Continued at...Freedom, Responsibility, and Human Values: Mathematician Lilian 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.
“The joy of existence must be asserted in each one, at every instant,” Simone de Beauvoir wrote in her paradigmshifting 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 outofprint 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 nonEuclidean 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!
[…]
Furthermore,
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:
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!
[…]
Euclid’s system
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
from which,
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
simple statements
(called “postulates” in mathematics)
as:
“It shall be possible to draw
a straight line joining
any two points,”
and others like it.
From these
he derived many complicated theorems
(the “consequences”)
like the wellknown
Pythagorean Theorem,
and many, many others.
And, as we all know,
to “prove” any theorem
one must show how
to “derive” it from the postulates —
that is,
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”?
Obviously they
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
Mathematics
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 “selfevident 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.
Freedom, Responsibility, and Human Values: Mathematician Lilian Lieber on How the Greatest Creative Revolution in Mathematics Illuminates the Core Ideals of Social Justice and Democracy – Brain Pickings
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01172018, 06:09 AM #1
Human Values and Science, Art, and Mathematics
"Ce que nous connaissons est peu de chose, ce que nous ignorons est immense."
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