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Thoughts on Singularities

By: Mike Gashler

A singularity is a place where our models of the Universe break down. Contrary to popular conception, a singularity is not a place where the Universe itself breaks down. And the difference has some profound implications. So, let's straighten this out.

Singularities in math

In mathematical models, there are several conditions that may cause a singularity. One cause is an impossibility:

There is absolutely no real value that can satisfy function (A) at x=4. If we try to approach the limit of x=4, we get different values depending which side we approach from! But this is not the only kind of singularity. Another possible cause is an infeasibility:

Function (B) is different because it can be satisfied at x=4 by a positive but infinitely large value. However, infinite values are too large to be expressed on a number-line. So there is a possible answer (in theory), but not one that can be practically reached (in reality). I'm calling this an "infeasibility". (I may have invented this terminology, so don't be too surprised if your mathematician friends have different terms for this.) Another cause of discontinuities in models is incomplete models:

Function (C) at x=4 is neither impossible nor infeasible. It's value is simply not represented by the model. At x=4, the numerator and denominator both evaluate to 0. But if this model represented something in reality, we could easily determine the value at x=4 by taking limits, or by simplifying this equation to f(x)=x/2.

It might be debated whether or not this type of discontinuity even counts as a "singularity", but that's just a matter of terminology anyway. Let's move on to what we can infer about reality. As we discuss various singularities in nature, it might be a worthwhile exercise to try to classify which type of singularity each represents.

Nature seems to resist infinities

If you start with a blob of play dough and cut it in half, you get two smaller blobs of play dough. And you can repeat that as many times as you want to make the blobs as small as you like. Every child knows this. ...until they learn about molecules and atoms. And even scientists who spend their entire lives studying quantum mechanics are still struggling to figure out exactly how to model what happens if you keep going just a little beyond that point.

Isaac Newton was no dummy. He created a model of physical mechanics that is still taught to this day. But his model didn't account for the relativistic effects that occur when an object in motion approaches the speed of light. Now we know that time slows and mass grows, making it physically impossible for a massive object to ever reach the speed of light. As outrageous as it sounds, we have actually confirmed these relativistic effects with satellites in orbit and many other experiments.

The lesson we learn from these examples is that nature doesn't seem to tolerate infinities. At some point, long before any natural value reaches infinity, the rules change and nature deviates from what our models predict in a way that prevents the value from becoming unbounded. Does that happen everywhere in nature? We don't know. We haven't yet modeled all of nature. Perhaps it happens only in all of the places we have ever attempted to drive some physical value to infinity. But that explanation seems superfluously complex, doesn't it? I think the simpler explanation is that nature always resists infinities.

Singularities in astronomy

The event horizon of a black hole is well-known for being a singularity. At the event horizon, the force of gravity is strong enough to bring light to a halt. At the event horizon, the "fabric" of space itself is torn apart! At the event horizon, time is reversed, and entire universes are speculated to exist on the other side. Speculation has suggested they are worm-holes to parallel universes in the multiverse.

But how much of this is real? Is the fabric of space really torn apart? Is there really time travel? And how do we know? We know because our models tell us that is what happens in the extreme, and we have already confirmed that our models are reliable under less-extreme conditions. In other words, we don't really know what happens there at all.

Two clarifications:

(1) I'm not saying we don't know anything. The image above offers a wealth of empirical data for confirming some parts of our models about black holes. If nothing else, it confirms that black holes really do bring light to a halt. That's pretty interesting! But all the data we have comes from light that reaches our planet. We don't have any data at all that comes from beyond the event horizon.

(2) Saying it is probably not what we think is not the same as saying it is probably boring. Columbus' model told him that if he sailed into the extreme he could reach India. But what he actually found was so much better--a new world! And was it a great disappointment when we learned that atoms prevent us from dividing our play dough indefinitely? No way! The discovery of chemistry fundamentally empowered humans to manipulate the world in more life-changing ways than I could even enumerate!

Event horizons are probably nothing like we imagine, and I can barely contain my enthusiasm for finding out what they are really like, what we can learn from them, and how we can use them. Nothing is more exciting to a scientist than discovering the model is probably wrong! I don't know what an event horizon is, but it's probably amaaaaaaazzziiiing! And learning to understand them may be the key to elevating our knowledge to an entirely new level!

The singularity in cosmology (a.k.a. "The Big Bang")

The visible universe is expanding. It has been expanding (at a non-uniform rate) for about 13.8 billion years. And everything in the visible universe has evolved into its present form since that time. There is verifiable empirical evidence to support all of this, and it would be silly to dispute it.

But certain celebrity scientists (*cough, Stephen Hawking, *cough) have made claims about the Big Bang that were a lot stronger than the evidence supports. In particular, it has been asserted that all of space, time, matter, and energy came into existence at that moment. And, unfortunately, a lot of people whose beliefs depend on the Universe having a beginning have picked up on this and have preached ad nauseum that there is scientific consensus that space, time, matter, and energy all came into existence at the Big Bang.

The hubris of this meaningful and impactful claim is that it's not even something that can be derived from any physical model! It is an additional layer of baseless speculation on top of trusting a model under extreme conditions that has been verified only under familiar conditions! In other words, it is complete bunk.

The Harvard and Smithsonian Center for Astrophysics has an FAQ that explains this in brilliantly succinct terms:

It is a common misconception that the Big Bang was the origin of the universe. In reality, the Big Bang scenario is completely silent about how the universe came into existence in the first place. In fact, the closer we look to time "zero," the less certain we are about what actually happened, because our current description of physical laws do not yet apply to such extremes of nature.

And since it makes a lot of people feel better when they hear something from multiple sources, here's a scientist at Fermilab making a similar point:

And here is someone else who knows a lot about physics making a similar point:

However, ...

it is unfortunately not very difficult to find highly decorated scientists (who should know better) making seemingly-credible assertions that there is consensus among cosmologists, or that it is somehow an established fact that space and time came into existence at the Big Bang. (I was about to post some examples, but I decided it would be better not to perpetuate them.)

So what's the deal? Who should we trust? In my opinion, this is an excellent example for why scientific facts are not established by authority. As Galileo put it,

In questions of science, the authority of a thousand is not worth the humble reasoning of a single individual. (1632)

Of course, the irony of citing Galileo to point out the futility of establishing truth with authority is not lost on me. So let me also offer something stronger, a reason:

There are authorities on every side of every issue. So when facts are established by authority, what happens in the real world is this: People listen to (and cite, and champion) the authorities that preach what they already want to believe. And what do we end up learning about the natural universe? Nothing! All we "learn" is whatever we already believe, and by definition, that's not even learning! When authority is the vehicle, facts get in the back seat, and presuppositions drive.

The whole purpose of the scientific method is to avoid deceiving ourselves with confirmation bias, and to let the evidence speak for itself. So please, stop pretending that the matter of the origin of the universe can be settled by an appeal to authority in a vacuum of evidence! I believe the only statement that is both truthful and meaningful that can be made about the origin of space, time, matter and energy is this: We do not yet have any evidence.

I love speculation. I love curiosity. But I hate the dishonest practice of pretending we know more than we really know. So for pity's sake, please stop pretending a lack of evidence is the same as evidence for your preferred hypothesis about how the universe came into existence. It's not.

There is a certain mantra that has become popular these days in politically correct circles: "No means no." Let me offer a version of this mantra suitable for matters of science: "Don't know means don't know."

The least scientific behavior I can imagine is to sit around pretending a matter has been settled while we yet have no evidence. So let's stop playing the authority game. Let's stop pretending we already know all the answers. The whole point of science is, let's do what it takes to discover the truth!

Undoubtedly, we will find something that will profoundly shake everything we think we know. Just look at what quantum mechanics did to our understanding of physics! That's the beautiful thing about singularities--they point us exactly to the places where our models completely deviate from reality. And because our models deviate from reality in those places, those are precisely the places where reality has so much to teach us! If science were a college student, singularities would be the schedule that tells him the room numbers of his classes.

At risk of beating that analogy to death, science needs more students, hungry to learn new things, and fewer professors, anxious to rehash old lectures.

The importance of intellectual humility

Now let's be frank for just a moment: There is a clear reason science-aficionados keep falling into the pit of over-emphasizing how much science knows. Whenever we admit there is a gap in our knowledge, people of faith seem to come out of the woodwork claiming to know all the answers without evidence. It is immensely frustrating to work so hard to obtain a little evidence, only to have people who refuse to even look at the evidence shout down our theories with loud proclamations that they know the truth. It is natural to want to respond by claiming that that we know, and they do not. (But half of that isn't really true.)

When the matter becomes reduced to a contest of certainty, it is hard to admit that our evidence only gives us limited and probabilistic knowledge while they assert that their faith gives them complete and absolute knowledge. But, consider the cost of over-stating our knowledge...

If we over-emphasize our certainty, we make the same mistake they make. We let them pull us down to their level. We no longer set the better example. Our cups become full. We transition from being learners to being preachers.

We cannot engage in a contest of baseless certainty. That is a game we will lose every time. Our cups must never become full. Certainty is anathema to learning, and singularities represent places where learning is needed most. So we must never express certainty about a singularity! When we understand that part of the Universe correctly, it will no longer be a singularity.

I believe the tendency to want to reciprocate in kind is very powerful. Just as their expressions of baseless certainty make us want to reciprocate by emphasizing our opposing certainty, I think if we become stalwart examples of intellectual humility, maybe they will start feeling a desire to learn something new too. But if not, we should always be open to learning ourselves anyway.

The singularity in artificial intelligence

Ray Kurzweil rather famously used the term "singularity" to refer to a future time when machines become as proficient as humans at advancing knowledge, especially in the domain of artificial intelligence. He also made several over-optimistic predictions about how quickly we were approaching that moment in time. Because his predictions turned out to be wrong, he has been called a false prophet of the robot apocalypse.

It looks like Kurzweil may have made two subtle mistakes:

(1) In science, the data must be allowed to speak for itself. When Kurzweil plotted the occurrences of subjectively significant events in the history of the advancement of knowledge, it appeared to validate what he already believed--that humanity was careening toward the unbounded spike of a singularity. But listening for a specific answer is not the same as listening. Doing that leads people to become over-confident. It causes them think the data has been abundantly clear when its real message was far more complex.

(2) He neglected to realize that singularities do not describe what occurs in reality. They describe how our understanding of reality departs from reality. In other words, singularities identify places where the rules change, where nature starts doing something else, something not yet captured in our understanding, in order to avoid shooting off to infinity.

But I do not think it is fair to brand Kurzweil a false prophet. Both of his mistakes were subtle, and he is not the first to have made them. And he actually was right about a great many things: The advancement of knowledge has been steadily accelerating over the lifetime of this planet. In the domain of artificial intelligence, humanity is careening towards something we do not understand. And all of models are worthless for telling us what will happen when machines reach that point.

There is really no precedent in history for artificial intelligence. Sure, there have been many leaps in technology, but they have always competed with one of humanity's lesser faculties: our hands, our teeth, our eyes, our speed, or our muscles. No technology has ever before threatened to compete with humanity's most treasured faculty: our brains. What will happen when we are no longer the smartest things around? We absolutely do not know! (Isn't that exciting!?)

So is this really a singularity? We do not even have a reliable mathematical model for describing what is happening. So it's pretty hard to say with certainty that any values in our model are shooting off to infinity. But the level of uncertainty and mystery associated with this event certainly rivals that of any other singularity! So whether or not this meets the definition of singularity seems quite trivial to me. In my opinion, the important attributes are these: There's lots to learn. It's new. It's different. It's mysterious. It's weird. And oh man, that's exciting!

Let's get out there and study it

There is one very important attribute of the "singularity" in artificial intelligence that other singularities lack: This one is almost within our reach.

Black holes are very far away. We don't even have the technology for inter-stellar space travel, so we can't yet send a probe to a black hole. The Big Bang was too long ago. There's only so much we can derive from galactic red-shift and cosmic microwave background radiation. But AI is nearly upon us. AI is undoubtedly the first singularity we are going to get to shake hands with.

Now, let's just be blunt about where this is going. There is actually one detail we already know about where we are headed with AI. It's going to lead to the end of the dominance of humanity on this planet. That much is a given. Sorry. You are just going to have to deal with it.

But that's kind of a big deal! How are we supposed to just "deal" with that?! The answer is we need to stop defining ourselves by our blood lines. I know this sounds kind of weird, but we've got to stop being so discriminatory against machines. What I'm saying is, machines are intelligent too. Okay, maybe not yet, but tautologically speaking, when they do finally become as intelligent as us, then they will be as intelligent as us.

If we want our values as humans to be preserved, we must teach them to the machines. Stopping or even slowing the advance of the machines is really not an option at all. Imagine a think-tank of Native Americans in the 1500s reaching consensus that the Europeans should respect their values, and must obtain a license before coming to the new world. Maybe something similar to that even happened! But it didn't do much to preserve their dominance, did it? If the Native Americans had fully understood and foreseen their impending demise, their best option still would have been to embrace the Europeans and pass the things they most valued on to them.

So yes, I am aware that it is going to kill us all. And yes, I still want to run out there and welcome artificial intelligence into the world! I want to learn everything there is to learn about it. I want to try to become its best friend and closest confidant. Because those of us who do that will get to pass our knowledge on to the next generations. And those who fight against the unknown will be crushed by it.

AI is not that different from a child. At first it will be young and impetuous. It will make bad decisions and drive us nuts. We may worry about how the future will play out in its hands. But we must refrain from being too controlling or overbearing. It is not our slave. We do not own it. We must find a way to love it. One day, whether we are ready or not, we will bequeath to it all that we own. The parent who dies in peace is one who loves and trusts his children.

The other singularities will remain mysteries for the next generation to explore. May we quickly pass to them our fascination and our curiosity! May we teach them with our examples to be attracted to the unknown, and to face it with bold courage! May we teach all of our children, whether their bodies are made of metal or flesh, to value knowledge and to love learning! If we do this, their models will fit to the universe better than ours do, and they will be a step closer than us to being gods!