Newtonmass fractals 2018

It is Newtonmass, and that means doing some fun math. I try to invent a new fractal or something similar every year: this is the result for 2018.

The Newton fractal is an old classic. Newton’s method for root finding iterates an initial guess $z_0$ to find a better approximation $z_{n+1}=z_{n}-f(z_{n})/f'(z_{n})$. This will work as long as $f'(z)\neq 0$, but which root one converges to can be sensitively dependent on initial conditions. Plotting which root a given initial value ends up with gives the Newton fractal.

The Newton-Gauss method is a method for minimizing the total squared residuals $S(\beta)=\sum_{i=1}^m r_i^2(\beta)$ when some function dependent on n-dimensional parameters $\beta$ is fitted to $m$ data points $r_i(\beta)=f(x_i;\beta)-y_i$. Just like the root finding method it iterates towards a minimum of $S(\beta)$: $\beta_{n+1} = \beta_n - (J^t J)^{-1}J^t r(\beta)$ where $J$ is the Jacobian $J_{ij}=\frac{\partial r_i}{\partial \beta_j}$. This is essentially Newton’s method but in a multi-dimensional form.

So we can make fractals by trying to fit (say) a function consisting of the sum of two Gaussians with different means (but fixed variances) to a random set of points. So we can set $f(x;\beta_1,\beta_2)=(1/\sqrt{2\pi})[e^{-(x-\beta_1)^2/2}+(1/4)e^{-(x-\beta_1)^2/8}]$ (one Gaussian with variance 1 and one with 1/4 – the reason for this is not to make the diagram boringly symmetric as for the same variance case). Plotting the location of the final $\beta(50)$ (by stereographically mapping it onto a unit sphere in (r,g,b) space) gives a nice fractal:

It is a bit modernistic-looking. As I argued in 2016, this is because the generic local Jacobian of the dynamics doesn’t have much rotation.

As more and more points are added the attractor landscape becomes simpler, since it is hard for the Gaussians to “stick” to some particular clump of points and the gradients become steeper.

This fractal can obviously be generalized to more dimensions by using more parameters for the Gaussians, or more Gaussians etc.

The fractality is guaranteed by the generic property of systems with several attractors that points at the border of two basins of attraction will tend to find their ways to other attractors than the two equally balanced neighbors. Hence a cut transversally across the border will find a Cantor-set of true basin boundary points (corresponding to points that eventually get mapped to a singular Jacobian in the iteration formula, like the boundary of the Newton fractal is marked by points mapped to $f'(z_n)=0$ for some n) with different basins alternating.

Merry Newtonmass!

A bit of existential hope for Christmas (and beyond)

Existential hope is in the air. The term was coined by my collegues Toby and Owen to denote the opposite of an existential catastrophe: the chance that things could turn out much better than expected.

Recently I had the chance to attend a visioning weekend with the Foresight Institute where we discussed ways of turning dystopias into utopias. It had a clear existential hope message, largely because  it was organised by Allison Duettman who is writing a book on the topic. I must admit that I got a bit nervous when I found out since I am also writing my own grand futures book, but I am glad to say we are dealing with largely separate domains and reasons for hope.

Now I extra am glad to add a podcast to the list of hopeful messages: the Future of Life Institute had me on the podcast Existential Hope in 2019 and beyond. It includes not just me and Allison, but also Max Tegmark, Anthony Aguirre, Gaia Dempsey, and Josh Clark (who also interviewed me for his podcast series End of the World).

I also participated in the Nexus Instituut event “The Battle between Good and Evil”. I assume the good guys won. I certainly had fun. I ended up arguing that good is only weak compared to evil like how water is weak compared to solid object – in small amounts it will deform and splash. In larger amounts it is like the tide or a tsunami: you better get out of the way. In retrospect that analogy might have been particularly powerful in the Netherlands. They know their water and how many hands (and windmills) can reshape a country.

Do we really have grounds for existential hope?

A useful analysis of the concept of hope can be found in Jayne M. Waterworth’s A Philosophical Analysis of Hope. He defines that hoping for something requires (1) a conception of an uncertain possibility, (2) a desire for an objective, (3) a desire that one’s desire be satisfied, and (4) that one takes an anticipatory stance towards the objective.

One can hope for things that have a certain or uncertain probability, but also for things that are merely possible. Waterworth calls the first category “hope because of reality” or probability hope, while the second category is “hope in spite of reality” or possibility hope. I might have probability hope in fixing climate change, but possibility hope in humanity one day resurrecting the dead – in the first case we have some ideas of how it might happen and what might be involved, in the second case we have no idea even where to begin.

Outcomes can also be of different importance: hoping for a nice Christmas present is what Waterworth calls an ordinary hope, while hoping for a solution of climate change or death is an extraordinary hope.

We may speak of existential hope in the sense that “existential eucatastrophes” can occur, or that our actions can make them happen. This would represent the most extraordinary kind of hope possible.

But note that this kind of hope is potentially “hope because of reality” rather than “hope in spite of reality”. We can affect the future to some extent (there is an interesting issue of how much). There doesn’t seem to be any law of nature dooming us to early existential risk or a necessary collapse of civilization. We have in the past changed the rules for our species in very positive ways, and may do so again. We may discover facts about the world that greatly expand the size and value of our future – we have already done so in the past. These are good reasons to hope.

Hope is a mental state. The reason hope is a virtue in Christian theology is that it is the antidote to despair.

Hope is different from optimism, the view that good things are likely to happen. First, optimism is a general disposition rather than directed at particular hoped for occurrences. Second, hope can be a very small and unspecific thing: rather than being optimistic about everything going the right way, a hopeful person can see the overwhelming problems and risks and yet hope that something will happen to get us through. Even a small grain of hope might be enough to fend of despair.

Still, there may be a psychological disposition towards being hopeful. As defined by Snyder in regarding motivations towards goals this involves a sense of agency (chosen goals can be achieved) and pathways (successful plans and strategies for those goals can be generated). This trait predicts academic achievement in students beyond intelligence, personality, and past achievement. Indeed, in law students hope but not optimism was predictive for achievement (but both contributed to life satisfaction). This trait may be more about being motivated to seek out good future states than actually being hopeful about many things, but the more possibilities are seen, the more likely something worth hoping for will show up.

If there is something I wish for everybody in 2019 and beyond it is having this kind of disposition relative to existential hope. Yes, there are monumental problems ahead. But we can figure out ways around/through/over them. There are opportunities to be grabbed. There are new values to be forged.

The winter solstice has just passed and the days will become brighter and longer for the next months. Cheers!