Rule #5: Get Back to the Rules


Reality is elusive. Truth hides beyond our reach. Break-these-rules makes it clear that we simply are looking for GSOT, not truth. Let’s not make the mistake of those who tried too hard to grasp reality.

Truth is supposed to be the same from the beginning to the end of time. GSOT is dynamic, in the moment, demanding courage as well as commitment, urging us on impulse to break the rules that guided well thus far. As we reach for GSOT, then, will the last word be break-these-rules?

When Rule #4 bids us to break these rules, which rules are meant, 1 through 3, or 1 through 4? Does break-these-rules refer to itself as well? The only possible answer is yes.

Break-these-rules, tripping on itself, cannot be the last rule. Break-these-rules led us to infinity, but infinity was not the last word. Now it’s time to breathe consciously and slow, to look around this small room in which so many hours have passed. The last rule is an humble one.

Rule #5:  Get back to the rules.

Explorers cannot survive without supplies. Our supplies are the words, relations, arguments built into the rules – we cannot leave them behind. Hunger and chaos fall upon those who sail too long in uncharted seas.

Get back to the rules. But what were they? Where have we been? Let’s review.

We began by looking at two curiously similar forms of GSOT, positivism and religious fundamentalism, both of which adhere to a principle that the best starting point is that which maximizes confidence. For positivism, confidence is gained from agreement alone. Any proposition that does not compel agreement is judged to be inconsequential. For religious fundamentalism, confidence in shared belief is a principal means by which a person seeks to glorify God.

Both positivism and religious fundamentalism are expressions of modernism, which features confidence as a key operational principle. Their starting points are simple, clear, and decisive…and they are completely different. To justify the founding principle of his creed, either the positivist or the fundamentalist will argue the worthlessness – or the nothingness – of alternatives. That is to say, the founding principle is chosen because it appears to be better than nothing.

Now we live in a time beyond modernism. The byline of our postmodern era is suspicion rather than confidence. Let our GSOT be that which survives under withering suspicion. Ours shall be a grudging, gritty kind of confidence. Our concern is not with soaring brilliance. We protest when those who would rule from such heights demean the value of everyday life. We begin the trek toward GSOT in the midst of busy days and nights. We shall start from where we are.

Our take on reality will be anchored in time, place, and person. As for the time of GSOT, it may be momentary or prolonged. As for the place, as small as a point of touch or as large as the physical universe. As for person, either singular or plural, either living or dead or yet to live.

Early in this series we contrasted the immediate, the personal, and the particular against that which is enduring, public (anonymous), reproducible (universal). But the purpose of that contrast was never to choose one pole to the neglect of the other. Instead the purpose is to emphasize the full range of phenomena versus sole attention to a single pole of enduring, public, and reproducible – whether that single pole happens to positivistic science or religious fundamentalism.

Positivism attempts to elevate science by casting a nihilistic judgment on everything outside of science, but this maneuver ends up leaving science barren, out of contact with human hopes and needs. Because we attend to the full range, however, science is included and embraced within our goals. Religious fundamentalism is not the opposite of science, but has its own definition of enduring, public, and reproducible truth – God’s truth accessible through a bestowed text and a single interpretation. Enforcement of one interpretation, though meant to glorify God, may leave no room to value the immediate, personal, and particular, where perhaps God may touch us more recognizably than through any sacred image of unblinking, stonehard truth.

To declare that “Every sentence is first person” is to assert that every judgment, question, and exclamation has an immediate, personal, and particular aspect. This aspect is implicit in the pronouns “I” and limited “we.” To deny the validity of the overarching viewpoint is to refuse the presumption of Godlike wisdom by one human mind or many. The “universal we” of science is really an “almost-universal we,” which fails to provide an overarching viewpoint. Because there is no overarching viewpoint, no interpretation of GSOT can command or require agreement from everyone.

Pragmatism reminds us that the search for truth operates through human decision-making. When the search for truth arcs back on itself to ask what is the truth about human decision-making (most crucially is it free or not?), pragmatism gives no firm answers. Because there are no firm answers, philosophy cannot exclude “inner factors” which might be called operations of will. These factors, the existence of which cannot be studied reproducibly, nevertheless have tangible effects. Operations of will cannot be excluded for lack of tangible effects.

The denial of free will generally claims to show a lack of evidence that would compel affirmation of free will. This strategy only signifies that free will cannot be affirmed in a positivist framework. When we ask the question of free will without preconceptions, our asking reveals a choice, which is the choice to ask the question.

It is impossible to tie choosing down within a philosophically chosen axiomatic system. When this is attempted, free will may often be proven false by the axioms. Free will flashed, then disappeared at the moment of choosing the axioms. The only way to dismiss free will is to forget that moment of choosing. The problem is mainly a confusion of past, present, and future. That which has been chosen is no longer free. Free will can be understood only in the expression of first person voice and only in the moving moment. Reproducibility of events and interchangeability of observers do not apply to free will, and therefore free will is not accessible to scientific testing. According to pragmatism the function of will, understood as axiomatic itself, is the filter through which all apparent truth comes to comprehension.

Free will, expressed in first person voice in the moving moment, is capable of self-reference. Positivism and fundamentalism are not capable of self-reference. Let’s ask Epimenides to come help us remove the plague of modernist philosophy.

The ancient Pythagoreans, who gave us the word “mathematics,” believed that secrets of the universe could be learned by the study of numerical relations. Their dream re-emerged in the 19th century as George Boole and other mathematicians developed systems of thought that founded statistics, quantified every field of science, and foreshadowed the binary logic of computers. As the 20th century began, Principia Mathematica or simply PM, the epic work of Whitehead and Russell, enticed students of logic with the promise of a starting point for the true understanding of reality. But in 1931 Kurt Gödel showed that no advanced system of logic based on the principles of PM could be both complete and consistent. The Pythagorean dream blew away like smoke in a breeze.

Emphasis on the first-person origin of every sentence calls attention to the limited reach of every conversation. Recognizing the limits, daily we pursue our interests under a set of intersecting and variously sized domes (introduced in Rule #4). For the individual, the cranium. For small groups, the rooms, houses, and buildings in which we interact. An arc of dirty air over a city by day, a glow of light by night. The sky above a nation, where guardians plan for desperate battles that should never be fought.

Now let’s realize that breaking the rules means moving, joining, or expanding some combination of domes. At best, it means discovering you. You are no small achievement; you are the joiner of worlds, the telos of creation. You signify the meeting of peers, until inescapably some form of rank and authority takes shape, and every joining of domes tends to dissolve into a single larger one, as you and I begin to say us.

Break-these-rules severs the causal chain by which the past maintains totalitarian control over present and future. Break-these-rules detaches the ropes that moor the present to history’s solid ground, sets a person swirling in the tide, swept away by a fluid present somewhere between the advancing past oranges-mango-squareand the receding future. If there is any anchor at all, it’s a sea anchor dragging behind in the water that keeps no fixed location, but only a direction relative to the prevailing wind.

A person cannot stay too long at sea. We need produce from the ground, Lind’s oranges, before gums bleed and arms weaken. Get back to the rules, then.

Break these rules. Get back to the rules. Break these rules again. Get back again. Here is a to-and-fro repetition which, far from being meaningless, gets to the heart of life, a beating pulse.

Let’s recognize in the cycling repetition an image of play. The most serious of all endeavors is that from which the sense of seriousness lifts away. Hans-Georg Gadamer understood this, and a long quote from him is appropriate. In the following, he provided the italics:

Here the primacy of play over the consciousness of the player is fundamentally acknowledged and, in fact, even the experiences of play that psychologists and anthropologists describe are illuminated afresh if one starts from the medial sense of the word “playing.” Play clearly represents an order in which the to-and-fro motion of play follows of itself. It is part of play that the movement is not only without goal or purpose but also without effort. It happens, as it were, by itself. The ease of play – which naturally does not mean that there is any real absence of effort but refers phenomenologically only to the absence of strain – is experienced subjectively as relaxation. The structure of play absorbs the player into itself, and thus frees him from the burden of taking the initiative, which constitutes the actual strain of existence. This is also seen in the spontaneous tendency to repetition that emerges in the player and in the constant self-renewal of play, which affects its form (e.g., the refrain). [1]

In retrieving this crucial paragraph from Gadamer, my eye was drawn (as it usually is) to the parenthesis. What does “the refrain” refer to? Gadamer gives no explanation, and I almost left it out of the quote. But if it is like the refrain of a song, could you allow that it also means getting back to the rules? There are moments in almost all sporting games when the players or the teams re-set according to fixed rules – think of the football teams lining up, the softball pitcher and the batter eyeing each other from a certain distance – and these moments remind us of getting back to the rules, recalling the theme, before the next chaotic effort begins.

Break these rules, but get back to the rules. Rules are qualified first by an adjective, then by an article. Why? First, it just seemed to fit. Perhaps these rules feel more modest, controllable, personal; the rules more expansive. The rules to which one returns need not be exactly those which were broken. Perhaps the subjective dome, I or we, will be constituted differently. It may be that something was learned from jumping into the unknown, from dancing in the ether. On returning, like Kate Smallwood coming home from China, you will be not be the same person. The words of the rules might be the same, but they could mean something different. Self-reference calls for reversal, built into the rules that tell us to break and get back. The rules and the person belong to each other. Over time, can they be distinguished? Over many rounds of breaking and returning, both the rules and the person may change.

Recall what Martin Buber said, that the world is two-fold to us. At the heart of the universe is a pulsing urge that beats continually from striving to reflecting and back again, creating this two-fold existence in which we play and work and love.


Next post:  Pragmatic Free Will for Individuals and Groups – pending

Previous post:  Saying Thou to the Universe

Searching for GSOT outline:  Home

Own photos. CC0 Public Domain.

[1] Gadamer, H.-G. Truth and Method. Transl. Weinsheimer J. and Marshall D.G. London, Continuum, 1989, p. 105.

Saying Thou to the Universe

Can a person live in I-Thou relation to the universe? Martin Buber thought so. But what did he mean?

The universe, too quickly confronted, overwhelms human inspection. Let’s examine the I-Thou relation in smaller places first.

We live daily beneath a set of intersecting and expanding domes, as presented in an earlier blog. The domes assume various physical forms – the cranium housing each human brain, the ceilings of rooms in which groups of people gather, the arcs of oxygen depletion and exhaled gas and smog shared by people who live in cities, the circles of light-speed information bouncing through the internet among connected groups (such as this one, you participating), and even the sky extending as far as the orbit of the international space station, which is the current physical limit of human life and intelligence.

“Every sentence is first-person” (our Rule #1) means that all discourse emanates from these domes. When we “break these rules” (according to Rule #4), our actions can provoke the emergence of a new dome, a new we-group, and a new first-person plural voice.

Communication moves back and forth between domes, thanks to the beautiful presence of you.

Communication also occurs between larger and smaller domes. Children go to school to funnel the wisdom of their forebears into their skulls. They grow up smart and later collectively and individually add to the cultural endowment – a larger dome extending in time as well as space.

Can we distinguish the I-Thou relation from the I-It relation, as proposed by Martin Buber, in the context of communication between the various domes in which we live? I think yes. If that distinction means anything at all, certainly it applies here.

Consider humanism as described by Morris Storer, Paul Kurtz, and others in the latter part of the 20th century.[1] Humanists turn to science as the means to interrogate nature objectively through I-It relations. Humanists find no Thou in nature.

If the question is asked whether nature, which created us through evolution, cares about us, humanists stoically answer no. The physical universe is uncaring. From this cold fact, they draw an imperative that we must care for each other. Then the human species becomes the focus, the archetype, the existential hero.

What is the relation between an individual person and humanity? The relation is I-Thou. Humanists indeed believe in a higher power, named humanity. They exhort each of us to pledge life and fealty to our species.

Is it possible that even science could involve an I-Thou relation? How do scientists relate to science. Consider that science is much more than an assembly of facts, more than a puzzle of interlocking pieces. Science should be viewed as the practice of an ongoing community of like-minded researchers, who pledge to accept only that which answers to public observation and experiment, compelling agreement within the community. I, an individual scientist, meet my Thou in this like-minded community pledged to impartiality.

Gazing at the stars, many have reminded us how small are humanity, science, and all that passes for knowledge and effort on this earth. What a tiny fraction of carbon in the universe occupies human bodies. How infinitesimal are the molecules of life compared to hydrogen clouds and fiery nuclear reactions in the galaxies. In the time scale of the stars, terrestrial humanity has appeared and will disappear in the blink of a cosmic eye.

Can we dare now consider the largest dome, the one that seems to overarch all others? Call it the universe, or call it natural law. We might with my friend Karl call it the grand scheme of things, GSOT. Some of us might say God.

Can there be an I-Thou relation between my own small hive of activity and the largest dome?

An ordinary believer in God, a person who prays to God, simply takes an I-Thou relation for granted. Sometimes the believer in God views herself as a child of God. Her sacred text tells her that she is made in the image of God. Generally she views herself as having free will, and she views the creator of the universe as having free will. Searching for GSOT, she perceives a spirit kindred to her own, and she speaks Thou.

Under the night sky, she may sing with the psalmist –

The heavens declare the glory of God;
the skies proclaim the work of his hands.
Day after day they pour forth speech;
night after night they reveal knowledge.
They have no speech, they use no words;
no sound is heard from them.
Yet their voice goes out into all the earth,
their words to the ends of the world. [2]

If the telephone rings and I hear someone say, “Hello, Johnny,” those four syllables are usually enough to let me recognize the voice of one of my brothers or sisters. I know their voices by heart. Likewise, we may look at the sky and our hearts say, “God, I recognize your voice.”

This is a theme that Immanuel Kant advanced when he wrote –


“Two things fill the mind with ever new and increasing admiration and awe, the more often and steadily reflection is occupied with them: the starry heaven above me and the moral law within me. Neither of them need I seek and merely suspect as if shrouded in obscurity or rapture beyond my own horizon; I see them before me and connect them immediately with my existence.” [3]

When I was young, I remember hearing or reading that Kant was the greatest of all philosophers. That opinion, I later learned, is by no means unanimous. Nevertheless,  let’s acknowledge the parallel between inner self and the universe in this famous quote. In his own terms, Kant says Thou to the universe.

A similar recognition might also be found in science. Think about the Cosmos television series with Carl Sagan, playing weekly in the Nova slot on public television. Sometime in each program to my recollection, Sagan appeared on the foredeck of a spaceship peering rapturously under magnificent eyebrows through a large curving window at the heavens, expressing wonder at the “billions and billions of stars” in front of him.

Carl Sagan viewed free will as a concept with no useful meaning, and he certainly viewed the universe as having no free will. Yet his response to the universe was conscious and intentional, by his own understanding of those words. His response involved recognition and embrace.

With just a couple of modifications, the 19th Psalm quoted above reads like this:

The heavens declare the glory of nature;
the skies proclaim the rule of constant law.
Day after day they pour forth speech;
night after night they reveal knowledge.
They have no speech, they use no words;
no sound is heard from them.
Yet their voice goes out into all the earth,
their words to the ends of the world.

Can you allow that Sagan answered this voice with his own voice? Let me propose that the relation of Carl Sagan to the universe resembles I-Thou much more than I-It. For Sagan, subject and predicate of conscious action are not categorically different as I-It demands. In the first episode of his television series on Nova, Sagan said this:


…the cosmos is also within us. We’re made of star stuff. We are a way for the cosmos to know itself. [4]

“We’re made of star stuff,” the same stuff that builds the universe. Therefore, subject and object are categorically similar, marking the relation as I-Thou in his own terms.

Sagan and others like him do speak Thou to the universe. They recognize in the universe something kindred to themselves. At times they may describe that something as nominally minimal, but their lives – work, play, and relations – emphatically define a far greater notion.

Most scientists are not astronomers engaging the universe. What about the others, such as those whose professional life might be devoted to a single class of molecules and their interactions? In the mid-90s, I was struggling to get new funding for research. A kind friend, much more successful in the lab than I, invited me to lunch. He simply wanted to give me encouragement, and I was grateful.

My friend told me about a biochemist in our school whose discoveries had advanced his field greatly, bringing recognition and top awards. In his 70s the man developed leukemia, putting him in the hospital time after time. Finally the treatment was only palliative. He had weeks to live. His response? To get back into the lab for another experiment. My friend said he once asked his esteemed colleague which of his discoveries he deemed the best. The old man’s answer came quickly. “The next one,” he replied.

That is the appeal of science and the response of a scientist. He seeks the feeling of being the first to unwrap a discoverable mystery, the first person in all the world to know something new. Nature calls through a set of clues. With the right design, the right instruments, the right controls and buffers, he answers the call, says Thou to nature.

Albert Einstein knew the feeling well. Here is how he said Thou to the universe:


Although it is true that it is the goal of science to discover rules which permit the association and foretelling of facts, this is not its only aim…. Whoever has undergone the intense experience of successful advances made in this domain is moved by profound reverence for the rationality made manifest in existence. By way of the understanding he achieves a far-reaching emancipation from the shackles of personal hopes and desires, and thereby attains that humble attitude of mind toward the grandeur of reason incarnate in existence, and which, in its profoundest depths, is inaccessible to man. This attitude, however, appears to me to be religious, in the highest sense of the word. [5]


Next post:  Rule #5. – pending

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Header image:  Galaxy and stars, CC0 Public Domain, Pixabay.  Immanuel Kant, frontispiece to Prolegomena and Metaphysical Foundations of Natural Science, transl. Earnest Belfort Bax, 1883, public domain. Carl Sagan with planets, CC0 Public Domain, Pixabay. Albert Einstein, CC0 Public Domain, Pixabay.

[1] Storer, M., ed. Humanist Ethics. Prometheus, New York, 1980.

[2] Psalm 19:1-4a. The Bible. New International Version, 1955.

[3] Kant, I. From the concluding remarks in Critique of Practical Reason. In Theoretical Philosophy, 1755–1770, trans. and ed. D. Walford, in collaboration with Ralf Meerbote, Cambridge: Cambridge University Press, 1992,  pp. 161-162. Quoted by Paul Guyer in Immanuel Kant, Routledge Encyclopedia of Philosophy,, accessed 9/17/2016.

[4] Sagan, C. from the first episode of Cosmos in the Nova television series, 1980. See and Youtube link, accessed 9/17/2016.

[5] Einstein, A. Ideas and Opinions. ed. by Seelig, C., transl. by Bargmann, S. New York: Bonanza, 1954, pp. 48-49.

Ich und Du

Learn this: “The I of the primary word I-Thou is a different I from that of the primary word I-It.”

I first read Martin Buber’s I and Thou (Ich und Du in the original German) some 45 years ago. It sat on my shelf with little attention until I made my way back to a new understanding of Buber’s central premise – the importance of the relation expressed when I acknowledge you in meeting. Buber probes this relation in far greater depth than I managed in a previous blog.

The book is short, just 120 pages in the small paperback translation that I read. It costs about $7.50 as an e-book, and I highly recommend that you read it entirely. It begins with these words:

To man the world is twofold, in accordance with his twofold attitude.

The attitude of man is twofold, in accordance with the twofold nature of the primary words which he speaks.

The primary words are not isolated words, but combined words.

The one primary word is the combination I-Thou.

The other primary word is the combination I-It wherein, without a change in the primary word, one of the words He and She can replace It.

Hence the I of man is also twofold.

For the I of the primary word I-Thou is a different I from that of the primary word I-It.

Buber is concerned with the kind of person each of us will become. I am different, he says, when I encounter Thou, from what I am when I experience It. It signifies a world that we experience and use. People can and do become for us objects in that world, when we only recognize those people as others, signifying things apart from myself, rather than Thou of the relation I-Thou.

A less obvious point is that Thou, for Buber, is not always a human person. He glances into the eyes of a housecat, and for the briefest moment he senses connection with the cat.

The beginning of this cat’s glance, lighting up under the touch of my glance, indisputably questioned me: “It is possible that you think of me? Do you really not just want me to have fun? Do I concern you? Do I exist in your sight? Do I really exist? What is it that comes from you? What is it that surrounds me? What is it that comes to me? What is it?”

In the following passage, Buber extends the I-Thou relation to primordial chaos as the crucible of personhood:

Every child that is coming into being rests, like all life that is coming into being, in the womb of the great mother, the undivided primal world that precedes form. From her, too, we are separated, and enter into personal life, sleeping free only in the dark hours to be close to her again; night by night this happens to the healthy man. But this separation does not occur suddenly and catastrophically like the separation from the bodily mother; time is granted to the child to exchange a spiritual connexion, that is, relation, for the natural connexion with the world that he gradually loses. He has stepped out of the glowing darkness of chaos into the cool light of creation.

Buber tells us that the world of Thou is not set “in the context of space and time.”[1] But he will not allow the common interpretation of timelessness. Too often otherwise competent philosophers or theologicans yield to the lazy thought that timeless signifies eternally the same, and a concept initially conceived as unbound by time becomes something static, effectively less than time. Instead Buber envisions the world of Thou as discontinuous when viewed from the world in which we must live, the world of things that we experience and use. “Every Thou in the world,” he says, “is by its nature fated to become a thing, or continually to re-enter into the condition of things.”[2]

The depth of Martin Buber’s thought exceeded my ability to understand when I first read him many years ago, as I approached adulthood. It still does.

Somewhat similar ideas, more accessible to me then and formative to my development, were expressed by Paul Tournier, a Swiss physician and psychiatrist. Tournier made a critical distinction between a person and a personage. We think we know the person, but generally it is the personage whom we see and evaluate. Tournier compared the personage to the driver of a car whose hands move the steering wheel left and right to stay within a driving lane as the road curves around fields and hills. At a certain moment, however, the car approaches an intersection, and the driver becomes a person who slows down and decides whether to take a crossroad left or right, or to move farther down the main road.

As Tournier put it, “We have the same driver, making the same movement with his steering-wheel, but this time the significance of his action is of quite a different order. In the first case his action is automatic and recurrent; in the second it is an isolated act of the will…. We can understand then how it is that the person always eludes objective investigation, that it is always the personage that one finds. Science comprehends only the automatic aspects of the living being, which thus appears to it to be nothing more than a collection of automatic phenomena.” Tournier emphasized that the person operates discontinuously and irreproducibly.[3]

By the time Tournier wrote down those ideas in Switzerland in the early 1950s, the work of Martin Buber had been available in German for more than 2 decades. One must consider that Tournier had been influenced by Buber, as were many others.

Buber believed strongly in the capacity to will. I’ll close this blog with his description of human choice –

The fiery stuff of all my ability to will seethes tremendously, all that I might do circles around me, still without actuality in the world, flung together and seemingly inseparable, alluring glimpses of powers flicker from all the uttermost bounds: the universe is my temptation, and I achieve being in an instant, with both hands plunged deeply in the fire, where the single deed is hidden, the deed which aims at me – now is the moment! [4]


Next post:  Saying Thou to the Universe

Previous post:  An Island Rising

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Header photo:  cc0 Public Domain from Pixabay.

[1] Buber, M. I and Thou. Charles Scribner’s Sons, New York, 1958, p. 33.

[2] Ibid, p. 17.

[3] Tournier, P. The Meaning of Persons. Harper & Row, New York, 1957, quote from page 93.

[4] Buber, pp. 51-52.

An Island Rising

An island rising
from a limitless sea.

All the beauty and providence of nature are here,
all bodily needs sustained,
all materials for art to flourish.

Challenge for the spirit is here,
exploration, storms,
occasional hunger and hurt,
and beasts of sea and land.

A far horizon beckoning.

What thought turns this paradise
into a place of unbearable pain?

I am alone.

Day and night follow each other
in endless succession.

No one appears to rescue me.

Why do I need rescue?

I am alone.

I begin to study the birds of the island.
Perhaps I can tame one or two,
even teach language.
What would it say if it could speak?

I have hunted the moderate-sized lizards
found in abundance on the island,
none so large as to threaten me,
a fine source of meat.

Around a rocky outcrop
a small clan of lizards lives,
and I begin to observe their actions,
their social life.

I do not hunt this group,
because I want to observe them closely.

How do they communicate,
signaling danger or perhaps good fortune?

Do they cooperate in seeking their prey?

How do they raise their young?

What kind of teaching occurs?

And will they accept my presence among them?

Will one of the lizards turn and look at me
and recognize my existence?

Could that look be something
other than a look of fear and caution?

Will some one among them come forward
to accept the fish I am offering,
by chance to feel the touch of my hand?

If I provide some kind of help or comfort,
could one of them express
some hint of thanks to me?


As indeed I am comforted by
and thankful for their presence.


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Header image:  Diomede island, by Dave Cohoe, CC by 3.0, Wikimedia Commons. Marine iguanas (c) Steveheap  |

The Most Beautiful Word in Any Language

The search for GSOT is in many ways a search for beauty. What does this mean?

Beauty moves human choosing in several ways. Beauty dwells in the object of attention and simultaneously in the experience of attending. We pause and linger to enjoy beauty. In its presence attention finds rest and delight. When it departs, we try to recall how or where it might be found. When beauty peeks at us, we turn toward it, hoping to rest in its embrace again. Beauty is more, but it has at least this essential character, that it calls a person to attend, to dismiss other cares, and to appreciate.

Among all the experiences of life, what are those that may be called most beautiful? If the class of most beautiful experiences could be named, what would the name be? That name, I warrant, would deserve to be called the most beautiful word one could speak or write.

Let me suggest that people of various tongues may have this in common – that the class of most beautiful experiences is similar across language and culture – so that the name of the class of most beautiful experiences translates well from any language into any other. Would the name of that class not deserve to be called the most beautiful word in any language? And what would that name, that word, be?

Art is the name for beautiful objects or movements of human design. Yet art is the endeavor as much as the result. We live in a beautiful world, and the best artists find the grace to guide our sight, hearing, and imagination to the beauty around us – through paintings, icons, symphonies, poems, novels; through Alhambra, Taj Mahal, Chartres, sunbeams of Aton, gardens of Suzhou.

Humble Administrator’s garden in Suzhou

We read the great authors, and our own lives pause as we enter the experience of other minds, even fictional minds. Moving between each writer’s world and our own, we find new ways to appreciate the stories of those around us as well as ourselves.

We go to galleries to learn how to see the world – even the most ordinary things – with the eye of an artist. We attend concerts and symphonies, entering music as dormant inner chords awake and resonate in song. Togetherness forms as individual self dissolves in the experience – composers, performers, and listeners all attending to the same moments and movements of sound. In dance, the play of music and form becomes our play. We become children again.

Sum together all the minutes of enchantment in your life. When you are lucky enough to find yourself in such a passage, you can feel as if you are 3 years old, and life is new. Art and beauty belong with each other.

Art is no more than a reflection, a reformulation of nature, some would say. Nature, they say, bears the gift of beauty to us even more than art. Behold the gifts: mountains and beaches, rain forest, arctic tundra, orcas and manatees, orchids and roses, chambered nautilus, a fly’s eye.

Through color and form, sound, smell, touch, taste, and expanse, nature greets us. Through mathematics and measurement, science apprehends nature, and the result is no less beautiful – the gravitational constant, Maxwell’s equations, e=mc2, natural selection.  Elegance – a word physicists use for the simplicity and profundity of universal law – describes it well. One feels humbled and thrilled at the same time. Beside the faithfulness of natural law, the most resolute human will darts and dives like a butterfly.

Art commands a high price, but nature is priceless. Nature cannot be possessed. It has to remain little more than a century to outlast any human person. Families, extending to clans and nations, may stake the boundaries of their claims, pretending to possess fields, hills, streams, and lakes, but the land outlasts all pretensions and endures. Nature remains a gift for future generations to share. Nature has a givenness and givingness that surpasses human experience.

But the durability of nature describes strength more than beauty. Perhaps there is something more beautiful than nature.


Among all creations of nature, human beings form the apex. Some have named humanity most beautiful, but I hold back from that choice. I look at myself – a tiny fraction, but a representative of what is called humanity. Thinking of myself, it perplexes me to give the name “most beautiful” to humanity. Walt Whitman may have had a soul large enough to embrace all people for a while. But the song of myself, Whitman’s song, meets contradictions, gradually grows old and frail, and at last falls silent.

Walt Whitman caught a glimpse of something. I’m just not sure he named it. Let us cast a broad net on human experience and sort the catch. Among all kinds of human experience, those suggesting will and spirit seem most alive and attractive…and we may judge these most desirable, most beautiful. Indeed look back at the descriptions of art and nature. The language is spiritual, putting into words the reception and the projection of things attending the human mind.

Could spirit be the most beautiful word? If it conjures up a shadow world of ghosts or mystical events, then spirit usually lacks beauty. Let it refer instead to personal experiences that can be described as filled with spirit. It might include religious moments – serene mindfulness, closeness to God, visions of life hereafter – but not necessarily and by no means exclusively. Spirit is so beautiful that even its ordinary manifestations bring delight. In the school-age years, think of pep rallies, bus trips, athletic competitions, and campfire skits and songs. Family reunions renew the ties of kinship. Who can forget the shoulder-to-shoulder closeness of a funeral where thoughts of grief and loss and life were shared?

The best stories from our favorite authors are those that demonstrate human spirit. In our most intimate experience with nature, resting from the climb up Enchanted Rock and surveying the hills around, it might occur to us that the essence of GSOT is spirit.

Science also can be spiritual in this sense: the scientist spends long hours in the laboratory, building on the discoveries of equally dedicated predecessors and peers. Confirmation, refinement, and application of her ideas may require the effort of future generations. Her reward includes participation in the great community of science – a spirit of common endeavor. 

Art, nature, humanity, spirit. These words describe so much beauty that it’s hard to choose. But something is missing…or someone. Martin Heidegger called it “dasein” – to be there. It is to find yourself in the midst of beauty. Unless you are there, beauty is sterile, even nonexistent.

It may not require a special word like “dasein.” Much earlier in this series we gave attention to the personal pronouns I, me, we, us, and you, the little words that Daniel Dennett proposed might someday disappear from human speech. To which let us answer, no.

I, me, we, us, and you – which of these pronouns should we examine most deeply? A strong case can be made for we (along with us). Togetherness is a beautiful feeling, a life-affirming spirit. The experience we share is part of the appeal of music and dance. To travel or vacation alone is good; to travel with someone is grand. We celebrate the sacraments and ceremonies of life, which would have little meaning if attempted alone.

But the case for you is stronger yet. You reach across the barrier of individuality and make us possible, at least for a moment and often longer. You, whether singular or plural, implies at least two, as I address you, totaling a special us. Too often we comprise a group dominated by one or a few, a hierarchical corpus. When you are addressed, a whole person meets another whole person, and status disappears. You respect individual sovereignty, as we sometimes fail to do. You do not demand a single path; you make multiplicity possible.

I/we and you are sometimes considered opposites, but this is not true. The antithesis of I/we is other. Saying you is respectful, while saying other lacks respect. Thinking you looks toward another; thinking other looks away. To address another person or another group as you invokes the idea of I/we, the I/we of meeting. To say you, one must recognize a similarity of type, a kinship.

The importance of you comes so early in life that the first feelings usually cannot be remembered, as outgrown as sucking milk and soiling diapers. You – the word that infants and toddlers hear more than any other. What do you want? What did you say? Look at you! I love you. Remember? You, welcome to this life. From infancy until death, the voice of mother echoes in the sound of “you.”

The most beautiful word in any language is you. Natural wonders and human creative works may be filled with beauty. Without you, they are almost as nothing. Everything expands many-fold when we explore the world with each other and in each other. The words I have written here signify nothing in isolation, but they are offered in meeting that you may read and respond.

You are the doorway the threshold of friend and family.
You are the shoreline of a world to explore.
You are the minutes of the board once hidden.
You are my jailbreak my tunnel to freedom.
You are my sunrise.
You are the portal opening to new horizons.
You provoke my release from a tower of sufficiency.
You prompt my escape from the gated community.
You are the recess bell.
You are the reason love cascades and rolls me under
Until you rescue me.

In the richness of you, I discover an image of GSOT, an image that suggests not solely the lawgiver who made the rules that govern reality, but more fully the creator who became a person when you were created. Of all mighty works, the most significant is you, the birth of separate will.


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Header image: CC0 own photo, public domain. Humble Administrator’s Garden in Suzhou © Frenta | Hornworms: own photo, CC0 public domain. Vitruvian man, Leonardo da Vinci, Public Domain, Wikimedia Commons.

Principia Mathematica and Kurt Gödel

Is reality mathematical? The ancient Pythagoreans thought so.

In the last blog, we saw how George Boole in the mid-1800s combined logic and set theory in a new algebraic formulation, which prefigured subsequent advances in statistics and digital computing. Boole was not alone, however, in moving the field of mathematical logic rapidly forward. The German philosopher and mathematician Gottlob Frege at the University of Jena founded a system of symbolic logic that covered some of the same topics as Boole, but served as a more rigorous basis for the merging of mathematics and philosophy. Frege explained, for example, what constitutes a “definition” and what constitutes a “proof,” and he provided a more concise notational system for logical analysis. Guiseppe Peano at the University of Turin contributed axioms supporting the operations of addition and subtraction on natural numbers (positive or non-negative integers).

The most ambitious achievement combining formal logic with mathematics was the 3-volume book, Principia Mathematica, published between 1910 and 1913 by the English mathematicians-cum-philosophers Alfred North Whitehead and Bertrand Russell. Beginning with simple axioms, which they termed primitive premises, the authors carefully worked out theories of hierarchical types, classes, relations, logical products and sums, cardinal arithmetic, mathematical series, and quantity and measurement. Their mathematical derivations, seemingly endless rows of equations, comprise the bulk of Principia Mathematica. They intended to produce a logical treatment of geometry as well, but could not continue as their intellectual energy and partnership were depleted after 10 years of work together. Principia Mathematica was hailed in any case as a monumental achievement, deserving in some subsequent publications the shorthand notation, PM.

If mathematics does not occupy a separate arena of thought, but grows out of basic logic…what might the next discovery be? The subtext of PM was that the world seemed several steps closer to being logically understandable. At the same time, physics, chemistry, and even biology were coming to be viewed as mathematically based. The work of Whitehead and Russell thus appeared to provide a framework, or the beginning of a framework, for the understanding of everything.

Symbols on the pages of PM revealed, line by line, truth-statements beginning with utterly simple axioms and aiming toward ever more complex consequences. Moreover, the operations that unfolded before the eye were driven by simple and automatic rules. A machine might produce such line-by-line derivations as valid as the derivations scribbled by a human hand. Therein was the allure of PM. All nature might be just such a machine, producing quite logically all the sights and sounds of the world before our eyes. PM re-ignited the flame tended so reverently more than 2 millenia earlier by the ancient Pythagoreans – the flaring intuition that reality itself is mathematical.

Enter a 25 year old Austrian, Kurt Gödel, who received his Ph.D. from the University of Vienna in 1930, just a year before publishing the most important research paper in mathematics of the 20th century.

Gödel used the methods of mathematical logic to prove that any system such as PM proceeding from logical axioms cannot be both complete and consistent. “Complete” means that every statement known to be true and expressible within the symbolic language of the system must be provable from the given axioms. “Consistent” means that one cannot derive from the axioms results that are contradictory.

The conclusion provided by Gödel thus undercut any pretension that the methods of PM can ever lead to a final mathematically and logically based theory of knowledge.

How did he do it? Gödel knew that theorems in PM could be self-referencing. Statements of the kind “This theorem is proven from the axioms and steps shown in the sequence above” were common. Importantly PM could also answer questions about the provability of certain theorems. (This is somewhat akin to showing the impossibility of trisecting an angle using only a compass and ruler.) These capabilities of mathematical logic allowed Gödel to use a variation of the Liar’s Paradox from Epimenides of Crete (who said, “Cretans, always liars…”). Gödel found that he could express within the symbolic language of PM the statement “This theorem is not provable” as a theorem.

Let’s consider the statement “This theorem is not provable.” If by some sequence of steps a proof of this statement were claimed, then the statement itself would not be true. If not true, then certainly it is not proven. The same logic applies to any sequence of steps claiming to be a proof. Hence the statement cannot be proven. The impossibility of proving the theorem is a solid result from the logic. Yet this is exactly what the theorem asserts. Thus it is true, but not provable.

As written above, Gödel’s Incompleteness Theorem looks like nothing more than a play on words. “Clever” or “cute,” one might remark, before turning to some more substantial topic.

The genius of Gödel, however, was to express the logic of his self-referencing liar’s theorem in the rigorous symbolic language of the theory of natural numbers described in PM. To appreciate his work, let’s take a very brief look at that language and the brilliant numerical transformation that exposed it to its own severe logic. I am guided here by the clear explanation of Gödel’s theorem given by Ernest Nagel and James Newman.[1]

Here is a formula of elementary logic:

(1)                                      (→ r ) → (( q     (( p V q )   ))

The logical symbol    signifies “if … then,” and the symbol V means “either … or.” The formula therefore can be expanded to read

(2)                                     If (if p then r ), then if [if (if q then )
.                                         then (if (either 
p or q ) then )]

and this can be re-stated

(3)                                       If p implies r, and if q also implies r, then
 .                                         if either p or q is true, then r is true

Of course the formula is valid according to usual assumptions (axioms).

Note that formula (1) is a more succinct statement than either (2) or (3). Logical statements written with symbols as in (1) can be developed and transformed (by substitution rules, symmetry rules, etc.) much more readily than word statements such as (2) or (3).

Gödel took the schematizaton and brevity to a new level in order to prove his Incompleteness Theorem. He showed how any formula such as (1) can be reduced to a single positive integer, its Gödel number, as long as the axioms and variables of the mathematical system are pre-specified.

To understand how the Gödel number is derived for a given formula, the reader is best referred to the wonderful short book by Nagel and Newman. Here is a key paragraph:

Gödel described a formalized calculus, which we shall call “PM,” within which all the customary arithmetical notations can be expressed and familiar arithmetical relations established. The formulas of the calculus are constructed out of a class of elementary signs, which constitute the fundamental vocabulary. A set of primitive formulas (or axioms) are the underpinning, and the theorems of the calculus are formulas derivable from the axioms with the help of a carefully enumerated set of Transformation Rules (or rules of inference).[2]

Nagel and Newman describe the derivation of Gödel numbers by postulating just 12 elementary constant signs (there are more in PM itself), including for our purposes the left parenthesis “(“, the right parenthesis “)”, the sign for “if … then” that we show as “  ”, and the sign for “either … or” that we show as “V”. The Gödel numbers 1 to 12 are assigned in a fixed manner to the constant signs. For the setential or “sentence” variables – p, q, and r – Gödel assigned the square of successive prime numbers greater than 12. Thus p could be signified by 132 = 169, q by 172 = 289, r by 192 = 361, and so on for as many such variables as are needed by the calculus. (Nagel and Newman provide a clear description of the difference between numerical variables, usually designated by x, y, and z, and setential variables p, q, and r. George Boole had noted the necessity of such a distinction. A setential variable can represent a formula or a logical premise; a numerical variable, commonly encountered in ordinary algebra, stands for a number.)

Now we are ready to assign a Gödel number to the formula (1) shown above. Each position in the formula is represented by a prime number beginning with 2. The Gödel number of the corresponding sign at that position is placed in the exponent. Thus the Gödel number for the formula (1) could be

(1)                                      (→ r ) → (( q     (( p V q )   ))

(4)                                     m = 21 × 3169 × 53 × 7369 × 112 × 133 × 171 ×
.                                          19
× 23289 × 293 × 31369 × 372 × 413 × 431
.                                          × 471 × 53169 × 574 × 59289 × 612 × 673 ×
.                                          71369 × 732 × 772

Galaxy_Hubble M100 spiral 2
M100 spiral galaxy as seen by the Hubble Space Telescope

We won’t try to calculate the product of the enormously large number shown above. It is far larger than the number of elementary particles in the universe. Gödel himself wasn’t interested in doing any such calculations. Yet he was careful to demonstrate that it is a distinct number which completely describes the logical formula (1). He further demonstrated that in theory any such number can be reduced to a product of prime numbers raised by discoverable exponents by exploiting the rules according to which Gödel numbers are derived. Thus in our example the number displayed in (4), even as a single gargantuan number, can be translated by factoring back to the formula (1), in theory at least, and that was all the mattered for the purposes of proving the Incompleteness Theorem.

Gödel did not stop with formulas, but went on to describe how his numbering system similarly can be applied to sequences of formulas beginning with primitive formulas (axioms) and leading via transformation rules to a concluding formula. These sequences, of course, are called theorems. Certain theorems, called demonstrations, state whether a particular formula under consideration is provable or not. A Gödel number can also be assigned to such a demonstration.

In the blog titled Emergence of Mathematics, I showed how the set of all rational numbers can be “mapped” onto the set of natural numbers – that is, zero and all the positive integers. This was done to understand the concept of orders of infinity, but it was also an example of mapping one set of mathematical entities onto another, in order to understand the first set of entities better. It should be clear that Gödel’s process was one of mapping the logic that produces mathematics onto natural numbers. Yet the natural numbers are the subject matter of mathematics. Therefore, Gödel’s method was well suited to the task of discerning whether mathematics could describe itself – that is, whether mathematics is complete, consistent, and “self-contained” as a logical system of thought.

One might wonder why completeness and consistency in mathematics ever drew such interest, or came to be regarded as critical in science and philosophy. Remember that mathematics, already the underpinning of physics at the beginning of the 20th century, was being attached at a rapid pace to every other field of science from chemistry to sociology. Much of the success was due to the advances in the field of statistics, owing much to pioneers like George Boole. Decisions about data henceforth would be determined mathematically.

Therefore questions about the completeness and consistency of mathematics, as the language of science, were considered crucial. The German mathematician George Hilbert defined the criteria for proving completeness and consistency. Within certain small, well circumscribed axiomatic systems such as logical tautologies, Hilbert’s criteria were achieved (see Nagel and Newman for an explanation). The monumental effort of Russell and Whitehead’s PM aimed at a complete, consistent description of quantitative mathematics.

Against this background, the outcome of Kurt Gödel’s 1931 research paper was that any axiomatic logical system – mathematics being the prime example – of more than minor complexity must be incomplete (that is, capable of formulating true statements not provable within the system), inconsistent (that is, flawed in the sense of leading logically to two or more contradictory results), or both.

Let’s look once more at the Gödel statement, “This theorem is not provable.” Although the statement is recognizably true, its truth is not shown through the axiomatized system of logic itself. Recognizing that axiomatized system of logic as mathematics, the truth of the Gödel statement can be described as metamathematical. It moves beyond mathematics (beyond that which can be summarized in a Gödel number) and applies to mathematics a criterion from outside.

In physics the concept of Elegance applies to the mathematical description of observed phenomena, when that description can be summarized simply, sometimes within a few key equations relating physical quantities, and yet can be applied to describe a vast array of real events in the physical world. I think of Maxwell’s equations, 4 in number, which can be used to predict the behavior of electromagnetic waves ranging up the frequency scale from infrared heat to radio waves to light to x-rays. Think about how many engineers’ calculations have brought how much benefit from those equations!

I wish there were a word like Elegance to describe what Kurt Gödel’s discovery could eventually mean to science and philosophy. It should be a word that suggests Openness or Going Beyond. There are truths not provable within the parameters of any closed, consistent logical system. Even if the system is addended to encompass a new truth, Gödel’s Incompleteness Theorem can be applied again. One cannot, therefore, construct a closed (complete), consistent system. Our minds are free to roam outside the walls. Gödel has stamped our passport for the journey. It may be a journey to a realm beyond, or a journey to self, or both. We’ll look at one destination next week.

One of the authors of Principia Mathematica, Bertrand Russell, subscribed to the philosophy of positivism, which held that notions of truth and reality should be limited to that which can demonstrated scientifically – that is, demonstrated reproducibly in space-time and among observers.

Because mathematics is the language of science, positivism in some ways revives the ancient Pythagorean creed that reality is mathematical.

Gödel tells us no.

How can reality be mathematical?

Not even math is consistently and completely mathematical.


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Header image: Left, medieval illuminated manuscript, Graduel d’Aliénor – Entrée dans Jérusalem, Wikimedia Commons Public Domain. Right, Principia Mathematica, by Alfred North Whitehead and Bertrand Russell, University of Michigan Historical Math Collection. M100 spiral galaxy from Hubble Legacy Archive, NASA, ESA, Wikimedia Commons Public Domain.

[1] Nagel E and Newman JR. Gödel’s Proof. New York University Press, New York, revised edition, 2001, Kindle edition.

[2] Ibid., Kindle location 862.


Boolean Thinking

Pure logic and mathematics began to come together in the mid to late 1800s in Europe. The publication in 1853 of a book titled An Investigation of the Laws of Thought, by the remarkable mathematician George Boole, established a foundation rapidly accepted and extended by others through the remainder of the century and beyond.[1]

Boole developed a kind of algebra in which the variables x, y, z… take on only the values zero and one. A curious equality, which he introduces early and to which he returns repeatedly in the book, is the following relation:

(1)                                      x = x2

The equation (1) is easily verified by substituting zero or one for x. Of course, (1) departs from our customary algebra of continuous quantities. Some, not all, of the usual algebraic rules were carefully transferred by Boole into his new system. Simple rearrangement of (1) gives

(2)                                     x (1-x) = 0

and setting either factor equal to zero, we return to

(3)                                     x = 0  or  x = 1

Boole recognized that such a variable x could stand for a logical proposition with a value of false (zero) or true (one). The dual statement (3) above then means “x is either false or true, but not both” and this can also be incorporated in a single quadratic equation such as (1) or (2). Boole named this the Law of Duality.

Letting various letters such as x, y, and z stand for propositions, Boole the master teacher summarized the process as follows:

…any system of propositions may be expressed by equations involving symbols x, y, z, which, whenever interpretation is possible, are subject to laws identical in form with the laws of a system of quantitative symbols, susceptible only of the values 0 and 1. But as the formal processes of reasoning depend only upon the laws of the symbols, and not upon the nature of their interpretation, we are permitted to treat the above symbols, x, y, z, as if they were quantitative symbols of the kind above described. We may in fact lay aside the logical interpretation of the symbols in the given equation; convert them into quantitative symbols, susceptible only to the values 0 and 1; perform upon them as such all the requisite processes of solution; and finally restore to them their logical interpretation. [Boole’s italics]

In a stunning synthesis of math, logic, and practical application, Boole then developed additional meanings for the variables x, y, z and the values 0 and 1. He identified x with the class of objects having a property signified by x. Thus x = 1 if an individual object belongs to the class, and x = 0 if an individual object does not belong to the class. Similarly y, z, or any other variable you might wish to name can identify objects according to their membership in a class. The value 1 serves a further purpose. He recognized 1 as standing for the universe of objects under consideration, the “class in which are found all the individuals that exist in any class.”[2] The value 0 correspondingly comprises the empty class or “Nothing.”

Here is the way Boole symbolically represents negation –

…let x represent the class men, and let us express, according to the last Proposition, the Universe by 1; now if from the conception of the Universe, as consisting of “men” and “not-men,” we exclude the conception of “men,” the resulting conception is that of the contrary class, “not-men.” Hence the class “not-men” will be represented by 1 − x. And, in general, whatever class of objects is represented by the symbol x, the contrary class will be expressed by 1 − x. [chapter III, p. 34]

In the new system, algebraic multiplication is transformed into successively more refined definitions of classes of objects. In his words,

…if x represents opaque substances, y polished substances, z stones, we shall have,

xyz = opaque polished stones;

xy(1 − z) = opaque polished substances which are not stones;

x(1 − y)(1 − z) = opaque substances which are not polished, and are not stones;

and so on for any other combination. [chapter IV, p. 39]

George Boole, circa 1860

We do not have space or time to develop Boole’s algebra step by step. His book is a marvel of clear writing, and I highly recommend it. I do think that summarizing an example which he gives in Chapter 8 can be helpful. Boole wants to show how his new algebra of logic can clarify an argument presented in Dr. Samuel Clarke’s book titled “Demonstration of the Being and Attributes of God.” There was no incongruity in this example, because Boole thought deeply about religion.[3]

In this particular train of reasoning, Clarke seeks to demonstrate that “Something has existed from eternity.” Boole distills Clarke’s written paragraph into a series of propositions, each of which may be found either true or false, as follows:

x =  Something is.
y =  Something always was.
z =  The things which now are have risen from nothing.
p =  It exists in the necessity of its own nature.
            (i.e., the something spoken of above)
q =  It exists by the will of another Being.

Among these propositions, y is the conclusion sought, or to be more specific, y = 1, which will be interpreted as “It is true that something always was.”

Clarke’s reasoning began with the following 5 premises, which I show first in words and then in Boolean symbols:

1st.  “Something is.” translates to     x = 1

2nd.  “If something is, either something always was, or the things that now are have risen out of nothing.”
translates to     x = v (1–x) + (1–y) ]

3rd.  “If something is, either it exists in the necessity of its own nature, or it exists by the will of another being.”
translates to    x = (1–q) + (1–p) ]

4th.  “If it exists in the necessity of its own nature, then something always was.”
translates to    p = vy

5th.  “If it exists by the will of another being, then the hypothesis, that the things which now are have risen out of nothing, is false.”
translates to   q = v (1–z)

Thus the premises have been transformed into these 5 equations:[4]

x = 1
x = v [ y (1–x) + x (1–y) ]
x = v [ p (1–q) + q (1–p) ]
p = vy
q = v (1–z)

Using the rules of his own algebra, Boole solves these equations in 12 steps, of which I show only (11) and (12) below:

(11)                 1 – y = 0, or,  y = 1.

(12)                 x = 0.

Finally, the interpretation of (11) is that it is true that something always was, and the interpretation of (12) is that it is false that the things which now are have risen from nothing.[5]

Boole’s new algebra could be used to solve either practical or philosophical problems. Its application to “the Being and Attributes of God” was, somewhat surprisingly, not the end or goal of the book. He went on to show how his new algebra could answer questions of probability, which we now recognize as belonging to the field of statistics. From the historic distribution of rainfall on the fields, Boolean algebra could be used to calculate the probability of a good harvest.

But I doubt that even George Boole would have predicted that his new algebra would underpin the binary logic of digital computers.

Think about your cell phone, listen to digitally recorded music, check out the spreadsheet data behind a chart of progress in your field, whatever it is. As you do, remember the words of George Boole…

…any system of propositions may be expressed by equations involving symbols x, y, z, which, whenever interpretation is possible, are subject to laws identical in form with the laws of a system of quantitative symbols, susceptible only of the values 0 and 1….



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Images:  Long Hall, University College Cork, Cork, Ireland, by Bjørn Christian Tørrissen –, CC BY-SA 3.0, from Wikimedia Commons. Color portrait of George Boole, Wikimedia Commons, public domain.

[1] Boole, George. An Investigation of the Laws of Thought, on Which Are Founded the Mathematical Theories of Logic and Probabilities. 1853, is available online through Project Gutenberg, accessed at on June 27, 2014.

[2] Ibid, chapter III, p. 34.

[3] George Boole was born to a working class family in Lincolnshire, England. He cared about and conversed with common folk, and he engaged repeatedly in efforts at social reform. See this link on the website of University College Cork.

[4] The astute reader may notice in the equations that form the premises an extra notation not previously defined, that is, v. This is a special symbol that Boole found necessary to incorporate the meaning of “some” or “sometimes”as in “either none, some, or all,” or “either never, sometimes, or always.”

[5] Perhaps your own assessment will not judge kindly the conclusions brought forward by Clarke and Boole. If that is the case, don’t blame the logic. Instead think hard about the 5 premises with which Clarke began his deductive process. Today one person or another might choose different premises. A positivist, for example, might begin with the premise “What is true is only that on which rational observers can agree.” A pragmatist might hold the premise “Unless it makes a difference in somebody’s disposition to act, then it makes no difference.” Elsewhere we examined the premises of personal viewpoint – “Every sentence is first person” and “The overarching viewpoint is not allowed.” Boolean analysis has this virtue – It brings clarity to the deductive consequences of our first principles. It may tell us, then, that the 4 premises I have just listed produce a contradiction, that 3 of them may be sufficient to negate the 4th. What do you think?