The main events in the history of science have involved new ideas overthrowing conventional wisdom. The notion that the earth was the center of the universe was upended by Copernicus. Species were thought to be permanent and fixed until Darwin. Physics was thought to be completely understood at the end of the nineteenth century and then came relativity theory and quantum mechanics to mess everything up. Godel overthrew the notion that mathematics was infallible. This story has been repeated so many times that people now seem to instinctively look for the counterintuitive answer to every problem. There are countless books on thinking outside of the box. However, I think that the supplanting of “linear” thinking with “nonlinear” thinking is not always a good idea and sometimes it can have dire consequences.
A salient example is the current idea that fiscal austerity will lead to greater economic growth. GDP is defined as the sum of consumption, investment, government spending and exports minus imports. If consumption or investment were to decline in an economic contraction, as in the Great Recession, then the simple linear idea would be that GDP and growth can be bolstered by increased government spending. This was the standard government response immediately after the financial crisis of 2008. However, starting in about 2010 when the recovery wasn’t deemed fast enough instead of considering the simple idea that the stimulus wasn’t big enough, the idea that policy makers, especially in Europe, adopted was that government spending was crowding out private spending so that a decrease in government spending would lead to a net increase in GDP and growth. This is very nonlinear thinking because it requires a decrease in GDP to induce an increase in GDP. Thus far this idea is not working and austerity has led to lower GDP growth in all countries that have tried it. This idea was reinforced by a famous, now infamous, paper by Reinhart and Rogoff, which claimed that when government debt reaches 90% of GDP, growth is severely curtailed. This result has been taken as undisputed truth by governments and the press even though there were many economists who questioned it. However, it turns out that the paper has major errors (including an Excel coding error). See here for a summary. This is case where the nonlinear idea (as well as conflating correlation with causation) is probably wrong and has inflicted immense hardship on a large number of people.
Scientific Clearing House 
The Austerian thing is “dumber” than the other famous examples. For instance, for some things a model of where the earth is at the center of the universe isn’t an unreasonable approximation. And at many time scales, regarding the roster of species as fixed is a good approximation.
That government austerity will fix a depression is based on no good approximation whatsoever.
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I admit I’m attracted to those (every?) theories which attempt to embed high dimensional problems into lower dimensional ones (eg ‘center manifold theory’). From this view, everything is (piecewise) linear—you have a straight line, then a miracle happens, and you have another straight line with a different slope, and so on. You can use fourier series, or wavelets, etc to smooth out (or hand wave over) miracles. (I can’t recall whether a Turing machine is linear—I think so (Darryl McCullough i think wrote on that), but those can describe n-degree polynomials (eg Hilbert’s 10 th problem—davis, Robinson, Matyjevich (sic)).
In the 80’s a ton of books were written on how the ‘new nonlinear science’ was replacing the old ‘newtonian’ paradigm; some did note that Newton’s laws for gravity and e-m are actually nonlinear, and the 3 body problem is chaotic. Many of the linear paradigms seem (like the gaussian-markov assumption) to be due to the fact that people could only solve problems if they truncated the Taylor’s series at the second term. (I note in my ‘into to quantum physics’ coursera.org course this is pointed out explicitly for derivation of the path integral). (And I hold, as I think Cosma Shalizi of CMU appears to, everything can be embedded into a 1st order markove chain, though its as conveniant as writing everything in machine code. An article in the New Yorker from Dec 21 2012—the year the world ended—describes Ithuikal language developed by a Mootor Vehicles employee (john quijada) shows a similar precise formalism for common language. Unfortunately, I guess, Noam Chimpsky led us on the wrong evolutionary path—Wittgenstein, James Joyce (finnegan’s wake), etc complement Goedel here ).
There is also ‘supply side economics’—which appears to be the idea behind Rogoff etc. (My mom said the U Mass grad student was on John stewart show). I wonder if there is a ‘demand side economics’—‘we demand…’ ( i was wondering how this austerity was going to play out locally—but I did get mail today saying i am now on the low income health care plan, though I’m going to try to avoid pneumonia.) But following Keynes, can we really inflate our way out of a recession or depression? In the ‘Big Bang Never happened’ eric lerner suggested that the inflationary universe was concocted to make physics more realistic—Occam’s razor—harmonizing it with the supply side economics (eg David Stockkman of Reagen era who was ranting on the radio a few weeks ago) of the period. Physics after all is dependent on economics—from E=mc**2 one realizes making rest mass (Higgs) costs money. Perhaps this is an example of ‘top-down causation’ (George Ellis of S africa, P Davies). Some werer worried that making Nukes would set the atmosphere on fire (which would affect the FIRE economy—finance, real estate, etc.) Otto Rossler had similar conCERNs about the higgs (get your esp on). But what if government austerity meant no more mass was created in the expanding universe? I guess one might get a big bounce.
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@ishi Turing machine is definitely nonlinear. A linear operator is defined by L(x1 + x2) = Lx1 + Lx2. This is certainly not true for a Turing machine. Also, a Turing machine can simulate any nonlinear function, which a linear machine cannot do.
The evolution of the wave function in quantum mechanics is linear but the collapse of the wave function is nonlinear.
Demand side economics is Keynesian economics. It does not suggest that you inflate out of a depression but that the government increase spending to compensate for lack of demand.
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i mentioned i was not sure, darryl mccoulugh emeretis math at uc davis had something on this in ‘beyond the turing machine’ (book).
i know that about quantum mechanics (s weinberg—i think he has a phD and is at Harvard (now Austin) like Rogoff—-tried a nonlinear shrodinger to avoid born’s rule but it didnt work. e nelson (a christian who tried to prove peano arithmatic is inconsistant—at princeton also, like bohm, tried something like that—hydrodynamics.).
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i thik it depends how you define linear. like a ‘hamiltonian’. supposedly you can’t have a hamiltonian for a non conservative system like diffusion/fokker-planck but as even in you your own papers you cite martin siggia rose from 77 or so (robert graham is my religion, and there is eysink of jhu or q a wang or the group at u wash—ao, ping–or annila–prsl b…). you can get a quasi-hamiltonian.
i might even be able to write down a set of linear equations (linear or tensor algebra) for a turing machine.
going to the gogo show (music). now. austerity. in the long run we’re all dead (keynes/boltzmann)–tipler had a different view. im out. good blog.
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ps its actually ‘universal turing machine–half century later’ by rolf herken published in 88 or 91. the paper i mentioned is not in there–lost on the web
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@ishi A Hamiltonian system must have a symplectic structure, (i.e. dynamics preserves the symplectic norm) which nonconservative systems do not. However, you can always define Hamiltonian-like objects for such systems.
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yeah—i looked around the web for that turing paper i mentioned and cant find it (andy vogt of georgetown u had a claim in a paper on putting nonlinear systems into high dimensional linear spaces but he didnt give me a reference, years ago)—i think any linear representation is ‘locally linear’ (taylor’s series, or ‘piecewise continuous’).
Morse and Feschbach 1953 (methods of math physics) have a hamiltonian for the diffusion occasion (or equation)—
arxiv.org/abs/cond-mat/9910039v1.pdf
on the other hand, maybe no systems are conservative (eq quanta as emergent phenomena—turtles all the way down, or solitons).
ola
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ps–i looked up ‘quanta as emergent phenomena’ on the web—and franklin fetzer institute (austria) comes up with a conference (along with ‘entropic gravity—verlinde).
(also http://arxiv.org/abs/1304.6295 tho vigier who worked with bohm had similar if not better ideas).
arnold mandell (chaos in the brain) who’s on the Fetzer board was the guy i saw talk at NIH sponsored by philip morris). grossing is also there (classical quantum mechanics). qed or so it goes
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[…] seem to downplay the importance of additive genetic effects. To me this is a perfect example of discounting the obvious as I blogged about before. Most people seem to believe that the interaction of genes or epistasis […]
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