Mathematician Lillian Lieber on Infinity, Art, Science, the Meaning of Freedom, and What It Takes to Be a Finite But Complete Human Being
Mathematics and poetry converge in an ode to the “sweet reasonableness” at the heart of a psychologically balanced character.
By Maria Popova
“We’re all intrinsically of the same substance,” astrophysicist Janna Levin wrote in her exquisite inquiry into whether the universe is infinite or finite. “The fabric of the universe is just a coherent weave from the same threads that make our bodies. How much more absurd it becomes to believe that the universe, space and time could possibly be infinite when all of us are finite.” How, then, do we set aside this instinctual absurdity in order to grapple with the concept of infinity, which pushes our creaturely powers of comprehension past their limit so violently?
That’s what the mathematician and writer Lillian R. Lieber (July 26, 1886–July 11, 1986) set out to explore more than half a century earlier in the unusual and wonderful 1953 gem Infinity: Beyond the Beyond the Beyond (public library) — one of seventeen marvelous books she published in her hundred years, inviting the common reader into science with uncommon ingenuity and irresistible warmth. Emanating from Lieber’s discussion of infinity is a larger message about what it means, and what it takes, to be a finite but complete and balanced human being.
Lieber belongs to the “enchanter” category of great writers and was among the first generation of women mathematicians to hold academic positions in her role chairing the Department of Mathematics at Long Island University. She had a peculiar style resembling poetry, though she insisted it was not free verse but, rather, a deliberate way of breaking lines in order to speed up reading and intensify comprehension. (Curiously, I find her style to have precisely the opposite effect, which is why I’ve enjoyed it so tremendously — it does what poetry does, which is slow down the spinning world and dilate the pupil of attention so that the infinite becomes comprehensible.)
Populating her books is the character of T.C. Mits, “the Celebrated Man-in-the-Street,” and his mate, Wits, “the Woman-in-the-Street.” Accompanying Lieber’s writing are original line drawings by her own mate, the illustrator Hugh Gray Lieber.
Lieber’s work was so influential in elevating the popular science genre that even Albert Einstein himself heartily praised her book on relativity, yet many of her books have fallen out of print — no doubt because the depth, complexity, and visionary insurgency of her style don’t conform to the morass of formulaic mediocrity passing for popular science writing today.
Lieber frames the premise of Infinity in the charming opening verse — or, as she insisted, decidedly not-verse — of the second chapter:
Of course you know that
is a subject which
has always been of the deepest interest
to all people —
to the religious,
as well as to
(The Celebrated Man-in-the-Street)
and to his mate,
And it probably interests you,
or you would not be reading this book.
But it is in the first chapter, titled “Our Good Friend, Sam,” that Lieber’s genius for science, metaphor, and wordplay shines most brilliantly as she takes on everything from the symbiotic relationship between art and science to free will to the vital difference between common sense and truth to the evils of antisemitism and all exclusionary ideologies. (It is self-evident to point out that Lieber, a Jewish woman writing shortly after WWII in a climate of acute antisemitism and sexism, was, like any artist, bringing all of herself to her art.)
For those who have not met SAM before,
I wish to summarize
what his old acquaintances
may already know,
and then to tell to all of you
MORE about him.
In the first place,
the name “SAM”
was first derived from
Science, Art, Mathematics;
but I now find
the following interpretation
much more helpful:
the “S” stands for
OUR CONTACT WITH THE OUTSIDE WORLD;
please note that
I do NOT say
that “S” represents “facts” or “reality”,
the only knowledge we can have of
the outside world
is through our own senses or
“extended” senses —
like microscopes and telescopes et al
which help us to see better,
or radios, etc., which
help us to hear sounds
which we would otherwise
not be aware of at all,
and so on and so on.
But of course
there may be
many, many more things
in the world
which we do not yet perceive
either directly through our senses
or with the aid of
our wonderful inventions.
And so it would be
to speak as if we knew
what the outside world “really” is.
That is why I wish to give to “S”
the more modest interpretation
and emphasize that
it represents merely
that PART of the OUTSIDE world
which we are able to contact, —
and therefore even “S” has
a “human” element in it.
the “A” in SAM represents
our emotions, —
loves, hates, fears, etc. —
and of course is also
a “human” element.
And the “M” represents
our ability to draw inferences,
and hence includes
mathematics, logic, “common sense”,
and other ways in which
we mentally derive the “consequences”
before they hit us.
So the “M” too is
a “human” element.
Thus SAM is entirely human
though not an individual human being.
a Scientist utilizes the SAM within him,
for he must make
he must use his “intuition” (“A”)
to help him formulate
a good set of basic postUlates,
from which his “reasoning powers” (“M”)
will then help him to
which in turn must again be
“tested” (“S” again!) to see
if they are “correct”.
Perhaps you are thinking that
SAM and the Scientist
are really one and the same,
and that all I am doing is
to recommend that we all become
But you will soon see that
this is not the case at all.
in the first place,
it too often happens, —
alas and alack! —
that when a Scientist is
not actually engaged in doing
his scientific work,
he may “slip” and not use
his “S”, his “A”, and his “M”,
will bear watching,
like the rest of us.
In a sentiment which physicist and poet Alan Lightman would come to echo decades later in his beautiful meditation on the creative sympathies of art and science, Lieber adds:
So, you see,
being a SAMite and being a Scientist
are NOT one and the same.
a SAMite may not be a Scientist at all,
but an Artist!
For an Artist, too, must use
his “S” in order to “observe” the world,
his “A” (“intuition”) to sense
some basic ways to translate his
and his “M”
to derive his “results” in the form of
drawings, music, and so on.
Thus an Artist, too,
WHEN AT HIS BEST,
is a SAMite.
Perhaps Lieber’s most interesting, layered, and timelessly relevant discussion is of the concept of freedom, its misconceptions and mutations, and its implication for our private, public, and political lives:
Now consider a person
who is SOMETIMES or OFTEN like this:
He is evidently relying very heavily on
his “intuition”, his “hunches”, his “emotions”,
hardly checking to see whether
the “observations” of the outside world (“S”)
and his own reasoning powers (“M”)
show his “hunch” to be correct or not!
precious as our “intuition” may be,
it can go terribly “haywire”
if not checked and double-checked
by “S” and “M”.
Thus, a person who
habitually behaves like this
is allowing his “S” and “M” to
become practically atrophied,
and is the wild, “over-emotional” type,
who is not only a nuisance to have around,
but is hurting himself and
not allowing himself to become
adjusted to the world he lives in.
Such a person,
with an exaggerated “A”,
and atrophied “S” and “M”,
has a feeling of “freedom”,
of not being held down by “S” and “M”
(“facts” and “reason”) ;
but, as you can easily see
this makes for Anarchy,
for a lack of “self-control” —
and can lead
to fatty degeneration from
feeling “free” to eat all he wants;
to the D.T.’s from
feeling “free” to drink all he wants;
to accidents because
he feels “free” to drive as fast as he wants
and to “hog” the road;
to a sadistic lack of
consideration for others
by feeling “free” to
kick them in the teeth for “nuttin'”;
to antisocial “black market” practices
from a similar feeling of “freedom”,
giving “free” reign to the “A”
without the necessary consideration of “facts” (“S”) and “reason” (“M”).
Needless to say this is a
a NORMAL, HEALTHY FREEDOM of
the well-balanced SAM
which is so necessary in society
in which EACH individual
must be guided by the SAM within himself
in order to avoid conflict with
the SAM in someone else.
This is something that
a bully does not understand —
that if he acts like a pathological sAm,
he induces sAmite-ism in others,
as in mob violence;
this is indeed a horrible “ism”
that can destroy a society as well as
individuals in it.
Lieber proceeds to build on this taxonomy of psychological imbalances, reminiscent of neuroscience founding father Santiago Ramón y Cajal ‘s taxonomy of the “diseases of the will.” She turns to the next imbalance — the person blinded by isolated facts, unable to integrate them into an understanding of the big picture:
there is the Sam type:
he may be called the “tourist” type —
running around seeing this and that
but without the “imagination” (“A”)
or the reasoning power (“M”)
to put his observations together
with either heart (“A”) or mind (“M”),
but is concerned only with
ISOLATED BITS OF INFORMATION:
he is like the man who,
seeing a crowd had gathered,
wanted to know what happened.
and, when someone told him
“Ein Mann hat sich dem Kopf zerbrochen”
(It happened to be in Germany),
corrected the speaker’s grammar
and said “DEN Kopf!”
He knew his bit of grammar,
but what an inadequate reaction
under the circumstances.
don’t you think?
Next comes the flawed rationalizer, who misuses the tools of logic against reason:
And there is also the saM type —
one who can reason (“M”)
but starts with perhaps
some postulate (“A”) favoring murder.
Such a man would make
a wonderfully “rational”
homicidal maniac or crook
who could plan you a murder
calmly and rationally enough
to surprise any who are not familiar with
this sAM type of pathological case.
Lieber returns to the core purpose of her SAM metaphor and its relationship to the central question of the book:
Thus SAM gives us a way of
examining our own behavior
and that of others,
taking into account the “facts” (“S”),
and using imagination and sympathy (“A”)
in a rational way (“M”).
Are you perhaps thinking,
“Well, this may be interesting,
why all this talk about SAM,
when you are writing a book about
To which the answer is:
The yearning for Infinity,
is an “intuitive” yearning (“A”):
we look for support for it
in the physical world (“S”),
we try to reason about it (“M”), —
but only when we turn
the full light of SAM upon it
are we able to make
genuine progress in considering
In a brilliant and necessary caveat reminiscent of mathematician Kurt Gödel’s world-changing incompleteness theorems, which unsettled some of our most elemental assumptions by demonstrating the limits of logic turned unto itself, Lieber adds:
There is only one more point
I must make here:
even being a well-balanced
and not a pathological anti-SAMite
like SAM, etc. etc. —
is NECESSARY but NOT SUFFICIENT.
You will probably agree that
it is further necessary
to have our SAM up-to-date.
For he is a GROWING boy,
and what was good enough for him in 1800
is utterly inadequate in 1953;
and unless the “S” is up-to-date
and the postulates (“A”)
and reasoning (“M”)
are appropriately MODERN,
we cannot make proper
ADJUSTMENT in the world TODAY.
And ADJUSTMENT is what we must have.
For adjustment means
and that is a MINIMUM demand —
for, without survival
we need not bother to study anything
we just won’t be here to tell the tale.
In a passage of piercing pertinence today, as we watch various oppressive ideologies and tyrannical regimes engulf the globe, Lieber concludes by returning to the subject of freedom, its malformations, and its redemptions:
And so let me summarize
by saying that the
hurt not only themselves,
by getting “ulcers”, nervous breakdowns,
drinking excessively, etc. etc.,
but hurt others also,
for from their ranks are recruited
those who advocate war and destruction,
the homicidal maniacs, the greedy crooks,
the gamblers, the drunken drivers,
the liars, et al.
Just a word more about
you have seen above
the pathological idea of freedom,
but when you consider this important concept
from SAM’s WEll-BALANCED viewpoint,
you will see that,
from this point of view,
the “feeling” of freedom (“A”),
being supported on one side by “S”
(the “facts” of the outside world),
and on the other by “M”
(“sweet reasonableness”) —
is definitely NOT the
ANARCHICAL freedom of SAM,
but is a sort of
CONTROLLED FREEDOM —
controlled by facts and reason
and is therefore SELF-controlled
(by the SAM within us)
and hence implies
VOLUNTARY COOPERATION rather than FORCE.
Thus anyone who demands
“freedom unlimited” as his right,
is a pathological SAM,
a destructive creature;
you will find the
CONTROLLED FREEDOM of SAM
and you will feel refreshed to see
how genuine progress can be made
with this kind of freedom.
Infinity: Beyond the Beyond the Beyond is a thoroughly magnificent read in its totality. Pair it with the lovely children’s book Infinity and Me, then complement this particular fragment with Simone de Beauvoir, writing shortly before Lieber, on art, science, and freedom, and James Baldwin, writing shortly thereafter, on freedom and how we imprison ourselves.