So, what is complexity?
What I'm going to try and do is section the concept:
to look at it from a multitude of different angles,
which I think is appropriate for any area of inquiry.
The way we'll do this is we'll look at complexity as a discipline,
the way we might look at physics as a discipline.
We'll look at the domain that complexity is investigating.
We'll also look at its methods, its epistemology,
the kinds of mathematics that you use in complexity science,
and then relate it to one of the more fundamental concepts
related to complexity, and that's emergence.
So, complexity and the disciplines -
if I asked you to define biology or geography, how would you do it?
It's extremely hard to do, and it's no easier for complexity science.
So the way to do this, I think,
Is to break up a discipline into its constituent parts,
or its preferred methods and approaches.
And we're going to do that for a number of disciplines,
and then finally do it for complexity.
So here are some disciplinary traits:
how quantitative is a field?
Is it obsessed with measurement and calculation,
as the natural sciences have tended to be,
or is it more satisfied with a qualitative narrative type of account?
How reductionist is it?
Now, reductionist relates to explanations that feel fulfilling
by virtue of presenting the parts of a system.
So, you go down levels to account for the level of interest -
that's what we mean by reductionism.
So, a physicist might say,
"to understand gravity, I need to understand gravitons,
or to understand electricity, electrons," and so forth.
Another question which gets confounded with reduction is compression.
Which is - are the fields amenable to compressive mathematical description?
Can you write down short equations that capture the essence
of the phenomenon?
That's not the same thing as reductionism.
It's a different kind of reduction -
a reduction to short, compressed equations.
And finally, how historicist is the field?
How important is history in accounting for phenomenon?
How far back do you have to go?
And of course in biology, we feel you have to go way back.
To understand traits and organisms, we have phylogenetic explanations,
less so in physics.
And so that - if you like, quartet of characteristics
helps us to define a discipline.
So lets consider physics.
So physics is obviously a very quantitative field.
Everything is presented in terms of numbers and measurements and so forth.
It's also very reductionist,
because the search for grand unified theories
or fundamental theories,
typically consist in looking for elementary constituents.
For example, the standard model in physics:
the minimum number of particles in fields.
And having done that, presenting that in mathematical terms,
using very elegant, very compressed mathematical formula:
F=ma, Maxwell's Laws, and so on.
So, physics has that characteristic
that's sometimes described as sort of back-of-the-envelope-like calculations:
short calculations based on fundamental constituents
that are highly quantitative.
Okay, so systems biology is highly quantitative.
It's also very reductionist.
You try to understand traits in terms,
for example, of their genetic or epigenetic factors.
And it's also very historicist.
You understand things in a taxinomical or phylogenetic framework.
But it's not very compressive.
It doesn't present regularities in terms of very short, elegant
mathematical equalities.
Then there's something like biological anthropology,
or biological linguistics,
and here, they're slightly less quantitative.
There's less data than there would be, for example, in genomics.
They tend to be reductionist, trying to understand things, again,
in terms of biological factors that contribute to behavior.
Very historicist, phylogenetic, and also somewhat compressive,
that is using fundamental evolutionary theories, like kin selection,
to understand behavior.
And having explained all of those, to try and illustrate that
all these fields should be understood in terms of how much they weight
different factors or traits, where does complexity science fit?
Well, complexity science is very quantitative, by and large,
it's very historical; we're studying, you know, adaptive agents,
and it's very compressive because we're looking for
mathematical theories that capture essential irregulatires.
But what we are not is reductionist.
Like, we are not looking down levels to explain the level of interest.
And that's one of its defining features that we'll come back to at some point
when we talk about emergence.
So, having talked a little bit about what complexity might mean
as a discipline,
let's talk about what complexity science studies;
that is the domain, the territory, of inquiry, that establishes
the way it looks and feels.
So if we look at classical mechanics, physics,
this is the study of very ordered processes.
And, you can write down equations that describe the orbits of the planets
and the stars
in a very compressive, compact form. That means low complexity.
In this case, complexity relates to, in some sense,
the number of pages of equations required to describe
the regularity of interest.
So, in that sense, Newton's Laws are very compressed.
When you get to quantum mechanics, that introduces more stochasticity,
more randomness, in the equations, even though they're still classical,
are correspondingly slightly more complicated.
Interestingly, if you introduce lots of randomness,
you can also write down a very compressed description
in terms of statistical mechanics and thermodynamics.
So these two limits are, in some sense, the limits of the physical world,
and that's why physics has been so effective at theorizing about phenomena.
But if you now look at the domain where noise and regularity compete -
the complex domain - what happens?
We don't really know. We need entirely new kinds of theories
to describe this intersection where frozen accidents dominate.
That is the world of nature, or of culture.
And so here are some examples.
On the left, classical mechanics, just a little bit more randomness,
the wave equation in quantum mechanics,
on the far right where you get a lot of randomness,
the description of the entropy of a system,
and again in the middle where that C is written,
some new mathematics, some new description is required
that respects the complex domain.
Now, what's happened in the 21st century is that two very distinct approaches
have evolved to deal with complexity.
On the one hand, you have machine learning AI
that encodes whole libraries of big data sets
with billions of parameters that produce, within a very circumscribed range,
highly predictive solutions.
On the other hand, you have complexity science,
which tries to do something closer to what physics was trying to do.
That is, a smaller number of essentially processes and equations
which describe regularities but don't predict.
So it looks as if we've reached this point of bifurcation
where you have to make a decision.
I can either go down the path of prediction
and lose understanding and comprehensibility,
or go down the path of mechanism and understanding,
and lose prediction.
And I think the open question that we're all dealing with
is could we reconcile these two different approaches to the complex domain?
Here are some examples of methods and frameworks in complexity science
that have been invented to deal with the complex domain,
and here's three examples:
Scaling theory, that is, what patterns of irregularities span multiple different
orders of magnitude in space and time;
agent based models, which takes seriously the idea of agency or reflexivity,
that is, the things that we study in the complex domain
have teleology, they have purpose, they have function,
and that's not true in physics;
and network theory, that takes seriously the collective dynamics
of complex systems.
And, of course, one of the interesting things about these three
is that they find application.
So, in scaling theory, we can explain how long organisms live,
how many species we typically find in a unit area,
we can even apply scaling theory to social phenomena,
where we're interested in how patterned production, for example,
scales as a function of city size.
Network theory is used ubiquitously,
in this particular case, to study political polarization,
or the spread of disease.
And agent based models are the prefered computational tools for looking at
things like swarming, flocking, and congestion in cities.
So, of interest here, is that even though the complex domain
doesn't yield to these highly compressive formalisms,
they prove to be extremely useful in studying real-world problems.
So when we talked about the complex domain,
what we were talking about was the structure of reality,
what we call ontology.
But then there's the question of how we understand that reality,
how we describe it, how we mathematize it:
the structure of knowledge itself, and we call that epistemology.
And complexity science has a very interesting epistemology.
In 1960, a prominent physicist working in quantum mechanics,
Eugene Wignor, wrote a paper
called "The Unreasonable Effectiveness of Mathematics
in the Physical Sciences."
And Wigner was very interested in this perplexing observation
that you can invent mathematics, freely, through your imagination,
and yet somehow that imagination, that imaginary object,
can predict regular patterns in the natural world
that have nothing to do with you.
So how is it that mathematics is so effective
at explaining and predicting the real world?
And we can place that in a slightly more mathematical framing
by saying that what amazed Wigner is that models with very few parameters,
that is, highly compressed, very parsimonious models,
could predict phenomena very precisely,
and that's what's represented on these two axes here:
the x-axis showing the number of parameters,
and the axis coming out towards you, how predictive that is,
and at the top there is an example of what he was amazed by:
Maxwell's Equations.
Here's another example, from the founder of the Santa Fe Institute,
working in particle physics, is Murray Gell-Mann,
and Murray wrote down the algebra, mathematical formalism,
to explain symmetries,
in this case, eight-fold symmetries captured by something called SU(3).
And by manipulating these lea-groups, he was able to predict particles
that had never been observed before.
So, exactly to Wigner's point,
the mathematics generated a solution that didn't seem to be present
in the mathematics to begin with.
Another example is the work of Paul Dirac,
this is Dirac's equation, it's a relativistic wave equation.
It takes quantum mechanics and special relativity and merges them,
and he solved this equation and discovered negative energy states ,
and he used those solutions to infer the existence of antimatter.
So, no one has seen antimatter;
these equations were derived to describe the ordinary world that we can measure
and observe,
and yet, they predicted something extraordinary.
So that's the world, if you like, the epistemological world,
that's made possible by the simple domains of physics, right?
The two edges of that graph I showed earlier
which are either perfectly regular or perfectly random.
That's what they allow us to do.
But most of the world we care about,
the social world, the biological world, and so on,
isn't like that.
It has a different ontology, right? It combines noise and order,
and so what do we do now?
It's not the unreasonable effectiveness of mathematics,
it's the unreasonable ineffectiveness of mathematics,
in dealing with the complex domain.
The key to effective coarse-graining is that you don't lose
predictive efficacy by losing degrees of freedom,
by losing parameters.
So, there are special domains where averaging is actually permitted,
and one very good example of that is the domain of scaling.
So in scaling theory, you get equations that look a bit like
equations from physics.
This looks a bit like F=ma. This is basal metabolic rate,
scales as mass raised to the three-quarter powers,
and those threes and fours there
are actually the dimensions of space, three,
divided by the dimensions displaced plus one fractal dimension,
so it's very physical.
And you can derive these equations through mathematics
of perturbation theory,
that would be very familiar to the world of physics.
And here's an example of what that looks like.
And so scaling theory gives us insights into the complex domain
by using the concept of coarse-graining very effectively.
But again, as I said, for many phenomena that's not an option.
And - so, much of complexity science does something different.
Instead of trying to find parsimonious models,
like F=ma, or B=M raise to the three-quarter power,
it asks, "what gives rise to those structures in the first place?
What allows for the possibility in the complex domain of coarse-graining?
Or what doesn't? What produces the structure that we want to theorize about?"
So let me make that quite explicit now with an example.
If you think about machine learning and the performance of algorithms
like AlphaGo, AlphaZero,
underlying all those lines of code, nd all those hundreds of millions,
if not billions, of parameters, is a very simple idea:
the idea of reinforcement learning.
And that can be written down in just a few lines of code;
That's just a few lines of mathematics.
So in that sense, this is highly compressive.
It's not the particular model that finally is instantiated,
but how the parameters are tuned.
And the same thing goes for biology.
People sometimes say, "evolutionary theory is not predictive."
Well, it's not predictive in the sense that you could predict
a giraffe, or a flea, or a bacterium,
but all of them were subject to the same optimization principle
in their local environment: natural selection and drift.
And so, what we're looking for is a parsimonious description
of the algorithm, or the process, that produces the object.
Physical science theorizes about the object parsimoniously,
we're theorizing, in some sense, about a process that gives rise to an object:
A process that gives rise to a theory.
So in that sense, complexity science is metatheoretic.
One of the concepts that one hears a lot about,
when talking about complexity is emergence.
It's its nearest relative, in a certain sense.
And just like complexity, it generates a lot of perplexity.
So I want to explain what emergence is now.
One very simple way of defining emergence is,
you're dealing with an emergent phenomenon,
when there's no need to look under the hood.
And I use that in the following sense:
if your car stops, you're not quite sure why,
and the most natural thing to do is to check whether you've run out of gas.
So there's a kind of reductionism there,
because you're saying to understand why a car stops, you need to look at its parts.
And where phenomenon is strongly emergent, you don't need to look under the hood,
and let me give an example.
This equation is the so-called Fermat Conjecture,
and it took hundreds of years to be solved,
and this is Andrew Wiles, who finally solved it,
between 1993 and 95,
in fact the first solution had an error in it,
which freaked him out, and then he was able to correct it.
But you can ask, you know, how did he do it?
How did he solve this theorem?
And let me show you quickly some pages from his proof.
Here's one page, where he establishes the relationship
between the Fermat Conjecture and elliptical forms.
He goes through a whole series of ingenious deductive steps,
recruiting unexpected errors of mathematics
until he finally arrives at the conclusion,
which is the proof of the Fermat Conjecture.
Now, this proof is presented to us only in terms of mathematics.
The language of mathematics is sufficient to establish the credibility of the result;
you don't have to look under the hood of Andrew Wiles to determine
whether or not this proof is right or not.
For example, we don't need to do brain science on Andrew Wiles,
we don't have to say, you know, "the reason that the proof is correct
is because he was expressing a lot of serotonin or dopamine," or,
"this particular neural circuit was being recruited."
That would be interesting; that would be something that you might want to know
but it has nothing to do with the correctness of the proof.
Similarly, Wiles's economic circumstances,
or the particular market that he's working in and the university,
Princeton, who's paying his salary.
None of this is relevant to the correctness of the proof,
and neither is his nationality or his ideology.
So, here's an example where correctness operates entirely
at the level of mathematics,
and moving below mathematics, for example, doing sort of particle physics
on Andrew Wiles's brain, might be interesting,
but is illuminating with respect to whether the theorem
has been proved or not.