10. Mathematics of Informed Choice

• The Potential for Informed Choice: the Final Requirement
• The Historical Context of Newtonian Physics
• Physics' Continuous Analog vs. Living Algorithm's Discontinuous Digital
• Living Algorithm’s Fresh & Free Data vs. Physics’ Hard Data
• Living Algorithm transforms Meaningless Instants into Meaningful Moments
• Potential for Informed Choice
• Closed Systems of Matter vs. Open Systems of Life

The Potential for Informed Choice: the Final Requirement

Strict Requirements for a Mathematics of Living Data Streams

In Data Stream Mathematics, the initial article of this volume, we argued that biological systems require some type of mathematical system to digest the ongoing flow of environmental information. This data stream mathematics of living systems must satisfy some precise requirements. This mathematical system must be able to address the immediacy of the moment as well as the ongoing relationship between moments. Further, the mathematics must provide some type of fungible interpretative mechanism that sacrifices precision for meaning. This meaning must be both descriptive and predictive.

Living Algorithm fulfills these Requirements; Probability does not.

In the volume’s subsequent articles, we demonstrated that the Living Algorithm fulfills these requirements for a mathematics of living systems. Conversely, Probability’s mathematical system is unable to satisfy these requirements. There is an innate reason for these differences in ability. Probability asks the question: what is the mathematical nature of fixed data sets? Conversely, the Living Algorithm asks the question: what is the mathematical nature of dynamic data streams? The question that is asked determines the nature of the answer. Consequently Probability delivers answers that are related to general characteristics of fixed data sets, while the Living Algorithm provides answers that are related to the individual characteristics of dynamic data streams.

Data Stream Mathematics must also incorporate the possibility of Informed Choice

The data stream mathematics of living systems must incorporate yet one more requirement. The system must include the possibility of interaction with the environment. This interaction is an essential feature of living systems as it enables the ability to monitor and adjust to external circumstances in order to survive. In other words, the mathematics of living systems must also incorporate the possibility of Informed Choice.

Can the Living Algorithm fulfill this crucial requirement? Traditional Physics, i.e. Mechanics, also specializes in characterizing data streams. What are the differences in the approaches of these 2 mathematical systems to data streams? Does Mechanics incorporate the possibility of Informed Choice?

Digital Living Algorithm vs. Continuous Equations of Physics

The fundamental difference between the two systems is rooted in the basic equations that each employ to characterize existence. The equations of classical Newtonian Physics utilize an infinite and continuous stream of numbers. As such, the focus is upon a number line. The Living Algorithm is digital, in the sense that her number stream is comprised of discrete points. As such, her focus is upon individual numbers. This seemingly small difference is the hairbreadth that leads to entirely different conclusions regarding life and the fundamental nature of the Universe. To see why, let us establish a historical context.

The Historical Context of Newtonian Physics

Matter, not alive, but governed by Universal Laws

In the ancient world, human culture in general tended to attribute animistic characteristics to matter. In other words humans tended to anthropomorphize their world. The animate and inanimate universe was infused with the same undifferentiated spirit. In this sense the ancient world believed that even matter was alive, or at least had life-like properties.

Then about a half millennia ago, along came Galileo. By rolling a variety of balls down an inclined plane, he found that each ball, regardless of size or weight, had the same constant acceleration. It was evident that instead of having a motive force of their own that matter obeyed universal laws of motion.

Newton's Laws of Motion bridge Heaven & Earth

A century later, Newton further articulated these universal laws of motion. It was found that these universal laws governed both the celestial and the terrestrial worlds. That was certainly a mindblower. Prior to this point, most philosophers and theologians were firmly convinced that Heaven and Earth obeyed different laws, if they obeyed any laws at all. Now, not only had Newton formulated some laws that governed the motion of matter upon Earth, but these same laws also governed the revolutions of the celestial bodies in the heavens as well.

Newton's calculus reflects the continuous space and time of the material world

To solve these mind-boggling riddles, Newton derived a new mathematical method called calculus. Calculus’ specialty is multiplying the infinitely small by the infinitely large, to come up with a normal everyday number. For instance, the mathematics of calculus enables scientists to add up an infinite number of infinitely small points to get the length of a line. This unusual approach was instantly ratified because these computational results correspond with material reality. In other words, Newton’s ‘calculus’ computes planetary position as well as the time it takes for an apple to fall to the earth.

This type of computation also creates the notion of a continuous unbroken space and time. This notion syncs up perfectly with the everyday world in which we live. None of us worries about tears or breaks in the fabric of space and time. Further, the equations were able to precisely predict the behavior of matter. These exact correspondences reinforced the notion that reality was essentially continuous. As a further affirmation of this continuous worldview Physics, in the Newtonian sense, has successfully employed continuous equations to precisely describe the behavior of matter. And this ability has enabled humans to transform the world in unimaginable ways.

Physics generalized his findings about Matter to Life

Intoxicated with this extraordinary success regarding the material world, philosopher/scientists generalized these findings to the biological world of living organisms. Their reasoning was straightforward. The equations of Physics can predict the behavior of matter with nearly absolute accuracy. Life is composed of matter. Thus, these universal laws of motion also determine living behavior. In other words, we live in an automatic material world where everything is predetermined. Choice is but an illusion concludes this logical chain. The continuous equations of Physics lead to the philosophical faith in scientific determinism.

Because of the proliferation and dominance of these continuous equations of classical Physics, it was further believed that this was the only type of equation. To indicate the importance of continuity to this perspective, Dirac won the Nobel Prize for uncovering a technique that transformed discontinuous functions into continuous ones. As we shall see and have seen, the Living Algorithm is quite different from the classical continuous equations of Physics. Physics specializes in number lines, while the Living Algorithm specializes in numbers. Accordingly, we say that the Living Algorithm’s method of digesting information is digital in nature.

Physics' Continuous Analog vs. Living Algorithm's Discontinuous Digital

The Sine Wave: A Classic Continuous Equation of Physics

Let's examine the differences between the traditional continuous analog equations of Physics and the digital feature of the Living Algorithm from a visual perspective. At right is one of the fundamental (perhaps even quintessential) graphs of Physics – the classic sine wave. The sine wave is the basis of such universal phenomena as electromagnetic waves and spring action. It is based upon initial conditions and is continuous, as are virtually all the equations of classical Physics.

Triple Pulse & Sine Wave, apparently similar, entirely different

Although the three alternating pulses of the sine wave and the Triple Pulse have many apparent similarities, the method employed to generate these two graphs is as different as night and day. Although both are seemingly flowing curves, the first is based upon a continuous analog equation, while the second is based upon a discontinuous digital equation. In fact, the two types of graphs seem so visually similar that it took the Author over 8 years to realize that there is a fundamental difference between the two.

Difference Distinguishes Traditional from New Perspective

He was further amazed to find that this mathematical difference articulates a key point of departure between the traditional and the new scientific perspective The traditional approach emphasizes automatic processes, while the new scientific approach stresses the potential for informed choice. (These ideas are detailed more completely in A New Age of Science.) As we shall see in the following paragraphs, the continuous equations of Physics inherently exclude the possibility of choice. In contrast, the potential for choice is inherent to the Living Algorithm System.

No wiggle room in Analog Equations (Automatic Future)

The continuous analog equations of Physics have no wiggle room. No matter how much the graphs of these equations are enlarged, they remain a smooth curve. As an example, let’s view a close-up of the classic sine wave of Physics (shown at the right). Note the curve remains unbroken. No matter how many times the graph is blown up, the curve will remain continuous.

Living Algorithm’s digital nature produces Data Streams of distinct Measures

In contrast, the Living Algorithm System contains an abundance of wiggle room. This is due to the digital nature of the Living Algorithm’s method of digesting information. With each iteration (repetition of the digestive process), a new piece of data enters the Living Algorithm System. The union of the Living Algorithm and the Raw Data Stream produces an ongoing Family of Measures. These Measures represent a smoothing out of the Data’s potential roughness. However, no matter how many times this smoothing out process is performed, the resultant measures remain discrete. As contrasted with Physic’s invariable automatic continuity, there is absolutely no connection between the points in the ongoing data streams that the Living Algorithm generates. In fact, the elements of the data streams, like data sets, are inherently distinct.

Graph: Triple Pulse Close-up

A close-up of the Living Algorithm’s Triple Pulse is visualized in the graph at the right. The image illustrates how she is made up of distinct, rather than continuous parts. The apparent continuity in her classic representation is just an illusion. The illusion of continuity is due to the large number of iterations (repetitions). In similar fashion, the characters in movies and cartoons appear to move continuously, but are instead based upon distinct frames that are shown rapidly enough that our visual processor turns them into a moving picture. In the case of the Triple Pulse, the bars create a similar effect to the cartoon frames. When there are enough of them, they give the image of the Triple Pulse a continuous appearance.

Living Algorithm’s digital method of digesting data incorporates potential for Informed Choice

As we’ve seen, the Living Algorithm digests external input in a digital fashion. As such, each data point in the stream of information is discrete/individual. In other words, there is ‘space’ between each data point. This ‘space’ provides ‘time’ to evaluate the meaning of the signal and respond. Accordingly, the Living Algorithm’s digestion method incorporates the ability of an organism to monitor and adjust to the environment. The capacity to monitor and adjust is an essential ingredient of the ability to choose – the essence of informed choice. It is evident that Living Algorithm mathematics incorporates the possibility of Choice, an inherent ability of living systems. In contrast, the continuous equations of Physics provide no opportunity for an interaction with the environment.

Living Algorithm’s Fresh & Free Data vs. Physics’ Hard Data

To further assist our understanding of how the Living Algorithm’s method of digesting data enables the potential for choice, let us contrast the relation the Living Algorithm and Physics have to their Data.

Living Algorithm requires ongoing flow of Fresh & Free Data

The Living Algorithm requires an ongoing flow of fresh raw data to fuel her System. Further this data is ‘free’, in the sense that it is not predetermined by the Living Algorithm. In fact, the Data is entirely independent of the Living Algorithm. Instead of describing her Data, the Living Algorithm organizes her data. This is of great use to Life, as the process provides the fresh & free Data with meaning.

Physics needs Data to generate Equations, but then abandons it.

In contrast, Physics has an entirely different relation with his Data Streams. Instead of organizing his Data Streams, he dominates them with his continuous, automatic equations. Physics only needs Data to determine his absolute formulas. Once Physics gives birth to his magnificent equations, he abandons his Data.

Physics Equations only need one Data Point – Initial Conditions

After the derivation of the mathematical formula, Data is unnecessary (except to maybe check results). Physics only needs the starting point (the initial conditions). Once the initial conditions are determined, any computer can crank out the results of continuous (analog) equations. The results of this amazing form of analysis can include the position, velocity, acceleration and force of virtually any material system. Plug in the initial conditions and out comes the results. The graphic visualization of these results generally includes a continuous curve from the distant past into the infinite future. ‘Everything’ follows automatically according to the immutable laws of the equations derived by the immortals (Newton et al).

Physics dominates Data Streams

This model drives the philosophy of scientific determinism. Under this way of thinking the initial conditions at the moment of the Big Bang determined everything that has transpired since - music, civilization, even your relationship with your dog - everything. God/Science knows all. The only choice is to set the initial conditions. Once these are set, every point in the data stream is predetermined. No more choices. This is why we say that Physics dominates his Data Streams with continuous Equations – a Master/Slave relationship. (Unbelievably, a significant group of humans actually believe that Physics will eventually dominate every data stream with his marvelous equations.)

Hard Data must be Accurate & Precise

Because the equations of Physics are derived from an examination of the Data, the accuracy and precision of the Data is of paramount importance. To indicate this importance Physics refers to his data, as hard data – the harder the better. Because his equations are able to absolutely dominate this hard data, Physics is referred to as a hard science, perhaps the quintessential hard science.

Co-evolution of Hard Data & Hard Science

Another indication of the importance of Hard Data is that theory and accuracy of measurement have moved hand-in-hand – one or the other leading. As an example of data leading theory, the accurate mapping of the position of the stars by ancient civilizations eventually led to Ptolemy’s planetary theory. An example of theory leading data: Copernicus’ revolutionary idea that the earth revolves around the sun was finally validated a few centuries later, when advances in technology made it possible to more accurately measure planetary position. In the case of modern Physics, unusual experimental discoveries that were made possible by advances in measurement technology led to the brilliant theoretical formulations of Einstein, Heisenberg, Feynman et al. This co-evolution of Hard Data and Theory characterizes the Hard Sciences.

Living Algorithm’s Free Data encompasses Potential for Choice

In contrast to the equations of hard sciences, as epitomized by Physics, the Living Algorithm is not derived from the Data. Her function is to digest Data Streams - crunch an ongoing flow of numbers into an ongoing flow of central measures. This process organizes data streams to reveal their meaning. As mentioned, the Living Algorithm does not predetermine the values of the instants in her Data Streams in any way. Instead, she waits patiently for the next Data Point. Her suggestive descriptors give an idea what the value might be, but do not in any way determine it. There is nothing automatic about her suggestive predictions. Her data is free.

But if the data in the stream is truly free, what can be said about it that is meaningful? Let’s see.

Living Algorithm transforms Meaningless Instants into Meaningful Moments

Definitions

The Living Algorithm digests a data point each step (iteration) along the data stream. We choose to characterize the individual data point as an instant. (Please note: the measurement of the length of our notion of an instant is necessarily arbitrary.) The Living Algorithm process digests data points by creating a context that relates the current instant (data point) to the immediately preceding instants (data points). Each time the Living Algorithm repeats her relating process, she determines what we define as a moment.

Living Algorithm provides meaning to Raw Data

According to our definition, each instant is independent. The Living Algorithm’s ongoing Family of Measures determines the ongoing relationship between these instants. Without this context, instants are inherently devoid of meaningful pattern. Due to the context provided by the Living Algorithm, moments are inherently filled with mathematical ‘meaning’. In essence, the Living Algorithm transforms ‘meaningless’ instants into ‘meaningful’ moments. Or yet another articulation of this process: the Living Algorithm's digestive system provides meaning to Raw Data.

Organism translates Living Algorithm Measures to derive Meaning

But this analysis is slightly misleading. The Living Algorithm is only able to transform the instants of a raw data stream into meaningful moments if there is biological organism to interpret or translate her message. The Living Algorithm only determines the mathematical nature of each ongoing moment. The organism must interpret these essential clues to give them meaning. The statement: “The Living Algorithm provides meaning”, is just shorthand for the above analysis. It is important to always remember that a biological intermediary is required to interpret the Living Algorithm measures and give them meaning. This factor becomes very important when we discuss the ‘mass’ of attention.

Potential for Informed Choice

To see how the Living Algorithm process incorporates the possibility of choice, let’s again contrast her tao with the tao of Physics.

No choice in Physics

Let us suppose that an organism is given the continuous equations of Physics to determine the nature of the next moment in the data stream. The organism may as well put everything on automatic, as all future points are predetermined with no possibility of choice.

Living Algorithm provides potential for Interactive Feedback Loop

Now let us suppose that the organism utilizes the Living Algorithm to determine the nature of the next moment. The Living Algorithm, according to her nature, digests the data stream and provides the organism with ongoing, evolving measures that describe the trajectories of each moment. These descriptions also provide estimates of future performance. With each iteration (repetition), the organism has the opportunity to evaluate the moment according to the Living Algorithm’s measures and then make an educated choice as to what to do next, based upon this information. This opportunity for informed choice exists at every point in the data stream. This is called an interactive feedback loop. Stimulus/response, monitor/adjust, give and take are other ways of referring to this omnipresent phenomenon that occurs every moment that an organism is alive.

Living Algorithm’s abundant wiggle room provides for Informed Choice.

Reiterating: in the Living Algorithm System there is the potential for choice from moment to moment. Accordingly there is abundant wiggle room in this digital system, which does not exist in an analog system such as Physics. Further, the potential for choice is informed, not random, because the Living Algorithm’s measures provide estimates regarding the future.

Choice affects Future

In the case of the Triple Pulse System, which we previously referred to, the choice is simple, whether to participate or not – On or Off - Yes or No. This choice has a direct effect upon the future. Each choice ripples through the subsequent system. The multiplicity of effects includes: increasing intensity, killing the intensity, allowing the system to settle down, or even disturbing the peace. This living action, these effects are illustrated in the graphic visualizations of the Triple Pulse computer experiments, chronicled in Triple Pulse Results. Each choice in the Triple Pulse System has a significant impact upon subsequent developments.

Closed Systems of Matter vs. Open Systems of Life

Physics Equations impervious to External World: a Closed System

The traditional analog system of Physics has a very different relationship to the potential for choice. Once the starting point (initial conditions) of the equations are set, the relationships between the variables in the equation determine all future circumstances. An external force is required to change the state of the system. This verbalization is one of Newton’s marvelous laws of motion. The equations of Physics accurately determine all material behavior. Without an external force, behavior is automatic. Once the initial conditions have been set, the results of the equations of Physics are predetermined. No new information can be incorporated. The equations operate in a closed system.

Living Algorithm porous to the Outside, an Open System

There is no such thing as a closed system in the Living Algorithm world. This is inherent to the methodology (tao) of the Living Algorithm. The Living Algorithm’s method of processing is porous to the external world. With each iteration (repetition of the process) a new byte of ‘info energy’ from the outside enters the Living Algorithm System. Each new info byte stimulates change to the System and therefore provides energy. (We will deal with info energy in the Data Stream Dynamics notebook.) For example, in the case of the binary (1 or 0) Triple Pulse System, each data bit (1 or 0) may or may not be a pulse of info energy depending on which number it is. Further, the individual value of each elemental bit has an effect upon the future of the System. This porousness to the outside indicates that the Living Algorithm System is not closed.

Open System of Living Algorithm vs. Closed Systems of Physics

The material systems of the hard sciences epitomized by Physics are always closed. In contrast, the info system of the Living Algorithm is never closed. This difference has some important implications. Because Physics operates in a closed system, it is possible to formulate the automatic laws of matter. Because the Living Algorithm operates in an open system, it is impossible to determine any definitive laws. The closed systems of hard science mathematics are intertwined with the conservation of energy – a core principle of material sciences. In other words, material energy is conserved in a closed system. Because variable amounts of info energy enter the Living Algorithm System with each iteration, the conservation of energy cannot possibly apply. Closed systems that characterize the material world are inherently not sensitive to environmental context. In the Living Algorithm’s open system context is an omnipresent feature. Finally, closed systems by nature cannot possibly incorporate the potential for choice. In contrast, the potential of choice is inherent with each repetition of the Living Algorithm process.

Analog Equation's Content & Living Algorithm's Context determines Result

To cement understanding let's look at this difference from another perspective. The basic equation for the analog graph comes in this form: y = f {x}. This has a simple meaning. If x, a variable, is assigned any number, then y, another variable, is automatically assigned another specific number. Further, x can be any number, real, imaginary, or complex. Each pairing of x and y is true forever and always, no matter what went before or after. The content is of paramount importance. In contrast, while the Living Algorithm could be written in same form y = f {x}, the first variable, x, can't just be any value. It is a specific value in an ordered data stream. Further the value of y, the other variable, is totally dependent upon what went before. The context is of paramount importance in determining the result.

Life also an Open System

One of the amazing discoveries of Science is that the material world operates in a closed system, where energy is conserved. Due to this closed nature, abstract equations can be written which accurately characterize the relationships of matter. The bound nature of the system enables mathematicians to specialize in ever-increasing complex equivalencies between abstractions and their derivatives. Of course, living systems have a material component that obeys these same laws. Intoxicated, however, by the success of this approach to the material world, some scientists infer that hard core rules dominate all features of existence (the root of scientific determinism). However, Life, like the Living Algorithm, exists as an Open Information System. There is an inherent permeability between the internal and external world that characterizes all biological systems, from a single celled amoeba on up. In living systems there is a constant give-and-take (stimulus/response) that, enables the possibility of informed choices.

Possibility of Choice: another Similarity between Living Algorithm & Life

Newtonian Physics does not incorporate the potential for stimulus and response. The traditional analog equations of Physics combined with initial conditions are the sole determiner of future events. Conversely the Living Algorithm, like Life, incorporates the possibility of choice with her digital equations. This give and take relation to data streams is yet one more similarity between Life and the Living Algorithm.

Link: Living Algorithm & Physics bound by a super highway.

Despite their many differences, the mathematics of the Living Algorithm and Physics are bound together by a common thread – which could more accurately described as a super highway. The two forms of mathematics, with their uniquely different fields of action (life’s informed choice & material determinism), are bound together by classic Newtonian concepts, such as force, work and power, mass, space, and even time. These concepts, while providing a common element between the two polar systems, have radically different manifestations. To see where these complementary systems merge and diverge, check out the next notebook – Data Stream Dynamics.

To begin to understand this intersection between two orthogonal planes of existence we must first get to know the Living Algorithm a little better. We have seen her in action, but we have yet to meet her. The initial article in the stream sets the stage with an exploration of the Living Algorithm’s algebra. Don’t worry; the discussion doesn’t require a mathematician. The Living Algorithm is a simple equation, only requiring a basic knowledge of arithmetic. To better understand the underlying patterns of this unique equation’s innate nature, check out the Living Algorithm, her Instantaneous Self.

 

Article List    Previous    Next    Comments