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"The Approximate Present Does Not Approximately Determine The Future"
Chaos Theory turns 50 years old this year, celebrating half a century of flapping butterfly wings in Brazil creating tornadoes in Texas. That most famous example is especially appropriate, since it was a meteorologist named Edward Lorenz who first outlined why seemingly consistent and knowable systems can still go wildly wrong. As it turns out, as ConvergEx's Nick Colas reminds us, small errors in measurement or observation at the start of a time series can significantly change how things look at the end. In the current low volatility, one-variable central bank driven global equity markets, Chaos Theory may seem a quaint relic of past crises. However, its central lesson – that complex interrelated systems create unexpected outcomes from seemingly benign inputs – is still relevant. Students of economics like to think of their discipline as scientific, just like physics or other hard sciences. They would do well to embrace the intellectual honesty neatly encapsulated by the central lessons of Chaos Theory.
Via ConvergEx's Nick Colas:
If I asked you to name a famous weatherman, I doubt you’d come up with Dr. Edward Lorenz of the Massachusetts Institute of Technology. No, he’s no Al Roker or Jim Cantore in terms of fame or fortune. He never stood in the middle of a hurricane to report for the Weather Channel or walked through a devastated trailer park after a tornado.
Dr. Lorenz’s contributions, however, have a far wider reach because he is the researcher who came up with what we know today as ‘Chaos Theory’. Here’s brief history of the man and his discovery:
- Edward Lorenz was born in West Hartford Connecticut in 1917, attending Dartmouth (BA 1935) and Harvard (Master’s 1940). He served as a meteorologist for the U.S. Army Air Corp. during World War II and earned two degrees from the Massachusetts Institute of Technology during and after the war.
- At the time, weather forecasting was considered pretty simple stuff. Take enough inputs from today’s climate and you should be able to forecast tomorrow’s weather pretty closely. Simple, but not especially effective. And potentially deadly for an air force during times of war, even the “Cold” one which followed the armistices of 1945.
- Lorenz thought that the linear approach was wrong, and started working on non-linear algorithms to forecast the weather from his new seat as a MIT professor. In the mid 1950s, he started to use an early computer – the Royal McBee LGP-30 – to help with the calculations. Its clock speed was 120 kHz, about 100,000 slower than an iPhone 5. It weighed 740 pounds. But it was better than doing thousands of calculations by hand.
- To speed up the calculations of the many iterations required for his research, Lorenz truncated the number of decimal places for the inputs to his model. He then went back and added more detail to those same inputs – 2.212 became 2.212175 – to see if he got a more fine-tuned response. To his surprise, those little tweaks created very different outcomes in his models. Small changes to the “Base state” – today’s weather conditions, for example – could result in radically different expectations for the weather in just a few days time.
- Lorenz published a paper on this phenomenon in 1963 – 50 years ago – titled “Deterministic Nonperiodic Flow.” Yeah, not a very catchy name… And according to a summary about the 50th anniversary of the paper in Physics Today, it garnered fewer than 20 citations in the dozen years after its publication.
- Lorenz’s observations begin to catch real traction only in the mid-1970s, when a paper titled “Period Three Implies Chaos” (Li and Yorke, 1975) gave his ideas a catchy name and a new audience: mathematicians and physicists.
In 1972, Lorenz gave a talk which he titled “Predictability: Does the flap of a butterfly’s wings in Brazil set off a tornado in Texas?” This famous observation is almost as well-known as the term “Chaos Theory” itself, and serves to explain the basics of Lorenz’s discovery. Three quick points here:
- Lorenz’s 1962 paper summarized his findings this way: “For those systems with bounded solutions, it is found that nonperiodic solutions are ordinarily unstable with respect to small modifications, so that slightly different initial states can evolve into considerably different states.”
- He later summarize “Chaos” more concisely as “When the present determines the future, but the approximate present does not approximately determine the future.
- In totally laymen’s language, Chaos Theory says that if you want to forecast the future you need to know everything about the present. And by “Everything” we mean all knowable characteristics of today, in infinite detail. Even if you have a great set of formulas in a comprehensive model about how those many variables interrelate, your predictions will run afoul of ‘Chaos’ – the ability of an overlooked (and typically small) characteristic of the starting point to have a large effect on the outcome.
In the world of economics and capital markets, Chaos Theory clearly resonates. The Financial Crisis of 2007-2008 and its follow-on travails all have the gentle flutter of the butterfly’s wings somewhere at their core. Small issues – a marginal residential development in Scottsdale, a deal by Greek monks for some Athenian commercial real estate – quickly cascade to become the tornado in Texas. Or New York, or London, for that matter.
What I find most striking is the current market psychology that seems to think all the butterflies are dead, or at least safely in their pupae. Observed volatility for stock prices, as measured by the S&P 500 Index, are trending lower over the last 10 and 20 days, even as the market itself reaches new highs. Implied volatility in options contracts are trending lower as well.
It not just the math of volatility that I find most puzzling, but the notion that central bank policy is all that matters to economic and market outcomes. I get the fact that the last few years have been humbling for everyone from risk-averse investors (who missed the move in risk assets) to policymakers who shovel liquidity into an economic system which still struggles to create jobs or growth. But it seems very much like commentators and market participants desperately want to believe the world behaves according to simply rules. “Just buy the equity market whose central bank has the largest bond buying program” is essentially the only piece of investment wisdom needed for the last 48 months. And counting…
I keep coming back to Lorenz’s statement that "The approximate present does not approximately determine the future." We know our approximate present very well, at least in the U.S.:
- A slow growth economy
- An accommodative central bank
- Only one other large economy (Japan) with the appetite to follow our lead in buying large quantities of long dated bonds
- A seemingly “Self sustaining” rally in stocks, where there is enough momentum to pull at least a few new buyers into the mix. And low enough interest rates to provide few alternatives to investors.
At the same time, Chaos Theory is clear: this does not approximately determine the future. There are more than enough variables out there – the butterflies flapping away – which can change outcomes. Don’t get me wrong – this is not meant to be a doom and gloom closing thought. If stock markets exhibited ‘normal’ volatility, it would be far easier to defend current price levels. You could leave the butterfly net at home. The problem is that current market price action –that slow steady grind higher – indicates marginal buyers don’t fret very much about the future. No matter how little we really know about it.
Additional Reading
Original 1963 paper: http://eaps4.mit.edu/research/Lorenz/Deterministic_63.pdf
50th Anniversary in Physics Today: http://www.physicstoday.org/resource/1/phtoad/v66/i5/p27_s1?bypassSSO=1
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War is Peace
Freedom is Slavery
Ignorance is Strength
Chaos is Order
"It's all relative."
first it is all relative, then it is absolute
Chaos Theory > Chaos Reality
When Chaos Theory bumbed into Chaos Reality, Obummer was pushed out of the vortex nine months later.
If a pyramid is sitting on a bedrock base it can stand for 5000 years. If it is balanced on its point a butterfly can push it over. Chaos theory applies to unstable systems. Economic systems become unstable when given over to short term thinking, scams, and Ponzi schemes. When based on honesty and genuine production of useful goods and services they are more stable.
Brace for turbulence.
Maybe "Brace for Impact" might be more appropriate!!
Boris is choose "turbulence" because is enigma to engineer and physicist. "Impact" is fun, but done and over, kaput.
welcome to the human race.
-snake plissken
"That's it man, game over man, game over."
- Marine in Aliens
The idiots have taken over the asylumn.
And black is white.
"Small issues – a marginal residential development in Scottsdale, a deal by Greek monks for some Athenian commercial real estate – quickly cascade to become the tornado in Texas."
Hindsight is 20/20.
A biologist studying one type of butterfly isn't going to understand the flapping of a different species' wings.
DSGE modeling and Hedge, these are the rotational magnetic gravities that will propel our global economic axis the next 10 years.
Chaos theory is for people who haven't graduated to quantum theory.
Quantum Theory is for those who don't know a wave from a particle and crowd all my measurements with their stinking vectors penetrating my space reducing my uncertainty to zero about you, math separatists void suckers all.
Quantum Theory of Banking... cannot know where dollar is and how much is worth at same time. Same theory is apply to shell game. Same is thing!
The sociopaths have taken over the government.
And wrong is right.
And the oligarchy beat goes on :
http://www.businessinsider.com/financial-weapons-of-mass-destruction-2013-5
Interesting "recommended" article showed up from the one you referenced:
http://www.businessinsider.com/bubonic-plague-saved-14th-century-europe-...
Lowering the denominator - heh.
YOUR government was taken over by sociopaths long, long before you were born.
Question is, why do you still support the beast by claiming ownership of it?
'THE' government is not 'MY' government any more than it's anyone's government other than 'THEIR' government.
Still you have to throw a couple of ducats at it every now and then to just to keep the dogs at bay...
I wish I had read this before I cloned all those dinosaurs...
One of the most informative ZH articles ever.
Apparently some people didn't like the extra reading assignment at the end of the article.
;-)
Nonetheless, great article for exposing a whole new audience to chaos theory.
Reminds me of Overbar transcription error:The most expensive hyphen in history
The error had occurred when a symbol was being transcribed by hand in the specification for the guidance program. The writer missed the superscript bar (or overline) by which was meant "the nth smoothed value of the time derivative of a radius R". Without the smoothing function indicated by the bar, the program treated normal minor variations of velocity as if they were serious, causing spurious corrections that sent the rocket off course. It was then destroyed by the Range Safety Officer
[Bottom of page]
This article makes you realise just how naff all that Pheoneix, Grant, Testosterone, etc. offal is. Kudos to ZH. One great article a week is enough to keep the pot boiling.
zn+1 = zn2 + c, Bitches
It's a big club, you ain't in it. Slainte.
And let's not forget what is perhaps the best corollary to Chaos Theory: Murphy's Law
"Anything that can go wrong will. And at the worst possible moment"
Morpheus : This is a sparring program, similar to the programmed reality
of the Matrix. It has the same basic rules, rules like
gravity. What you must learn is that these rules are no
different than the rules of a computer system...some of them
can can be bent. Others...can be broken. Understand?
https://www.youtube.com/watch?v=j82GKTgVDkw
so true---lie about the butterfly and lie about the outcome-- its really not real--
When you're invited to shoot pool on someone else's table for the first time, keep your money in your pocket.
and keep your eye on their balls...
in 50 yrs we will probably have some distopian nightmare and a helluva lot less people.
10 or less.
Food aid to Africa/India/China when TSHTF anyone?
Remember this? > http://www.ft.com/cms/s/0/485c93ae-a06f-11df-a669-00144feabdc0.html#axzz...
"Markets" seem like chaos but only because they are micromanaged to appear that way.
Economics is not in any way science. Most people and almost all economists seem to think that science is about applying big complex mathematics to describe things. That is not what science is. Science is simply an approach to the world which says that "You are trying to convinve me that this thing behaves according to this principle? Then Prove it! And I won't believe it until I've seen years and years of rigorous hypothesis testing which is unable to disprove the hypothesis".
That's all science is. It's formulating hypotheses and then performing rigorous attacks on them to see if those hypotheses can hold up. If they can continue to be un-disproven (they can never actually be proven, only un-disproven) after several years of this scientific method then they are elevated to the ranks of "scientific theory".
Then, economists come along and think that because they can apply the same sorts of complex mthematics and pretty charts that scientists use, then their "theories" (in actuality, economic "theories" are nothing more than hypotheses) can be scientific too!
Economics is not a dismal science, because it isn't a science at all. Unfortunetely, however, the average person doesn't understand the distinction and this is very unfortunate because when the economic system collapses people are going to (rightly) blame it on the academic economists, and then (wrongly) lump scientists in with them. Economics is the laughing stock of the academic world, it is tarnishing everything to to with higher education, it is a total sham. Yet this farce of a discipline is also the one that has free reign over our societies.
Archimedes
Galileo
DaVinci
Copernicus
Newton
Linnaeus
Faraday
Maxwell
Pasteur
Mendeleyev
Bell
Einstein
Krugman
Which one does not belong in the list?
ROFL... Let me guess...
Your point would've been better made by using Keynes.
The answer is "Krugman", which you misspelled. It should have read "Klugman", as in Jack.
+1000000000
Somewhere, Karl Popper just smiled.
not bad Mark_BC .... and I am no 'academic lover' but think you sum it up well, at least for ecnomics.
2 comments:
Yeah I agree ReactionToClose, there are limits to what science can do. Regarding Point 2, that "measurement problem" is a critical idea regarding the scientific method, and a perfect example of why science can never prove anything, it can only non-disprove things. If the world had "proven" the validity of Newtonian or Einsteinian mechanics then there would have been no room left for quantum mechanics. But if science merely un-disproved Einsteinian and Newtonian mechanics then that leaves open tons of opportunity for other explanations of the world to be developed, that are applicable to size and time scales that Newton and Einstein didn't encompass.
Back in the Bronze ages and before, it was the the prophets, oracles and associated fortune tellers that used to be the economists of the day.
Just like today you would be amazed at what you can predict with a bucket-full of goat entrails...
Your logic is weird.
You argue that economics is not a science at all, and associate this with the premise that modern economists have foolishly pursued complex mathematics as a means to achieving the scientific end, but then you state (correctly I would add) that this is not what science truly is.
So in that case, why is economics not a science? I don't see the connections you are making?
In my opinion, economics definitely is a science - not one like physics in the sense of precise mathematical equations, but as a rational study of human behavior along with a development of certain key laws, based on exactly your definition of science:
"It's formulating hypotheses and then performing rigorous attacks on them to see if those hypotheses can hold up."
It might just be me, but I find it an invalid argument.
But economists do not engage in the scientific method. They don't postulate hypotheses and then subject them to years of attack using observable evidence, controlling the methods to factor out unrelated inputs. It seems they take some simple hard rules like supply and demand, and then expand on these basied on political ideology to develop all of their more complex models (taken to the extreme with Keynesian economics). This is why, today, you can find just about any economic "theory" being flogged by someone, all the way from radical right wing anarchists to left wing communists. And all purport to be more accurate in how their particular beliefs represent the real world. The problem is, none of them do, otherwise the evidence would weed out the invalid economic principles. You'd never see such a wide divergence in beliefs in science.
The problem is economics can't go through a formal scientific method because it is our economies and societies that would be the guinea pigs. And the experiment would have to span more than 50 years to get any really meaningful data, since this is the length of time the US dollar has been the global reserve currency which would grossly skew any results obtained from observing how economies react to certain policies. Plus the world is running out of energy so that, over 50 years, would be an uncontrollable factor that could no be isolated. Economics is not predicatable.
You are right it is not practical to carry out 50 year experiments comparing different economic or political systems. This is where history comes in. Nearly everything has been tried at one time or another. History can help figure out the cause/effect of different policies.
Except that we...humans...write history.
And we lie. A lot.
Oh looky looky
Top IRS Official to Plea the Fif:
http://www.latimes.com/news/politics/la-pn-top-irs-official-fifth-amendm...
Nice find Doc. Will Obama be next?
Hire a special prosecutor and seat Lois in front of a Federal Grand Jury.
What makes me want to projectile vomit is the fact that NO ONE will ultimately be held to account for this act of tyranny! Yet Steve Miller the "acting" more like "prosecuting" director resigns as a patsy, although his time was up anyway. How about the forfeit of his pension that we will be paying for? The only consulation is that he will hopefully be paid in inflated fiat currency and living in a hovel due to his diminished value fixed income. We should be so lucky! BAH!!!
Story that Obama was incapacitated while Benghazi was taking place. He'll plead he was drinking a fifth.
"Those who surrender freedom for security will not have, not do the deserve, either one" Thomas Jefferson Behold the impending age of financial and civil CHAOS!
It won't be chaos. The controlled demolition has been well planned ... as has the next phase. Central Planning™ is about to move up to the next level of centralisation ... rule by the World Politburo™.
If we say eternity is not infinite duration of time but timelessness. Eternal QE belongs to the ones who buys the moment.
BUY NOW PAY LATER....
(Slightly modified Ludvig Wittgenstein quote)
Has anyone else noticed the complete disconnect between the propaganda media and alt-media?
Each hour one can find on DrudgeReport and Zerohedge.com that the current scandals are widening in scale and scope and are being revealed to be even more criminal than anyone thought. Yet one would barely be aware of the scandals at all if he only read propaganda media sites.
Of course they are called the propaganda media for a reason, but in viewing both one is left with a strange feeling of living in two worlds at once.
hujel
Electronic heroin
http://www.jayhanson.us/page21.htm
It is frankly amazing but most likely has to do with the revenue from advertisers which the prop media cannot afford to lose as they are already losing their viewership, due to their not offending their advertisers.
Unfortunately about 99.9% of the public does not really give a shit about any of the things which we here discuss. As an example of that unintelligence I would point out who we recently elected as president and who he was running against.
Tell me about it. What is truly astonishing is how seemingly intelligent people can swallow the propaganda whole. It happens I am involved in a thread on another site, that involves Mayor Rob Ford of Toronto who is accused of smoking crack. The evidence:
A drug dealer claims to have a video on his cell phone. The video has been seen by only 3 people. One of them from Gawker, a scandalous web site whose motto is "the gossip of today is the news of tomorrow". The other 2 are Toronto Star reporters. The Star is notorious for its long standing vendetta against the mayor.
So, we have the testimony of 2 so called reporters with an axe to grind who claim to have seen a video on a 6 square inch screen. A scandal monger and a drug dealer.
I've seen more convincing evidence in a supermarket tabloid story on Bigfoot. Yet everyone believes he is guilty. What can you say to such people?
Whomever the dude is around here that wrote
"Even in mildly complex systems, an outcome is the wrong thing to look at, with the process being where the focus should be"
I salute you, sir...for that is fking golden, and exactly how fundamentals should be applied.
your emphasis sounds like that of a very good sports coach ........ maintain high standards and the outcome part will take care of itself ....
That would have been 'Urban Redneck'. He's one of the commenters worth listening to.
the means is the end.....namaste
This is one of those articles that's above all others... it articulates in simple terms an outstanding point of view.
Obviously this isn't written by one of the Tylers, that wrote about Marc Faber's recent treatise on young boys and whores.
Great writing. +1000
Everyone on ZH needs to check out the movie "Visioneers" with zack galifinakis. It relates to this thread and it pretty fkn funny! Cheers
Edit its on netflicks
Check the trailor http://www.imdb.com/title/tt0833557/
Low volatility is a gift that must be purchased. Remember as vol rises VAR dictates size must fall as vol rises, hence a self feeding sell off. That is why vol is falling, and mde to fall, because it must in order to justify higher prices.
Chaos theory does not apply to stock market indexes at this point in time. But it does play a part in global socio-economic systems.
Central banks have removed stock markets from socio-economic systems through FRAUD, access to inifinite counterfeit cash and lack of transparency. The rest of the world's real systems are not at the mercy of Central banksters.
And, central banksters can only take things so far before there is an Assault on Wall Street.
A huge credit boom worldwide is hardly a flutter of a butterfly's wing. This was all easily predictable and not the result of some random factor.
Hear you, but a minor HTF glitch can unravel the hole shit in a nanosecond!
removed
Excellent and timely article. While some people can accept randomness, few can get their head around a system being both deterministic (meaning that known forces gov ern the process) and unpredictable. The statistics we learned in school are an impoverished way to look at the world - most events do not fit into typical random distributions, and especially not Gaussian.
I speak with university students in economics frequently and am dismayed to learn that even now complex system's theory is seldom part of their education. Far-from-equilibrium systems are as different from equilibrium systems as a living person is from a corpse. The concepts require study, but in the end are not too difficult to learn.
Think of the mousetrap / ping pong ball videos you have seen, usually put on by a science teacher to demonstrate a nuclear reaction. Now every mousetrap and the ball on top of it is going to do exactly the same thing as every other one. Therefore if you had 100 of these set up it should be a simple matter to predict which will be the 30th. trap to go off. Right? And that is when you are working with exactly the same parts within the system all doing the same thing.
....and central planners think they can forecast really complex systems?
In another dimensional reality, the eternal moment is the past, present & future and absolutely. Man knows nothing of this, but in hubris thinks he knows all about it.
Ya man, now we are talking. DMT anyone...
I could never predict the future. All I can do is describe the present as best as I am able to...and it may, may give me a hedge on the next 30 minutes.
I hope some of the readers went to the Physics today article and read it. Interesting that Lorentz used a simple model of convective heat transfer due to non-uniform solar heating from the earths surface to demonstrate the first choatic attractor. Thus, such a simple system cannot be used to, for, example predict the temperature at the earths surface due to solar heating at any point in phase space. So, now you know why not only can't you predict the temperature 1 day or 1 week from now (and think how true that is), much less 100 years from now. And, this non-linear system with feedback, (i.e. the climate), is just as unpredictitable for fundamental reasons as when he discovered it. Yet, we are expected to take future predictions of earth's temperature seriously? No real scientist thinks climate modelers can even get the sign right, much less the magnititude.
TD, great article. Even if the material is too 'challenging'/academic for most readers.
J. Rickards recently remarked that Bernanke et al are quite clueless on the matter of Complexity Theory. Probably. If so, it seems that he/they have decided to circumvent complexity/chaos by throwing out a host of variables, such that they get a reduced mathematical subset. They use a Reduced Instruction Set Computing (RISC) model of money and the economy, to borrow from the Computer Sciences. Enter Dr. Krugman et al, prognosticating to TPTB with his KISS* model.
In plain English: They rigged the game and are manipulating the system to such an extent, that no advanced modeling is required. Or wanted. Collateral effects are to be ignored or treated symptomatically with OTC Rx, i.e. 95% (2 sigma) of you are on your own.
They will stay with their game plan until they can't. By which time their sheep will be high & dry, and they either start the Game anew or with a whole new Game.
*While most readers may know KISS as "Keep It Simple, Stupid", it is also used politically as "Kiss Up, Step Down" in the human jungle. There seems to be a Whole Lotta Love/KISSing going on. STDs are bound to follow.
Remember computers are tools of Order. They cannot participate in True Chaos. What seems chaotic to us is a pattern of pseudo-random Order on a long enough timeline. When the Systems collapse and True Chaos reigns then you will really see some sh-t!
Chaos Theory is a beautiful science.
Economics is the last of the deterministic sciences. i.e man can determine all probabilities.
Corrected to read as :
Economics is the last of the deterministic sciences. i.e Ben will determine all outcomes
For all here who liked this article, may I suggest reading Taleb's books, beginning with his first...Fooled by Randomness.
Just another example of the futility in trying predict the outcome of complex systems. Years ago Gene Amdahl (a computer genius ofhis time) put together a company named Trilogy and got immense V/C backing for it. It was a big deal and everyone in SV talked about it.The goal was quite simple: build an entire computer on a wafer. All the processes used were known, the materials all available, the architecture defined and Gene figured that they would build in heavy redundancy toovercome some circuits not working. Trilogy's failure to ever make it work just shows how poorly even the most brilliant people underestimate risk and what will happen in very complex systems.
Chaos is one of the most important discoveries in the history of science. James Glieck's book (Chaos: The Making of a New Science) is still the best popular introduction. Lorenz later himself released a set of popular lectures on the origins of the chaos idea in modern physics, with hat tips to important precursors, especially the great French mathematician Poincaré, who anticipated most essentials of chaos over a hundred years ago. He just lacked the computing power to rapidly demonstrate it in a computational way.
The Lorenz equations are a reduction of the notorious Navier-Stokes equations that govern the dynamics of real (viscous) fluids, like the atmosphere and oceans. Lorenz simplified them into something that captures the essential chaotic behavior, but still not so complex that they were impossible to understand. The full equations are notorious because they cannot be solved in most realistic cases. The chaotic feature that appears in fluids is turbulence, still one of the great unsolved problems of mathematics and physics.
The best path to understanding for most people interested in the practical impact of chaos is NOT the "butterfly effect," but the essential "aperiodic" or "nonperiodic" nature of chaos. It means that, however much a system may evolve by repeating itself (in general, at a spectrum of various frequencies), chaos introduces an infinite stream of "one-offs" that elude periodic analysis. This is a different way of seeing why chaotic systems are deterministic, yet not predictable. You would have to wait more than an infinity of time for the system to fully repeat itself.
Of course, it's trivially possible for unbounded systems to behave this way. What chaos does is to show that this is also possible even for bounded systems -- systems in a "box," so to speak. It's ultimately rooted in the continuous infinity of irrational numbers (which have no finite or repeating representation as numbers) that manifests itself even for a line of finite length. There's still an infinity of rational numbers on it, and an even "bigger" infinity of irrationals. See "transfinite number theory" to learn how mathematicians deal sensibly with such infinities.
P.S. It's not "chaos theory," just "dynamical systems theory." Chaos is one part of this theory, but far from the whole.
Seems to me the quantum theory route is more applicable to the current market.
Is it boom or bust? Is it a bull or is it a bear?
It depends on who's looking.
BTW, thanks for posting about chaos!
invalidating Seldon's assumption that no single individual could have a measurable effect on galactic socio-historical trends on their own, due to the plan relying on the predictability of the actions of very large numbers of people.