The Intellectual Revolution
That You’ve Never Heard of:
The Discovery of
Non-Euclidean Geometry
in the Age of Enlightenment
Revolutions You Have Heard Of
Non-Euclidean Geometry
Revolutions You Have Heard Of
Copernicus and Galileo: 16th–17th C.
Non-Euclidean Geometry
Revolutions You Have Heard Of
Copernicus and Galileo: 16th–17th C.
Darwin: Mid 19th C.
Non-Euclidean Geometry
Revolutions You Have Heard Of
Copernicus and Galileo: 16th–17th C.
Darwin: Mid 19th C.
Einstein: Early 20th C.
Non-Euclidean Geometry
One More!
Copernicus and Galileo: 16th–17th C.
Non-Euclidean Geometry: Early 19th C.
Darwin: Mid 19th C.
Einstein: Early 20th C.
Non-Euclidean Geometry
So What?
Non-Euclidean Geometry
So What?
The physical and biological sciences have repeatedly had
revolutions, but mathematics usually builds upon what was
already proved—this revolution is one of the few cases in
mathematics where previously held views were overturned
Non-Euclidean Geometry
So What?
The physical and biological sciences have repeatedly had
revolutions, but mathematics usually builds upon what was
already proved—this revolution is one of the few cases in
mathematics where previously held views were overturned
You might say this revolution is just about mathematics,
whereas the others are science, and as such are about the real
world
Non-Euclidean Geometry
So What?
The physical and biological sciences have repeatedly had
revolutions, but mathematics usually builds upon what was
already proved—this revolution is one of the few cases in
mathematics where previously held views were overturned
You might say this revolution is just about mathematics,
whereas the others are science, and as such are about the real
world
To the thinkers of early part of the Age of Enlightenment, and
before, mathematics is not separate from the sciences and
philosophy
Non-Euclidean Geometry
Plato, “Republic” book VII
Non-Euclidean Geometry
Plato, “Republic” book VII
“Then, my noble friend, geometry will draw the soul towards truth,
and create the spirit of philosophy, and raise up that which is now
unhappily allowed to fall down.”
Non-Euclidean Geometry
Plato, “Republic” book VII
“Then, my noble friend, geometry will draw the soul towards truth,
and create the spirit of philosophy, and raise up that which is now
unhappily allowed to fall down.”
“Nothing will be more likely to have such an effect.”
Non-Euclidean Geometry
Plato, “Republic” book VII
“Then, my noble friend, geometry will draw the soul towards truth,
and create the spirit of philosophy, and raise up that which is now
unhappily allowed to fall down.”
“Nothing will be more likely to have such an effect.”
“Then nothing should be more sternly laid down than that the
inhabitants of your fair city should by all means learn geometry.”
Non-Euclidean Geometry
Galileo
Non-Euclidean Geometry
Galileo
You have learned about his role in supporting the heliocentric
view
Non-Euclidean Geometry
Galileo
You have learned about his role in supporting the heliocentric
view
“Dialogue Concerning the Two Chief World Systems”
Non-Euclidean Geometry
Galileo
You have learned about his role in supporting the heliocentric
view
“Dialogue Concerning the Two Chief World Systems”
He did something even more important
Non-Euclidean Geometry
Galileo
You have learned about his role in supporting the heliocentric
view
“Dialogue Concerning the Two Chief World Systems”
He did something even more important
Though it is not as well known, because there was no
confrontation with the church
Non-Euclidean Geometry
Galileo
You have learned about his role in supporting the heliocentric
view
“Dialogue Concerning the Two Chief World Systems”
He did something even more important
Though it is not as well known, because there was no
confrontation with the church
“Discourses and Mathematical Demonstrations Relating to
Two New Sciences”
Non-Euclidean Geometry
Galileo
You have learned about his role in supporting the heliocentric
view
“Dialogue Concerning the Two Chief World Systems”
He did something even more important
Though it is not as well known, because there was no
confrontation with the church
“Discourses and Mathematical Demonstrations Relating to
Two New Sciences”
Galileo virtually invented modern science
Non-Euclidean Geometry
Galileo, “Opere Il Saggiatore”
Non-Euclidean Geometry
Galileo, “Opere Il Saggiatore”
“[The universe] cannot be read until we have learnt the
language and become familiar with the characters in which it
is written. It is written in mathematical language, and the
letters are triangles, circles and other geometrical figures,
without which means it is humanly impossible to comprehend
a single word.”
Non-Euclidean Geometry
Descartes
Non-Euclidean Geometry
Descartes
In an appendix to “Discourse on Method,” Descartes develops
what we now call analytic geometry, using x and y coordinates
to describe the plane, and relates geometry to algebra
Non-Euclidean Geometry
From “The Autobiography of Charles Darwin”
Non-Euclidean Geometry
From “The Autobiography of Charles Darwin”
During the three years which I spent at Cambridge my time
was wasted, as far as the academical studies were concerned,
as completely as at Edinburgh and at school. I attempted
mathematics, and even went during the summer of 1828 with
a private tutor (a very dull man) to Barmouth, but I got on
very slowly. The work was repugnant to me, chiefly from my
not being able to see any meaning in the early steps in
algebra. This impatience was very foolish, and in after years I
have deeply regretted that I did not proceed far enough at
least to understand something of the great leading principles
of mathematics, for men thus endowed seem to have an extra
sense.
Non-Euclidean Geometry
Non-Euclidean Geometry
Shelley, “Frankenstein” Chapter 3
Non-Euclidean Geometry
Shelley, “Frankenstein” Chapter 3
“ ‘I am happy,’ said M. Waldman, ‘to have gained a disciple;
and if your application equals your ability, I have no doubt of
your success. Chemistry is that branch of natural philosophy
in which the greatest improvements have been and may be
made: it is on that account that I have made it my peculiar
study; but at the same time I have not neglected the other
branches of science. A man would make but a very sorry
chemist if he attended to that department of human
knowledge alone. If your wish is to become really a man of
science, and not merely a petty experimentalist, I should
advise you to apply to every branch of natural philosophy,
including mathematics.’ ”
Non-Euclidean Geometry
Mathematics in the Age of Enlightenment
Non-Euclidean Geometry
Mathematics in the Age of Enlightenment
Great progress was made in mathematics in this period
Non-Euclidean Geometry
Mathematics in the Age of Enlightenment
Great progress was made in mathematics in this period
The greatest advance in mathematics in this period—and one
of the greatest achievements of the human mind ever—was
the invention of the calculus in the late 17th C. by Newton
and Leibniz
Non-Euclidean Geometry
Mathematics in the Age of Enlightenment
Great progress was made in mathematics in this period
The greatest advance in mathematics in this period—and one
of the greatest achievements of the human mind ever—was
the invention of the calculus in the late 17th C. by Newton
and Leibniz
The invention of the calculus was the culmination of an
evolutionary process, not a revolution
Non-Euclidean Geometry
Mathematics in the Age of Enlightenment
Great progress was made in mathematics in this period
The greatest advance in mathematics in this period—and one
of the greatest achievements of the human mind ever—was
the invention of the calculus in the late 17th C. by Newton
and Leibniz
The invention of the calculus was the culmination of an
evolutionary process, not a revolution
The roots of the calculus go back to Ancient Greece, and in
the process of inventing calculus, the strict Euclidean
approach was sidestepped, though not overturned
Non-Euclidean Geometry
Mathematics in the Age of Enlightenment
Great progress was made in mathematics in this period
The greatest advance in mathematics in this period—and one
of the greatest achievements of the human mind ever—was
the invention of the calculus in the late 17th C. by Newton
and Leibniz
The invention of the calculus was the culmination of an
evolutionary process, not a revolution
The roots of the calculus go back to Ancient Greece, and in
the process of inventing calculus, the strict Euclidean
approach was sidestepped, though not overturned
The discovery of non-Euclidean geometry was a true
revolution
Non-Euclidean Geometry
The Context for the Non-Euclidean Revolution
Non-Euclidean Geometry
The Context for the Non-Euclidean Revolution
Just as Galileo and Darwin fought against a doctrine based in
a text—the Bible—that was viewed as without error, the
non-Euclidean revolution fought against another such
text—Euclid’s “Elements”
Non-Euclidean Geometry
The Context for the Non-Euclidean Revolution
Just as Galileo and Darwin fought against a doctrine based in
a text—the Bible—that was viewed as without error, the
non-Euclidean revolution fought against another such
text—Euclid’s “Elements”
Euclid wasn’t viewed as divinely inspired, presumably, but was
viewed as nonetheless absolutely true, both in terms of what
he said and how he argued
Non-Euclidean Geometry
The Context for the Non-Euclidean Revolution
Just as Galileo and Darwin fought against a doctrine based in
a text—the Bible—that was viewed as without error, the
non-Euclidean revolution fought against another such
text—Euclid’s “Elements”
Euclid wasn’t viewed as divinely inspired, presumably, but was
viewed as nonetheless absolutely true, both in terms of what
he said and how he argued
It might seem amazing that people took geometry so seriously
back then, but ideas were valued back then—perhaps they
should be today as well—and mathematics was viewed as an
integral part of our conception of the world
Non-Euclidean Geometry
The Origins of Geometry
Non-Euclidean Geometry
The Origins of Geometry
Geometry was not discovered by the Ancient Greeks
Non-Euclidean Geometry
The Origins of Geometry
Geometry was not discovered by the Ancient Greeks
Geometrical ideas are found in a variety of ancient cultures,
for example Egypt, Babylon and China prior to the rise of
ancient Greece
Non-Euclidean Geometry
The Origins of Geometry
Geometry was not discovered by the Ancient Greeks
Geometrical ideas are found in a variety of ancient cultures,
for example Egypt, Babylon and China prior to the rise of
ancient Greece
Geometry was originally an empirical matter, and arose in
order to solve practical problems
Non-Euclidean Geometry
The Origins of Geometry
Geometry was not discovered by the Ancient Greeks
Geometrical ideas are found in a variety of ancient cultures,
for example Egypt, Babylon and China prior to the rise of
ancient Greece
Geometry was originally an empirical matter, and arose in
order to solve practical problems
The singular contribution of the ancient Greeks to the study
of geometry is the understanding that empirical justification of
geometric statements is not sufficient, and that it is necessary
to prove statements if we want to be sure that they are true
Non-Euclidean Geometry
Ancient Greek Geometry
Non-Euclidean Geometry
Ancient Greek Geometry
The first person generally attributed with proving theorems in
geometry is Thales of Miletus, about 624–547 BCE
Non-Euclidean Geometry
Ancient Greek Geometry
The first person generally attributed with proving theorems in
geometry is Thales of Miletus, about 624–547 BCE
The next great figure is Pythagoras of Samos, about 569–475
BCE
Non-Euclidean Geometry
Ancient Greek Geometry
The first person generally attributed with proving theorems in
geometry is Thales of Miletus, about 624–547 BCE
The next great figure is Pythagoras of Samos, about 569–475
BCE
Ancient Greek mathematics was put into its ultimate
deductive form by Euclid of Alexandria, about 325–265 BCE
Non-Euclidean Geometry
Euclid
Non-Euclidean Geometry
Euclid
In his work “The Elements,” Euclid took an already developed
large body of geometry, and gave it logical order by isolating a
few basic definitions and axioms, and then deducing
everything else from these definitions and axioms
Non-Euclidean Geometry
Euclid
In his work “The Elements,” Euclid took an already developed
large body of geometry, and gave it logical order by isolating a
few basic definitions and axioms, and then deducing
everything else from these definitions and axioms
Euclid, and the ancient Greeks generally, viewed geometry
rather differently than we currently do
Non-Euclidean Geometry
Euclid
In his work “The Elements,” Euclid took an already developed
large body of geometry, and gave it logical order by isolating a
few basic definitions and axioms, and then deducing
everything else from these definitions and axioms
Euclid, and the ancient Greeks generally, viewed geometry
rather differently than we currently do
The only quantities that were of interest were geometric ones,
and of those, geometric quantities that could be constructed
with straightedge and compass were of particular interest
Non-Euclidean Geometry
Example: The Pythagorean Theorem
Non-Euclidean Geometry
Example: The Pythagorean Theorem
In contemporary language, the Pythagorean Theorem states
that if three numbers a, b and c correspond to the two sides
and hypotenuse of a right triangle, then a2 + b2 = c2
Non-Euclidean Geometry
Example: The Pythagorean Theorem
In contemporary language, the Pythagorean Theorem states
that if three numbers a, b and c correspond to the two sides
and hypotenuse of a right triangle, then a2 + b2 = c2
To Euclid, such a statement about numbers would have made
no sense
Non-Euclidean Geometry
Example: The Pythagorean Theorem
In contemporary language, the Pythagorean Theorem states
that if three numbers a, b and c correspond to the two sides
and hypotenuse of a right triangle, then a2 + b2 = c2
To Euclid, such a statement about numbers would have made
no sense
Euclid’s version: “In right-angled triangles the square on the
side subtending the right angle is equal to the squares on the
sides containing the right angle.”
Non-Euclidean Geometry
Example: The Pythagorean Theorem
In contemporary language, the Pythagorean Theorem states
that if three numbers a, b and c correspond to the two sides
and hypotenuse of a right triangle, then a2 + b2 = c2
To Euclid, such a statement about numbers would have made
no sense
Euclid’s version: “In right-angled triangles the square on the
side subtending the right angle is equal to the squares on the
sides containing the right angle.”
When Euclid says that two planar figures (such as squares)
are equal, he is not making a statement about the numerical
values of their areas, but is saying that one figure can be cut
up into triangles, and reassembled into the other figure
Non-Euclidean Geometry
Non-Euclidean Geometry
The Importance of Euclid
Non-Euclidean Geometry
The Importance of Euclid
“The Elements” is one of the most important, and influential,
works of mathematics ever written—it is arguably one of the
most influential intellectual achievements of human
civilization as a whole, not just of mathematics
Non-Euclidean Geometry
The Importance of Euclid
“The Elements” is one of the most important, and influential,
works of mathematics ever written—it is arguably one of the
most influential intellectual achievements of human
civilization as a whole, not just of mathematics
Euclid’s treatment of geometry became the universally
accepted method of doing geometry for almost two millenia,
up till the 19th century
Non-Euclidean Geometry
The Importance of Euclid
“The Elements” is one of the most important, and influential,
works of mathematics ever written—it is arguably one of the
most influential intellectual achievements of human
civilization as a whole, not just of mathematics
Euclid’s treatment of geometry became the universally
accepted method of doing geometry for almost two millenia,
up till the 19th century
Not only was Euclidean geometry accepted as an
unquestionably true description of the real world, but Euclid’s
method of deductive reasoning was considered a model of
logical argumentation, and an example of reasoning that
produced theorems that were unquestionably true
Non-Euclidean Geometry
However
Non-Euclidean Geometry
However
Without discounting from the enormous intellectual and
historical importance of Euclid’s work, from a modern vantage
point we can identify three fundamental flaws in Euclid, the
resolutions of which did not take place until roughly two
millenia after Euclid’s time
Non-Euclidean Geometry
The Flaws in Euclid
Non-Euclidean Geometry
The Flaws in Euclid
Euclid tried to give precise definitions for geometric concepts
Non-Euclidean Geometry
The Flaws in Euclid
Euclid tried to give precise definitions for geometric concepts
Euclid tried to give a set of axioms that describe geometry
Non-Euclidean Geometry
The Flaws in Euclid
Euclid tried to give precise definitions for geometric concepts
Euclid tried to give a set of axioms that describe geometry
Euclid tried to prove all other results in geometry in a rigorous
fashion based only on his definitions and axioms
Non-Euclidean Geometry
The Flaws in Euclid
Euclid tried to give precise definitions for geometric concepts
Euclid tried to give a set of axioms that describe geometry
Euclid tried to prove all other results in geometry in a rigorous
fashion based only on his definitions and axioms
There are flaws in Euclid’s definitions
Non-Euclidean Geometry
The Flaws in Euclid
Euclid tried to give precise definitions for geometric concepts
Euclid tried to give a set of axioms that describe geometry
Euclid tried to prove all other results in geometry in a rigorous
fashion based only on his definitions and axioms
There are flaws in Euclid’s definitions
Euclid’s axioms are neither complete nor necessarily true
Non-Euclidean Geometry
The Flaws in Euclid
Euclid tried to give precise definitions for geometric concepts
Euclid tried to give a set of axioms that describe geometry
Euclid tried to prove all other results in geometry in a rigorous
fashion based only on his definitions and axioms
There are flaws in Euclid’s definitions
Euclid’s axioms are neither complete nor necessarily true
Some of Euclid’s proofs have gaps
Non-Euclidean Geometry
The Flaws in Euclid
Euclid tried to give precise definitions for geometric concepts
Euclid tried to give a set of axioms that describe geometry
Euclid tried to prove all other results in geometry in a rigorous
fashion based only on his definitions and axioms
There are flaws in Euclid’s definitions
Euclid’s axioms are neither complete nor necessarily true
Some of Euclid’s proofs have gaps
From a modern point of view, Euclid did not really achieve
the level of rigor that has traditionally been ascribed to him
Non-Euclidean Geometry
The Flaws in Euclid
Euclid tried to give precise definitions for geometric concepts
Euclid tried to give a set of axioms that describe geometry
Euclid tried to prove all other results in geometry in a rigorous
fashion based only on his definitions and axioms
There are flaws in Euclid’s definitions
Euclid’s axioms are neither complete nor necessarily true
Some of Euclid’s proofs have gaps
From a modern point of view, Euclid did not really achieve
the level of rigor that has traditionally been ascribed to him
None of this is to deny the greatness—and importance—of
Euclid’s achievement
Non-Euclidean Geometry
Some Definitions From Book I of “The Elements”
Non-Euclidean Geometry
Some Definitions From Book I of “The Elements”
1. A point is that which has no part
Non-Euclidean Geometry
Some Definitions From Book I of “The Elements”
1. A point is that which has no part
2. A line is breadthless length
Non-Euclidean Geometry
Some Definitions From Book I of “The Elements”
1. A point is that which has no part
2. A line is breadthless length
3. The extremities of a line are points
Non-Euclidean Geometry
Some Definitions From Book I of “The Elements”
1. A point is that which has no part
2. A line is breadthless length
3. The extremities of a line are points
4. A straight line is a line that lies evenly with the points on itself
Non-Euclidean Geometry
Nice Try
Non-Euclidean Geometry
Nice Try
Euclid wanted define all the mathematical terms that he uses,
such as point and line
Non-Euclidean Geometry
Nice Try
Euclid wanted define all the mathematical terms that he uses,
such as point and line
By modern standards, Euclid’s definitions are meaningless
Non-Euclidean Geometry
Nice Try
Euclid wanted define all the mathematical terms that he uses,
such as point and line
By modern standards, Euclid’s definitions are meaningless
And no other definitions would have accomplished what
Euclid wanted
Non-Euclidean Geometry
The Modern Approach
Non-Euclidean Geometry
The Modern Approach
In the modern axiomatic approach to geometry, we start with
some undefined terms (for example, “point” and “line”),
though we hypothesize various axiomatic properties for these
undefined terms (for example, we assume that every two
distinct points are contained in a unique line)
Non-Euclidean Geometry
The Modern Approach
In the modern axiomatic approach to geometry, we start with
some undefined terms (for example, “point” and “line”),
though we hypothesize various axiomatic properties for these
undefined terms (for example, we assume that every two
distinct points are contained in a unique line)
Although Euclid’s definitions do not work as stated, there are
modern axiom schemes, such as the ones by Hilbert in 1899
and Birkhoff in 1932, in which the definitions are worked out
properly, with the caveat that some terms are left undefined
Non-Euclidean Geometry
The Common Notions From Book I of “The
Elements”
Non-Euclidean Geometry
The Common Notions From Book I of “The
Elements”
1. Things that are equal to the same things are equal to one
another
Non-Euclidean Geometry
The Common Notions From Book I of “The
Elements”
1. Things that are equal to the same things are equal to one
another
2. If equals be added to equals, the wholes are equal
Non-Euclidean Geometry
The Common Notions From Book I of “The
Elements”
1. Things that are equal to the same things are equal to one
another
2. If equals be added to equals, the wholes are equal
3. If equals be subtracted from equals, the remainders are equal
Non-Euclidean Geometry
The Common Notions From Book I of “The
Elements”
1. Things that are equal to the same things are equal to one
another
2. If equals be added to equals, the wholes are equal
3. If equals be subtracted from equals, the remainders are equal
4. Things that coincide with one another are equal to one
another
Non-Euclidean Geometry
The Common Notions From Book I of “The
Elements”
1. Things that are equal to the same things are equal to one
another
2. If equals be added to equals, the wholes are equal
3. If equals be subtracted from equals, the remainders are equal
4. Things that coincide with one another are equal to one
another
5. The whole is greater than the part.
Non-Euclidean Geometry
The Postulates From Book I of “The Elements”
Non-Euclidean Geometry
The Postulates From Book I of “The Elements”
1. To draw a straight line from any point to any point
Non-Euclidean Geometry
The Postulates From Book I of “The Elements”
1. To draw a straight line from any point to any point
2. To produce a finite straight line continuously in a straight line
Non-Euclidean Geometry
The Postulates From Book I of “The Elements”
1. To draw a straight line from any point to any point
2. To produce a finite straight line continuously in a straight line
3. To describe a circle with any center and radius
Non-Euclidean Geometry
The Postulates From Book I of “The Elements”
1. To draw a straight line from any point to any point
2. To produce a finite straight line continuously in a straight line
3. To describe a circle with any center and radius
4. That all right angles are equal to one another
Non-Euclidean Geometry
The Postulates From Book I of “The Elements”
1. To draw a straight line from any point to any point
2. To produce a finite straight line continuously in a straight line
3. To describe a circle with any center and radius
4. That all right angles are equal to one another
5. That, if a straight line falling on two straight lines makes the
interior angles on the same side less than two right angles, the
straight lines, if produced indefinitely, meet on that side on
which are the angles less than the two right angles
Non-Euclidean Geometry
What Do the Common Notions and Postulates
Say?
Non-Euclidean Geometry
What Do the Common Notions and Postulates
Say?
The common notions are not about geometry per se
Non-Euclidean Geometry
What Do the Common Notions and Postulates
Say?
The common notions are not about geometry per se
The first three postulates involve straightedge and compass
constructions
Non-Euclidean Geometry
What Do the Common Notions and Postulates
Say?
The common notions are not about geometry per se
The first three postulates involve straightedge and compass
constructions
The First Postulate states that, given two different points, we
can construct (using a straightedge) a line segment from one
point to the other
Non-Euclidean Geometry
What Do the Common Notions and Postulates
Say?
The common notions are not about geometry per se
The first three postulates involve straightedge and compass
constructions
The First Postulate states that, given two different points, we
can construct (using a straightedge) a line segment from one
point to the other
The Second Postulate states that if we are given a line
segment (which is finite in length), we can extend the line
segment
Non-Euclidean Geometry
What Do the Common Notions and Postulates
Say?
The Third Postulate states that given a point, and given a
radius, we can draw the circle that has the given point as its
center, and has the given radius
Non-Euclidean Geometry
What Do the Common Notions and Postulates
Say?
The Third Postulate states that given a point, and given a
radius, we can draw the circle that has the given point as its
center, and has the given radius
The Fourth Postulate says that any two right angles, no
matter where they are located in the plane, are equal to one
another
Non-Euclidean Geometry
The Fifth Postulate
Non-Euclidean Geometry
The Fifth Postulate
The Fifth Postulate is much more complicated than the other
four
Non-Euclidean Geometry
The Fifth Postulate
The Fifth Postulate is much more complicated than the other
four
Suppose we are given a line, say k, and two lines that
intersect k, say m and n
Non-Euclidean Geometry
m
A
B
k
Non-Euclidean Geometry
n
The Fifth Postulate
The Fifth Postulate is much more complicated than the other
four
Suppose we are given a line, say k, and two lines that
intersect k, say m and n
Non-Euclidean Geometry
The Fifth Postulate
The Fifth Postulate is much more complicated than the other
four
Suppose we are given a line, say k, and two lines that
intersect k, say m and n
Let α and β be the angles shown in the figure
Non-Euclidean Geometry
The Fifth Postulate
The Fifth Postulate is much more complicated than the other
four
Suppose we are given a line, say k, and two lines that
intersect k, say m and n
Let α and β be the angles shown in the figure
The Fifth Postulate states that if α + β < 180◦ , then the lines
m and n will eventually intersect on the same side of k as α
and β
Non-Euclidean Geometry
More Problems
Non-Euclidean Geometry
More Problems
It turns out that the Euclid’s axioms are not sufficient to
prove his propositions
Non-Euclidean Geometry
More Problems
It turns out that the Euclid’s axioms are not sufficient to
prove his propositions
However, two thousand years after Euclid, mathematicians
have showed that the problems with Euclid’s definitions and
axioms can be fixed, and that Euclid’s arguments were
essentially correct, just missing some details
Non-Euclidean Geometry
More Problems
It turns out that the Euclid’s axioms are not sufficient to
prove his propositions
However, two thousand years after Euclid, mathematicians
have showed that the problems with Euclid’s definitions and
axioms can be fixed, and that Euclid’s arguments were
essentially correct, just missing some details
The real problem in Euclid is with the Fifth Postulate
Non-Euclidean Geometry
More Problems
It turns out that the Euclid’s axioms are not sufficient to
prove his propositions
However, two thousand years after Euclid, mathematicians
have showed that the problems with Euclid’s definitions and
axioms can be fixed, and that Euclid’s arguments were
essentially correct, just missing some details
The real problem in Euclid is with the Fifth Postulate
The first four are simple to state, and immediately believable
Non-Euclidean Geometry
More Problems
It turns out that the Euclid’s axioms are not sufficient to
prove his propositions
However, two thousand years after Euclid, mathematicians
have showed that the problems with Euclid’s definitions and
axioms can be fixed, and that Euclid’s arguments were
essentially correct, just missing some details
The real problem in Euclid is with the Fifth Postulate
The first four are simple to state, and immediately believable
The fifth is much longer to state, and, while certainly
believable, does not seem as immediately obvious as the first
four
Non-Euclidean Geometry
The Fifth Postulate, Again
Non-Euclidean Geometry
The Fifth Postulate, Again
Mathematicians throughout the centuries after Euclid noticed
this problem
Non-Euclidean Geometry
The Fifth Postulate, Again
Mathematicians throughout the centuries after Euclid noticed
this problem
If Euclid is to be viewed as demonstrating definitively true
knowledge, then his axioms must be indisputably true
Non-Euclidean Geometry
The Fifth Postulate, Again
Mathematicians throughout the centuries after Euclid noticed
this problem
If Euclid is to be viewed as demonstrating definitively true
knowledge, then his axioms must be indisputably true
The first four postulates seem quite convincing—The Fifth
Postulate, by contrast, does not seem quite as indisputable
Non-Euclidean Geometry
The Fifth Postulate, Again
Mathematicians throughout the centuries after Euclid noticed
this problem
If Euclid is to be viewed as demonstrating definitively true
knowledge, then his axioms must be indisputably true
The first four postulates seem quite convincing—The Fifth
Postulate, by contrast, does not seem quite as indisputable
It cannot simply be dropped—it is definitely used in some of
Euclid’s proofs
Non-Euclidean Geometry
The Fifth Postulate, Again
Mathematicians throughout the centuries after Euclid noticed
this problem
If Euclid is to be viewed as demonstrating definitively true
knowledge, then his axioms must be indisputably true
The first four postulates seem quite convincing—The Fifth
Postulate, by contrast, does not seem quite as indisputable
It cannot simply be dropped—it is definitely used in some of
Euclid’s proofs
A number of people after Euclid tried to deduce the Fifth
Postulate from the other four, and some claimed success
Non-Euclidean Geometry
The Fifth Postulate, Again
Mathematicians throughout the centuries after Euclid noticed
this problem
If Euclid is to be viewed as demonstrating definitively true
knowledge, then his axioms must be indisputably true
The first four postulates seem quite convincing—The Fifth
Postulate, by contrast, does not seem quite as indisputable
It cannot simply be dropped—it is definitely used in some of
Euclid’s proofs
A number of people after Euclid tried to deduce the Fifth
Postulate from the other four, and some claimed success
They were all mistaken
Non-Euclidean Geometry
A Good Mistake
Non-Euclidean Geometry
A Good Mistake
One person who attempted to prove the Fifth Postulate from
the other four was Giovanni Girolamo Saccheri 1667–1733
Non-Euclidean Geometry
A Good Mistake
One person who attempted to prove the Fifth Postulate from
the other four was Giovanni Girolamo Saccheri 1667–1733
He attempted to prove the Fifth Postulate by assuming that it
was false, and trying to reach a logical contradiction
Non-Euclidean Geometry
A Good Mistake
One person who attempted to prove the Fifth Postulate from
the other four was Giovanni Girolamo Saccheri 1667–1733
He attempted to prove the Fifth Postulate by assuming that it
was false, and trying to reach a logical contradiction
He proved many interesting things about geometry with the
Fifth Postulate false
Non-Euclidean Geometry
A Good Mistake
One person who attempted to prove the Fifth Postulate from
the other four was Giovanni Girolamo Saccheri 1667–1733
He attempted to prove the Fifth Postulate by assuming that it
was false, and trying to reach a logical contradiction
He proved many interesting things about geometry with the
Fifth Postulate false
He thought that he had eventually reached a logical
contradiction, but he was mistaken
Non-Euclidean Geometry
The Revolution will not be Publicized
Non-Euclidean Geometry
The Revolution will not be Publicized
Saccheri’s problem was conceptual—he couldn’t admit that
by assuming the Fifth Postulate is false, he obtained an
alternative, but perfectly good, geometry
Non-Euclidean Geometry
The Revolution will not be Publicized
Saccheri’s problem was conceptual—he couldn’t admit that
by assuming the Fifth Postulate is false, he obtained an
alternative, but perfectly good, geometry
The first person who seems to have realized that negating the
Fifth Postulate could lead to a perfectly good geometry was
Carl Friedrich Gauss, 1777–1855
Non-Euclidean Geometry
Non-Euclidean Geometry
The Revolution will not be Publicized
Saccheri’s problem was conceptual—he couldn’t admit that
by assuming the Fifth Postulate is false, he obtained an
alternative, but perfectly good, geometry
The person who realized that negating the Fifth Postulate
could lead to a perfectly good geometry was Carl Friedrich
Gauss, 1777–1855
Non-Euclidean Geometry
The Revolution will not be Publicized
Saccheri’s problem was conceptual—he couldn’t admit that
by assuming the Fifth Postulate is false, he obtained an
alternative, but perfectly good, geometry
The person who realized that negating the Fifth Postulate
could lead to a perfectly good geometry was Carl Friedrich
Gauss, 1777–1855
Gauss is widely viewed as one of the three greatest
mathematicians of all time
Non-Euclidean Geometry
The Revolution will not be Publicized
Saccheri’s problem was conceptual—he couldn’t admit that
by assuming the Fifth Postulate is false, he obtained an
alternative, but perfectly good, geometry
The person who realized that negating the Fifth Postulate
could lead to a perfectly good geometry was Carl Friedrich
Gauss, 1777–1855
Gauss is widely viewed as one of the three greatest
mathematicians of all time
He seems to have had this realization about geometry by
1817, and he worked out what a non-Euclidean geometry
would be like
Non-Euclidean Geometry
The Revolution will not be Publicized
Saccheri’s problem was conceptual—he couldn’t admit that
by assuming the Fifth Postulate is false, he obtained an
alternative, but perfectly good, geometry
The person who realized that negating the Fifth Postulate
could lead to a perfectly good geometry was Carl Friedrich
Gauss, 1777–1855
Gauss is widely viewed as one of the three greatest
mathematicians of all time
He seems to have had this realization about geometry by
1817, and he worked out what a non-Euclidean geometry
would be like
He kept his views about geometry secret
Non-Euclidean Geometry
Letter of Gauss to Bessel, 27 January 1829
Non-Euclidean Geometry
Letter of Gauss to Bessel, 27 January 1829
...There is another topic, one which for me is almost 40 years
old, that I have thought about from time to time in isolated
free hours, I mean the first principles of geometry; I dont
know if I have ever spoken to you about this. Also in this I
have further consolidated many things, and my conviction
that we cannot completely establish geometry a prioir has
become stronger. In the meantime it will likely be quite a
while before I get around to preparing my very extensive
investigations on this for publication; perhaps this will never
happen in my lifetime since I fear the cry of the Boetians if I
were to voice my views. ...
Non-Euclidean Geometry
Meanwhile
Non-Euclidean Geometry
Meanwhile
Gauss’ ideas were independently discovered by two young
mathematicians: János Bolyai, 1802–1860, and Nikolai
Ivanovich Lobachevsky, 1792–1856
Non-Euclidean Geometry
Non-Euclidean Geometry
Non-Euclidean Geometry
Bolyai
Non-Euclidean Geometry
Bolyai
Bolyai made his discoveries by 1824, though it was published
only in 1831 as an appendix to a book by his father, a friend
of Gauss
Non-Euclidean Geometry
Bolyai
Bolyai made his discoveries by 1824, though it was published
only in 1831 as an appendix to a book by his father, a friend
of Gauss
One the one hand
Non-Euclidean Geometry
Bolyai
Bolyai made his discoveries by 1824, though it was published
only in 1831 as an appendix to a book by his father, a friend
of Gauss
One the one hand
Letter of Gauss to Gerling, 14 February 1832
Non-Euclidean Geometry
Bolyai
Bolyai made his discoveries by 1824, though it was published
only in 1831 as an appendix to a book by his father, a friend
of Gauss
One the one hand
Letter of Gauss to Gerling, 14 February 1832
I consider this young geometer, v. Bolyai, to be a genius of
the first class....
Non-Euclidean Geometry
Bolyai
Bolyai made his discoveries by 1824, though it was published
only in 1831 as an appendix to a book by his father, a friend
of Gauss
One the one hand
Letter of Gauss to Gerling, 14 February 1832
I consider this young geometer, v. Bolyai, to be a genius of
the first class....
On the other hand
Non-Euclidean Geometry
Letter of Gauss to Bolyai’s father, 6 March 1832
Non-Euclidean Geometry
Letter of Gauss to Bolyai’s father, 6 March 1832
....Now for some remarks about the work of your son. If I
start by saying I cannot praise it then you will most likely be
taken aback; but I cannot do otherwise; to praise it would be
to praise myself; the entire contents of the work, the path
that your son has taken and the results to which it leads, are
almost perfectly in agreement with my own meditations, some
going back 30 35 years. In truth I am astonished. My
intention was not to release any of my own work in my
lifetime.
Non-Euclidean Geometry
Lobachevsky
Non-Euclidean Geometry
Lobachevsky
Lobachevsky published his work in an obscure Russian journal
in 1829, which neither Bolyai nor Gauss saw
Non-Euclidean Geometry
Lobachevsky
Lobachevsky published his work in an obscure Russian journal
in 1829, which neither Bolyai nor Gauss saw
He published his work in French in 1837, and in German in
1840, at which point it because more widely known
Non-Euclidean Geometry
Lobachevsky
Lobachevsky published his work in an obscure Russian journal
in 1829, which neither Bolyai nor Gauss saw
He published his work in French in 1837, and in German in
1840, at which point it because more widely known
Gauss was impressed
Non-Euclidean Geometry
Simultaneity
Non-Euclidean Geometry
Simultaneity
What do we make of the fact that three people independently
made the same astonishing discovery?
Non-Euclidean Geometry
Simultaneity
What do we make of the fact that three people independently
made the same astonishing discovery?
Nothing
Non-Euclidean Geometry
Simultaneity
What do we make of the fact that three people independently
made the same astonishing discovery?
Nothing
It is not very unusual in mathematics
Non-Euclidean Geometry
Simultaneity
What do we make of the fact that three people independently
made the same astonishing discovery?
Nothing
It is not very unusual in mathematics
Sometimes the field has developed to a point where new ideas
are waiting to be discovered
Non-Euclidean Geometry
Simultaneity
What do we make of the fact that three people independently
made the same astonishing discovery?
Nothing
It is not very unusual in mathematics
Sometimes the field has developed to a point where new ideas
are waiting to be discovered
And perhaps it is not a coincidence that this mathematical
revolution happened in a generally revolutionary period
Non-Euclidean Geometry
What did Gauss, Bolyai and Lobachevsky
Accomplish?
Non-Euclidean Geometry
What did Gauss, Bolyai and Lobachevsky
Accomplish?
There had never been a proof that Euclid’s axioms are
consistent—that is, that they have no internal contradictions
Non-Euclidean Geometry
What did Gauss, Bolyai and Lobachevsky
Accomplish?
There had never been a proof that Euclid’s axioms are
consistent—that is, that they have no internal contradictions
Gauss, Bolyai and Lobachevsky also did not prove that their
type of geometry was consistent, they simply asserted that
there is such geometry, and deduced various consequences
Non-Euclidean Geometry
What did Gauss, Bolyai and Lobachevsky
Accomplish?
There had never been a proof that Euclid’s axioms are
consistent—that is, that they have no internal contradictions
Gauss, Bolyai and Lobachevsky also did not prove that their
type of geometry was consistent, they simply asserted that
there is such geometry, and deduced various consequences
Later in the 19th C., it was proved that non-Euclidean
geometry is no more or less consistent than Euclidean
geometry, so that if one accepts Euclidean geometry, one has
to accept the alternatives as well
Non-Euclidean Geometry
Playfair’s Axiom
Non-Euclidean Geometry
Playfair’s Axiom
If m is a line, and A is a point not on m, then there is one
and only one line through A that is parallel to m
Non-Euclidean Geometry
Playfair’s Axiom
If m is a line, and A is a point not on m, then there is one
and only one line through A that is parallel to m
“Playfair’s Axiom” in not an axiom at all, but a theorem in
Euclidean geometry
Non-Euclidean Geometry
Playfair’s Axiom
If m is a line, and A is a point not on m, then there is one
and only one line through A that is parallel to m
“Playfair’s Axiom” in not an axiom at all, but a theorem in
Euclidean geometry
Playfair’s Axiom is not only a theorem in Euclidean geometry,
but, more strongly, it is equivalent to Euclid’s Fifth Postulate
Non-Euclidean Geometry
A
m
Non-Euclidean Geometry
A
n
m
Non-Euclidean Geometry
Non-Euclidean Geometry
Non-Euclidean Geometry
Non-Euclidean Geometry
Non-Euclidean geometry results from taking Euclid’s first four
postulates, but replacing the Fifth Postulate by something else
Non-Euclidean Geometry
Non-Euclidean Geometry
Non-Euclidean geometry results from taking Euclid’s first four
postulates, but replacing the Fifth Postulate by something else
It is easier to understand what happens in non-Euclidean
geometry if we consider alternatives to Playfair’s Axiom
Non-Euclidean Geometry
Non-Euclidean Geometry
Non-Euclidean geometry results from taking Euclid’s first four
postulates, but replacing the Fifth Postulate by something else
It is easier to understand what happens in non-Euclidean
geometry if we consider alternatives to Playfair’s Axiom
Suppose that m is a line, and A is a point not on m
Non-Euclidean Geometry
Non-Euclidean Geometry
Non-Euclidean geometry results from taking Euclid’s first four
postulates, but replacing the Fifth Postulate by something else
It is easier to understand what happens in non-Euclidean
geometry if we consider alternatives to Playfair’s Axiom
Suppose that m is a line, and A is a point not on m
If Playfair’s Axiom were not true, then there are two possible
cases
Non-Euclidean Geometry
Non-Euclidean Geometry
Non-Euclidean geometry results from taking Euclid’s first four
postulates, but replacing the Fifth Postulate by something else
It is easier to understand what happens in non-Euclidean
geometry if we consider alternatives to Playfair’s Axiom
Suppose that m is a line, and A is a point not on m
If Playfair’s Axiom were not true, then there are two possible
cases
(1) There is more than one line through A that is parallel to m
Non-Euclidean Geometry
Non-Euclidean Geometry
Non-Euclidean geometry results from taking Euclid’s first four
postulates, but replacing the Fifth Postulate by something else
It is easier to understand what happens in non-Euclidean
geometry if we consider alternatives to Playfair’s Axiom
Suppose that m is a line, and A is a point not on m
If Playfair’s Axiom were not true, then there are two possible
cases
(1) There is more than one line through A that is parallel to m
That is called hyperbolic geometry
Non-Euclidean Geometry
Non-Euclidean Geometry
Non-Euclidean geometry results from taking Euclid’s first four
postulates, but replacing the Fifth Postulate by something else
It is easier to understand what happens in non-Euclidean
geometry if we consider alternatives to Playfair’s Axiom
Suppose that m is a line, and A is a point not on m
If Playfair’s Axiom were not true, then there are two possible
cases
(1) There is more than one line through A that is parallel to m
That is called hyperbolic geometry
(2) There is no line through A that is parallel to m
Non-Euclidean Geometry
Non-Euclidean Geometry
Non-Euclidean geometry results from taking Euclid’s first four
postulates, but replacing the Fifth Postulate by something else
It is easier to understand what happens in non-Euclidean
geometry if we consider alternatives to Playfair’s Axiom
Suppose that m is a line, and A is a point not on m
If Playfair’s Axiom were not true, then there are two possible
cases
(1) There is more than one line through A that is parallel to m
That is called hyperbolic geometry
(2) There is no line through A that is parallel to m
That is called spherical geometry
Non-Euclidean Geometry
Non-Euclidean Geometry Redux
Non-Euclidean Geometry
Non-Euclidean Geometry Redux
Euclid’s Fifth Postulate implies that the sum of the angles in
any triangle is 180◦
Non-Euclidean Geometry
Non-Euclidean Geometry Redux
Euclid’s Fifth Postulate implies that the sum of the angles in
any triangle is 180◦
In hyperbolic geometry, the sum of the angles in a triangle is
always less than 180◦ (the precise sum can vary from triangle
to triangle)
Non-Euclidean Geometry
Non-Euclidean Geometry Redux
Euclid’s Fifth Postulate implies that the sum of the angles in
any triangle is 180◦
In hyperbolic geometry, the sum of the angles in a triangle is
always less than 180◦ (the precise sum can vary from triangle
to triangle)
In spherical geometry, the sum of the angles in a triangle is
always greater than 180◦ (again, the precise sum can vary
from triangle to triangle)
Non-Euclidean Geometry
Non-Euclidean Geometry Once Again
Non-Euclidean Geometry
Non-Euclidean Geometry Once Again
It is hard to imagine how hyperbolic or spherical geometry
would work if we use the familiar sort of straight lines found
in the plane
Non-Euclidean Geometry
Non-Euclidean Geometry Once Again
It is hard to imagine how hyperbolic or spherical geometry
would work if we use the familiar sort of straight lines found
in the plane
But why restrict ourselves only to the most familiar situation?
Non-Euclidean Geometry
Non-Euclidean Geometry Once Again
It is hard to imagine how hyperbolic or spherical geometry
would work if we use the familiar sort of straight lines found
in the plane
But why restrict ourselves only to the most familiar situation?
On the surface of a sphere, where “straight lines” are great
circles, it is seen that the sum of the angles in a triangle is
greater than 180◦ , and hence is spherical geometry
Non-Euclidean Geometry
Non-Euclidean Geometry Once Again
It is hard to imagine how hyperbolic or spherical geometry
would work if we use the familiar sort of straight lines found
in the plane
But why restrict ourselves only to the most familiar situation?
On the surface of a sphere, where “straight lines” are great
circles, it is seen that the sum of the angles in a triangle is
greater than 180◦ , and hence is spherical geometry
Models for hyperbolic geometry can also be found
Non-Euclidean Geometry
Non-Euclidean Geometry
Non-Euclidean Geometry
Geometry Liberated
Non-Euclidean Geometry
Geometry Liberated
If geometry is not necessarily as Euclid said, then many other
possibilities open up
Non-Euclidean Geometry
Geometry Liberated
If geometry is not necessarily as Euclid said, then many other
possibilities open up
Even staying within the realm of Euclidean geometry, new
approaches were developed soon after the discovery of
non-Euclidean geometry
Non-Euclidean Geometry
Geometry Liberated
If geometry is not necessarily as Euclid said, then many other
possibilities open up
Even staying within the realm of Euclidean geometry, new
approaches were developed soon after the discovery of
non-Euclidean geometry
One new approach was via the recently developed theory of
groups
Non-Euclidean Geometry
Geometry Liberated
If geometry is not necessarily as Euclid said, then many other
possibilities open up
Even staying within the realm of Euclidean geometry, new
approaches were developed soon after the discovery of
non-Euclidean geometry
One new approach was via the recently developed theory of
groups
This approach led to a thorough study of symmetry
Non-Euclidean Geometry
Geometry Liberated
If geometry is not necessarily as Euclid said, then many other
possibilities open up
Even staying within the realm of Euclidean geometry, new
approaches were developed soon after the discovery of
non-Euclidean geometry
One new approach was via the recently developed theory of
groups
This approach led to a thorough study of symmetry
Three-dimentional crystallographic symmetry was classified in
the 1880’s
Non-Euclidean Geometry
Variable Geometry
Non-Euclidean Geometry
Variable Geometry
Euclidean, spherical and hyperbolic geometries are constant
geometries
Non-Euclidean Geometry
Variable Geometry
Euclidean, spherical and hyperbolic geometries are constant
geometries
It is also possible to have variable geometry
Non-Euclidean Geometry
Variable Geometry
Constant Geometry
Non-Euclidean Geometry
Higher Dimensions
Non-Euclidean Geometry
Higher Dimensions
Euclid discusses planar and spatial geometry
Non-Euclidean Geometry
Higher Dimensions
Euclid discusses planar and spatial geometry
Soon after the discovery of non-Euclidean geometry, the idea
of higher dimensional geometry arose in the work of a few
authors
Non-Euclidean Geometry
Higher Dimensions
Euclid discusses planar and spatial geometry
Soon after the discovery of non-Euclidean geometry, the idea
of higher dimensional geometry arose in the work of a few
authors
The most important contributor to the development of
higher-dimensional geometry was Georg Friedrich Bernhard
Riemann, 1826-1866, a student of Gauss
Non-Euclidean Geometry
Non-Euclidean Geometry
Higher Dimensions
Euclid discusses planar and spatial geometry
Soon after the discovery of non-Euclidean geometry, the idea
of higher dimensional geometry arose in the work of a few
authors
The most important contributor to the development of
higher-dimensional geometry was Georg Friedrich Bernhard
Riemann, 1826–1866, a student of Gauss
Non-Euclidean Geometry
Higher Dimensions
Euclid discusses planar and spatial geometry
Soon after the discovery of non-Euclidean geometry, the idea
of higher dimensional geometry arose in the work of a few
authors
The most important contributor to the development of
higher-dimensional geometry was Georg Friedrich Bernhard
Riemann, 1826–1866, a student of Gauss
Riemann’s ideas on higher-dimensional geometry were
delivered in the inaugural lecture “Über die Hypothesen
welche der Geometrie zu Grunde liegen” (On the hypotheses
that lie at the foundations of geometry), delivered on 10 June
1854
Non-Euclidean Geometry
Riemann’s Inaugural Lecture
Non-Euclidean Geometry
Riemann’s Inaugural Lecture
Riemann introduced the notion of variable geometry in higher
dimensions
Non-Euclidean Geometry
Riemann’s Inaugural Lecture
Riemann introduced the notion of variable geometry in higher
dimensions
Riemann introduced the idea of the metric tensor and the
curvature tensor
Non-Euclidean Geometry
Riemann’s Inaugural Lecture
Riemann introduced the notion of variable geometry in higher
dimensions
Riemann introduced the idea of the metric tensor and the
curvature tensor
We now refer to Riemann’s type of geometry as Riemannian
geometry
Non-Euclidean Geometry
Riemann’s Inaugural Lecture
Riemann introduced the notion of variable geometry in higher
dimensions
Riemann introduced the idea of the metric tensor and the
curvature tensor
We now refer to Riemann’s type of geometry as Riemannian
geometry
“Among Riemann’s audience, only Gauss was able to
appreciate the depth of Riemann’s thoughts. ... The lecture
exceeded all his expectations and greatly surprised him.
Returning to the faculty meeting, he spoke with the greatest
praise and rare enthusiasm to Wilhelm Weber about the depth
of the thoughts that Riemann had presented.”
Non-Euclidean Geometry
The Impact Continues into the 20th C.
Non-Euclidean Geometry
The Impact Continues into the 20th C.
Geometry and its descendants are flourishing—the golden era
of mathematics research is now!
Non-Euclidean Geometry
The Impact Continues into the 20th C.
Geometry and its descendants are flourishing—the golden era
of mathematics research is now!
Differential geometry
Non-Euclidean Geometry
The Impact Continues into the 20th C.
Geometry and its descendants are flourishing—the golden era
of mathematics research is now!
Differential geometry
Algebraic geometry
Non-Euclidean Geometry
The Impact Continues into the 20th C.
Geometry and its descendants are flourishing—the golden era
of mathematics research is now!
Differential geometry
Algebraic geometry
Topology
Non-Euclidean Geometry
The Impact Continues into the 20th C.
Geometry and its descendants are flourishing—the golden era
of mathematics research is now!
Differential geometry
Algebraic geometry
Topology
Hyperbolic three-manifolds and the Poincaré Conjecture
Non-Euclidean Geometry
The Impact Continues into the 20th C.
Geometry and its descendants are flourishing—the golden era
of mathematics research is now!
Differential geometry
Algebraic geometry
Topology
Hyperbolic three-manifolds and the Poincaré Conjecture
String theory and other aspects of mathematical physics
Non-Euclidean Geometry
The Impact Continues into the 20th C.
Geometry and its descendants are flourishing—the golden era
of mathematics research is now!
Differential geometry
Algebraic geometry
Topology
Hyperbolic three-manifolds and the Poincaré Conjecture
String theory and other aspects of mathematical physics
One person in the 20th C. who thought about the nature of
the real world geometrically was Albert Einstein, 1879–1955
Non-Euclidean Geometry
Non-Euclidean Geometry
Einstein
Non-Euclidean Geometry
Einstein
There are two parts to the theory of relativity: special
relativity in 1905, and general relativity in 1916
Non-Euclidean Geometry
Einstein
There are two parts to the theory of relativity: special
relativity in 1905, and general relativity in 1916
Special relativity does not use sophisticated mathematics
Non-Euclidean Geometry
Einstein
There are two parts to the theory of relativity: special
relativity in 1905, and general relativity in 1916
Special relativity does not use sophisticated mathematics
Einstein was not a big fan of mathematics
Non-Euclidean Geometry
Einstein
There are two parts to the theory of relativity: special
relativity in 1905, and general relativity in 1916
Special relativity does not use sophisticated mathematics
Einstein was not a big fan of mathematics
“I don’t believe in mathematics.”
Non-Euclidean Geometry
Einstein
There are two parts to the theory of relativity: special
relativity in 1905, and general relativity in 1916
Special relativity does not use sophisticated mathematics
Einstein was not a big fan of mathematics
“I don’t believe in mathematics.”
“As far as the laws of mathematics refer to reality, they are
not certain; and as far as they are certain, they do not refer to
reality.”
Non-Euclidean Geometry
Einstein
There are two parts to the theory of relativity: special
relativity in 1905, and general relativity in 1916
Special relativity does not use sophisticated mathematics
Einstein was not a big fan of mathematics
“I don’t believe in mathematics.”
“As far as the laws of mathematics refer to reality, they are
not certain; and as far as they are certain, they do not refer to
reality.”
“Reading after a certain time diverts the mind too much from
its creative pursuits. Any man who reads too much and uses
his own brain too little falls into lazy habits of thinking.”
Non-Euclidean Geometry
General Relativity
Non-Euclidean Geometry
General Relativity
General relativity deals with gravity
Non-Euclidean Geometry
General Relativity
General relativity deals with gravity
Einstein at first had the intuitive idea for general relativity,
which involves the idea of curved space-time, but he did not
have the mathematical tools to formulate his ideas
Non-Euclidean Geometry
General Relativity
General relativity deals with gravity
Einstein at first had the intuitive idea for general relativity,
which involves the idea of curved space-time, but he did not
have the mathematical tools to formulate his ideas
He sought help from a mathematician, who brought
Riemannian geometry to his attention
Non-Euclidean Geometry
General Relativity
General relativity deals with gravity
Einstein at first had the intuitive idea for general relativity,
which involves the idea of curved space-time, but he did not
have the mathematical tools to formulate his ideas
He sought help from a mathematician, who brought
Riemannian geometry to his attention
General relativity was then formulated in the framework of
Riemannain geometry, which Riemann had initiated 50 years
earlier
Non-Euclidean Geometry
General Relativity
General relativity deals with gravity
Einstein at first had the intuitive idea for general relativity,
which involves the idea of curved space-time, but he did not
have the mathematical tools to formulate his ideas
He sought help from a mathematician, who brought
Riemannian geometry to his attention
General relativity was then formulated in the framework of
Riemannain geometry, which Riemann had initiated 50 years
earlier
Without the non-Euclidean revolution, the mathematical tools
for general relativity would not have existed for Einstein to use
Non-Euclidean Geometry
The Nature of the Non-Euclidean Revolution
Non-Euclidean Geometry
The Nature of the Non-Euclidean Revolution
Up till the non-Euclidean revolution, it was thought that there
was only one type of geometry, and that this geometry
described the real world
Non-Euclidean Geometry
The Nature of the Non-Euclidean Revolution
Up till the non-Euclidean revolution, it was thought that there
was only one type of geometry, and that this geometry
described the real world
If there are actually many possible geometries, then they
cannot all describe the real world
Non-Euclidean Geometry
The Nature of the Non-Euclidean Revolution
Up till the non-Euclidean revolution, it was thought that there
was only one type of geometry, and that this geometry
described the real world
If there are actually many possible geometries, then they
cannot all describe the real world
In his inaugural lecture, Riemann raised the question of the
relation between mathematical space and physical space
Non-Euclidean Geometry
The Nature of the Non-Euclidean Revolution
Up till the non-Euclidean revolution, it was thought that there
was only one type of geometry, and that this geometry
described the real world
If there are actually many possible geometries, then they
cannot all describe the real world
In his inaugural lecture, Riemann raised the question of the
relation between mathematical space and physical space
Mathematics can describe the possible geometries of the world
Non-Euclidean Geometry
The Nature of the Non-Euclidean Revolution
Up till the non-Euclidean revolution, it was thought that there
was only one type of geometry, and that this geometry
described the real world
If there are actually many possible geometries, then they
cannot all describe the real world
In his inaugural lecture, Riemann raised the question of the
relation between mathematical space and physical space
Mathematics can describe the possible geometries of the world
To determine which geometry describes our world, Riemann
concludes his lecture by saying: “This leads us into the
domain of another science, of physic, into which the object of
this work does not allow us to go to-day.”
Non-Euclidean Geometry
The Shape of the Universe
Non-Euclidean Geometry
The Shape of the Universe
The original non-Euclidean revolution was about
2-dimensional geometry
Non-Euclidean Geometry
The Shape of the Universe
The original non-Euclidean revolution was about
2-dimensional geometry
There are also a variety of 3-dimensional geometries
Non-Euclidean Geometry
The Shape of the Universe
The original non-Euclidean revolution was about
2-dimensional geometry
There are also a variety of 3-dimensional geometries
We assume that the universe has three physical dimensions
Non-Euclidean Geometry
The Shape of the Universe
The original non-Euclidean revolution was about
2-dimensional geometry
There are also a variety of 3-dimensional geometries
We assume that the universe has three physical dimensions
Mathematics can say something about the possible types of
geometry that the 3-dimensional universe can have
Non-Euclidean Geometry
The Shape of the Universe
The original non-Euclidean revolution was about
2-dimensional geometry
There are also a variety of 3-dimensional geometries
We assume that the universe has three physical dimensions
Mathematics can say something about the possible types of
geometry that the 3-dimensional universe can have
Would it be possible to know the geometry of the universe?
Non-Euclidean Geometry
The Shape of the Universe
The original non-Euclidean revolution was about
2-dimensional geometry
There are also a variety of 3-dimensional geometries
We assume that the universe has three physical dimensions
Mathematics can say something about the possible types of
geometry that the 3-dimensional universe can have
Would it be possible to know the geometry of the universe?
Some people, such as Jeffrey Weeks, are actively trying to
find out
Non-Euclidean Geometry
The Shape of the Universe
The original non-Euclidean revolution was about
2-dimensional geometry
There are also a variety of 3-dimensional geometries
We assume that the universe has three physical dimensions
Mathematics can say something about the possible types of
geometry that the 3-dimensional universe can have
Would it be possible to know the geometry of the universe?
Some people, such as Jeffrey Weeks, are actively trying to
find out
If they succeed, it will be an achievement on par with the
work of Copernicus
Non-Euclidean Geometry
The Nature of the Non-Euclidean Revolution
Part II
Non-Euclidean Geometry
The Nature of the Non-Euclidean Revolution
Part II
It led to a flourishing of new approaches to geometry
Non-Euclidean Geometry
The Nature of the Non-Euclidean Revolution
Part II
It led to a flourishing of new approaches to geometry
It changed our understanding of the relation of mathematics
to the physical world
Non-Euclidean Geometry
The Nature of the Non-Euclidean Revolution
Part II
It led to a flourishing of new approaches to geometry
It changed our understanding of the relation of mathematics
to the physical world
And yet, it did not undermine the belief in the certainty of
mathematics, in the knowability of mathematics, nor of the
value of mathematics in the study of the physical world
Non-Euclidean Geometry
The Nature of the Non-Euclidean Revolution
Part II
It led to a flourishing of new approaches to geometry
It changed our understanding of the relation of mathematics
to the physical world
And yet, it did not undermine the belief in the certainty of
mathematics, in the knowability of mathematics, nor of the
value of mathematics in the study of the physical world
The non-Euclidean Revolution was a product of the
Enlightenment mindset
Non-Euclidean Geometry
The Next Revolution in Mathematics
Non-Euclidean Geometry
The Next Revolution in Mathematics
There was a revolution that undermined our faith in the
knowability of mathematics, due to Kurt Gödel, 1906–1978
Non-Euclidean Geometry
Non-Euclidean Geometry
The Next Revolution in Mathematics
There was a revolution that undermined our faith in the
knowability of mathematics, due to Kurt Gödel, 1906–1978
Non-Euclidean Geometry
The Next Revolution in Mathematics
There was a revolution that undermined our faith in the
knowability of mathematics, due to Kurt Gödel, 1906–1978
Gödel showed, essentially, that in any mathematical system
that contains arithmetic, there are facts that can be proved
neither true nor false
Non-Euclidean Geometry
The Next Revolution in Mathematics
There was a revolution that undermined our faith in the
knowability of mathematics, due to Kurt Gödel, 1906–1978
Gödel showed, essentially, that in any mathematical system
that contains arithmetic, there are facts that can be proved
neither true nor false
Gödel’s result, which was a very precise mathematical
theorem, seems to fit in with other questions that have been
raised in the 20th C. about the limits of knowledge
Non-Euclidean Geometry
And Yet
Non-Euclidean Geometry
And Yet
The non-Euclidean revolution, which did not change our
understanding of the limits of knowledge of mathematics,
caused a great flourishing of new mathematical activity
Non-Euclidean Geometry
And Yet
The non-Euclidean revolution, which did not change our
understanding of the limits of knowledge of mathematics,
caused a great flourishing of new mathematical activity
The revolution of Gödel, which had great philosophical
impact, had little practical effect upon the work of research
mathematicians
Non-Euclidean Geometry
And Yet
The non-Euclidean revolution, which did not change our
understanding of the limits of knowledge of mathematics,
caused a great flourishing of new mathematical activity
The revolution of Gödel, which had great philosophical
impact, had little practical effect upon the work of research
mathematicians
What is the relation between our understanding of how
knowledge is gained with the actual pursuit of knowledge?
Non-Euclidean Geometry
And Yet
The non-Euclidean revolution, which did not change our
understanding of the limits of knowledge of mathematics,
caused a great flourishing of new mathematical activity
The revolution of Gödel, which had great philosophical
impact, had little practical effect upon the work of research
mathematicians
What is the relation between our understanding of how
knowledge is gained with the actual pursuit of knowledge?
An important question, but, to paraphrase Riemann
Non-Euclidean Geometry
And Yet
The non-Euclidean revolution, which did not change our
understanding of the limits of knowledge of mathematics,
caused a great flourishing of new mathematical activity
The revolution of Gödel, which had great philosophical
impact, had little practical effect upon the work of research
mathematicians
What is the relation between our understanding of how
knowledge is gained with the actual pursuit of knowledge?
An important question, but, to paraphrase Riemann
This leads us into the domain of another science, of
philosophy, into which the object of this work does not allow
us to go to-day.
Non-Euclidean Geometry
The End