Does Economics Suffer From Physics Envy?


Time is Money - 003


Every year when the Central Bank of Sweden hands out its “Nobel Memorial Prize in Economic Sciences”, there are always snorts of derision and cries of protest that this is “not a real Nobel Prize” and economics is “not a real science”…

The Oxford Dictionary definition of science is “the systematic study of the structure and behaviour of the physical and natural world through observation and experiment”.

Economics is sort of unique among the “social sciences” in that it attempts to apply mathematical rigor to the study of human behaviour.  This would seem to imply that economics is an exact science, like physics or chemistry; which further implies that economists are in the business of discovering “fundamental truths”…

Critics of “economic science” argue that the use of the paraphernalia of physics, like dense mathematical models, is in reality, purely “for show”; a vain attempt to give economics the elevated aura of physics (without unfortunately the same power of prediction). In fact some people would go so far as to say that “economic science” is really about as scientific as astrology and voodoo.

A bit Harsh Maybe !?…


Linear Dynamics

Physicists like to describe their science as the hardest of “hard science” because physics can claim to be governed by hard and fast scientific “Laws”.  This of course would seem to imply that many of the so-called “soft sciences” are in some way not quite as elevated, not quite as good.

In truth however we could say that physics is an “easy science”, and the soft sciences are “difficult” because the “laws” of physics only really work in the absence of “noise”, and yet the everyday world of the soft sciences is full of noise because most everything is continually battered and buffeted by “constantly changing feedback” which can generate wild “nonlinear dynamics”.

In reality all dynamics have feedback (and resultant nonlinearity), it is just that some dynamics have much less feedback than others.  Physics is, in a sense, the science of “dynamics with negligible feedback”, the science of linear dynamics” — or in other words it is the science of the nonlinear stuff that can be safely “compressed” into neat linear differential equations which express neat linear “cause and effect”.

All the mathematical laws of physics are linear approximations of nonlinear behavior.  In fact the reason that these “Laws of Physics” are actually considered to be “Universal Laws” is precisely because the dynamics behind these laws can be linearized (which then allows us to repeat experiments over and over again, with the same initial conditions, and always get the same predictable result).

Nonlinear Dynamics

Ironically, it is our inability to express the dynamics of a system in a linear form that results in us referring to that system’s behavior as being “nonlinear dynamics”.  In the simplest possible terms, linear dynamics are dynamics where the effect is proportional to the cause, and nonlinear dynamics are where the effect is disproportional to the cause.  In any system of any type, the universal causes of nonlinearity are

  1. The inability to damp down constant adaptation causes the emergence of diversity (i.e. deviations from the uniformity of equilibrium).
  2. The inability to damp down positive feedback causes the emergence of bias or “skew” (i.e. some random things can arbitrarily get reinforced).

Economics is highly nonlinear and subject to both incompressible adaptation and incompressible feedback.  This means that there are no, and never will be any “fundamental laws of economics”.  Even  the slightly move away from linearity is akin to moving away from a “law” to a “strong probability”; and economics has moved a long way away from linearity.

Economics is highly nonlinear because economies and economic activity are riddled with adaptation and feedback, which means that even probabilistic predictions can become difficult; but this is not necessarily a bad thing, because this unpredictable nonlinearity can lead to the emergence of unpredictable complexity”…

The Nature of Emergence

In an economy it is innovation that drives adaptation, and it is investment that drives positive reinforcement.

In any nonlinear system, it is hard to know what adaptations are going to occur and which of them will get reinforced through investment; and the more nonlinear the system is, the less amenable it will be to even probabilistic prediction. However this lack of predictability does not mean that we cannot have a “Qualitative Understanding” of the potential behavior of the system.

Both adaptation and positive reinforcement can independently drive emergent behavior, but it is the interplay between the two that determines “the nature of emergence”.  While diverse innovation without positive reinforcement can lead to the emergence of chaos, and reinforcement without diversity to the emergence of a bubble, the right balance between the two will always drive “the emergence of complexity”.

Matrix of EconoComplexity Dynamics


Aggregated Behaviour

Economies are effectively aggregated behaviour.  Some would argue that, left to its own devices, a society’s aggregate behaviour will self-organise the optimal economic equilibrium; but this is obviously not true, for without laws society’s aggregate behaviour can often disintegrate into chaos.

Okay, you might say, we need laws but we don’t need government nor bureaucratic interference in the economy; but this is not true either.  When it comes to financial matters, in general humans (and banks for that matter) can often behave like complete idiots, displaying a herd-like follow the crowd mentality.

The raison d’être of rational capital markets is that valuable investment funds are allocated to a diverse number of innovative business ideas, thereby generating economic complexity to the benefit of society as a whole.  But with the emergence of herding these funds (and even borrowed funds) are often almost blindly redirected and allocated into one particular area or asset class thereby generating a financial asset bubble to the detriment of society as a whole.

So when it comes to managing the economy, a hands off approach is not always the greatest idea….

Managing an Economy

When it comes to managing an economy, the question that needs to be asked is not what are the driving forces unique to economic systems, but “what the universal forces that drive all complex systems?…”

All complex systems are driven by an interplay of diversity and combination, or in other words, “complexity emerges” as a result of the diversification of uniformity and how this emergent diversity best fits together”.

So in reality, it is not really possible to truly understand economics without understanding complexity, and it is not possible to understand complexity unless we understand the interplay between  order and chaos.

Managing an economy, like managing any other complex system, is an art, or more aptly a talent, like cooking, and like cooking it needs a good chef.  A good chef knows how much of each thing he needs and how long to cook it for (to achieve optimal integration).

Too much reinforcement without diversity of innovation and we get inequality and lock-in. Too much innovation and no reinforcement and we get the chaos of many fragmented ideas (that can quickly form, but just as quickly disappear).

What we need is to ensure that “the demand for investment opportunities is, as much as possible, always in sync with the supply of good ideas”; we need to ensure just the right mixture between innovation and reinforcement to achieve the perfect dish of integrated diversity…

Economics vs. Physics

Both physicists and economists need to realise that economics is indeed a science, just not a linear science; which means it cannot rely on linear mathematics, and thus is not amenable to mathematical prediction.

Economics is a nonlinear science, a “complex system science”, and so although not predictable it is still deterministic, governed by the same universal forces that drive all complex systems.

Economics as a disciple should cease attempting to be like Physics, because although natural systems are predictable, economics systems have vast potential for “unpredictable creativity”, and with the right complexity management we harness this potential, to the benefit of all…

Revamp…

Today (20th January 2016) I revamped this website.  I did this because over the last number of months it became obvious to me (from feedback) that “Coarse Damping” was too contorted a term for most people to grasp, and it would be better to highlight how

Chaos and Complexity are simply

“dynamics that are difficult to compress”…

The primary result of this revamp therefore is that, within the permanent pages, I have expanded on the idea that “Chaos and Complexity” are different manifestations of systems that “coarse damp to equilibrium” by explaining that this “coarse damping” is always the result of “Incompressible Dynamics”.

Furthermore, in order to quantify the unpredictability of these incompressible dynamics, I have borrowed from Computer Science the concept of “Information Entropy”

Matrix of Entropy Dynamics - Copyright - Kieran D. Kelly


Now that I have revamped the permanent pages on this website, I intend to start blogging in earnest.

The idea behind the permanent pages is that they will hopefully act as a knowledge base, that the blog posts can reference.

The subject matter of this blog site  going forward will be “Emergent Complexity in the 21st Century”…

What is Euler’s Secret Identity?


Eulers Secret Identity - 001

Is there some connection between the Mathematics of Euler’s Identity and the Physics of a Photon?


Euler’s Formula (eix = cos(x) + i sin(x) ) is usually thought of as a mathematical description of how a “vector rotates” through an angle in the complex plane; and consequently Euler’s Identity is usually considered simply a rotation through an angle of π radians.  But as long ago as 1581 Galileo had already realized something amazing; “rotation” has a secret identity — he is also an “oscillation”…

Galileo’s startling realization that uniform circular motion and periodic motion were different sides of the same coin, meant that he saw that angles could also be used to describe an oscillation.  And this, it turns out, was a truly brilliant insight!…

So although most people would probably still associate angles with geometry and trigonometry, angles can also be associated with the amount of “energy in an oscillation” –-  which means that a function that defines an amount of  rotation in the complex-plane ( y=(-1+i0))  can also be thought of as a “definable angular frequency” expressed as x% of (-1+i0), or in other words: “ x% of a half-cycle”.


Now, what is really interesting about this idea is that:  the amplitude of all “normalized” oscillations have a magnitude of 0.25 units of measurement, or ¼ of the normalized cycle.  So

  • In our formula ( y=(-1+i0)), when we set  x=0.5, we get “a quarter-rotation in the complex plane” 

(-1+i0)0.5   =   (0+i1)

  • But in setting x=0.5 we are also, in effect, setting an angular frequency of π/2 radians for the equivalent oscillation on the real-line; and so we could also say that (0+i1) is “a quarter-oscillation on the real-line”, which means that

(0+i) is the amplitude of all normalized oscillations

  • And this means that

(-1) is the “amplitude of an normalized oscillation squared (i2)” 

  • And this means that 

all frequencies can be defined as some % of (i2)

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So, any angular frequency of oscillation, defined in terms of a single half-cycle ( y = (-1)x ), can now also be defined in terms of the normalized amplitude squared  ( y = (i2)x ).  And this, to anyone who has studied even a small amount of physics is strangely reminiscent of the “energy in simple harmonic motion”.

This would seem to suggest that an angular frequency (i2)x  might be equivalent is some way to the energy in an oscillation (½kA2), and that (x) might represent some form of “restoring variable”.

Furthermore since we know that the equation for the energy in a photon of light is

E  =  hf  =  ħω

This further suggests that there might be some form of a connection between (eiπx) and (ħω).  So let’s take a closer look at ( ei )…


We can use Euler’s Formula to define a smallest “unit of angular change”.  Let’s define this unit to have the same magnitude as the value of Planck’s Reduced Constant.

(1 ħ-radian = 1.054571726  ×  10-34 radians)

euler -4

eix   =   =   cos(x) +  i sin(x)

ei   =   cos(1) +  i sin(1)

eiħ   =   cos(ħ) +  i sin(ħ)

(-1)ħ   =   (0.99999999 +  iħ)

Given this defined “unit of angular change”, we can now think of every other size of angular change as an integer number(n) of units of (1 +iħ);

(eiħ)n   =   (ei)ħn   =   einħ   =   (1 + iħ)n   =   n ħ-radians   (where n = 1,2,3….)

And when n = Ħ =  9.48252238653324  x  10+33 (i.e. capital-h-bar : the inverse of  ħ) we get an angle of

(eiħ)Ħ   =   (ei)ħĦ   =   ei   =   (1 + iħ)Ħ   =   Ħ ħ-radians =   1 radian

And when n = Ħπ we get an angle of

(eiħ)Ħπ   =   (ei)ħĦπ   =   eiπ   =   (1 + iħ)Ħπ   =   π radians  = (-1+i0)


Equivalence 003

So if (-1)x  is indeed equivalent to  (½kA2) we could now say that since

(-1)x   =   (eiπ)x   =   (1+ iħ) Ħπx

And since

½kA2   =   hf

Therefore

Equivalence 004


So maybe different photon frequencies simply represent different amounts of restoring force in the elastic fabric of space-time.  If so then different frequencies would simply represent different percentages of the ultimate elasticity of space-time.

So what is Euler’s Secret Identity? 

Maybe he is the speed of light’s cousin;

“The Angular Speed of Mass” – the fastest frequency of all…