I recently wrote about the question “When do you understand?“. In this post I outlined a pattern that I observed how I end up with a deep understanding of a given topic. However, there is also a second path that I totally missed in this post.

The path to understanding that I outlined requires massive efforts to get to the bottom of things. I argued that you only understand something when you are able to explain it in simple terms.

The second path that I missed in my post doesn’t really lead to understanding. Yet, the end result is quite similar. Oftentimes, it’s not easy to tell if someone got to his understanding via path 1 or path 2. Even worse, oftentimes you can’t tell if you got to your own understanding via path 1 or path 2.

**So what is this second path?**

It consists of reading something so often that you start to accept it as a fact. The second path makes use of repetition as a strong argument.

Once you know this, it is shocking to observe how easy oneself gets convinced by mere repetition.

However, this isn’t as bad as it may sound. If dozens of experts that you respect, repeat something, it is a relatively safe bet to believe them. This isn’t a bad strategy. At least, not always. Especially when you are starting, you need orientation. If you want to move forward quickly, you can’t get to the bottom of every argument.

Still, there are instances where this second path is especially harmful. Fundamental physics is definitely one of them. If we want to expand our understanding of nature at the most fundamental level, we need to constantly ask ourselves:

Do we really understand this? Or have we simply accepted it, because it got repeated often enough?

The thing is that physics is not based on axioms. Even if you could manage to condense our current state of knowledge into a set of axioms, it would be a safe bet that at least one of them will be dropped in the next century.

Here’s an example.

## Hawking and the Expanding Universe

In 1983 Stephen Hawking gave a lecture about cosmology, in which he explained

“*The De Sitter example was useful because it showed how one could solve the Wheeler-DeWitt equation and apply the boundary conditions in a simple case. […] However, if there are two facts about our universe which we are reasonably certain, one is that it is not exponentially expanding and the other is that it contains matter*.”

Only 15 years later, physicists were no longer “reasonable certain” that the universe isn’t exponentially expanding. On the contrary, we are now reasonable certain of the exact opposite. By observing the most distant supernovae two experimental groups established the accelerating expansion as an experimental fact. This was a big surprise for everyone and rightfully led to an Nobel prize for its discoverers.

The moral of this example isn’t, of course, that Hawking is stupid. He only summarized what everyone at this time believed to know. This example shows how quickly our most basic assumptions can change. Although most experts were certain that the expansion of the universe isn’t accelerating, they were all wrong.

## Theorems in Physics and the Assumptions Behind Them

If you want further examples, just have a look at almost any theorem that is commonly cited in physics.

Usually, the short final message of the theorem is repeated over and over. However you almost never hear about the assumptions that are absolutely crucial for the proof.

This is especially harmful, because, as the example above demonstrated, our understanding of nature constantly changes.

Physics is never as definitive as mathematics. Even theorems aren’t bulletproof in physics, because the assumptions can turn out to be wrong through new experimental findings. What we currently think to be true about physics will be completely obsolete in 100 years. That’s what history teaches us.

An example, closely related to the accelerating universe example from above, is the Coleman-Mandula theorem. There is probably no theorem that is cited more often. Most talks related to supersymmetry mention it at some point. It is no exaggeration when I say that I have heard at least 100 talks that mentioned the final message of the proof: “space-time and internal symmetries cannot be combined in any but a trivial way”.

Yet, so far I’ve found no one who was able to discuss the assumptions of the theorem. The theorem got repeated so often in the last decades that it is almost universally accepted to be true. And yes, the proof is, of course, correct.

However, what if one of the assumptions that go into the proof isn’t valid?

Let’s have a look.

An important condition, already mentioned in the abstract of the original paper is Poincare symmetry. This original paper was published in 1967 and then it was reasonable certain the we are living in a universe with Poincare symmetry.

However, as already mentioned above, we know since 1998 that this isn’t correct. The expansion of the universe is accelerating. This means the cosmological constant is nonzero. The correct symmetry group that preserves the constant speed of light and the value of a nonzero cosmological constant is the De Sitter group and not the Poincare group. In the limit of a vanishing cosmological constant, the De Sitter group contracts to the Poincare group. The cosmological constant is indeed tiny and therefore we aren’t too wrong if we use the Poincare group instead of the De Sitter group.

Yet, for a mathematical proof like the the one proposed by Coleman and Mandula, whether we use De Sitter symmetry or Poincare symmetry makes all the difference in the world.

The Poincare group is a quite ugly group and consist of Lorentz transformations and translations: $ \mathbb{R}(3,1) \rtimes SL(2,\mathbb{C}) .$ The Coleman-Mandula proof makes crucial use of the inhomogeneous translation part of this group $\mathbb{R}(3,1)$. In contrast, the De Sitter group is a simple group. There is no inhomogeneous part. As far as I know, there is no Coleman-Mandula theorem if we replace the assumption: “Poincare symmetry” with “De Sitter symmetry”.

This is an example where repetition is the strongest argument. The final message of the Coleman-Mandula theorem is universally accepted as a fact. Yet, almost no one had a look at the original paper and its assumptions. The strongest argument for the Coleman-Mandula theorem seems to be that is was repeated so often in the last decades.

Maybe you think: what’s the big deal?

Well, if the Coleman mandula no-go theorem is no longer valid, because we live in a universe with De Sitter symmetry, a whole new world would open up in theoretical physics. We could start thinking about how spacetime symmetries and internal symmetries fit together.

## The QCD Vacuuum

Here is another example of something people take for given only because it was repeated often enough: The structure of the CP vacuum. I’ve written about this at great length here.

I talked to several PhD students who work on problems related to the strong CP problem and the vacuum in quantum field theory. Few knew the assumptions that are necessary to arrive at the standard interpretation of the QCD vacuum. No one new where the assumptions actually come from and if they are really justified. The thing is that when you dig deep enough you’ll notice that the restriction to gauge trafos that satisfy $U \to 1$ at infinity is not based on something bulletproof, but simply an assumption. This is a crucial difference and if you want to think about the QCD vacuum and the strong CP problem you should know this. However, most people take this restriction for granted, because it has been repeated often enough.

## Progress in Theoretical Physics without Experimental Guidance

The longer I study physics the more I become convinced that people should be more careful about what they think is definitely correct. Actually there are very few things we know for certain and it never hurts to ask: what if this assumption everyone uses is actually wrong?

For a long time, physics was strongly guided by experimental findings. From what I’ve read these must have been amazing exciting times. There was tremendous progress after each experimental finding. However in the last decades there were no experimental results that have helped to understand nature better at a fundamental level. (I’ve written about the status of particle physics here).

So currently a lot of people are asking: **How can there be progress without experimental results that excite us?**

I think a good idea would be to take a step back and talk openly, clearly and precisely about what we know and understand and what we don’t.

Already in 1996, Nobel Prize winner Sheldon Lee Glashow noted:

“*[E]verybody would agree that we have right now the standard theory, and most physicists feel that we are stuck with it for the time being. We’re really at a plateau, and in a sense it really is a time for people like you, philosophers, to contemplate not where we’re going, because we don’t really know and you hear all kinds of strange views, but where we are. And maybe the time has come for you to tell us where we are. ‘Cause it hasn’t changed in the last 15 years, you can sit back and, you know, think about where we are.”*

The first step in this direction would be that more people were aware that while repetition is a strong argument, it is not a good one when we try to make progress. The examples above hopefully made clear that only because many people state that something is correct, does not mean that it is actually correct. The message of a theorem can be invalid, although the proof is correct, simply because the assumptions are no longer up to date.

This is what science is all about. We should always question what we take for given. As for many things, Feynman said it best:

“*Science alone of all the subjects contains within itself the lesson of the danger of belief in the infallibility of the greatest teachers in the preceding generation. . . Learn from science that you must doubt the experts. As a matter of fact, I can also define science another way: Science is the belief in the ignorance of experts.*“