Humility in the Face of the Stochasm

In my introduction to a news feed early last month I confounded some of our readers by using the word “stochastic”. In its context it meant “indeterminate”. The most commonly used synonym for it would be “random”, but that’s not exactly what it means. More on that later.

I first encountered the word in about 1970, when I was looking at the course catalog for the Math Department of the College of William and Mary. Back then Computer Science was a subdivision of the Math Department (a few years later it was elevated to the status of a separate department), so while I was browsing through the advanced math courses I came across an entry for “Discrete Stochastic Systems”, probably a 400-level course. I understood “discrete”, but I had to look up “stochastic”. In its computer context it referred to the fact that some events within a complex electronic computer are indeterminate, since they are affected by quantum fluctuations at the lowest level. One of the major tasks in the development of modern digital computers was to control for stochastic events as much as possible — via parity checks and so on.

The “discrete” in the course title referred, of course, to the fact that digital computers were under discussion. Analog computers had been fairly widespread in the early days (late ’40s and ’50s), but for reasons that are too complicated for this discussion, digital computation eventually won out and became the preferred method used in modern computers.

I duly incorporated the useful word “stochastic” in my vocabulary. A few years later I even wrote a poem entitled “The Intelligence of Discrete Stochastic Systems” (it wasn’t any good). In the decades that followed I forgot how specialized the word was; that’s how I ended up confusing the readers of the news feed.

Interestingly enough, in both of my major dictionaries (which are very old), the specialized meaning for the word is not listed. The definition given is “pertaining to conjecture”. The 1921 Webster’s lists it as “rare”, while the Oxford (more precisely: The Shorter Oxford English Dictionary on Historical Principles) says it is “obsolete or rare”.

But not anymore. Now it’s simply a geek word.

It comes from the Ancient Greek word στοχαστικός (stokhastikós), which is formed from στοχάζομαι (stokházomai, “to take aim, as at a target, or to guess”), from στόχος (stókhos, “an aim, a guess”). I don’t know how it eventually acquired its current mathematical meaning; perhaps indeterminate events were things that geeks had to guess at. The Wikipedia entry has a lot more information on stochastic processes, for those who are interested.

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The most common synonym for “stochastic” (as you can see from the Wiki) is “random”, but that’s not really accurate, and I wouldn’t use it myself. I prefer “indeterminate”, or more fully: “a stochastic event is one whose cause cannot be determined.”

In my computer science courses it was drummed into us that there is no such thing as a random number in digital computing. What are commonly called “random number generators” are actually algorithms that output pseudo-random numbers. Every number produced by them is generated by a process that is totally determined, and not at all random. A programmer finds them useful to employ when it is necessary to simulate a series of random numbers.

In mathematical terms, randomness doesn’t exist. It has no meaning. Try to define it yourself — you’ll end up in a circular chain of references, without an external phenomenon to peg the word to. It’s not like “light”, or “vacuum”, or “hydrogen”. It has no meaning in a scientific or mathematical sense. It’s just a quaint linguistic relic, and is no more real than phlogiston or the aether.

What is commonly called a random event is simply one for which a cause has not been determined. Believing in “randomness” is therefore a form of religious faith, like believing in God, or Satan, or the Flying Spaghetti Monster. We are continuously bathed in events that seem random, but accepting their randomness is a leap of faith on our part.

The discipline of Statistics was devised to help understand the patterns and mathematical structures behind large quantities of “random” data. Events that may be indeterminate on an individual basis are totally predictable when a large enough quantity of data is aggregated. But data on totally determined events can also be analyzed — the behavior of thousands of vehicles on a freeway, for example, can be statistically evaluated, even though the causes of individual drivers’ choices are unknown to the statisticians. That doesn’t mean that those decisions are random, but rather that they are indeterminate from the statistician’s point of view.

Darwinian evolution is a statistical model that relies on indeterminate events at the micro level to explain the process of natural selection. However, to call the underlying genetic changes “random” is to take a leap of faith that is no different, scientifically speaking, from believing in Intelligent Design or the Garden of Eden. All three are forms of religious faith. A scientific observer doesn’t observe randomness, but only events whose causes have not been determined.

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Two and a half centuries ago the paradigm of the Clockwork Universe had its day. Newtonian physics provided the tantalizing possibility that the mechanism of the universe could eventually be understood in its entirety. Given enough data, and enough careful calculation, there would be no indeterminacy left. The causes of all phenomena would be known and understood.

It was a heady time. Newtonian ideas gave rise to Deism, which posited a Creator who created the universe the way a clockmaker builds a clock. Once the clock was completed and wound, all observed processes would follow from it without any further intervention by the Creator.

Later the Creator was abandoned, and the clock just kind of came into being, with the Big Bang serving as the ultimate “explanation” for how everything eventually happened.

But the 20th century threw a spanner into all that elegant clockwork — first Einstein at the macro level, and then Heisenberg at the micro level. With Kurt Gödel’s Incompleteness Theorem, the Newtonian clockwork was finally melted like one of Salvador Dalí’s watches. Not only were some causes indeterminate, but they couldn’t be determined. There are limits to what man can know or deduce. There will always remain “unknown unknowns”.

Once the digital computer became complex and powerful enough, the science of indeterminacy could advance even further, producing what is now known as Chaos Theory. Fractal structures allow us to map the infinitely complex interface where the known leaves off and the unknown begins. The Mandelbrot set cannot be fully determined — the calculations would of necessity be infinite — but the process of approximating it produces infinitely complex structures of exquisite beauty.

In the physical universe, it appears that the basic components of reality are the result of stochastic processes. Some of the most important constants used in mathematics and physics — for example, e (the base of the natural or Naperian logarithms) and pi (the ratio of the circumference of a circle to its diameter) — were established by quantum fluctuations during the first nano-second after the Big Bang. The geometry of the very fabric of our existence was thus created through indeterminacy. And it is not known whether a universe could even exist with other, different basic constants that might be formed by slight variations in the earliest quantum flux.

All of this leads us to conclude, per Hamlet, that “There are more things in heaven and earth, Horatio, than are dreamt of in your philosophy.” Whether terrifying or exquisite, there is so much that we don’t know and, more importantly, cannot know.

As Walt Whitman wrote in “Leaves of Grass”:

Hurrah for positive science! Long live exact demonstration!
Gentlemen, to you the first honors always!
Your facts are useful, and yet they are not my dwelling,
I but enter by them to an area of my dwelling.

One would think that a basic acquaintance with the Stochasm — the ubiquitous indeterminacy in which we all reside — would induce a sense of humility among those who contemplate it. Yet humility seems to be in short supply here in Post-Modernity; hubris is the order of the day.

We’re all aware how that eventually ends. As Robert Hunter put it in his song “Book of Daniel” (which is a gloss on the biblical original):

And when the king, your father, achieved humility
He was restored his kingdom and his sanity.
And though you knew all this to start, you humbled not your heart.
The writing on the wall commands your fall!

37 thoughts on “Humility in the Face of the Stochasm

  1. That second image, a detail from a Mandelbrot set, looks like neuronal patterns, dendrites…

    The further down one delves, the more the sense of free-falling without a parachute.

    This is why I think “climate science” is dismally dumb.

  2. Brilliant précis, Baron. I have to still think of the religious finis of your gem, though.

    Greetings from a dude who studied CS in the 70s and 80s as well, just on the other side of the Curtain.

    • Nope. The causes of their being there are not at all indeterminate.

      You can believe in “random”, just like you can believe in the transmigration of souls. But that doesn’t mean that it’s real.

      • Thanks. I didn’t say I believed in “random” though I do for reasons having only a basis in the non-mathematical view of reality that I have.

        I’m hard pressed to think that ccs. of coolant that leak from my ancient automobile every week, the number of times my dog will bark every morning, the number of times the name “Gertrude” will be used somewhere on the internet each day, the windspeed every day at 1000 in my front yard, or the number of ounces of birdseed my feathered friends will eat every day are predictable or can be calculated.

        This fellow believes in randomness.

        This source agrees supposedly random numbers generated by an algorithm are not secure. I assume random number generators have precisely that flaw. However, the author indicates that the cypher can’t be broken if it’s “generated by a non-deterministic, non-repeatable process.” I don’t know what such a process is or if the author thinks it can be used to generate a key. The gent at the first link above thinks he can describe such a process.

        Other restrictions on the proper use of OTPs involve never using the particular pad more than once and, of course, the enemy’s never obtaining access to the OTP used. Practial limits on the use of OTPs along with the use of algorithms.

        The number of times Trump says he will withdraw troops from Syria seems like it would be a number that meets all tests for randomness, but maybe that’s just me.

        • I think what we may have run aground on here is the definition of the word “random”.

          To me, a random event is one that has no cause. My assertion is that science is unable to identify any single event that is random — an event either has an identifiable cause, or its cause (if any) is not known. In some cases (e.g. quantum events), causes cannot be known.

          The roll of a die is not random — its final resting position is entirely a result of its material structure, the force of gravity, and the musculature of the hand that throws it. Yet the number it shows cannot be predicted. In other words, it is stochastic.

  3. May I suggest that what is perceived as being stochastic is the result of not being able to see the whole picture, but only our imperfect part of it. In keeping with tomorrow being the Feast of First Fruits, and with Jesus Christ being the first fruits of those who are risen from the dead, may I offer the following hymn:

    Christ Jesus lay in death’s strong hands for our offenses given;
    But now at God’s right hand He stands, and brings us life from heaven;
    Wherefore let us joyful be, and sing to God right thankfully,
    Loud songs of Alleluia! Alleluia!

    It was a strange and dreadful strife when Life and Death contended,
    The victory remained with Life; the reign of Death was ended;
    Stripped of power no more he reigns, an empty form alone remains;
    his sting is lost forever! Alleluia!

    So let us keep the festival whereto the Lord invites us;
    Christ is Himself the joy of all, The Son that warms and lights us;
    By His grace He doth impart eternal sunshine to the heart;
    The night of sin is ended! Alleluia!

    Then let us feast this Easter Day on the true Bread of heaven;
    The Word of grace hath purged away the old and wicked leaven;
    Christ alone our souls will feed; He is our meat and drink indeed;
    Faith lives upon no other! Alleluia!

    Martin Luther

    • As much as I emotionally like the hymn, and as much as I defend Christianity as a movement of enormous civilizational significance, I fear that the said stochasticity interpreted in religious terms is not going to “do it” …

      Einstein’s famous statement “Raffiniert ist der Herr Gott, aber boshaft ist Er nicht” can be discussed here as well …

      Anyhow: we are digressing from a treatment dedicated to randomness, hidden truths and imperfections to invoking Good, Death and Evil – categories a tad different.

  4. “Not only were some causes indeterminate, but they couldn’t be determined. ”

    “There are limits to what man can know or deduce.”

    The first sentence was attempted to be disputed by David Boehm – basically, he says that there are “underlying causes” behind the quantum phenomena, and we just do not know those – his “hidden variables”. And of course John Bell with his non-locality – perhaps the most shocking breakthrough in the post-Einstein physics.

    The second statement nicely summarizes Goedel’s finding – original results related to deductions in formal systems (arithmetic in his case), all can be generalized as “there are true statements whose veracity cannot be determined within the framework of the system where they are expressed/defined” (my simplification).

    Kurt Goedel was born in Brno, Moravia – the city where I studied CS … well, looong time ago. The spirit of Goedel was somehow still palpable in Brno of my college years, as well as the ghost of Kafka can be still sensed in Praha.

    But my point here is – thanks, Baron – very nice essay, it has brought my mind 35 years back, to Brno.

    • …basically, he says that there are “underlying causes” behind the quantum phenomena, and we just do not know those…

      That is exactly my point. If the cause of event is not known, it is indeterminate, but not random. Asserting the existence of “randomness” is an act of religious faith. It is not based in empiricism or science.

  5. I too have a problem with the word random and what it is supposed to mean. I too agree that a better word is indeterminate.

    What I believe these words represent is a measure of man’s knowledge. Random or randomness exists at one end of a continuum, with certainty at the other end. Other words that associate with randomness are disorder, irregularity, and ignorance. Words associated with certainty are order, regularity (or pattern), and knowledge. So I would say that randomness is the opposite of certainty, and that both are measures of man’s knowledge.

    If we accept the Principle of Sufficient Reason to be true, then we accept that nothing happens without a cause. Everything has a cause and effect. Consequently, even if we infer an event to be random or indeterminate, we should still hold that event to be determined (caused), though indeterminate at present time.

    Randomness (and certainty) is a property we impose on (or infer from) our observations. With humility we could refocus of our observations away from the thing and toward ourselves. In that way, indeterminate is a property not of the observations themselves, but of our shortcoming in knowledge.

    What if we conclude something to be indeterminate? Could we at some point also say it to be indeterminable?

    • Gödel demonstrated that some things must remain indeterminable. And Heisenberg showed that there are things we cannot possibly know in their entirety.

      • All this takes me back to discussions with one naively deterministic question: is there any limit in our quest to understand what the world around truly is?

        Sadly , the answer is: yes, there seem to be such sentinels – Heisenberg uncertainty principle being one, Goedel’s finding another. But: we had these discussions in the time where the Great Fermat Theorem was basically deemed undecidable: now we know it is true.

        Our journey from the dark caves to quantum computers is always causing me vertigo.

    • Elena, excellent reference – Lem hugely influenced my teen years with his incredible intellect yet I fear he remains virtually unknown to the West … perhaps except his Solaris, clooneyzed in the recent Hollywood movie. Tarkowski is much deeper in my view …

      Anyway – Lem and his monumental intellectual legacy should be part of our discussion here, for sure !

  6. There is also the non-deterministic nature of Quantum Mechanics to take into consideration, but considered opinion is that ‘deterministic’ is not t quite the opposite of ‘stochastic’. Deep metaphysical waters, these.

  7. “Some of the most important constants used in mathematics and physics — for example, e (the base of the natural or Naperian logarithms) and pi (the ratio of the circumference of a circle to its diameter) — were established by quantum fluctuations during the first nano-second after the Big Bang.”

    Uh, not so for the constant pi, which is derived from the definition of a circle and does not depend on what happened at the Big Bang. A Big Bang-dependent physical constant is, for example, the
    (approximately 1/137).

    • That’s not true. The value of pi depends upon the curvature of space.

      The analogy is to a circle drawn on the surface of a sphere. The ratio of the circumference of that circle to its diameter (also drawn on the surface of the sphere) is not 3.14159265[…].

      I can’t remember if it was Riemann or someone else who established the curvature of the three-space we live in, but it is there, and it affects the value of pi.

  8. Stochastic is a word I associate with my training in (fire) tactics and strategy. I have seen the word only used in that context until today with this essay and thread. The concept is that you commit to a plan of action, analyse the result of your efforts and adjust and readjust to the constantly changing situation as you advance or retreat toward your objective. It is similar to the OODA loop.

    From wikipedia. “The OODA loop is the cycle observe–orient–decide–act, developed by military strategist and United States Air Force Colonel John Boyd. Boyd applied the concept to the combat operations process, often at the operational level during military campaigns. It is now also often applied to understand commercial operations and learning processes. The approach explains how agility can overcome raw power in dealing with human opponents. It is especially applicable to cyber security and cyberwarfare”

    • I read the Boyd biography some years ago. His OODA Loop should have been widely adopted/adapted by our military, but it wasn’t.

      From the Full wiki (without the embedded links):

      Foundation of theories

      Boyd never wrote a book on military strategy. The central works encompassing his theories on warfare consist of a several hundred slide presentation entitled Discourse on Winning & Losing and a short essay entitled Destruction & Creation (1976).

      In Destruction & Creation, Boyd attempts to provide a philosophical foundation for his theories on warfare. In it he integrates Gödel’s Incompleteness Theorem, Heisenberg’s Uncertainty Principle, and the Second Law of Thermodynamics to provide a context and rationale for the development of the OODA Loop.

      Boyd inferred the following from each of these theories:

      Gödel’s Incompleteness Theorem: any logical model of reality is incomplete (and possibly inconsistent) and must be continuously refined/adapted in the face of new observations.
      Heisenberg’s Uncertainty Principle: there is a limit on our ability to observe reality with precision.
      Second Law of Thermodynamics: The entropy of any closed system always tends to increase, and thus the nature of any given system is continuously changing even as efforts are directed toward maintaining it in its original form.
      From this set of considerations, Boyd concluded that to maintain an accurate or effective grasp of reality one must undergo a continuous cycle of interaction with the environment geared to assessing its constant changes. Boyd, though he was hardly the first to do so, then expanded Darwin’s theory of evolution, suggesting that natural selection applies not only in biological but also in social contexts (such as the survival of nations during war or businesses in free market competition). Integrating these two concepts, he stated that the decision cycle was the central mechanism of adaptation (in a social context) and that increasing one’s own rate and accuracy of assessment vis-a-vis one’s counterpart’s rate and accuracy of assessment provides a substantial advantage in war or other forms of competition.

      The whole of it is interesting, especially his influence on the Marine Corps.

      • “Gödel’s Incompleteness Theorem: any logical model of reality is incomplete (and possibly inconsistent) and must be continuously refined/adapted in the face of new observations.”

        “Heisenberg’s Uncertainty Principle: there is a limit on our ability to observe reality with precision.”

        I fear these wordings are (gross) simplifications and especially in the GIT’s case the real substance of Goedel’s finding is obscured.

        I checked Wikipedia and that’s the exact wording used there – alas.

        • I fear these wordings are (gross) simplifications and especially in the GIT’s case the real substance of Goedel’s finding is obscured.

          What is this ‘real substance’ of Goedel’s finding? I’m not being snarky, just curious about your take on Goedel.

  9. As a believer in ‘Intelligent Design’ behind the theory of Evolution,
    I wonder just how ‘random’ are the apparently random genetic mutations which may lead to evolutionary changes.

    Perhaps there are quantum mechanics or other mechanisms behind it that can be explained by some mathematical equations?

    I find it interesting that the human brain always tries to find patterns, sequences, connections or something familiar when we are shown something chaotic or random such as a random collection of dots on a page.

  10. We are almost able to find randomness, but not quite; you are correct, Baron.

    “ Although randomness seems to be present in the world (eg the perceived randomness in the weather, Geiger counters, Zener diodes, real coin flips, etc) it does not seem to be in the perfect form of the unbiased and independent coin tosses we have postulated.)” Princeton Companion to Mathematics, p599.

  11. “A scientific observer doesn’t observe randomness, but only events whose causes have not been determined.”

    This seems to entail that there can be no human freedom and that everything that happens was predetermined from the very beginning (though the causes to us are often unknowable and so we speak, the Baron says, of indeterminacy). From that perspective, if I understand the Baron, there is no first cause that is itself uncaused. But we have direct experience otherwise.

    Most of us most of the time have mostly unfree thoughts and feelings based in largely mindless habit, and we tend to engage in mostly unfree, habitual behavior. But human freedom, to whatever extent an individual has achieved it, appears to entail that a person has to some extent become his own first cause, which is to say some aspect of his being “causes” itself, in other words is uncaused. That is not the same as what is usually meant by “random,” but the consideration may nevertheless be relevant to whether randomness actually exists.

    The idea of an uncaused first cause, or self-causation, whether associated with “God” or with human action, boggles some minds, but I would argue that it can become direct experience and does not require a leap of faith.

    Probably everyone has heard the witticism about the wise man who is asked what the world rests upon and who answers “a turtle,” only to be asked “what does the turtle rest on?” The wise man then answers, “another turtle.” The questioner is about to ask what that second turtle rests on, but before the question is fully out, the wise man cuts the questioner short and says “it’s turtles all the way down.” The humor of his answer is in how it glibly tries to camouflage its own nullity. To my mind, an infinite regress of causes is more boggling and without sense than a first cause, especially since first causes, I think, can be directly experienced. The world is not “turtles all the way down.” Admittedly though, if one believes in advance that all experience is physical experience, then one precludes agreement with much of what I’ve been saying.

    To participate to some extent in self-causation (necessarily a primarily non-physical experience), or, what is the same thing, to participate in uncaused first causation, might bring in a real factor of randomness or arbitrariness, insofar as different free agents’ trajectories are not coordinated to begin with by any overarching order, though perhaps tend to become coordinated after the fact by overarching order. This way of looking at the world perhaps entails the view of Charles Hartshorne and Alfred North Whitehead, according to whom God exists but is neither omnipotent nor omniscient. Hartshorne argued that God was superlative in various respects (to do with feeling for creatures and perception of them), but that omniscience and omnipotence are not divine attributes. Hartshorne used formal logic to rehabilitate Anselm’s proof and claimed 4 or 16 proofs of God’s existence result, depending on how one counts. But leaving Hartshorne and his logical “proofs” aside, it seems to me that human experience of, and partial participation in, a realm of ongoing “first causation” — where the coming-into-being of being can be glimpsed right down to the bottom of things, where, in other words, to some extent nothing is hidden and all is glimpsed — does happen.

    • You misunderstand me. I don’t posit that all events have causes — to assert that would require full knowledge of the continuum, which none of us has. I’m simply saying that our knowledge and understanding are limited, and must remain limited, which the mathematicians and physicists of the 20th century have proven. Perhaps I should have been more explicit in what I wrote.

      I once had conversations with a very knowledgeable and thoughtful Anglican theologian about quantum physics and related matters, and he told me that in his opinion the quantum level was where our God-given free will was implemented. It made sense to me.

      • “I’m simply saying that our knowledge and understanding are limited, and must remain limited, which the mathematicians and physicists of the 20th century have proven.”

        I understand you better now.

        I’d like to add that we are not capable of omniscience and never will be omniscient. I follow Hartshorne in doubting that even God is omniscient. At the same time, my sense is that human beings can experience the evolving world foundation. We can sometimes dive into the living, becoming, ever new, eternal God and the nested hierarchy of spiritual beings in the spiritual world. So maybe some sort of self-contradictory expression is necessary to categorize such experience, say, “limited omniscience,” or “inklings of omniscience,” or “finite human awareness of the divine spiritual world.”

        • I follow Hartshorne in doubting that even God is omniscient.

          We need another word besides “omniscience”. Don’t know what that would be, but the Process theology (and Philosophy) folk would posit that if you say “God is love”, then you have to allow for the growth and maturing that any kind of love brings. IOW, God changes as he interacts with His creation.

          I’ve always liked this idea as it fits better with our current cosmology; at its heart is paradox. God is laughing…cf. the Persian poet, Hafez.

          • Yes. It is primarily in rare moments or periods when love (I don’t mean romantic love) engages the whole person — which is rare and perhaps more likely to happen in adolescence than in adulthood — that the person goes through metamorphosis, which is another word for a death and rebirth process, which when genuine makes the participant to some extent conscious of an immortal reality. The person’s whole being becomes for a time a coming-into-being, becomes an eye that is itself coming into being, making visible and to some extent transparent to the participant the actual moments of creation through which he himself is passing, moments whose mysteries are then to some extent laid open and revealed, because it is his own being that is going through the process. “Creation out of nothing” seems to be a redundant expression. All “creation,” if it really is creation, brings something truly new, something without precedent, something that in one sense comes “out of nothing”. Thus we can sometimes participate in a world of first causes, new realities, behind which no prior causes are hidden. Reality thus appears naked. If this occurs through the loving engagement of a person’s whole self (an engagement which can best be maximized the more one can approximate to the honesty and transparency of an innocent child or youth), the person can glimpse the very foundation of reality, can experience that foundation not just via concepts but with his total being. Or so it appears to me.

  12. “What is commonly called a random event is simply one for which a cause has not been determined. Believing in “randomness” is therefore a form of religious faith, like believing in God, or Satan, or the Flying Spaghetti Monster. We are continuously bathed in events that seem random, but accepting their randomness is a leap of faith on our part.”

    Hmmm …. not sure I follow da Baron here. While computers may not be able to do more than simulate randomness, that doesn’t mean there isn’t such a thing. I’m not going to try to explain it here, but quantum mechanics (QM) does describe truly random events- for example, which atom of U235 will decay next.

    This causes me to recall a conversation between Albert Einstein and Neils Bohr. Einstein, like the Baron (and who also? did not believe in a personal God), never quite reconciled with QM, and spent much of his life looking for “hidden determinants”. He famously said to Bohr, “God does not play dice”, to which the mere moral, Bohr, replied, “And how do you know what God does?”. Not sure if Einstein had a reply to that one.

    Anyway, I am pleased to know that the Baron and I share something of an intellectual background.

    • I don’t say that there is no such thing as randomness. I’m saying that to believe there is constitutes an act of religious faith, just like believing in God. An atheist may be a devout believer in randomness.

      Quantum mechanics, if you look at it closely, does not posit randomness, but simply indeterminacy.

  13. Also it is a great relief to discover, when we get old enough, that we don’t have to try to match a feature of reality with a word. We can just try to describe what we are seeing in our own words and/or using our own math.

    The inventors of the word could have been as imprecise as any of its users.

  14. This comment is from France.
    Thank you dear Baron for this magistral piece of writing. Now my brain feels like a a Salvador Dali watch !…
    I agree with you. Talking of ” randomness” for some events that cannot be explained by our logic or by a mathematical theorem is a little arrogant.
    Because we think, at our present position on the ” arrow of time” of human evolution, that something is “random” we quickly put it in a niche with that calling.
    A little more humble attitude would be to say : This looks random, but maybe our human brain is not ready to understand it yet… Maybe it is ” random”, but maybe not. I agree with Baron that “undetermined ” is a better word.
    From then on, we can either think that us human we will never comprehend these things. We will be stuck permanently with chaos theories, fractal structures and Mendelbrot sets. ( Which is already quite good for an evolved primate to have gone that far.)
    Or we can believe, observing what we know of the always moving past, that our human evolution is still going on, and that our understanding of things will continueto progress in the future.
    Things that are now ” undetermined” will be explained… Or will not..

    Thank you so much for this mind opening essay. I will now include ” stochastic” in my vocabulary !..

    • I guess we can sort of get to “random” if we say the Probability of the event=.5000000…., but we have to have an infinite number of zeros.

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