This episode shows how the specific values of the constants present a serious problem for the realization of physicists' dream of finding a Theory of Everything. It explains the mystery of constants, a problem which Richard Feynman called “one of the greatest damn mysteries of physics.” The mystery is that the numbers seem completely arbitrary with no seeming reason for their values. From the perspective of physics, these numbers could have taken on any value whatsoever. So how can physicists explain these numbers?
Essay Version of Episode 2
Let's start with a quote from one of the greatest physicists of all time, Richard Feynman, about one of the constants of nature: "It's one of the greatest damn mysteries of physics, a magic number that comes to us with no understanding by man. You might say the hand of God wrote that number. And we don't know how he pushed his pencil.” Before we try to explain exactly what he means by this, let's review what we covered in the last episode.
Summary of the Last Episode
In Episode 1, we made three basic points. First, we defined what fundamental physics refers to, and how it's the basis of all the different sciences, and we showed how fundamental physics breaks down into two different parts. It defines the different particles of the universe, the basic building blocks that the universe is made out of, like electrons and quarks. We also explained how there are fundamental laws of physics that govern the interaction between these particles.
Second, we showed how both the particles and the laws have qualities and quantities. The qualities of let's say, an electron, a particle, would be an electron has a mass. But the quantity would be how much mass does it have? And the quality of a law of physics would be the law that opposite charges attract - like an electron and a proton - and similar charges repel - like two electrons would repel each other. And the quantity would be how strong is the force of repulsion? How strongly do two electrons push each other away?
We showed that these quantities are called the constants of nature, values that are always the same everywhere in the universe. For example, two electrons always push each other away with the same force everywhere; and every electron everywhere has the same quantity of mass. And these are called constants of nature - the mass of an electron and the strength of the repulsion between two electrons.
And the third thing we explained was physicists' dream of finding a theory of everything - which is one set of laws that explains, in theory, everything in our universe. And this was the pursuit - the ultimate goal of all physics was to find this theory of everything.
Feynman's Quote On The Mystery of The Constants of Nature
In this episode, we'll show how these constants create a great mystery. To see this, let's see the full quote from 1985 where Feynman discusses the mystery. Don't worry about fully understanding the quote, just try to get a basic feel for the idea. We'll explain it more as we go along. Here's Feynman: "There is a most profound and beautiful question associated with the observed coupling constant. It is a simple number that has been experimentally determined to be close to 0.08542455."
Before we go further in the quote, let me interrupt for a second. One thing to keep in mind is that the coupling constant is a number that determines the probability of an electron interacting with light. And that's what determines the strength of the electromagnetic force between two electrons. We called this last time by its much more common name, the fine structure constant, and that number is 1 over 137.03599139. And it's just the square of the number Feynman is talking about.
Let's continue the quote:
It has been a mystery ever since it was discovered more than 50 years ago, and all good theoretical physicists put this number up on their wall and worry about it. Immediately, you'd like to know where this number for a coupling comes from? Is it related to pi, or perhaps to the base of natural logarithms? Nobody knows. It's one of the greatest damn mysteries of physics, a magic number that comes to us with no understanding by man. You might say the hand of God wrote that number, and we don't know how he pushed his pencil. We know what kind of dance to do experimentally to measure this number very accurately. But we don't know what kind of dance to do on the computer to make this number come out without putting it in secretly. - QED, page 127
Let's try to explain the mystery Feynman is referring to and understand why it's one of the greatest mysteries of physics.
Criteria for a Theory of Everything
To understand the mystery, we need to take a step back and appreciate the characteristics physicists are looking for in a theory of everything, a theory that is supposed to explain everything in the universe. To do that, let's look at another statement from Albert Einstein, who spent the last 30 years of his life pursuing the dream of a theory of everything. Here's Einstein:
It can scarcely be denied that the supreme goal of all theory is to make the irreducible basic elements as simple and as few as possible, without having to surrender the adequate representation of a single datum of experience.
In this quote, Einstein implicitly supplies us with four criteria by which we can judge the basic components of a theory of everything, what Einstein is referring to as the supreme goal of all theory. His four criteria are: (i) the basic components should be irreducible; (ii) they should be few; (iii) they should be simple; (iv) they should be representative of all the available data.
Let's explain what these criteria mean, one at a time.
(i) Irreducible: Irreducible is just another word for fundamental. It means it's the most basic thing, it can't be reduced to anything else. And in a theory of everything, you're looking for the foundation of all science, all physics, and therefore, you're looking for something that's irreducible. If it could be reduced to another theory, that other theory is your theory of everything, not this one. So a theory of everything must be fundamental, and therefore irreducible.
(ii) Few: By few, Einstein means that we don't want a thousand different laws to explain the different phenomena of nature. We're not looking for a final theory of nature that has one law for thunder, one for lightning, one for rain, and one for the sun. That's almost a pagan way of viewing the universe, where every single phenomenon of nature has a different god that represents it. Rather, scientists have shown, over the course of the whole scientific enterprise, that you can reduce the laws of nature and show how there's a unity behind them, and reduce the laws to fewer and fewer laws.
Currently, as it stands, there are two fundamental laws, quantum mechanics and general relativity. And the idea is to find a unified theory, to reduce it to even one. Einstein is saying, the fewer laws you have, the better your theory of everything is. And ideally, a unified theory, where there's only one law that encompasses everything else, would be the ultimate dream and goal.
(iii) Simple: The way physicists use the term simple is a little different than it's commonly used. Usually, people think about simple as opposed to complicated. Simple refers to something that's easy to understand, as opposed to something hard to understand.
Simple in physics is being contrasted with complex. Something simple has very few components, as opposed to something complex which has many different components, many different moving parts.
So for example, E equals mc squared. That's not a fundamental law of physics - it's close to it, but it's not a fundamental law. But it means energy is equal to mass times the speed of light squared. There's only a few different parts there - there's energy, which is equal to mass, and then you have the constant of the speed of light squared. As opposed to a law which you might have one unified law, but there might be 20 different terms and different variables which have complicated relationships to each other.
So the idea is to have not just a few laws of physics, ideally one, but you want that law to be simple and not to have a lot of moving parts and complex components. That's what makes a law simple.
(iv) Representative: Einstein said that we want these components of the theory of everything that are irreducible, few, and simple to also be representative of all the available data. He means that a theory of everything, the fundamental law of nature, should explain everything in the universe. Every observation, every data point that we have ever discovered, and hopefully that we will ever discover - if it's a complete theory of everything - should be explained by this theory.
Besides Einstein's four criteria, there's one more criteria that scientists often use to describe their theories. This is something that they would have never predicted. But they've discovered that all their theories are beautiful, at least to a physicists' judgment - that they somehow appeal to our aesthetic sense. Physicist Steven Weinberg discusses this abstract property of beauty in his book, "Dreams of a Final Theory." Here's Weinberg:
There is no logical formula that establishes a sharp dividing line between a beautiful explanatory theory and a mere list of data. But we know the difference when we see it. We demand the simplicity and rigidity in our principles before we are willing to take them seriously. Thus, not only is our aesthetic judgment, a means to the end of finding scientific explanations and judging their validity, it is part of what we mean by an explanation.
Beauty is a surprising feature we find in fundamental physics equations. And it's hard for a layman to fully appreciate what it means. Descriptively it means that something is simple, intuitive, compelling, as opposed to arbitrary and contrived. And there is just the sense you have when you're in the area, of a beautiful, elegant equation, as opposed to some arbitrary contrived equation with a lot of complexity - that things don't really fit together, they feel as if they’ve been put together. There's something about an equation and simplicity - which is interesting since simplicity was one of Einstein's criteria - that's part of what it means for something to be beautiful; something which is simple and elegant and fits together.
As physicists get the sense that there's something beautiful about this equation, that keys them that there's something true about it, something real about it. This observation, that physics equations on a fundamental level are beautiful, it's not something that we know a priori. We wouldn't have predicted in advance. It's something that we've discovered, that we find as scientists uncover these laws, and they go deeper into the underlying reality that makes up our universe, they discover that it gets simpler, it's more intuitive, there's a sense that's compelling, and they find that it becomes beautiful. And that's something that they know, really empirically, it's something they find out through the discovery of what the fundamental laws of physics are. It's not something that they would have known a priori has to be the case.
We ended up having a number of criteria with which we judge a theory of everything, which scientists are looking for when they find the theory of everything. To help us understand these criteria and show that they're intuitive, go back to our analogy from last time.
That is, we had an ethical system. And in the ethical system, we had a number of laws. For example, you can't steal, you can't slander, you have to give a certain amount to charity, you're not allowed to charge a certain amount of interest. And we had dozens of ethical laws. We realize that we could unify these laws - instead of looking at as a list of dozens of laws, we could boil it down to one theory of everything if you will. We had one law, "Do to your neighbor as you want unto yourself."
And we discovered that this one law could be looked at as the foundation, as the unified theory for all the laws of this ethical system, how everything in this ethical system could be derived from this one law. We'd like to show now how this law actually meets the criteria which Einstein has been talking about.
First of all, this law is irreducible, all the other laws - you can't steal, you can slander, you give charity - we showed they could be broken down, they could be reduced to this simpler law, "Do to your neighbor as you'd want unto yourself." But that law itself is the end of the line, we've hit the bottom. This is irreducible, it can't be explained based on something more simple, something more basic.
Secondly, the criteria of being few. We started with dozens of laws, but we end up with one law, "Do to your neighbor as you want done to you." This is a goal, to have few, one if possible.
Also, this law is simple. The laws which we started with may involve a lot of details. For example, the laws of interest might be formulated with complexity to them. And this law, "Do to your neighbor as you want them to yourself" is a simply formulated law.
Also, this law is representative of all the available data. It's truly a theory of everything. All the laws in our ethical system could be boiled down, could be explained by this one law, "Do to your neighbor as you want done unto yourself."
And lastly, it's a beautiful, it's an intuitive law, it appeals to our sense of what the foundation of an ethical system should sound like. It is intuitive, it does sound like it makes sense. And this is basically what scientists are looking for in physics. They're looking for an analog, something where we can have a theory of everything that meets all these criteria.
We would just add to that idea that "Do to your neighbor as you want unto yourself" is a very popular law that's known throughout many cultures over ancient periods of time. And the reason why it's been accepted for so long is because it meets these criteria. It is a foundational law to ethics. And it's viewed as such by all these cultures for such a long period of time because it has these characteristics of what a foundational law of ethics should really be.
The Problem with the Two Theories that Explain the Constants
Based on these criteria that scientists are looking for in a theory of everything, we can begin to see the immense challenge presented by the constants of nature.
Let's try to state the mystery. How can physicists incorporate the constants into the ultimate theory of everything? Meaning to say, how can physicists explain these numbers?
To understand this question, let's think about it as follows. Whenever you're trying to explain something - anything - you have the question whether that thing is fundamental or not. For example, if you're trying to explain water, is water a fundamental element in nature? Or is it comprised of other things? Some of the ancient philosophers thought water is the basic element in nature -that everything is made out of water. But now we know that that's not true. Water can be broken down into hydrogen and oxygen.
Now you get to hydrogen and oxygen - are those fundamental? Or can they be broken down to something else? And we say that no, they're not fundamental either. They can be broken down into electrons and quarks. But when we get to electrons and quarks, we say they're fundamental - there's nothing that they can be broken down into. And that makes them fundamental. That's how we conceive of them - as fundamental particles.
The same thing is with the constants. We're faced with the question: Are these constants fundamental or not? Do they have a deeper cause? Is there something else that's deeper than them, that explains them? Or are they fundamental? Are they uncaused? Are they just the basic numbers of reality?
And it turns out, either way you go trying to explain these constants - by treating them as fundamental or treating them as not fundamental and deriving from some deeper theory - either way you go, it presents an immense challenge to the ultimate goal of discovering a true theory of everything.
Let's explain the problem with each of these possibilities, one at a time. First, let's see the problem with suggesting that these numbers are fundamental, that they can't be explained in terms of anything else more fundamental.
If the constants are fundamental, if they're irreducible and there's nothing that they can be broken down into, then they're uncaused facts of nature - that's just the way reality is. Then you have the problem that they are part of your theory of everything. There's no other way to explain them, you have to incorporate them into the very theory of everything. And now your theory of everything becomes one or two qualitative, beautiful laws, and 25 complex, arbitrary numbers, like 1 over 137.035999139. That's an ugly list of data.
That's not what scientists are looking for in a theory of everything. They're looking for few simple, beautiful ideas and instead get 25 complex numbers of strings of digits. These numbers are not two or one or pi or things like that. It's these long strings of digits with many digits after the decimal point, and there's 25 of them. And if you have to incorporate those into your theory of everything, the theory of everything loses all those key characteristics that makes it a good theory of everything. That's the problem if the constants are fundamental.
Now let's consider the problem with the possibility that they're not fundamental - that they're reducible and could be explained based on something else. On the surface, this is a perfectly logical possibility - that there's some deeper theory, some deeper explanation, which naturally results in these 25 constants. The problem is when you think about that, is you're trying to derive 25 precise, definite quantities, numbers, from a qualitative law. And that just doesn't seem reasonable or plausible that you're going to be able to get from a qualitative law, which has no specific quantities built into it, and then say somehow from that qualitative law, these 25 constants come. It seems unreasonable that that's going to be possible, that you're going to derive exactly these 25 numbers.
Again, these are numbers which you have to determine to the exact decimal point. If it's a theory of everything, it has to define everything, everything has to be explained, which means every single last digit and all these constants. And it's perfectly determined, a logical consequence of this one qualitative law would be these 25 numbers. And it will be determined up to the last decimal point.
And if you could do that, that would be great. But when Feynman was talking about this, it seemed that people had been trying for decades, it didn't seem possible. This was over 50 years after, when Feynman wrote this after the fine structure constant had been discovered. And it's been over 100 years now. Nobody has gotten close at all. Nobody has any clue how this would be. And to some degree, physicists intuitively realized that it just seems highly implausible that you would ever be able to come up with some qualitative law which would determine these 25 constants down to the last decimal point.
So it turns out that there were two potential ways to explain the cause of these values of the constants and salvage the dream of a theory of everything. But neither of them really work. The first way is to say that these constants are uncaused. They're just built into the basic fabric of our universe. And the other is to say that these constants are a result of some master law. Neither of them seems to work, none of them really leads to a good explanation to fitting them into the theory of everything. And this is really the mystery.
Going back to our ethical system, let's give an analogy to the problem of constants which are such random strings of numbers. Remember, we had a system of ethics with certain laws - maybe one of the laws is that a person has to give charity, another one of the laws is the maximum amount of interest someone's allowed to charge. Let's say that we looked closely at the law and we saw that it said every person has to give at least 13.7436218742% to charity. Or another law said, one is not allowed to charge interest above 28.87416320771%. Imagine our laws had those features.
And let us think about our quest to find a unified theory of our ethical system. How would we explain these numbers? Previously, we had been successful at saying, we have one law: "Do unto others as you'd want done to yourself." But what about these numbers? How do you get these numbers?
So again, to parallel what we talked about within the two possibilities in our theory of everything in physics, we could imagine having our unified law of our ethical system be, "Do unto others, as you would want done to yourself, but also, by the way, plug in the number 13.743627, and the number 28.136274403." But that's ugly. We had such a beautiful unified theory of everything, the law, "Do unto others as you'd want unto yourself." And just to plug on these additional numbers doesn't fit in, it undermines everything we've done at finding a beautiful simple law which explains everything within our ethical system.
On the other hand, you might say, oh - so we're not just going to lop these numbers on. Rather, we're going to suggest that maybe these numbers could be derived from our beautiful foundational law, of our ethical system. But if we think about it, we say, how in the world is a law, for example, "Do unto others as you want unto yourself" How are you going to generate numbers like that? You might say, "Do unto others as you want done to yourself" - you could say, give half of your money to charity and half for yourself; or charge interest - don't charge double the amount of the loan in interest, or some sort of a number.
But to have such ridiculous numbers, such arbitrary, detailed, so many decimal spots, it's not reasonable to suggest that we would be able to derive these numbers from one law. Whether that law would be "Do unto others as you would want done to yourself," or any other qualitative law. These numbers just create a problem, when we have laws like that, laws that have such detailed numbers, it really threatens our ability to find a unified law of our system of ethics.
And the same way that's the case in our ethical system, that's exactly the mystery of the constants. Scientists have been so successful at formulating a theory of everything that could explain all the phenomena in the universe. And then at the bottom, they finally got to these 25 unwieldy numbers, and they have no way to possibly explain them. To say that they're just parts of the very laws of nature is unsatisfying, but to somehow hope that you'll be able to find the qualitative law that will explain them, also seems unreasonable. And this is the mystery of the constants.
There's one other point of understanding why it's so difficult to derive these physical constants from any deeper theory. It's because, like Feynman was saying, they're unrelated to even constants in mathematics, like pi.
Pi is a mathematical constant that has to do with the circumference of a circle. And for years, people were trying to use other numbers that come up in mathematics, and say, pi to the fourth power over two and see that comes out to something like 1/ 137. And people tried that. But then, every time they measured it to another decimal point, it turned out that it was wrong. And eventually, people started giving up and they realized that that's not a method. There's no combination of mathematical constants, numbers that come up in pure mathematics, that have any relationship whatsoever to these physical constants, the constants of nature.
And that's what furthers the mystery of the constants. These numbers seem completely arbitrary. They don't seem to have any relationship to anything else and they have no connection to mathematics. They just seem to be these numbers that are part of reality and we don't know how to explain them.
This last point is an advanced point. Don't worry it if you don't follow it. It'll be sufficient if you take that analogy in our law of ethics, how the idea of numbers which are so unwieldy don't quite work in the quest to find a Theory of Everything.
The Logical Fallacy in God of the Gaps
Let's mention one more really important idea. Notice, that at this point in the story of the constants, there were only two legitimate scientific approaches: that the constants are fundamental, or there is a master law which determines them. It would not be valid to say, "God did it." Even though Feynman said, "You might say the hand of God wrote that number," he's not seriously suggesting that God is a good theory at this point. That would be a mistake called God of the gaps, and proper scientific methodology at this point is to say that the two possible theories we have - it's true they're not very good - have problems with them. So we should be patient. And we should wait for more scientific knowledge to come and remove our ignorance, and not just try to plug a gap in our knowledge by saying, "God did it."
Let's explain a little bit what we means by God of the gaps. God of the gaps reasoning is a logical fallacy, sometimes known as an argument from ignorance. Let's illustrate by an example.
Imagine that physicists were very successful at explaining everything in our universe. They discovered laws of nature which could explain all the phenomenon that we observe. However, there was one thing that they couldn't explain: why is the sky blue? That's the big mystery. Imagine that were the case. So then imagine someone comes along and says: physicists, I got an answer for you - the sky is blue because God made it blue. That's not a good answer. That's just taking God and plugging a gap. You could always take any scientific theory and find the little holes in that theory and just plug it with God. But that's not a good methodology.
The idea is, science grows slowly. And science discovers laws, and formulates the laws and derives consequences from those laws. And sometimes there are gaps in those explanations. Sometimes there are details which can't quite be explained by the law. But the proper scientific methodology is to be patient, to see if we can explain the details. And more often than not, patience leads to being able to explain it. And just to say, whenever we get stuck and find some gap in our knowledge, to say, "Ah, that's the way it is - God did it." That's not a good explanation. It doesn't quite explain anything. There's nothing which indicates God. You're just saying that because you want to plug a gap.
That would be the case in this instance. To suggest that we have the mystery of the constants, we have great laws of physics, we have quantum mechanics, we have general relativity, yet we have these constants we can't possibly explain - to just say, "Oh, those constants are that way because God did it." That's a God of the Gaps mistake. There's nothing which indicates God. All you have is a mystery. And just to plug a gap, a mysterious gap, by saying God did it, that doesn't quite explain anything.
In a nutshell, because God is not a legitimate answer at this point, we're left with the great mystery of the constants. Meaning to say, we have these 25 constants which seem to be the antithesis of a beautiful, simple theory of everything. How can you reconcile this apparent contradiction? Next time, we'll present the solution to this great mystery.
You may be wondering the following: Last time we told you that we're going to see how the mystery of the constants will lead directly to God. But now we're saying that God isn't a good explanation for the constants. So which one is it?
Once again, the answer to this question will require some patience. We're almost there. This was just setting the stage to get us ready for the solution to the mystery. That is, the discovery of something called fine tuning, which shows that these mysterious values that drive physicists crazy actually point us from physics to God. But we can't appreciate the solution of fine tuning until we first fully appreciate the mystery. We'll talk all about fine tuning in our next episode.
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