax regarding attribution. Therefore, you should know that when text is in italics, those are the words of other people. I attempt to provide the source for all material in italics in context. When no source is specifically provided, it is from Roger Penrose's book. (Italics are also used to refer to titles of books and articles that I mention along the way). What follows is a gross (inexact, partial, sparse) summary of what I found to be beautiful in the book, intended as a note to myself to try to remember what I read. I urge you to read the original, all 1099 pages, for yourself.

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Roger Penrose, in his The Road to Reality, provides us with the above image illustrating the deep connections he perceives. Each of the circles can be interpreted as "worlds": The Platonic Mathematical, Physical, and Mental. A small portion of the Platonic Mathematical World "instantiates" the Physical World (as shown by the cone segment emanating from the Platonic Mathematical). In turn, the Physical instantiates the Mental, and the Mental instantiates the Platonic Mathematical. He writes:

Thus the entire physical world is depicted as being governed according to mathematical laws. ... On this view, everything in the physical universe is indeed governed in completely precise detail by mathematical principles. ... If this is right, then even our own physical actions would be entirely subject to such ultimate mathematical control, where ‘control’ might still allow for some random behaviour governed by strict probabilistic principles. ... I have represented the entire Platonic world to be within the compass of mentality. This is intended to indicate that—at least in principle—there are no mathematical truths that are beyond the scope of reason. ... it remains a deep puzzle why mathematical laws should apply to the world with such phenomenal precision. Moreover, it is not just the precision but also the subtle sophistication and mathematical beauty of these successful theories that is profoundly mysterious. There is also an undoubted deep mystery in how it can come to pass that appropriately organized physical material—and here I refer specifically to living human (or animal) brains—can somehow conjure up the mental quality of conscious awareness.

This is an interpretation that speaks to me. In my own thinking about "what is going on", I recognize three deeps mysteries: (1) The Big Bang. How is it that from something infinitesimal, a whole universe, as we perceive it, can arise? (2) The emergence of life. How is it that from inanimate matter, evolution happens to instantiate life as we know it? (3) The emergence of consciousness. How is it that life, once it begins, evolves to instantiate what we know and feel as consciousness?

In my view, having accepted that The Big Bang is the best explanation for the emergence of what we perceive as the interactions of matter and energy, how then, through the process of evolution, did life begin? It is no simple thing for such a thing to arise, and we have no coherent explanation for it. Similarly, evolution can be accepted as the driver for future change in the forms of life, but then, how do we explain consciousness? Consciousness is not a mere form of life; it is something different, something much more mysterious than what the evolution of life forms can account for.

In the rest of this post, I will leave those questions aside, for that is not what Roger Penrose's book, The Road to Reality, is about. I think we first need to understand what has been learned so far. For me, The Road to Reality is an attempt to synthesize what we already know, and there is much in the book I do not understand and do not know. But it seems clear that there is, as written in the Wikipedia article about The Unreasonable Effectiveness of Mathematics in the Natural Sciences:

Wigner begins his paper with the belief, common among those familiar with mathematics, that mathematical concepts have applicability far beyond the context in which they were originally developed. Based on his experience, he says "it is important to point out that the mathematical formulation of the physicist's often crude experience leads in an uncanny number of cases to an amazingly accurate description of a large class of phenomena". He then invokes the fundamental law of gravitation as an example. Originally used to model freely falling bodies on the surface of the earth, this law was extended on the basis of what Wigner terms "very scanty observations" to describe the motion of the planets, where it "has proved accurate beyond all reasonable expectations".

There is something deeply mysterious about mathematics and its ability to describe the universe. In the rest of this post, I will attempt to summarize what Roger Penrose writes about. Let's begin.

Commentary on the Chapters

Chapter 1 - The Roots of Science

It is this chapter that the above text is derived from. The graphic at the beginning of this post comes from this chapter and serves to illustrate the three great mysteries I think about. The graphic provides me with a synthesis of the three great mysteries. The rest of the chapter deals with the idea of 'mathematical truth'. Notable takeaways:

Mathematics itself indeed seems to have a robustness that goes far beyond what any individual mathematician is capable of perceiving. Those who work in this subject, whether they are actively engaged in mathematical research or just using results that have been obtained by others, usually feel that they are merely explorers in a world that lies far beyond themselves - a world which possesses an objectivity that transcends mere opinion, be that opinion their own or the surmise of others, no matter how expert those others might be.

It may be helpful if I put the case for the actual existence of the Platonic world in a different form. What I mean by this 'existence' is really just the objectivity of mathematical truth. Platonic existence, as I see it, refers to the existence of an objective external standard that is not dependent upon our individual opinions nor upon our particular culture. Such 'existence' could also refer to things other than mathematics, such as to morality or aesthetics, but I am here concerned just with mathematical objectivity, which seems to be a much clearer issue.

Of less obvious relevance here—but of clear importance in the broader context—is the question of an absolute ideal of morality: what is good and what is bad, and how do our minds perceive these values? Morality has a profound connection with the mental world, since it is so intimately related to the values assigned by conscious beings and, more importantly, to the very presence of consciousness itself. It is hard to see what morality might mean in the absence of sentient beings. As science and technology progress, an understanding of the physical circumstances under which mentality is manifested becomes more and more relevant. I believe that it is more important than ever, in today’s technological culture, that scientific questions should not be divorced from their moral implications. But these issues would take us too far afield from the immediate scope of this book. We need to address the question of separating true from false before we can adequately attempt to apply such understanding to separate good from bad.

There is, finally, a further mystery concerning Fig. 1.3, which I have left to the last. I have deliberately drawn the figure so as to illustrate a paradox. How can it be that, in accordance with my own prejudices, each world appears to encompass the next one in its entirety? I do not regard this issue as a reason for abandoning my prejudices, but merely for demonstrating the presence of an even deeper mystery that transcends those which I have been pointing to above. There may be a sense in which the three worlds are not separate at all, but merely reflect, individually, aspects of a deeper truth about the world as a whole of which we have little conception at the present time. We have a long way to go before such matters can be properly illuminated. I have allowed myself to stray too much from the issues that will concern us here. The main purpose of this chapter has been to emphasize the central importance that mathematics has in science, both ancient and modern. Let us now take a glimpse into Plato’s world—at least into a relatively small but important part of that world, of particular relevance to the nature of physical reality.

Chapter 2 - An Ancient Theorem and a Modern Question

Euclid's Postulates: 1) There is a unique straight line segment connecting any two points. 2) Any straight line segment has unlimited extendibility. 3) There exists a circle with any center and any radius. 4) All right angles are equal (in effect, this asserts congruence when a body is moved; ie. the isotropy and homogeneity of space). 5) For any straight line and any point not on the line, there is a unique straight line through the point which is parallel to the line.

The first 4 postulates assert the isotropy and homogeneity of two-dimensional space. Adding the fifth allows us to rigorously define the existence of a square.

Euclid’s pedantry is related to a deep issue that has a great deal to say about the actual geometry of the universe, and in more than one way. In particular, it is not at all an obvious matter whether physical ‘squares’ exist on a cosmological scale in the actual universe. This is a matter for observation, and the evidence at the moment appears to be conflicting.

The rest of the chapter discusses proofs of the Pythagorean Theorem, and establishes the existence of Euclidean geometry (which crucially depends on the 5th postulate). Then we consider the possibility that the 5th postulate might be false. This brings us to Hyperbolic geometry. While the sum of the angles of a triangle in Euclidean geometry sum to pi (180 degrees), we learn that in Hyperbolic geometry this is not the case. It turns out that the sum of the angles of a Hyperbolic triangle is less than 180 degrees, where the shortfall is proportional to the area of the triangle.

More explicitly, if the three angles of the triangle are a, b, and c, then we have the formula (found by Johann Heinrich Lambert 1728–1777) 180-(a+b+c)=CA where A is the area of the triangle and C is some constant. This constant depends on the ‘units’ that are chosen in which lengths and areas are to be measured. We can always scale things so that C=1. It is, indeed, a remarkable fact that the area of a triangle can be so simply expressed in hyperbolic geometry. In Euclidean geometry, there is no way to express the area of a triangle simply in terms of its angles, and the expression for the area of a triangle in terms of its side-lengths is considerably more complicated.

What follows in the rest of the chapter is some historical notes on the development of hyperbolic geometry, including formulas for distance and a discussion of some of the projections used to represent hyperbolic geometry (conformal, projective, pseudo-sphere). Unfortunately, this type of mathematics is beyond me. The author concludes the chapter with this:

What, then, is the observational status of the large-scale spatial geometry of the universe? It is only fair to say that we do not yet know, although there have been recent widely publicized claims that Euclid was right all along, and his Wfth postulate holds true also, so the averaged spatial geometry is indeed what we call ‘Euclidean’.12 On the other hand, there is also evidence (some of it coming from the same experiments) that seems to point fairly Wrmly to a hyperbolic overall geometry for the spatial universe.13 Moreover, some theoreticians have long argued for the elliptic case, and this is certainly not ruled out by that same evidence that is argued to support the Euclidean case (see the later parts of §34.4). As the reader will perceive, the issue is still fraught with controversy and, as might be expected, often heated argument. In later chapters in this book, I shall try to present a good many of the considerations that have been put forward in this connection (and I do not attempt to hide my own opinion in favour of the hyperbolic case, while trying to be as fair to the others as I can).

Chapter 3 - Kinds of Number in the Physical World

Here I will simply summarize the important points and curious things that are revealed in this chapter. I urge the interested reader to read the original for much more detail and for a full appreciation of the beauty of mathematics.

We are all familiar with whole numbers (integers) [1,2,3...] and ratios (fractions) [1/2,2/3,3/5...]. To the Pythagoreans, it was at first natural to believe that this was all that was needed, i.e. that any number could be expressed as a rational. So it came as a surprise when they discovered that their beloved Pythagorean Theorem revealed that this is not the case. Consider the Pythagorean Theorem:

c^2=a^2+b^2

Now take a right triangle whole sides are equal to 1, and the formula becomes:

c^2=1^2+1^2

c^2=1+1

c^2=2

c=2^(1/2)

What is this number, the square root of 2? The Pythagoreans would like to express it as a rational number (as a fraction): "the square root of two" is the number whose value is the fraction a/b where a and b are some integer. But it turned out that this cannot be done. There is no way to express "the square root of two" as a rational number (for the proof, reference Chapter 3 of The Road to Reality).

Chapter 4 - Magical Complex Numbers

I don't have much to say about this chapter. It deals with the concept of complex numbers. Take the equation:

x^2+1=0

x^2=-1

So what two numbers, multiplied together, give the value -1? This is quite a conundrum. That number, in modern mathematics, is denoted as i. Studying the properties of this number, one can then build up the complex plane and define the properties of complex numbers, which are denoted by the expression a+ib, where a and b are real numbers.

A more interesting aspect of complex numbers is their ability to give some insight into infinite series:

1+x^2+x^4+x^6+x^8... = (1-x^2)^(-1) and

1-x^2+x^4-x^6+x^8... = (1+x^2)^(-1)

It is remarkable that these infinite series converge. When looked at naively, these equations lead to absurdities such as (using 2 for the value of x):

1+2^2+2^4+2^6+2^8... = (1-2^2)^(-1) = (1-4)^(-1) = -1/3

But what do these complex singularities have to do with the question of convergence or divergence of the corresponding power series? There is a striking answer to this question. We are now thinking of our power series as functions of the complex variable z, rather than the real variable x, and we can ask for those locations of z in the complex plane for which the series converges and those which which it diverges. The remarkable general answer, for any power series whatever is that there is some circle in the complex plane, centred at 0, called the circle of convergence, with the property that if the complex number z lies strictly inside the circle then the series converges for that value of z, whereas if z lies strictly outside the circle then the series diverges for that value of z.

...

To find where the circle of convergence actually is for some particular given function, we look to see where the singularities of the function are located in the complex plane, and we draw the largest circle, centred about the origin z = 0, which contains no singularity in its interior.

...

In the particular case (1-z^2)^(-1) and (1+z^2)^(-1) that we have just been considering, the singularities are of a simple type called poles. Here these poles all lie at unit distance from the origin, and we see that the circle of convergence is, in both cases, just the unit circle about the origin. ... This explains why the two functions converge and diverge in the same regions - a fact that is not manifest from their properties simply as functions of real variables. Thus complex numbers supply us with deep insights into the behaviour of power series that are simply not available from the consideration of their real-variable structure.

Chapter 5 - Geometry of Logarithms, Powers, and Roots


Six Months of Postal Mail

Start Date: Friday, 27 November 2020

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Favorite Quotes

I see it all perfectly; there are two possible situations — one can either do this or that. My honest opinion and my friendly advice is this: do it or do not do it — you will regret both.

--- Søren Kierkegaard


In addition to my other numerous acquaintances, I have one more intimate confidant… My depression is the most faithful mistress I have known — no wonder, then, that I return the love.

--- Søren Kierkegaard


You have your way. I have my way. As for the right way, the correct way, and the only way, it does not exist.

--- Friedrich Nietzsche


None are more hopelessly enslaved than those who falsely believe they are free.

--- Johann Goethe


It is no measure of health to be well adjusted to a profoundly sick society.

--- J. Krishnamurti


The victor will always be the judge, and the vanquished the accused.

--- Hermann Goering


... but only the hypocrite is really rotten to the core.

--- Hannah Arendt


A casual stroll through the lunatic asylum reveals that faith does not prove anything.

--- Friedrich Nietzsche


Ninety percent of the time things will turn out worse than you expect. The other 10 percent of the time you had no right to expect so much.

--- Norman Augustine


What Orwell feared were those who would ban books. What Huxley feared was that there would be no reason to ban a book, for there would be no one who wanted to read one. Orwell feared those who would deprive us of information. Huxley feared those who would give us so much that we would be reduced to passivity and egoism. Orwell feared that the truth would be concealed from us. Huxley feared the truth would be drowned in a sea of irrelevance. Orwell feared we would become a captive culture. Huxley feared we would become a trivial culture, preoccupied with some equivalent of the feelies, the orgy porgy, and the centrifugal bumblepuppy. As Huxley remarked in Brave New World Revisited, the civil libertarians and rationalists who are ever on the alert to oppose tyranny "failed to take into account man's almost infinite appetite for distractions." In 1984, Orwell added, people are controlled by inflicting pain. In Brave New World, they are controlled by inflicting pleasure. In short, Orwell feared that what we fear will ruin us. Huxley feared that what we desire will ruin us.

--- Neil Postman


I have my opinions and beliefs, and I’ve engaged in more than my share of futile arguments. As far as I know, I’ve never changed anybody’s mind about anything. People believe what they want to believe and then find things that support and justify their beliefs.

--- Steven Papier


We live in a society exquisitely dependent on science and technology, in which hardly anyone knows anything about science and technology.

--- Carl Sagan


Wer etwas will, findet Wege, wer etwas nicht will, findet Gründe.

--- Jim Rohn


The really dangerous American fascist... is the man who wants do in the United States in an American way what Hitler did in Germany in a Prussian way. The American fascist would prefer not to use violence. His method is to poison the channels of public information. With a fascist the problem is never how best to present the truth to the public but how best to use the news to deceive the public into giving the fascist and his group more money or more power... They claim to be super-patriots, but they would destroy every liberty guaranteed by the Constitution. They demand free enterprise, but are the spokesmen for monopoly and vested interest. Their final objective, toward which all their deceit is directed, is to capture political power so that, using the power of the state and the power of the market simultaneously, they may keep the common man in eternal subjection.

--- Henry Wallace, New York Times, April 9, 1944


You're looking for three things, generally, in a person: intelligence, energy, and integrity. And if they don't have the last one, don't even bother with the first two.

--- Warren Buffet


The avocation of assessing the failures of better men can be turned into a comfortable livelihood, providing you back it up with a Ph.D.

-- Nelson Algren, "Writers at Work"


Above all, don't lie to yourself. The man who lies to himself and listens to his own lie comes to a point that he cannot distinguish the truth within him, or around him, and so loses all respect for himself and for others. And having no respect he ceases to love.

― Fyodor Dostoevsky, The Brothers Karamazov


I am an atheist, out and out. It took me a long time to say it. I've been an atheist for years and years, but somehow I felt it was intellectually unrespectable to say one was an atheist, because it assumed knowledge that one didn't have. Somehow, it was better to say one was a humanist or an agnostic. I finally decided that I'm a creature of emotion as well as of reason. Emotionally, I am an atheist. I don't have the evidence to prove that God doesn't exist, but I so strongly suspect he doesn't that I don't want to waste my time.

― Isaac Asimov


We must respect the other fellow's religion, but only in the sense and to the extent that we respect his theory that his wife is beautiful and his children smart.

― H.L. Mencken, Minority Report


If you do not respect your own wishes, no one else will. You will simply attract people who disrespect you as much as you do.

― Vironika Tugaleva