What this covers
The geometry of chaos can explain our uncertain world, from weather and pandemics to quantum physics and free will.
This talk was recorded at the Ri on 21 April 2023.
Join Tim Palmer as he explores how it provides the means to predict the world around us, and provides new insights into some of the most astonishing aspects of our universe and ourselves.
Watch the Q&A here: https://youtu.be/VZQnFQAJ6Io Subscribe for regular science videos: http://bit.ly/RiSubscRibe
00:00 Introduction 00:55 Illustrating Chaos Theory with pendulums (demo) 02:44 Fractal geometry: A bridge from Newton to 20th Century mathematics 08:43 The three great theorems of 20th Century mathematics 11:24 The concept of State Space 14:43 Lorenz State Space 19:24 Cantor's Set and the prototype fractal 22:52 Hilbert's Decision Problem 24:04 The link between 20th Century mathematics and fractal geometry 27:21 The predictability of chaotic systems 32:26 Predicting hurricanes with Chaos Theory 43:44 The Bell experiment: proving the universe is not real? 51:45 Counterfactuals in Bell's theorem 56:29 Applying fractals to Bell's theorem 01:03:57 The end of spatial reductionism
Buy Tim's book 'The Primacy of Doubt' here: https://geni.us/5bgfg
Tim Palmer is a Royal Society Research Professor in the Department of Physics at the University of Oxford. Following a PhD in general relativity theory, he spent much of his career working on the predictability and dynamics of weather and climate, developing probabilistic ensemble prediction systems across a range of weather and climate timescales. He also researches the foundations of quantum physics, in addition to applications of quantum and imprecise computing. He is a Fellow of the Royal Society and an International Member of the US National Academy of Sciences. Amongst other awards, he has won the Institute of Physics Dirac Gold Medal, and the top medals of the American and European Meteorological Societies.
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Chaos theory, via fractal geometry, bridges classical Newtonian mathematics with 20th-century mathematical discoveries and offers a reinterpretation of quantum mechanics's Bell experiment that preserves determinism without invoking non-locality or indeterminism.
- Fractal attractors generated by chaotic systems like the Lorenz equations exhibit self-similar structure at all scales, fundamentally different from Euclidean geometry that Newton knew
- Gödel, Turing, and Wiles's theorems about mathematical truth, computability, and number theory are formally linked to properties of fractal geometry
- Bell experiment can be reinterpreted: counterfactual measurement scenarios may lie in fractal gaps (gaps in the attractor structure) and thus be physically impossible rather than requiring spooky action at a distance
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The Lorenz equations, discovered by meteorologist Ed Lorenz in the early 1960s, describe a simple model of weather using three coupled non-linear differential equations, yet generate a geometry (now called fractal or attractor geometry) that would have been completely alien to Newton despite Newton's invention of differential calculus.
“these were equations um discovered by the guy on the top left Ed lorentz who was an American meteorologist who was actually trying to show a simple model of weather...Lorentz made use of a mathematical Discovery by Isaac Newton called the calculus...if one were able to kind of go back in time and show Newton the equations that lorentz had discovered...Newton...would have actually understood exactly what they were”
Climate change tilts the table on which the chaotic weather system operates: emitting CO2 changes the equilibrium, making warm-weather states more probable. The weather remains chaotic, but the statistics of the system change in a predictable way.
“what's happening with climate change is that we're tilting the table we we we're emitting carbon dioxide and that's making this thing now operate on a on a Surface which is no longer flat so when I now with my tilted surface I look at the statistics of these different types of weather we see it's no longer the same as when it was flat the warm weather states have now become more populated more likely a priori”
State space is an abstract multi-dimensional space in which each dimension represents a parameter or variable of a system, and a point in state space represents a complete state of that system.
“I have to introduce you to the concept of State space this is going to be an all-important concept...suppose I bought a pair of trousers...you have to decide...what leg length you are what your waist size is...and maybe the trousers come in different colors...so you have um three parameters that you have to decide on...a pair of trousers being represented as a point in a three-dimensional space”
Chaotic systems are often described as unpredictable, but this characterization is misleading; unpredictability is only about long-term behavior. Interestingly, the predictability of a chaotic system depends critically on where you start—some initial states are very predictable while others are very unpredictable.
“chaotic Dynamics or chaotic systems are often described as systems that are unpredictable and I even I use that word earlier but in some sense that's I mean whilst it's true it's not a very useful I mean if something's unpredictable there's not much you can you can do about it but actually this notion of unpredictability is only really about the long-term um you know predicting things perhaps for a long time in the future one of the more interesting aspects of chaotic systems and we can demonstrate this...is that the predictability of a chaotic system depends on where you start the system and some states are very predictable and some states are very unpredictable”
The Lorenz equations describe how the state of the system evolves over time—as if the state moves through state space, continually changing its position like trousers magically changing their waist size or length.
“what these equations that Ed Lawrence discovered um what they do is first of all uh we have not trials or state Facebook well called Laurent State space where a point in Laurent State's face corresponds to a value of these three uh variables x y and z and the equations actually describe how the state moves it's as if the trousers sort of magically start to change their waist size or their length”
A hierarchy of models is used in meteorology: from idealized models (like Lorenz's) to complex supercomputer models that produce actual tropical cyclone trajectories. All levels of this hierarchy are connected and mutually informing.
“this example of um uh the answer the ensembles and the um uh anticipatory action I mean this is quite a nice I think example of how in meteorology we use a hierarchy of models from from rather idealized ones the the one that lorentz discovers very idealized model um right through to the very complex models which need you know big supercomputers that produce the actual trajectories for the tropical Cyclones and hurricanes”
Chaos theory is one of the three great theories of 20th century physics, alongside quantum mechanics and relativity, because it describes how simple systems can exhibit extremely complex and unpredictable motion.
“well thank you very much um so Chaos Theory I'm going to uh assert is really one of the Great theories of 20th century physics um it essentially describes how relatively simple systems...can exhibit extremely complex and unpredictable motions”
For Hurricane Katrina, ensemble forecasts about a week before landfall showed multiple possible tracks with roughly equal probability; New Orleans was one possibility among several, so a disaster preparedness director would not have invested all resources into preparing New Orleans specifically.
“the middle one is is the famous Hurricane Katrina...we're looking at these ensembles of hurricane tracks about a week before landfall and you can see actually at that range there was clearly a possibility that Katrina would hit New Orleans which is halfway along the Gulf of Mexico there but it was by far not clear it was not and it was not the most perhaps the most likely scenario at that range so if you were governor of New Orleans I mean you know you or or indeed a national disaster preparedness you wouldn't perhaps put all your eggs in the basket of of making sure putting all your resources into preparing New Orleans”
Andrew Wiles's proof of Fermat's Last Theorem relied on p-adic numbers, a number system unfamiliar to most physicists and even most mathematicians, which are designed to describe fractal geometry rather than Euclidean geometry.
“proving fermatch theoremed...was an enormous achievement it used a type of number system which number theorists are very familiar with but for I think most people including actually most physicists are not familiar with and they're called periodic numbers...the ordinary numbers are the sorts of numbers you need to describe mathematically euclidean geometries...but if you want to describe the geometry really get down to the geometry of fractals then actually these periodic numbers are the are you know provide the means to actually talk kind of analytically and mathematically about these fractals”
Quantum mechanics and general relativity both affect our daily lives in practical ways: quantum mechanics underlies all modern electronics, and general relativity provides essential corrections to Newtonian gravity needed for GPS to function accurately.
“quantum mechanics affects our daily lives everything you know modern electronic gadgets and you know everything that we use in our daily lives has some link back to Quantum Mechanics um even actually general relativity...the fact that GPS is a useful tool for locating our positions um actually makes use of of General relativistic corrections to newtonian's gravity if we only use Newtonian gravity uh we would GPS's would be too inaccurate to be useful”
Palmer and colleagues implemented ensemble forecast methods at the European Center for Medium-Range Weather Forecasts, applying the Lorenz model to real-world weather prediction for extreme weather events like tropical cyclones and hurricanes.
“this is what I worked on for a number of years with colleagues...at the European Center for medium range weather forecasts kind of putting this idea into kind of weather forecast reality”
For cyclone Cider a week before landfall, ensemble forecasts predicted with high probability that it would hit Bangladesh; this high-confidence forecast allowed disaster preparedness agencies to give Bangladesh early warning and prepare effectively.
“this top one is uh of a forecast about a week ahead of a cyclone called Cider which as you can see was predicted with a pretty high probability most of the tracks are going up towards Bangladesh...clearly if you were um you know if you were leading uh some kind of disaster preparedness in Bangladesh you would have a pretty good heads up that this was a very high probability type of event and indeed that's exactly what happened”
For Hurricane Nadine, ensemble forecasts showed explosive instability—literally uncertain whether the storm would go east toward Europe or west back toward the Caribbean, representing an extreme unpredictability regime of a chaotic system.
“the bottom one which was a bit like that explosive instability uh in in Lawrence is called hurricane it was a thing called hurricane Nadine where you literally didn't even know whether it was going to go East towards Europe or west back towards the Caribbean”
Previously, humanitarian agencies waited until after a disaster occurred before sending aid (food, medicine, water, shelter), which often caused logistical delays of a week or more, especially to remote areas. Anticipatory action overcomes this by sending supplies ahead based on probabilistic forecasts.
“in the past as I'm sure you're all aware um you know these agencies didn't have like infinite amounts of money they had a limited amount of money...typically then what tended to happen is they went in after and after a hurricane had hit supplying food and and medicine and shelters and and water and so on to people afflicted but after it's hit you know often the logistics are really difficult uh and it's often you know when you have outlying islands and things it can take a week or more for people to get desperately needed food and medicine”
Seaman Dube showed that uncomputable problems like Hilbert's decision problem can be formulated as geometric properties of fractal attractors, and determining whether a line intersects a fractal is similarly uncomputable—the two problems are isomorphic (the same problem mathematically).
“in the 1990s a computer scientist called Seaman Dube showed...problems like Hilbert's decision problem can actually be formulated in a completely equivalent way in terms of geometric properties of these fractal...you can express problems like Hilbert's decision problem things that are known to be not computable not algorithmically improvable as a property of a fractal so just as Hilbert's decision problem cannot be uh cannot be proven by an algorithm so for example determining whether a line intersects a fractal or not is similarly algorithmic the uncomputable and the two are basically what a mathematician will call isomorphic they're the same problem”
A Lorenz section (cross-section through the folded surfaces of the Lorenz attractor) reveals a Cantor set, discovered by Georg Cantor in the 19th century, which is the prototype fractal.
“if you take a cross section through that fold now this is sometimes called a lorento lorentz section...you have a structure which was discovered by a mathematician called Georg Cantor in the 19th century and it's called a Cantor set and it is the Prototype fractal”
Fractal geometries never get boring when zoomed into at any scale—they retain structure at all magnifications, unlike Euclidean geometries (spheres, ellipses, hyperboloids) which become flat and featureless under sufficient zoom.
“if you kind of zoom into a sphere sufficiently...it kind of gets boring it just gets flat it looks flat...but these types of geometries which We Now call fractal geometries uh never get boring you can zoom in as much as you like you can zoom and zoom and zoom and zoom and you'll still see structure in this geometry”
Physics has not yet succeeded in unifying quantum mechanics with gravity; the conventional thinking is that we need to probe even smaller distances (the Planck length, ~10^-33 cm) to understand unification.
“but we haven't as I say we haven't yet managed to succeed in combining these ideas with gravity now the conventional thinking behind that is that we need to get to even smaller distances before we can really understand how quantum mechanics uh and gravity uh unify there's a thing called the Planck Mass the Planck sorry the Planck length of single plank mass as well but plank length which is absolutely tiny 10 to the minus 33 centimeters and the the thinking is we've got to get down to these even more fundamentally smaller lengths to understand that”
Skeptics of climate change sometimes argue that if weather is chaotic and unpredictable beyond a few days, how can century-long climate forecasts be trusted? However, this argument misunderstands the nature of climate change.
“I hear from time to time from let's say those who are skeptical about climate change the argument well if the if the weather is chaotic and you can't really make kind of reliable predictions more than a few days ahead how can we possibly ever trust a century-long forecast uh that the climate is going to change and my answer to that uh type of um assertion if you like is to say that you haven't really understood the nature of climate change”
Bell's theorem contains an implicit assumption rarely mentioned: it assumes that counterfactual measurements (hypothetical measurements that were not performed) are meaningful and have deterministic values assigned by the laws of physics.
“there is a literature but it's not a well-known literature on Bell's theorem which actually makes this point about counterfactual measurements or the the kind of well-defightedness of counterfactual measurements are critical to the interpretation of this experiment”
A Cantor set is constructed by starting with a line segment, removing the middle third, then removing the middle third from each remaining segment, and repeating this process infinitely. What remains is the Cantor set.
“you start with a line with a finite length and you remove a third the middle third of that line so you now have two pieces...you take these remaining two-thirds and you chop out the middle third of each of those so now you've got four smaller bit still...you chop out the middle third of those and now you've got eight and you chop out the middle third of those and you keep doing this literally forever and Cantor set is what is left over when you have continued chopping out middle thirds forever”
The Cantor set has measure zero (no size) from outside measurement, but contains infinitely many points—as many points as the original full line—making it paradoxical like the Doctor Who TARDIS: small on the outside, palatial on the inside.
“what is really weird about this...is that how big Cantor set is depends on whether you look at it from the inside or from the outside...Cantor set takes no is no size whatsoever it's uh the mathematician call it zero measure zero it can it has no size it's zero size and yet from the inside it contains not only an infinite number of points and this is the bizarre thing it contains as many points as you started with on the full uh line”
Kurt Gödel proved in the 1930s that the set of mathematical truths is much larger than the set of things provable by algorithms, meaning mathematical truth exceeds what computers can prove.
“Kurt girdle a kind of mathematical logician um uh Austrian in the 1930s proved this kind of remarkable result which is that the set of mathematical truths is actually much much larger than the set of things that can be proven by algorithms”
Three Nobel Prize winners in 2022 (Alan Aspect, John Clauser, and Anton Zeilinger) won for the Bell experiment—a quantum mechanics experiment that Palmer describes as conceptually the most difficult physics experiment to understand.
“2021 was Suki minabi 2022 Nobel Prize were three physicists Alan aspay John klauser and Anton seilinger who did one of these Quantum experiments which as I say uh I would say is is if you had to say one physics experiment which is just conceptually uh difficult to understand it is it is this one it's the it's the so-called Bell experiment”
In the Bell experiment, on each day (Monday through Thursday), Alice and Bob choose different orientations for their polarizers and measure correlations between the polarization outcomes. Bell's theorem shows that a traditional deterministic model cannot satisfy a certain inequality relating four correlations unless non-locality occurs.
“the experiment is run over four days just for the sake of argument uh and on each day Monday Tuesday Wednesday Thursday Alice and Bob choose different uh orientations for their polarizers...they can come together and basically calculate how well correlated the measurements were...this is a an equation for these correlation coefficients the four correlations and it turns out for a range of possible orientations the this particular mathematical sum and difference of correlations exceeds the value to the reason why that's significant is that Bell himself John Bell showed that it's impossible for a traditional deterministic model to satisfy that inequality unless this notion of spooky action at a distance is occurring”
Henry Poincaré discovered chaos in gravitationally bound systems at the start of the 20th century, demonstrating that a four-body gravitational system exhibits intermittent instability—bodies can orbit in stable ellipses for periods and then suddenly escape to infinity.
“it's gravity chaotic you bet it's chaotic in fact the very first example of chaos was actually shown by French mathematical physicists to Henry poincare right at the beginning of the 20th century and here's a really beautiful illustration of what Lauren sorry what poincare found... we're looking at four uh gravitationally bound bodies... they're orbiting around each other in four really nice ellipses and it actually looks extremely predictable doesn't it until whoa uh it's such a beautiful result... we've we've got five periods of oscillation where these bodies seem to almost describe perfect ellipses and then on this the sixth one or something they just zip off to Infinity”
Fractal geometry provides a bridge between the classical mathematical and physical world of Newton and the greatest discoveries in 20th century mathematics by Gödel, Turing, and Wiles.
“what I'm going to argue in this talk is that this geometry provides a bridge between the classical mathematical and indeed Physical World of Newton...and first of all the world of 20th century mathematics...I'm going to claim that this fractal geometry provides a link between the maths of Newton and and and the some of the the the greatest discoveries in 20th century mathematics that these three individual mathematicians um help show”
The Bell experiment, which won the 2022 Nobel Prize, seems to resolve the Einstein-Bohr debate in favor of Bohr's quantum mechanical interpretation, but fractal geometry might tilt the balance back toward Einstein's deterministic view.
“this type of geometry can also explain one of the most conceptually difficult experiments in in 21st century physics in fact it was an experiment that was performed by three uh come onto this three experimenters won the Nobel Prize in 2022 What's called the Bell experiment...It seemed to resolve the big debate that Einstein and Bohr Niels Bohr had in the early 20th century in favor of Niels Bohr...I'm going to claim that fractal geometry actually might tilt the balance back towards Einstein”
Physicists typically respond to concerns about counterfactuals by noting that counterfactuals are used routinely in physics (e.g., 'if a ball were thrown with half the velocity, would it reach the father?'), and the deterministic laws of physics allow counterfactually varying initial conditions.
“here's a boy he's throwing the ball the football where it is to his father he launches the ball up it goes through the air on a parabolic Arc the father catches it all well and good you might say well suppose the child had thrown the ball with half the initial velocity would it have reached the father well that sort of question you can answer with your high school physics...you have a formula which says you know for a certain initial velocity of the projectile and a certain angle it goes on a parabola”
This idea of global structure mattering as much as small scales may sound 'cranky' to many physicists, but Palmer cites Roger Penrose (2020 Nobel Prize winner) as supporting the idea that radical new physics is needed to understand quantum non-locality.
“this sounds this to a lot of physicists will sound extremely even cranky might be the word but I want to finish my talk with um I've I've already mentioned two Nobel Prize winners one in uh 2022 three Nobel Prize in 2022 I've mentioned tsuki minabi 2021 I want to finish with a couple of quotes from the Nobel Prize winner of 2020 Roger Penrose who was a great influence on me”
Approximately 99% of physicists believe that Bell experiments demonstrate either Einstein's 'spooky action at a distance' (non-locality) or that quantum reality is indeterminate (neither of which Einstein liked).
“now okay so you might think that's bizarre but this is what probably 99 of physicists believe that Bell these Bell experiments the experiment I say what I mean by Bell experience the experiments that these three guys showed independently...either mean that physics has what Einstein called spooky action at a distance...can affect uh an outcome of an experiment...or we have to say that Quantum reality somehow yeah is indeterminate...and Einstein also phrased coined the phrase to his dislike of uh he kind of very spooky action as a distance...and he also kind of phrased God doesn't play dice”
The fractal interpretation of Bell's experiment signals the end of a particular reductionism in physics—the principle that smaller scales are more fundamental to understanding nature.
“what I want to move to the into the next slide is what this is really saying about physics in in the big picture what is the real message behind the Bell experiment if this interpretation is correct I think it signals the end of some of a kind of principle that has served us extremely well in the past in science but maybe running its course and that's a kind of reductionism now I'm not claiming every type of reduction is wrong but a particular reductionism which says the smaller we look at things the more fundamentally we're probing the structure of Nature”
Suppose the entire universe is a deterministic dynamical system evolving on a humongously large state space (incorporating all particles in the universe) with a fractal attractor structure.
“suppose the whole universe is a deterministic dynamical system evolving on some now not weather or pendulum or something like that but some humongously big uh State space trouser space on steroids if you like it's every particle in the universe you know is contributing to this state space But nevertheless Suppose there is some kind of fractal attractor”
Roger Penrose suggested that the correct theory of quantum gravity might be deterministic but non-computable (uncomputable in the Turing-Gödel sense), and fractal attractors are formally of this non-computable type, suggesting this approach is in the right direction.
“it seems to me quite plausible that the correct theory of quantum gravity might be a deterministic but non-computable Theory non-computable in this Turing girdle sense of the word and again I would just remind you that these fractal attractors are formally of this non-computable type so that gives us a a clue that this might be in in the right direction”
If one of the counterfactual worlds (e.g., Wednesday's) lies in a fractal gap and is thus inconsistent with the assumption that realistic states of the universe lie on the fractal attractor, then that counterfactual is not physically real. This removes the basis for Bell's argument without invoking non-locality or indeterminism.
“I'm going to zoom into this because what I've done is deliberately put Wednesday's counterfactual in a fractal Gap in the sense that it now violates my assumption that any realistic states of the universe have to lie on the fractal attractor so what it means is that Wednesday's counterfactual according to this illustration is not a a plausible realistic state in other words it's inconsistent with this notion that the universe is a dynamical system a deterministic dynamical system evolving on a fractal attractor”
Ensemble prediction has enormous applications beyond weather and climate, including conflict, COVID-19, and economics. These systems also appear to have periods of high predictability alternating with periods of extreme intermittent instability.
“I think Ensemble prediction has enormous applications in other areas and in in the book which I uh which I wrote and is is out now I I discuss applications of Ensemble prediction to conflict covid and indeed the economy and it's a kind of Fascination to me about whether these sorts of systems also have periods and it seems that they do periods of very high predictability uh interspersed with periods of extreme intermittent instability”
Newton would have had no concept whatsoever of fractal geometry, despite understanding calculus, because Newton's education was in Euclidean geometry and fractals violate the fundamental properties of scale-invariance that Newton would expect.
“the equations generate a type of geometry that Newton would have been utterly utterly alienated to Newton he would have had no concept whatsoever of the type of geometry that these equations generate”
Higher-dimensional state spaces, while impossible to visualize, are conceptually coherent; adding a fourth dimension (e.g., cotton-to-polyester ratio for trousers) is conceptually manageable even though visualization is impossible.
“if you wanted to represent a point in this Extended State space the state space would have to have a fourth uh axis somehow orthogonal to these other ones...I hope what the point I want to make is that I hope from a kind of concept although we can't you know none of us can imagine what four dimensions is like but I hope from a conceptual point of view...it's you can't visualize it but it's conceptually it's okay”
Syukuro Manabe won the 2021 Nobel Prize in Physics for developing one of the first climate models for studying climate change. Palmer expresses sadness that Lorenz died around 2008 without winning the Nobel Prize, which he 'certainly should have' won.
“and all of our or myself and my my colleagues were delighted when our our colleague Suki manabi who developed really one of the first climate models for doing climate change uh won the Nobel physics prize in in 2021. um it's really a great I I sadness to me that Lawrence never won the Nobel Prize he died in I think around 2008 or so um uh certainly should have won the Nobel Prize”
Cantor's contemporaries thought he was insane for proposing fractals, and his ideas were not recognized in his lifetime, sending Cantor into depression. However, later mathematicians realized how brilliant Cantor's work was.
“when Cantor came up with this people thought he was complete in his contemporaries thought he was completely crazy and in fact it sent Cantor into into a depression because these ideas never got recognized in his lifetime it's only much later we realize how utterly brilliant uh Cantor was”
In proving his result about the halting problem, Turing had to define what a computer was; he devised what is now called a universal Turing machine, which is the concept of a general-purpose computer.
“one of the great remarkable things about uh turing's paper was that he to to prove this result he had to Define what a computer was in those days there weren't electronic computers so he kind of devised what's now called a universal turing machine which is really what we now call a general purpose computer”
Andrew Wiles proved Fermat's Last Theorem in the 1990s: the equation x^n + y^n = z^n has no solution in positive integers for any integer n greater than 2.
“Andrew Wiles in the 1990s proved the very kind of famous conjecture...fermat didn't actually prove it but he conjectured it that a certain equation this equation x to the N plus y to the N equals z to the n has no solution in integers whole numbers positive whole numbers if n another whole number is greater than two”
Reductionism has worked well for understanding atoms, nuclei, and quarks; smashing atoms apart revealed electrons, and smashing nuclei revealed protons, neutrons, and quarks.
“it's worked of course extremely well in terms of our discovery of atoms um and of course understanding atoms and the electrons that go around the at the nuclei has been critical for understanding chemistry...and to understand atoms you know we have built accelerators which smash atoms apart They smash nuclei apart and we can understand that really what goes on in a in a nucleus of an atom are collections of protons and neutrons and we can smash those and we see there there are quarks”
Palmer wrote a book discussing uncertainty in the context of how systems become uncertain and how understanding uncertainty helps us understand systems themselves better.
“I've taken a couple of things from this book um I believe there might be one or two copies outside if people are interested I certainly write in a lot more detail about stuff I've talked about and also a lot of stuff I haven't talked about broadly under the theme of uncertainty uncertainty um not only for helping us make better decisions but also understanding how systems become uncertain actually helps us understand the systems themselves better”