“There is scarcely any well-informed person, who, if he has but the will, has not also the power to add something essential to the general stock of knowledge.”
By Maria Popova
“It is always difficult to teach the man of the people that natural phenomena belong as much to him as to scientific people,” the trailblazing astronomer Maria Mitchell wrote as she led the first-ever professional female eclipse expedition in 1878. The sentiment presages the importance of what we today call “citizen science,” radical and countercultural in an era when science was enshrined in the pompous pantheon of the academy, whose gates were shut and padlocked to “the man of the people,” to women, and to all but privileged white men.
Two decades earlier, Mitchell had traveled to Europe as America’s first true scientific celebrity to meet, among other dignitaries of the Old World, one such man — but one of far-reaching vision and kindness, who used his privilege to broaden the spectrum of possibility for the less privileged: the polymathic astronomer John Herschel (March 7, 1792–May 11, 1871), co-founder of the venerable Royal Astronomical Society, son of Uranus discoverer William Herschel, and nephew of Caroline Herschel, the world’s first professional woman astronomer, who had introduced him to astronomy as a boy.
Several years before he coined the word photography, Herschel became the first prominent scientist to argue in a public forum that the lifeblood of science — data collection and the systematic observation of natural phenomena — should be the welcome task of ordinary people from all walks of life, united by a passionate curiosity about how the universe works.
In 1831, the newly knighted Herschel published A Preliminary Discourse on the Study of Natural Philosophy as part of the fourteenth volume of the bestselling Lardner’s Cabinet Cyclopædia (large chunks of which were composed by Frankenstein author Mary Shelley). Later cited in Lorraine Daston and Elizabeth Lunbeck’s altogether excellent book Histories of Scientific Observation (public library), it was a visionary work, outlining the methods of scientific investigation by clarifying the relationship between theory and observation. But perhaps its most visionary aspect was Herschel’s insistence that observation should be a network triumph belonging to all of humanity — a pioneering case for the value of citizen science. He writes:
To avail ourselves as far as possible of the advantages which a division of labour may afford for the collection of facts, by the industry and activity which the general diffusion of information, in the present age, brings into exercise, is an object of great importance. There is scarcely any well-informed person, who, if he has but the will, has not also the power to add something essential to the general stock of knowledge, if he will only observe regularly and methodically some particular class of facts which may most excite his attention, or which his situation may best enable him to study with effect.
Pointing to meteorology and geology as the sciences best poised to benefit from distributed data collection by citizen scientists, Herschel adds:
There is no branch of science whatever in which, at least, if useful and sensible queries were distinctly proposed, an immense mass of valuable information might not be collected from those who, in their various lines of life, at home or abroad, stationary or in travel, would gladly avail themselves of opportunities of being useful.
Herschel goes on to outline the process by which such citizen science would be conducted: “skeleton forms” of survey questions circulated far and wide, asking “distinct and pertinent questions, admitting of short and definite answers,” then transmitted to “a common centre” for processing — a sort of human internet feeding into a paper-stack server. (Lest we forget, Maria Mitchell herself was employed as a “computer” — the term we used to use for the humans who performed the work now performed by machines we have named after them.)
“If a man has any genuine talent he should be ready to make almost any sacrifice in order to cultivate it to the full.”
By Maria Popova
“Resign yourself to the lifelong sadness that comes from never being satisfied,” Zadie Smith counseled in the tenth of her ten rules of writing — a tenet that applies with equally devastating precision to every realm of creative endeavor, be it poetry or mathematics. Bertrand Russell addressed this Faustian bargain of ambition in his 1950 Nobel Prize acceptance speech about the four desires motivating all human behavior: “Man differs from other animals in one very important respect, and that is that he has some desires which are, so to speak, infinite, which can never be fully gratified, and which would keep him restless even in Paradise. The boa constrictor, when he has had an adequate meal, goes to sleep, and does not wake until he needs another meal. Human beings, for the most part, are not like this.”
Ten years earlier, the English mathematician and number theory pioneer G.H. Hardy (February 7, 1877–December 1, 1947) — an admirer of Russell’s — examined the nature of this elemental human restlessness in his altogether fascinating 1940 book-length essay A Mathematician’s Apology (public library).
In considering the value of mathematics as a field of study and “the proper justification of a mathematician’s life,” Hardy offers a broader meditation on how we find our sense of purpose and arrive at our vocation. Addressing “readers who are full, or have in the past been full, of a proper spirit of ambition,” Hardy writes in an era when every woman was colloquially “man”:
A man who is always asking “Is what I do worth while?” and “Am I the right person to do it?” will always be ineffective himself and a discouragement to others. He must shut his eyes a little and think a little more of his subject and himself than they deserve. This is not too difficult: it is harder not to make his subject and himself ridiculous by shutting his eyes too tightly.
A man who sets out to justify his existence and his activities has to distinguish two different questions. The first is whether the work which he does is worth doing; and the second is why he does it, whatever its value may be. The first question is often very difficult, and the answer very discouraging, but most people will find the second easy enough even then. Their answers, if they are honest, will usually take one or other of two forms; and the second form is a merely a humbler variation of the first, which is the only answer we need consider seriously.
Most people, Hardy argues, answer the first question by pointing to a natural aptitude that led them to a vocation predicated on that particular aptitude — the lawyer became a lawyer because she naturally excels at eloquent counter-argument, the cricketer a cricketer because he has a natural gift for cricket. In what may sound like an ungenerous sentiment but is indeed statistically accurate, Hardy adds:
I am not suggesting that this is a defence which can be made by most people, since most people can do nothing at all well. But it is impregnable when it can be made without absurdity, as it can by a substantial minority: perhaps five or even ten percent of men can do something rather well. It is a tiny minority who can do something really well, and the number of men who can do two things well is negligible. If a man has any genuine talent he should be ready to make almost any sacrifice in order to cultivate it to the full.
But while talent exists in varying degrees within each field of endeavor, Hardy notes that the fields themselves occupy a hierarchy of value — different activities offer different degrees of benefit to society. And yet most people, he argues, choose their occupation not on the basis of its absolute value but on the basis of their greatest natural aptitude relative to their other abilities. (Not to do so, after all, renders one the faintly smoking chimney in Van Gogh’s famous lament about unrealized talent: “Someone has a great fire in his soul and nobody ever comes to warm themselves at it, and passers-by see nothing but a little smoke at the top of the chimney.”) Hardy writes:
I would rather be a novelist or a painter than a statesman of similar rank; and there are many roads to fame which most of us would reject as actively pernicious. Yet it is seldom that such differences of value will turn the scale in a man’s choice of a career, which will almost always be dictated by the limitations of his natural abilities. Poetry is more valuable than cricket, but [the champion cricketer Don] Bradman [whose test batting average is considered the greatest achievement of any sportsman] would be a fool if he sacrificed his cricket in order to write second-rate minor poetry (and I suppose that it is unlikely that he could do better). If the cricket were a little less supreme, and the poetry better, then the choice might be more difficult… It is fortunate that such dilemmas are so seldom.
Presaging the ominous twenty-first-century trend of talented mathematicians and physicists swallowed by Silicon Valley for lucrative jobs ranging from the uninspired to the downright pernicious, Hardy adds:
If a man is in any sense a real mathematician, then it is a hundred to one that his mathematics will be far better than anything else he can do, and that he would be silly if he surrendered any decent opportunity of exercising his one talent in order to do undistinguished work in other fields. Such a sacrifice could be justified only by economic necessity or age.
Every young mathematician of real talent whom I have known has been faithful to mathematics, and not from lack of ambition but from abundance of it; they have all recognized that there, if anywhere, lay the road to a life of any distinction.
Ambition, he argues, has been the motive force behind nearly everything we value as a civilization — every significant breakthrough in art and science, “all substantial contributions to human happiness.” (George Orwell, too, pointed to personal ambition as the first of the four universal motives of great writers.) But while various ambitions can possess us, ranging from the vain and greedy to the most elevated and idealistic, Hardy points to one as the crowning achievement of the purposeful life:
Ambition is a noble passion which may legitimately take many forms… but the noblest ambition is that of leaving behind something of permanent value.
“Euclid alone has looked on Beauty bare,” Edna St. Vincent Millay wrote in her lovely ode to how the father of geometry transformed the way we see and comprehend the world. But although the ancient Alexandrian mathematician provided humanity’s only framework for understanding space for centuries to come, shaping both science and art, his beautiful system was wormed by one ineluctable flaw: Euclid’s famous fifth postulate, known as the parallel postulate — which states that through any one point not belonging to a particular line, only one other line can be drawn that would be parallel to the first, and the two lines, however infinitely they may be extended into space, will remain parallel forever — is not a logical consequence of his other axioms.
This troubled Euclid. He spent the remainder of his life trying to prove the fifth postulate mathematically, and failing. Generations of mathematicians did the same for the next two thousand years. It even stumped Gauss, considered by many the greatest mathematician of all time. It took a Hungarian teenager to solve the ancient quandary.
In 1820, more than two millennia after Euclid’s death, the seventeen-year-old János Bolyai (December 15, 1802–January 27, 1860) told his father — the mathematician Wolfgang Bolyai, who had introduced his son to the enchantment of mathematics four years earlier — about his obsession with the parallel postulate.
Don’t waste an hour on that problem. Instead of reward, it will poison your whole life. The world’s greatest geometers have pondered the problem for hundreds of years and not proved the parallel postulate without a new axiom. I believe that I myself have investigated all the possible ideas.
But the young man persisted. On November 3, 1823, the twenty-one-year-old mathematical maverick wrote to his father while serving as an artillery officer in the Hungarian army:
I have resolved to publish a work on the theory of parallels as soon as I have arranged the material and my circumstances allow it. I have almost been overwhelmed by them, and it would be the cause of constant regret if they were lost. When you see them, my dear father, you too will understand. At present I can say nothing except this: I have created a new universe from nothing. All that I have sent to you till now is but a house of cards in comparison with a tower. I’m fully persuaded that this will bring me honor, as if I had already completed the discovery.
The discovery in which he exults is one of humanity’s most groundbreaking insights into the nature of reality: Bolyai had laid the foundation of non-Euclidean geometry — a wholly novel way of apprehending space, which describes everything from the shape of a calla lily blossom to the growth pattern of a coral reef, and which would become a centerpiece of relativity; without it, Einstein couldn’t have revolutionized our understanding of the universe with his notion of spacetime, the curvature of which is a supreme embodiment of non-Euclidean geometry.
Impressed by his son’s tenacity and swayed by the significance of the breakthrough, Wolfgang pivoted 180 degrees and now urged his son to publish his findings as soon as possible in order to ensure priority of discovery:
If you have really succeeded in the question, it is right that no time be lost in making it public, for two reasons: first, because ideas pass easily from one to another, who can anticipate its publication, and secondly, [because] there is some truth in the fact that many things have an epoch, in which they are discovered at the same time in several places, just as the violets appear on every side in spring.
These were words of remarkable prescience. When János’s paper, completed in 1829 and published as an appendix to a book of his father’s in 1832, reached Gauss — an old friend of Wolfgang Bolyai’s — the great German mathematician was astonished. He responded that he couldn’t praise János’s work, for it would mean praising himself — the young mathematician’s breakthrough, from the central questions he had tackled to the path he had pursued in answering them to the results he had obtained, coincided “almost entirely” with what had been occupying Gauss’s own mind for more than thirty years, though he had resolved never to publish these meditations in his lifetime. With the selfless graciousness of a true scientist, who sets aside all personal ego and celebrates any triumph of knowledge, Gauss wrote to János’s father:
So far as my own work is concerned, of which up till now I have put little on paper, my intention was not to let it be published during my lifetime… On the other hand it was my idea to write down all this later so that at least it should not perish with me. It is therefore a pleasant surprise for me that I am spared this trouble, and I am very glad that it is just the son of my old friend who anticipates me in such a remarkable manner.
But the young Bolyai’s elation at having “created a new universe from nothing” was swiftly grounded when he realized that a third mathematician — Nikolai Lobachevsky in Russia — had preceded both him and Gauss in publishing a paper outlining the selfsame ideas. Lest we forget how information traveled in the pre-Internet era, it took Bolyai sixteen years to learn of Lobachevsky’s book. Once he read it, he reconciled himself to the loss of priority by rooting his ego in the animating principle of science, which he recorded in an uncommonly poetic and profound meditation in his notebook:
The nature of real truth of course cannot be but one and the same [in Hungary] as in Kamchatka and on the Moon, or, to be brief, anywhere in the world; and what one finite, sensible being discovers, can also not impossibly be discovered by another.
The discovery at which these three finite, sensible beings had arrived simultaneously and independently forever changed not only mathematics but our fundamental grasp of nature. For a fine complement, see mathematician Lillian Lieber’s 1961 masterpiece drawing on the non-Euclidean revolution to illustrate the building blocks of moral values like democracy and social justice, then revisit physicist Alan Lightman on the shared psychology of creative breakthrough in art and science.
An imaginative extension of Euclid’s parallel postulate into life, liberty, and the pursuit of happiness.
By Maria Popova
“The joy of existence must be asserted in each one, at every instant,” Simone de Beauvoir wrote in her paradigm-shifting treatise on how freedom demands that happiness become our moral obligation. A decade and a half later, the mathematician and writer Lillian R. Lieber (July 26, 1886–July 11, 1986) examined the subject from a refreshingly disparate yet kindred angle.
Einstein was an ardent fan of Lieber’s unusual, conceptual books — books discussing serious mathematics in a playful way that bridges science and philosophy, composed in a thoroughly singular style. Like Einstein himself, Lieber thrives at the intersection of science and humanism. Like Edwin Abbott and his classic Flatland, she draws on mathematics to invite a critical shift in perspective in the assumptions that keep our lives small and our world inequitable. Like Dr. Seuss, she wrests from simple verses and excitable punctuation deep, calm, serious wisdom about the most abiding questions of existence. She emphasized that her deliberate line breaks and emphatic styling were not free verse but a practicality aimed at facilitating rapid reading and easier comprehension of complex ideas. But Lieber’s stubborn insistence that her unexampled work is not poetry should be taken with the same grain of salt as Hannah Arendt’s stubborn insistence that her visionary, immensely influential political philosophy is not philosophy.
In her hundred years, Lieber composed seventeen such peculiar and profound books, illustrated with lovely ink drawings by her husband, the artist Hugh Lieber. Among them was the 1961 out-of-print gem Human Values and Science, Art and Mathematics (public library) — an inquiry into the limits and limitless possibilities of the human mind, beginning with the history of the greatest revolution in geometry and ending with the fundamental ideas and ideals of a functioning, fertile democracy.
Lieber paints the conceptual backdrop for the book:
This book is really about
Life, Liberty, and the Pursuit of Happiness,
using ideas from mathematics
to make these concepts less vague.
We shall see first what is meant by
“thinking” in mathematics,
and the light that it sheds on both the
CAPABILITIES and the LIMITATIONS
of the human mind.
And we shall then see what bearing this can have
on “thinking” in general —
even, for example, about such matters as Life, Liberty, and the Pursuit of Happiness!
For we must admit that our “thinking”
about such things,
without this aid,
often leads to much confusion —
mistaking LICENSE for LIBERTY,
often resulting in juvenile delinquency;
mistaking MONEY for HAPPINESS,
often resulting in adult delinquency;
mistaking for LIFE itself
just a sordid struggle for mere existence!
Embedded in the history of mathematics, Lieber argues, is an allegory of what we are capable of as a species and how we can use those capabilities to rise to our highest possible selves. In the first chapter, titled “Freedom and Responsibility,” she chronicles the revolution in our understanding of nature and reality ignited by the advent of non-Euclidean geometry — the momentous event Lieber calls “The Great Discovery of 1826.” She writes:
One of the amazing things
in the history of mathematics
happened at the beginning of the 19th century.
As a result of it,
the floodgates of discovery
were open wide,
and the flow of creative contributions
is still on the increase!
this amazing phenomenon
was due to a mere
CHANGE OF ATTITUDE!
Perhaps I should not say “mere,”
since the effect was so immense —
which only goes to show that
a CHANGE OF ATTITUDE
can be extremely significant
and we might do well to examine our ATTITUDES
toward many things, and people —
this might be the most rewarding,
as it proved to be in mathematics.
In order to fully comprehend a revolutionary change in attitude, Lieber points out, we need to first understand the old attitude — the former worldview — supplanted by the revolution. To appreciate “The Great Discovery of 1826,” we must go back to Euclid:
first put together
the various known facts of geometry
into a SYSTEM,
instead of leaving them as
isolated bits of information —
as in a quiz program!
has served for many centuries
as a MODEL for clear thinking,
and has been and still is
of the greatest value to the human race.
Lieber unpacks what it means to build such a “model for clear thinking” — networked logic that makes it easier to learn and faster to make new discoveries. With elegant simplicity, she examines the essential building blocks of such a system and outlines the basics of mathematical logic:
In constructing a system,
one must begin with
a few simple statements
by means of logic,
one derives the “consequences.”
We can thus
“figure out the consequences”
before they hit us.
And this we certainly need more of!
Thus Euclid stated such
(called “postulates” in mathematics)
“It shall be possible to draw
a straight line joining
any two points,”
and others like it.
he derived many complicated theorems
like the well-known
and many, many others.
And, as we all know,
to “prove” any theorem
one must show how
to “derive” it from the postulates —
every claim made in a “proof”
must be supported by reference to
the postulates or
to theorems which have previously
already been so “proved”
from the postulates.
Of course Theorem #1
must follow from
the postulates ONLY.
Now what about
the postulates themselves?
How can THEY be “proved”?
CANNOT be PROVED at all —
since there is nothing preceding them
from which to derive them!
This may seem disappointing to those who
thought that in
EVERYTHING is proved!
But you can see that
this is IMPOSSIBLE,
even in mathematics,
since EVERY SYSTEM must necessarily
START with POSTULATES,
and these are NOT provable,
since there is nothing preceding them
from which to derive them.
This circularity of certainty permeates all of science. In fact, strangely enough, the more mathematical a science is, the more we consider it a “hard science,” implying unshakable solidity of logic. And yet the more mathematical a mode of thinking, the fuller it is of this circularity reliant upon assumption and abstraction. Euclid, of course, was well aware of this. He reconciled the internal contradiction of the system by considering his unproven postulates to be “self-evident truths.” His system was predicated on using logic to derive from these postulates certain consequences, or theorems. And yet among them was one particular postulate — the famous parallel postulate — which troubled Euclid.
The parallel postulate states that if you were to draw a line between two points, A and B, and then take a third point, C, not on that line, you can only draw one line through C that will be parallel to the line between A and B; and that however much you may extend the two parallel lines in space, they will never cross.
Euclid, however, wasn’t convinced this was a self-evident truth — he thought it ought to be mathematically proven, but he failed to prove it. Generations of mathematicians failed to prove it over the following thirteen centuries. And then, in the early nineteenth century, three mathematicians — Nikolai Lobachevsky in Russia, János Bolyai in Hungary, and Carl Friedrich Gauss in Germany — independently arrived at the same insight: The challenge of the parallel postulate lay not in the proof but, as Lieber puts it, in “the very ATTITUDE toward what postulates are” — rather than considering them to be “self-evident truths” about nature, they should be considered human-made assumptions about how nature works, which may or may not reflect the reality of how nature work.
This may sound like a confounding distinction, but it is a profound one — it allowed mathematicians to see the postulates not as sacred and inevitable but as fungible, pliable, tinkerable with. Leaving the rest of the Euclidean system intact, these imaginative nineteenth-century mathematicians changed the parallel postulate to posit that not one but two lines can be drawn through point C that would be parallel to the line between A and B, and the entire system would still be self-consistent. This resulted in a number of revolutionary theorems, including the notion that the sum of angles in a triangle could be different from 180 degrees — greater if the triangle is drawn on the surface of a sphere, for instance, or lesser if drawn on a concave surface.
It was a radical, thoroughly counterintuitive insight that simply cannot be fathomed, much less diagramed, in flat space. And yet it wasn’t a mere thought experiment, an amusing and suspicious mental diversion. It bust open the floodgates of creativity in mathematics and physics by giving rise to non-Euclidean geometry — an understanding of curved three-dimensional space which we now know is every bit as real as the geometry of flat surfaces, abounding in nature in everything from the blossom of a calla lily to the growth pattern of a coral reef to the fabric of spacetime of which everything that ever was and ever will be is woven. In fact, Einstein himself would not have been able to arrive at his relativity theory, nor bridge space and time into the revolutionary notion of spacetime, without non-Euclidean geometry.
Here, Lieber makes the conceptual leap that marks her books as singular achievements in thought — the leap from mathematics and the understanding of nature to psychology, sociology, and the understanding of human nature. Reflecting on the larger revolution in thought that non-Euclidean geometry embodied in its radical refusal to accept any truth as self-evident, she questions the notion of “eternal verities” — a term popularized by the eighteen-century French philosopher Claude Buffier to signify the aspects of human consciousness that allegedly furnish universal, indubitable moral and humane values. Considering how the relationship between creative limitation, freedom, humility, and responsibility shapes our values, Lieber writes:
Even though mathematics is
only a MAN-MADE enterprise,
man has done very well for himself
in this domain,
where he has
FREEDOM WITH RESPONSIBILITY —
though he has learned the
HUMILITY that goes with
knowing that he does
NOT have access to
“Self-evident truths” and
that he is NOT God —
yet he knows also that
he is not a mouse either,
but a man,
with all the
HUMAN DIGNITY and the
needed to develop
the wonderful domain of
The very dignity and ingenuity driving mathematics, Lieber points out in another lovely conceptual bridging of ideas, is also the motive force behind the central aspiration of human life, the one which Albert Camus saw as our moral obligation — the pursuit of happiness.
In the final chapter, titled “Life, Liberty and the Pursuit of Happiness,” Lieber recounts the principle of metamathematics demanding that a set of postulates within any system not contradict one another in order for the system to be self-consistent, and considers mathematics as a sandbox for the subterranean morality without which human life is unlivable:
[This] means of course that
CANNOT SERVE as an instrument of thought!
Now is not this statement
usually considered to be
a MORAL principle?
without it we cannot have
ANY satisfactory mathematical system,
nor ANY satisfactory system of thought —
indeed we cannot even PLAY a GAME properly
with CONTRADICTORY rules!
In a similar way,
I wish to make the point that
there are other important MORAL ideas
BEHIND THE SCENES,
without which there cannot be
ANY MATHEMATICS or SCIENCE.
And therefore, in this sense,
Science is NOT AT ALL AMORAL —
any more than one could have
a fruitful and non-trivial postulate set
which is not subject to
the METAmathematical demand for
One of these “behind-the-scenes” moral ideas, Lieber argues, is the notion of taking Life itself as a basic postulate:
there can be
no living thing —
no human race —
also of course
no music, no art,
I am not suggesting that we consider here WHETHER life is worth living,
whether it would make more “sense”
to commit suicide,
whether it is all just
“Sound and fury, signifying nothing.”
I am proposing that
LIFE be taken as a POSTULATE,
and therefore not subject to proof,
just like any other postulate.
But I propose to MODIFY this
and take more specifically as
THE PRESERVATION OF
LIFE FOR THE HUMAN RACE
is a goal of human effort.
This does not mean that
we are to go about
wantonly killing animals,
but to do this only when
it is necessary to support
HUMAN life —
for prevention of disease,
Indeed a horse or dog or other animals,
through their friendliness and sincerity,
might actually HELP to sustain
Man’s spirit and faith and even his life.
And I interpret this postulate
also to mean that
or “ganging up” on one little fox —
a hole gang of men and women
(and corrupting even horses and dogs
to help!) —
is really a cowardly act,
so unsportsmanlike that it is amazing
how this activity could ever be called
All this is by way of interpreting
the meaning to be given to Postulate I:
ACCEPTANCE of LIFE for the HUMAN RACE.
Surely everyone will accept the idea that
is definitely present,
behind the scenes,
in science or mathematics.
But this is not all.
For, I take this postulate to mean also
that we are not to limit it to
only a PART of the human race,
as Hitler did,
because this inevitably leads to WAR,
and in this day of
and CBR (chemical, biological, radiological)
this would certainly contradict
would it not?
May I say at the very outset that
the “SYSTEM” suggested here
makes no pretense of finality (!),
remembering how difficult it is,
EVEN in MATHEMATICS,
to have a postulate set which is
Nevertheless, one must go on,
one must TRY,
one must do one’s BEST,
as in mathematics and sciences.
And so, let us continue, in all humility,
to try to make
what can only at best be regarded as
in the hope that the basic idea —
that there is a MORALITY behind the scenes
in Mathematics and Science —
may prove to be helpful
and may be further
improved and strengthened
as time goes on.
Drawing on the consequence of the Second Law of Thermodynamics, which implies for living things an inevitable degradation toward destruction, Lieber offers additional postulates for the moral system that undergirds a thriving democracy:
Each INDIVIDUAL HUMAN BEING
must fight this “degeneration,”
must cling to LIFE as long as possible,
must grow and create —
physically and/or mentally.
And for this we need
We must all have the LIBERTY to
so grow and create,
without of course interfering
with each other’s growth,
This Freedom or LIBERTY
must be accompanied by
if it is not to lead to
individuals or groups
which would of course
CONTRADICT the other postulates.
All this is of course very DIFFICULT to do,
accepting LIFE without whimpering,
growing without interfering with
the growth of others,
it involves what Goethe called
But how can this be done?
It seems clear that we must now add
The PURSUIT of HAPPINESS
is a goal of human effort.
For without some happiness,
or at least the hope of some happiness
(the “pursuit” of happiness)
it would be impossible
to accept “cheerfully”
the program outlined above.
And such acceptance leads to
a calm, sane performance of our work,
in the spirit in which a mathematician
accepts the postulates of a system
and accepts his creative work
based on these —
accepting even the Great Difficulties
which he encounters
and is determined to conquer.
And I finally believe that
the results of such a formulation
will re-discover the conclusions
reached by the
great religious leaders and the
Lieber distills from this conception of the system “some invariants and some differences,” drawing from science and mathematics a working model for democracy:
(1) Invariants: LIFE — which demands
(a) Sufficient and proper food;
(b) Good Health;
(c) Education — both mental and physical;
(d) NO VIOLENCE!
(a real scientist does NOT go
into his laboratory with an AXE
with which to DESTROY his apparatus,
but rather with a well-developed BRAIN,
and lots of PATIENCE
with which to CREATE new things
which will be BENEFICIAL to the
This of course implies PEACE,
and better still
(e) FRIENDSHIP between K and K’!
(f) Humility — remembering that
he will NEVER know THE “truth”
(g) And all this of course
requires a great deal of
(2) Differences which will
NOT PREVENT both K and K’
from studying the universe
WITH EQUAL RIGHT and EQUAL
which is certainly
the clearest concept of
what DEMOCRACY is:
(a) Different coordinate systems
(b) Differences in color of skin!
(c) Different languages — or
other means of communication.
And please do not consider this program
as an unattainable “Utopia,”
for it really WORKS in
Science and Mathematics,
as we have seen,
and can also work in
if we would only
put our BEST EFFORTS into it,
fighting WARS —
(HOT or COLD)
or even PREPARING for wars —
HATING other people,