The Hilbert Problems are a set of 23 mathematical challenges presented by the German mathematician David Hilbert in 1900. Aimed at defining future research directions, these problems span various areas of mathematics. While some were clearly stated, others were more open-ended. Over the decades, a majority of these problems have been resolved, significantly advancing mathematical knowledge. Some led to the establishment of entire new fields of study. Unresolved - as of AE53 - Hilbert problems:
Mathematical Treatment of the Axioms of Physics
Rigorous Foundation of Schubert's Enumerative Calculus
Topology of Algebraic Curves and Surfaces
Analysis of Solutions of Variational Problems
The
Hilbert
Problems
are
a
set
of
23
mathematical
challenges
presented
by
the
German
mathematician
David
Hilbert
in
1900
.
Aimed
at
defining
future
research
directions
''
these
problems
span
various
areas
of
mathematics
.
While
some
were
clearly
stated
''
others
were
more
open
-
ended
.
Over
the
decades
''
a
majority
of
these
problems
have
been
resolved
''
significantly
advancing
mathematical
knowledge
.
Some
led
to
the
establishment
of
entire
new
fields
of
study
.
Unresolved
-
as
of
AE53
-
Hilbert
problems
:<
p
class
=
„
fragment
“
>
Mathematical
Treatment
of
the
Axioms
of
Physics
p
><
p
class
=
„
fragment
“
>
Rigorous
Foundation
of
Schubert
'
s
Enumerative
Calculus
p
><
p
class
=
„
fragment
“
>
Topology
of
Algebraic
Curves
and
Surfaces
p
><
p
class
=
„
fragment
“
>
Analysis
of
Solutions
of
Variational
Problems
p
>
From Middle English probleme, from Middle French probleme, from Latin problema, from Ancient Greek πρόβλημα (próblēma, “anything thrown forward, hindrance, obstacle, anything projecting, a headland, promontory”), from προβάλλω (probállō, “to throw or lay something in front of someone, to put forward”), from προ- (pro-, “in front of”) + βάλλω (bállō, “to throw, to cast, to hurl”).
[note the etymologic similarity to word "project": from Latin projectum ‘something prominent’, neuter past participle of proicere ‘throw forth’, from pro- ‘forth’ + jacere ‘to throw’)]
1014
From
Middle
English
probleme
''
from
Middle
French
probleme
''
from
Latin
problema
''
from
Ancient
Greek
πρόβλημα
(
próblēma
''
“
anything
thrown
forward
''
hindrance
''
obstacle
''
anything
projecting
''
a
headland
''
promontory
”)''
from
προβάλλω
(
probállō
''
“
to
throw
or
lay
something
in
front
of
someone
''
to
put
forward
”)''
from
προ
-
(
pro
-''
“
in
front
of
”)
+
βάλλω
(
bállō
''
“
to
throw
''
to
cast
''
to
hurl
”).<
br
/><
br
/>[
note
the
etymologic
similarity
to
word
„
project
“
:
from
Latin
projectum
‘
something
prominent
’''
neuter
past
participle
of
proicere
‘
throw
forth
’''
from
pro
-
‘
forth
’
+
jacere
‘
to
throw
’)]
The Millennium Problems are seven mathematical challenges established by the Clay Mathematics Institute in 2000, each with a prize of one million dollars for a correct solution. These problems represent some of the most persistent and critical unsolved issues in mathematics, spanning areas such as number theory, geometry, analysis, and fluid dynamics. Only one of the problems, the Poincaré Conjecture, has been solved so far. Unresolved - as of AE53 - ones are:
Birch and Swinnerton-Dyer Conjecture
Hodge Conjecture
Navier–Stokes Existence and Smoothness
P vs NP Problem
Riemann Hypothesis
Yang–Mills Existence and Mass Gap
1039
The
Millennium
Problems
are
seven
mathematical
challenges
established
by
the
Clay
Mathematics
Institute
in
2000
''
each
with
a
prize
of
one
million
dollars
for
a
correct
solution
.
These
problems
represent
some
of
the
most
persistent
and
critical
unsolved
issues
in
mathematics
''
spanning
areas
such
as
number
theory
''
geometry
''
analysis
''
and
fluid
dynamics
.
Only
one
of
the
problems
''
the
Poincaré
Conjecture
''
has
been
solved
so
far
.
Unresolved
-
as
of
AE53
-
ones
are
:<
p
class
=
„
fragment
“
>
Birch
and
Swinnerton
-
Dyer
Conjecture
p
><
p
class
=
„
fragment
“
>
Hodge
Conjecture
p
><
p
class
=
„
fragment
“
>
Navier
–
Stokes
Existence
and
Smoothness
p
><
p
class
=
„
fragment
“
>
P
vs
NP
Problem
p
><
p
class
=
„
fragment
“
>
Riemann
Hypothesis
p
><
p
class
=
„
fragment
“
>
Yang
–
Mills
Existence
and
Mass
Gap
p
>
No Definitive Formulation
No Stopping Rule
Solutions Are Not True-or-False
No Immediate and No Ultimate Test
Every Solution is a 'One-Shot Operation'
No Enumerable (or Exhaustively Describable) Set of Potential Solutions
Every Wicked Problem is Essentially Unique
Every Wicked Problem Can Be Considered to Be a Symptom of Another Problem
The Existence of a Discrepancy Representing a Wicked Problem Can Be Explained in Numerous Ways
The Planner Has No Right to Be Wrong
1038
<
p
class
=
„
fragment
“
>
No
Definitive
Formulation
p
><
p
class
=
„
fragment
“
>
No
Stopping
Rule
p
><
p
class
=
„
fragment
“
>
Solutions
Are
Not
True
-
or
-
False
p
><
p
class
=
„
fragment
“
>
No
Immediate
and
No
Ultimate
Test
p
><
p
class
=
„
fragment
“
>
Every
Solution
is
a
'
One
-
Shot
Operation
'
p
><
p
class
=
„
fragment
“
>
No
Enumerable
(
or
Exhaustively
Describable
)
Set
of
Potential
Solutions
p
><
p
class
=
„
fragment
“
>
Every
Wicked
Problem
is
Essentially
Unique
p
><
p
class
=
„
fragment
“
>
Every
Wicked
Problem
Can
Be
Considered
to
Be
a
Symptom
of
Another
Problem
p
><
p
class
=
„
fragment
“
>
The
Existence
of
a
Discrepancy
Representing
a
Wicked
Problem
Can
Be
Explained
in
Numerous
Ways
p
><
p
class
=
„
fragment
“
>
The
Planner
Has
No
Right
to
Be
Wrong
p
>
This is the set of problem(s) to which I expose all my D&C students since AE50.
Hopefully, at least 1 will be "solved" during my lifetime, 2 during Your lifetime and 3 during lifetime of my children.
1040
This
is
the
set
of
problem
(
s
)
to
which
I
expose
all
my
D
&
C
students
since
AE50
.<
div
><
br
>
div
><
div
>
Hopefully
''
at
least
1
will
be
„
solved
“
during
my
lifetime
''
2
during
Your
lifetime
and
3
during
lifetime
of
my
children
.
div
>
Radioactive waste disposal
Liberation from fossil fuel dependency
Language extinction
Weltverständlichmachung
Unf*cking the Wohnungsmarkt
...
...
...
...
The Clock Reform
1054
<
p
class
=
„
fragment
“
>
Radioactive
waste
disposal
p
><
p
class
=
„
fragment
“
>
Liberation
from
fossil
fuel
dependency
p
><
p
class
=
„
fragment
“
>
Language
extinction
p
><
p
class
=
„
fragment
“
>
Weltverständlichmachung
p
><
p
class
=
„
fragment
“
>
Unf
*
cking
the
Wohnungsmarkt
p
><
p
class
=
„
fragment
“
>...
p
><
p
class
=
„
fragment
“
>...
p
><
p
class
=
„
fragment
“
>...
p
><
p
class
=
„
fragment
“
>...
p
><
p
class
=
„
fragment
“
>
The
Clock
Reform
p
>
1055
Unsolved problems are problems for which we neither have a solution, nor know whether such solution can be found.
1057
Unsolved
problems
are
problems
for
which
we
neither
have
a
solution
''
nor
know
whether
such
solution
can
be
found
.
1013
Used as a general rule in chemistry; the old alchemical principle "similia similibus dissolvuntur" meaning "like dissolves like" refers to the ability of polar or non polar solvents to dissolve polar or non polar solutes respectively.
1018
Used
as
a
general
rule
in
chemistry
;
the
old
alchemical
principle
„
<
strong
>
similia
similibus
dissolvuntur
strong
>
“
meaning
„
like
dissolves
like
“
refers
to
the
ability
of
polar
or
non
polar
solvents
to
dissolve
polar
or
non
polar
solutes
respectively
.
In engineering, the solution S is that vector of P which minimizes (resp. maximizes) the value of objective function F.
1069
In
engineering
''
the
solution
S
is
that
vector
of
P
which
minimizes
(
resp
.
maximizes
)
the
value
of
objective
function
F
.
The word "solution" has its roots in Latin, deriving from the word “solutio.” The etymology can be broken down as follows:
- Latin "solutio": Meaning a loosening or unbinding. This comes from the verb “solvere,” which means to loosen, untie, or dissolve.
- Latin "solvēre": This is a compound of “sol-,” a variant of “se-” (apart), and “luere” (to wash or to loosen).
Over time, the meaning of "solution" has evolved and broadened. In English, it has been used since the 14th century to refer not just to the act of dissolving a substance in a liquid, but also to the means of solving a problem or answering a question. The sense of finding an answer or explanation encapsulates the idea of resolving a complex situation, untangling its components to reach a clear understanding or resolutionIn summary, the etymology of "solution" reflects a journey from the physical act of loosening or dissolving, through to the more abstract notion of resolving problems and finding answers.
1017
The
word
„
solution
“
has
its
roots
in
Latin
''
deriving
from
the
word
“
solutio
.”
The
etymology
can
be
broken
down
as
follows
:<
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“
>
Latin
„
solutio
“
span
>:
Meaning
a
loosening
or
unbinding
.
This
comes
from
the
verb
“
solvere
,”
which
means
to
loosen
''
untie
''
or
dissolve
.
li
><
li
style
=
„
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:
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solid
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-
ring
-
color
:
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(
69
''
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164
,.
5
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--
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-
ring
-
offset
-
shadow
:
0
0
transparent
;
--
tw
-
ring
-
shadow
:
0
0
transparent
;
--
tw
-
shadow
:
0
0
transparent
;
--
tw
-
shadow
-
colored
:
0
0
transparent
;
font
-
weight
:
600
;
color
:
var
(--
tw
-
prose
-
bold
);
margin
-
top
:
1
.
25em
;
margin
-
bottom
:
1
.
25em
;
“
>
Latin
„
solvēre
“
span
>:
This
is
a
compound
of
“
sol
-,”
a
variant
of
“
se
-”
(
apart
)''
and
“
luere
”
(
to
wash
or
to
loosen
).
li
>
ul
>
Over
time
''
the
meaning
of
„
solution
“
has
evolved
and
broadened
.
In
English
''
it
has
been
used
since
the
14th
century
to
refer
not
just
to
the
act
of
dissolving
a
substance
in
a
liquid
''
but
also
to
the
means
of
solving
a
problem
or
answering
a
question
.
The
sense
of
finding
an
answer
or
explanation
encapsulates
the
idea
of
resolving
a
complex
situation
''
untangling
its
components
to
reach
a
clear
understanding
or
resolutionIn
summary
''
the
etymology
of
„
solution
“
reflects
a
journey
from
the
physical
act
of
loosening
or
dissolving
''
through
to
the
more
abstract
notion
of
resolving
problems
and
finding
answers
.
In multi-objective optimization, optimize for two or more objectives in the same time.
1077
In
multi
-
objective
optimization
''
optimize
for
two
or
more
objectives
in
the
same
time
.
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