From Euclid to Equality: Mathematician Lillian Lieber on How the Greatest Creative Revolution in Mathematics Illuminates the Core Ideals of Social Justice and Democracy
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.
Half a century before physicist Janna Levin wrote so beautifully about the limitations of logic in the pursuit of truth, Lieber zeroes in on a central misconception about mathematics:
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,
In a counterpoint to Camus, who considered the question of suicide the “one truly serious philosophical problem,” and with an allusional jab at Shakespeare, Lieber writes:
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
Echoing Alan Watts’s insistence that “Life and Reality are not things you can have for yourself unless you accord them to all others,” Lieber springboards from this postulate into the moral foundations of equality, human rights, and social justice:
All this is by way of interpreting
the meaning to be given to
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?
Lieber uses this first postulate as the basis for a larger, self-consistent “System of MORALITY,” just as Gauss, Lobachevsky, and Bolyai used the landmark revision of the parallel postulate as the basis for the revolution of non-Euclidean geometry. Four years before Joan Didion issued her timeless, increasingly timely admonition against mistaking self-righteousness for morality and a generation before physicist Richard Feynman asserted that “it is impossible to find an answer which someday will not be found to be wrong,” Lieber offers this moral model with conscientious humility:
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,
Complement Human Values and Science, Art and Mathematics with Carl Sagan on science and democracy, Robert Penn Warren on art and democracy, and Walt Whitman on literature and democracy, then revisit Lieber’s equally magnificent exploration of infinity and the meaning of freedom.