Mathematical Creation
Mathematics 15: Lecture 25
Dan Sloughter
Furman University
November 29, 2006
Dan Sloughter (Furman University)
Mathematical Creation
November 29, 2006
1/7
Henri Poincaré
I
1854 - 1912
Dan Sloughter (Furman University)
Mathematical Creation
November 29, 2006
2/7
Henri Poincaré
I
1854 - 1912
I
The last universalist: made contributions in analysis, probability,
topology, mathematical physics, thermodynamics, and more
Dan Sloughter (Furman University)
Mathematical Creation
November 29, 2006
2/7
Henri Poincaré
I
1854 - 1912
I
The last universalist: made contributions in analysis, probability,
topology, mathematical physics, thermodynamics, and more
I
One of the creators of special relativity (along with Albert Einstein
and Hendrik Lorentz)
Dan Sloughter (Furman University)
Mathematical Creation
November 29, 2006
2/7
Henri Poincaré
I
1854 - 1912
I
The last universalist: made contributions in analysis, probability,
topology, mathematical physics, thermodynamics, and more
I
One of the creators of special relativity (along with Albert Einstein
and Hendrik Lorentz)
I
Proved that the three-body problem was not exactly solvable (and
discovered the mathematical theory of chaos)
Dan Sloughter (Furman University)
Mathematical Creation
November 29, 2006
2/7
Why are there many who do not understand mathematics?
I
Isn’t mathematics just following logical rules, which any sane person
could do?
Dan Sloughter (Furman University)
Mathematical Creation
November 29, 2006
3/7
Why are there many who do not understand mathematics?
I
Isn’t mathematics just following logical rules, which any sane person
could do?
I
Is it that some do not have a good enough memory?
Dan Sloughter (Furman University)
Mathematical Creation
November 29, 2006
3/7
Why are there many who do not understand mathematics?
I
Isn’t mathematics just following logical rules, which any sane person
could do?
I
Is it that some do not have a good enough memory?
I
Is a special aptitude for mathematics due “only to a very sure
memory or to a prodigious force of attention?” (page 2042)
Dan Sloughter (Furman University)
Mathematical Creation
November 29, 2006
3/7
Just a series of syllogisms?
I
From pages 2042 - 2043:
Dan Sloughter (Furman University)
Mathematical Creation
November 29, 2006
4/7
Just a series of syllogisms?
I
From pages 2042 - 2043:
A mathematical demonstration is not a simple juxtaposition
of syllogisms, it is syllogisms placed in a certain order, and
the order in which these elements are placed is much more
important than the elements themselves. If I have a feeling,
the intuition, so to speak, of this order, so as to perceive at
a glance the reasoning as a whole, I need no longer fear lest
I forget one of the elements, for each of them will take its
allotted place in the array, and that without any effort of
memory on my part.
Dan Sloughter (Furman University)
Mathematical Creation
November 29, 2006
4/7
Just a series of syllogisms?
I
From pages 2042 - 2043:
A mathematical demonstration is not a simple juxtaposition
of syllogisms, it is syllogisms placed in a certain order, and
the order in which these elements are placed is much more
important than the elements themselves. If I have a feeling,
the intuition, so to speak, of this order, so as to perceive at
a glance the reasoning as a whole, I need no longer fear lest
I forget one of the elements, for each of them will take its
allotted place in the array, and that without any effort of
memory on my part.
I
That is: “It seem to me than, in repeating a reasoning learned, that I
could have invented it.”
Dan Sloughter (Furman University)
Mathematical Creation
November 29, 2006
4/7
What is mathematical creation?
I
Mathematical creation “does not consist in making new combinations
with mathematical entities already known.” (page 2043)
Dan Sloughter (Furman University)
Mathematical Creation
November 29, 2006
5/7
What is mathematical creation?
I
Mathematical creation “does not consist in making new combinations
with mathematical entities already known.” (page 2043)
I
Page 2043: “Invention is discernment.”
Dan Sloughter (Furman University)
Mathematical Creation
November 29, 2006
5/7
Conscious and unconscious work
I
How do the conscious and unconscious work of the mathematician
interact, at least in Poincaré’s example?
Dan Sloughter (Furman University)
Mathematical Creation
November 29, 2006
6/7
Conscious and unconscious work
I
How do the conscious and unconscious work of the mathematician
interact, at least in Poincaré’s example?
I
The unconscious phenomena which are susceptible of becoming
conscious “are those which, directly or indirectly, affect most
profoundly our emotional sensibility.” (page 2047)
Dan Sloughter (Furman University)
Mathematical Creation
November 29, 2006
6/7
Conscious and unconscious work
I
How do the conscious and unconscious work of the mathematician
interact, at least in Poincaré’s example?
I
The unconscious phenomena which are susceptible of becoming
conscious “are those which, directly or indirectly, affect most
profoundly our emotional sensibility.” (page 2047)
I
Page 2048: “The useful combinations are precisely the most
beautiful, I mean those best able to charm this special sensibility that
all mathematicians know, but of which the profane are so ignorant as
often to be tempted to smile at it.”
Dan Sloughter (Furman University)
Mathematical Creation
November 29, 2006
6/7
Is mathematical work simply mechanical
I
Poincaré argues that mathematical work “could not be done by a
machine, however perfect.” (page 2046)
Dan Sloughter (Furman University)
Mathematical Creation
November 29, 2006
7/7
Is mathematical work simply mechanical
I
Poincaré argues that mathematical work “could not be done by a
machine, however perfect.” (page 2046)
I
The number of possible combinations of ideas are too numerous, the
“true work of the inventor consists in choosing among these
combinations so as to eliminate the useless ones or rather to avoid
the trouble of making them, and the rules which must guide this
choice are extremely fine and delicate.” (page 2046)
Dan Sloughter (Furman University)
Mathematical Creation
November 29, 2006
7/7