Tag Archives: mathematics

Information, Part 2: Escape from the Mathematical to the Strategic

[Note: This title presents this escape in methodological terms.It can also be presented as a shift from theory to practice or as a shift from universal laws acting on identical units to information-guided choice within unique local configurations (i.e., strategic scenarios).]

A new wrinkle!
  • With biology, information is transformed from being structurally implicit (a simple readout of the interplay of universal laws with initial local configurations) to being encoded and transmitted through local lineages that begin to develop characteristics partly determined by the local determinations of species-specific DNA, and not simply by universal physical and chemical laws.
  • Biology creates local conditions that uniquely determine local configurations, namely those determined by the DNA of distinguishable local species and the habitats with which they interact.
  • Now, information of merely local significance matters: habitable planets, environmental habitats, lineages — and, where lineages face bottlenecks, even individuals can matter.
  • Biology oozes out of the zone of determinism by purely universal laws acting on identical units of matter or energy into a gray zone where unique local configurations partly determine outcomes.
  • But that’s not all, folks!
  • The next stage in the development of information is the emergence of biological species that survive within their habitats by decoding information.
  • In animals, motor capabilities allow the organism to react to decoded information about its environs (sensory stimuli) in real-time and (for practical purposes) with an immediacy beyond the reactions of plants (even Venus fly-traps are just vegetative mousetraps, not sentient hunt-enabled predators).
  • Initially, this animate hunting and fleeing behavior is only the decoding of transient information regarding obstacles, dangers and resources, predators and prey. But, more advanced neurological systems, respond not only to sensory stimuli, but also to stimuli not currently present but remembered, imagined or foreseen. 
  • And with human-level cognitive powers, organisms construct mental generalizations about classes of objects, and arts and crafts for dealing with them. Later, such beings can develop the methods and powers to decode both implicit physical information and encoded biological information, amplifying their powers by first creating scientific knowledge and then engineering its technological applications.

The View from the Top Down

  • I will now depart from bottom-up analysis to give a top-down view of the matter, to give the reader some idea of where all this is headed
  • Two caveats:
  • 1) I’m not trying to pull a fast one here, but you’ll be the judge of that.
  • 2) In critical thinking, the exploratory journey matters more than the destination, which too often becomes ossified into dogmatism. 
  • In reality, all you have is the journey, the successes and mistakes made along the way, and what you’ve learned from them. So, take from this exploration what works by the light of your highest standards of critical thinking. Treat the rest as leftovers, to be reheated, doctored or dumped, as you see fit.
  • I will begin by defining two polar extremes of definition in the free will vs. determinism controversy. My own position is best categorized as compatibilism, the halfway house between the two extremes, influenced by such bright lights as Aristotle, Kant, and Daniel Dennett (the last of whom still breathes as of this writing; I cavort with a somewhat larger society than that of dead white men). In particular, see Dennett’s Elbow Room: the Varieties of Free Will Worth Wanting.
  • Free will (arguably better designated by the Aristotelian proairesis, in English, choice) must be understood in reference to determinism, more precisely, the absolute determination of effects by their causes in the classic sense of physical causation.  This conception evolved from the sense of efficient cause — one of four causes in Aristotle’s schema for analyzing changes, both mechanical and mental, physical and historical — to the mechanical causality envisioned by early modern empirical science. That conception relied heavily on the analogy of clockwork (a cutting-edge artifact and mechanism of the time, geared to produce precise and predictable results, namely, an accurate model of the passage of time).
  • In either conception, the common element was that causation in the sense of an immediate trigger of physical motion was devoid of any indirect causation, especially that of purpose or intention.
    • (As modern empirical science first emerged in largely Christian lands, indirect intentional and purposeful causation was allowed, but only to God or his agents, and only by suspending the laws of causation that would otherwise apply, in the performance of a miracle.)
  • The world we know at the levels of physics and chemistry, the two most basic levels, is effectively described by modern empirical science.
  • In part, this is because the characteristics of physics and chemistry seem reducible to measurable units of space and time.
  • In part, this is because the scientific method — the method whose practice generates modern empirical science — was explicitly and intentionally designed to register as cause and effect only what could be observed and measured.
  • Consistent and recurring measurements, when observed, can be expressed as mathematical expressions (e.g., ratios, formulas, constants) which, in the more complex case of formulas, correlate variable inputs to outputs determined by them. 
  • So, hypotheses expressible as formulas can be tested and verified or refuted by measurements which do not vary from one observer to another. This allows confirmation between observers, and thus reduplication of observed results becomes the standard of proof.
  • This is all possible because the scientific method assumes, in order to operate, that observations are themselves determined by universal laws that apply similarly to all similar cases; this outcome is guaranteed by the method’s requirement that all credible results must be reduplicable by independent observers in independent labs.
  • And the similarity — strictly speaking the identity — among observed cases is underpinned by the assumption that all complex things observed can be reduced to ultimate units that are identical (except for their location on the grid of space and time) and not further reducible.
  • Caveat: Kant argues (Critique of Pure Reason, “The Antimony of Pure Reason, Kemp ed., pp. 384-484) that science can proceed empirically while leaving unresolved the question of whether matter reduces to ultimate indivisibles or not (the 2nd antinomy), along with 3 other unresolved questions: whether space and time have limits (the 1st antinomy), whether conditioned causality is without limits or free will exists (the 3rd antinomy), and whether the world is without limits or whether God, its Creator, exists (the 4th antinomy). Having broached this perspective, I’m not entirely sure where it leaves the thrust of this whole essay, which argues that free will has a meaningful use in a conditioned world. Let’s just say that multiple perspectives can be brought to bear on the matter, and leave it at that!
  • Physics and chemistry are almost totally explicable within the methodological limitations of the scientific method, as above described, which amount to the parameters within which the scientific method is operationally competent and effective.
  • Put differently, physics and chemistry are the foremost disciplines for which the paradigm of empirical science — defined, powered and limited by the scientific method — is both necessary and sufficient. 
  • Put yet another way, those two disciplines need nothing beyond the scientific method to complete their observational and predictive mission, and thus to serve their practical mission — producing applied  science and technology — the primary reasons that the practice of science is permitted and often subsidized by governments (power) and societies (money).
    • However, never forget that regimes (and demographic subcultures) that feel threatened by science have little compunction about controlling, directing, adulterating, ignoring, and even outlawing and punishing ,the practice of science itself or of those parts felt as threatening.
  • Physics and chemistry are fully explicable by the scientific method — geared, as it is, toward expounding universal laws producing the same results wherever they can be observed and tested — because they study the causes and effects that work upon aggregates of simpler units that can, in principle, be analyzed and reduced to identical units subject to universal laws.  Physics studies and expounds the universal laws that control the observable behavior of particles/waves/probability-distributions and force-fields in the aggregate.  The laws predicting outcomes of controlled experiments are universal laws because they apply to every simple unit, subject to identical forces and conditions, over the entire field of units being observed. 
  • Chemistry is similar, except that it deals with larger compounds (atoms and the molecular compounds of atoms) even though these compounds all follow, in some sense predictably, from analysis of the simples (protons, electrons, neutrons, and the ever-growing zoo of lower-level sub-atomic particles) of which they are composed.
  • Chemistry is thus merely a higher level of organization of the objects observed in physics, adding nothing unprecedented, just organized in more highly structured configurations, and following the limits inherent in those configurations (e.g., the number of electrons that can tolerate one another within a given orbital shell). 
  • Whether the epiphenomenal characteristics of atoms qualify as emergent properties – not fully understandable except in terms unique to their level of organization — is a matter of fine semantic disputation that need not concern us here.

Locally Significant Contexts

If this one shoots first, then that one goes down, but if…
  • Biology, however, represents a true break in this bottom-up explanatory chain, not because it escapes causal determination but because it, by its very existence, creates locally significant contexts, captured in and carried forward by self-replicating structures like genes and DNA. 
  • For the first time, it matters where, and with what species, and under what conditions of habitability calibrated to those species, the observed events are occurring.  Local differences are decisive, and very few universal laws apply in determining the rise and fall of species and of the individuals that comprise them.
  • Now the process by which all biological development occurs — its ultimate causation — is the thoroughly deterministic process of natural selection.
  • Crucially, however, the replication part of that process is imperfect, producing the DNA-transcription errors known as mutations, without which evolution would not produce the permutations that allow it to explore every possibility within the matrix of possible combinations that define the open pathways for evolutionary experimentation or exploration.
  • Whether these transcription errors are themselves predictably deterministic or not, is another semantic complexity we need not here concern ourselves with.
  • All that matters here is that such errors, when they by chance produce traits that survive natural selection, create viable new species with survival-enhancing new characteristics, and that these new traits, much like advances in military equipment or doctrine, determine the strategic outcome of the ongoing competition (both strategic and evolutionary) between biological species for available resources, and thus, for survival.
  • The introduction of a strategic element changes everything irrevocably, for now the accurate accumulation and processing of data is the biological weapon of mass destruction, and thus the key to survival and dominance.
    • Thus, homo sapiens made short work of megafauna, which survive today only in the ocean depths.
  • This strategic element may be more crucial for predators than for prey and for social than for solitary animals. Social omnivores may be the optimal showcase for natural selection for strategic capacities.
  • For the first time in the history of the planet, information and analysis, foresight  and strategy, become the crucial factors in natural selection.  More precisely, the history of the competition between human cultures displaces the biological chronology of competition between species as the fastest driver of biological changes on the planet.
So, how does this play out?…

Information, Part 1: Implicit vs. Encoded Information

Travelling in space and over time

Information Only Implicit in Physics and Chemistry

  • Information is inherent in the structure of all physical things.
  • Universal laws of nature are constrained by the configurations in which simples of matter and energy combine (e.g., atoms and molecules, waves and force fields).
  • These configurations and their changes leave physically observable traces from which can be drawn precisely measured data descriptions of the things observed.
  • A representative set of such observations can, in turn, be captured in mathematical formulas, that is, hypothetical reconstructions of the universal laws that produced the observed configurations, given their initial configurations.
  • This information — both of observed physical configurations and of the hypothetical laws that govern them — though structurally inherent in the configuration of physical things, is implicit only, and is not, by itself, expressed as information nor even as its constituent data. All such data and hypotheses are implicit but unexpressed.
  • One way to see this more clearly is to remember that data registered by any observer is a function of how that observer’s sense organs interact with the implicit data, of which there is far more than any one sense organ could capture, even as doing so would be a pointless waste of the organism’s resources that natural selection would never countenance.
  • Put differently, usable information is always less than the total information available to all forms of sensing, whether sense organs of natural organisms or man-made devices.
  • Information is superabundant, and consequently useless, until it is sifted and organized for particular purposes. (For usefulness means suitability to a purpose.)
  • Physics and chemistry are entirely determined by such structurally inherent yet merely implicit information.
  • Information inherent in matter is an observable and measurable configuration in space.
  • Information inherent in energy (vectors of force, force fields, chains of causation) is an observable and measurable configuration in space over time.
  • Both the spatial configuration of matter and the spatiotemporal configuration of energy are best described by mathematics, for mathematics correlates quantities measured along the grids and contours of space and time in determinate formulas, which can be used to generate testable hypotheses.
  • Therefore, aspects of reality readily reducible to configurations in space and time (e.g., protein folding) are completely explicable through the natural sciences, with physics and chemistry the ultimate foundations of such analysis.
  • However, aspects of reality not readily reducible to space and time — or that operate under conditions of uncertainty — like the strategic aspects of competition among organisms or agents — must draw on other kinds of analysis, in addition to physical, chemical, or even biological, reverse engineering (that is, guessing, formulating as a hypothesis, and then testing what known and deterministic processes could account for the observed outcome).
  • Engineering deals in certainties — or, at least, in uncertainties with knowable probability distributions — while the strategic analysis of competition deals with uncertainties, with unknown probability distributions, and with unforeseen innovations and and unforeseeable improvisations.
  • Put differently, strategic analysis deals not only with the known knowns and known unknowns (i.e., the variables in an equation) dealt with by scientific engineering, but also with the unknown unknowns (i.e., innovations and improvisations).

New Wrinkle in the World: Biological Information Encoded as DNA

Codebooks at the ready!
  • Biology introduces a new wrinkle in the world of information.
  • For biology only exists where complex configurations too rare and intermittent to occur in significant quantities by chance — that is, through the random configurations produced by the operation of natural laws of physics and chemistry over time — yet further by chance produce configurations capable of self-replication.
  • These configurations, in their present form, are the RNA and DNA from which life as we know it is built.
  • Biology diverges from physics and chemistry because it encodes the information that configures living beings into types known as species (comparatively stable types, though continuously open to change, driven by a mix of changing habitats and serendipitous chance mutations).
  • Further complications are captured by the distinction between genotype and phenotype — the difference between the encoded information (the genotype) and the particular form in which it develops (the phenotype) as its encoded blueprint information (its genotype) interacts with environmental influences (its individual experience of its environment — since experience is an idiosyncratic sampling of an environment, limited by the particular construction of an organism’s sense organs and by its attentional focus).
  • Thus biology blazes a path out of the narrowly deterministic world of universal laws acting on identical units that are the subject matter of physics and chemistry.
  • This is not to say that biology is not deterministic, only that it is deterministic in a novel way — a way in which unique habitats, particular species and even particular individuals matter.
  • Individuals do not matter in the general course of evolution: it’s a numbers game. Nature is careless of individuals.
  • But in the special case when a species enters an evolutionary bottleneck, individual outcomes may determine the survival or extinction of the species.
  • [A minor corollary of some import:] Homo sapiens has the distinction of being the first species able to create its own potential evolutionary bottlenecks — points at which the fate of the species hinges on a few individual choices.
Tell those stupid dinosaurs we’ll choose our own asteroid, thank you!
  • This concludes Part 1 of a projected 3-part series on Information. The projected titles of Information, Parts 2 and 3, are:
    • The Great Escape: From the Mathematical Realm to the Strategic Realm
    • Choice: Neither merely Deterministic nor merely Situational

Context & Certainty

Physical reality, captured in the math
  • The context of mathematics is one of abstract definitions generated by iterating operations (numbers and arithmetic), which make measurement possible, and by dividing space into abstract figures and measurable units of extension (length, distance, volume, proportion) and by dividing time into measurable units of extension (duration, sequence, simultaneity, periodicity and proportion).
  • Music draws on the mathematical context but adds sound (timbre, intensity, pitch) which combined with sequence and periodicity produces melody, harmonic chords, and rhythm, while adding the very non-mathematical aspects of taste, culture, and emotional resonance.
  • Physics adds matter (that which occupies space) and energy (that which gives motion to matter) to its ultimate context of space and time, making its ultimate context mathematics. This is what gives it its certainty.
  • Chemistry is about how matter combines into structures. It is mathematical to the extent that matter and energy define or limit its possible configurations.
  • For example, electrons orbiting an atomic nucleus repel one another, so they can only tolerate one another in orbital shells accommodating fixed maximums of electrons. This yields different kinds of atoms with different properties — the elements, along with their kin, isotopes and ions.

Why is life carbon-based? Because carbon ( atomic number 6) has slots in its outermost shell for 4 electrons, sufficient to allow for the variety of bonds needed to build complex polymer chains (like DNA and proteins) critical to organic chemistry and life.
  • Elements have differing measurable qualities and differing capacities for combining with other elements, yielding differing combinatorial possibilities — molecules.
  • Biology is about complex molecules (organic chemistry) which have the additional characteristic of exhibiting functionality when certain rare combinatorial structures emerge from random aggregations resulting from lower-level chemical and physical properties without regard to the resultant accidental functionality.
  • Because such biological products are in the above sense accidental, their origin and continuance can only be assessed in the context of their entire localized (and, thus, peculiar and non-universal) history, that is, their local evolution within their local environment.
  • Thus, the unique lineages that define biological species are the first instance in which locality trumps universality — the accidental sequence yielding the configuration tells us more about the species than any universally applicable laws can.
  • Biological lineages are the first context in which individuals matter, and they matter initially only insofar as they carry mutations that weaken or strengthen the lineage.
If a mutation works, it survives, breeding variant mutations, which, in turn, survive if they work.
Every day is a twisting road…

The Individual Self in the Social Context

  • The next stage in the development of individuality must wait for the characteristically accidental biological evolution of consciousness sufficient to support behavior localized not simply by the internal (genetic) and external (habitat) physical environments but also by the internalized behavioral environments of societies and selves.
  • In other words, only once behavioral characteristics can be carried by something other than physical genes can a truly social environment arise, and only within a truly social environment can a self arise.
  • The term memes is used to refer to this non-gene carrier of behavioral attributes, passed by imitation and communication, social rather than strictly physical events.
  • Conscious memory makes learning from experience possible, and learned behavior can be passed on by bottom-up mimicking, by top-down training, and ultimately by linguistic instruction. But all of these require, at a minimum, the society of the family unit with parental care.
  • And parental care is a noteworthy feature of birds and mammals, the two most intelligent classes of animals, with the most complex brains and behavioral patterns.
  • Culture likely came first, for culture is typical behavior copied throughout a social group, transmitted by parents and often enforced by the social group or by similar responses of the group’s members to individual behavior.
  • Selves likely emerged when individuals experienced themselves as split between an outward face presented to the group and an inward reserve aware of its own distinct interests, even when they diverged from those of the group.
  • Psychology and the social sciences are about the relations of selves and societies.
  • Because of their localized historical (thus, accidental) evolution, both genetic and cultural lineages are not determined by universal laws but by localized genealogies.
  • So, genealogies cannot be understood universally but only historically. Historical sequences routinely have gaps, which detract from certainty, and, even when intact, require interpretive hypotheses about why the history developed as it did and how its components — both inert objects and agent subjects — reacted to one another as they did.
  • Selves and societies interact with other selves and other societies in many ways ranging from friendly to hostile, from affection and kinship to trade and transfer (economic exchange and cultural imitation) to force and fraud (whether criminal or geopolitical).

Certainty & Context

  • Mathematics attains certainty because its ultimate context is one of definitions imposed upon the uniform extensions of space and time.
  • Logic attains certainty because its ultimate context is that of the identity, mutual exclusion or partial overlap of abstract definitions. (A logical map of concepts would look like a massive and intricate fabric of Venn diagrams. )
  • Physics attains certainty because its ultimate context is mathematics. To the extent that any non-mathematical properties of matter or energy (including their convertibility, origin, or variants like “anti-” or “dark” matter) are not derived from the context of mathematics, they must remain hypotheses and guesswork.
  • Inorganic chemistry attains certainty only where it draws on the context of mathematics. For example, filled outer electron shells of noble gases make bonding and forming compounds difficult. The math might not be enough to predict all the chemical properties of a new compound, but if those properties are consistent, statistical analysis will reveal such consistent properties after repeated observations, assuming that underlying conditions remain constant.
  • Organic chemistry will always remain incompletely certain, as so much depends on functionality within accidentally organized systems of functions. For example, food for one species may be toxic or indigestible for another. Yet how proteins fold can apparently be partially predicted by deep AI.
  • Biological species will always be uncertain and non-universal because: 1) species and their environments are continuously evolving, 2) the form a lineage takes is an accidental historical emergence of functionalities adequate for survival in its changing local habitat, and 3) individuals themselves are accidental products of random mutation, while their survivability is determined by their altered functionality within their local habitat.
  • Cultures and human nature and character types (the social sciences equivalent of species and socio-cultural categories, e.g., extroverted Russian ballerina) will be yet more uncertain and non-universal than species because: 1) cultures change faster than physical habitats, 2) the more complex the culture, the greater the division of labor and of distinctions, 3) information, memes, and cultural attitudes change and spread faster than genes, and 4) reasoned foresight and rationalized social organizations and technology accelerate purposeful change with top-down guidance (for human agents deliberately target and accelerate intended transformations more purposefully than random physical events (whose outcomes are aimless, accidental to any end), while agents’ conflicting purposes can trigger unintended consequences, whether beneficial (Adam Smith’s invisible hand) or devastating (war), whether immediately (peace treaties, nuclear holocaust) or cumulatively so (scientific progress or climate change).
  • The sciences of life (including the social sciences) may be uncertain, but for all that, they are the sciences of us. And since value is in the eye of the beholder, what knowledge we can glean from them is of most value to us.
  • And be warned that technical knowledge and know-how devoid of self-knowledge is as dangerous as a loose cannon on deck.
  • Life, in short, is where the action is, in all its unpredictable splendor and potential desolation. Ladies and gentlemen, place your bets!