The Most Advanced Mathematical Proof Used in Reasoning, Ranked

Voting rules: Choose the mathematical proof used you think is the most advanced!

Updated on May 4, 2024 06:16

Mathematics often serves as the backbone of innovation and understanding in various fields, from physics to finance. Ranking advanced mathematical proofs can throw light on how these foundational concepts are currently influencing reasoning and innovation. It gives a perspective on which mathematical ideas are proving most instrumental in solving complex problems. By engaging with this ranking, users participate in a collective recognition of intellectual achievements in mathematics. This dynamic assessment helps in identifying which proofs are resonating the most in current academic and professional circles, potentially guiding future educational and research focuses.

What Is the Most Advanced Mathematical Proof Used in Reasoning?

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65
votes
Fermat's Last Theorem

Pierre de Fermat

This proof was one of the most famous and long-standing unsolved problems in mathematics, which took more than 350 years to solve. It states that no three positive integers a, b, and c satisfy the equation an + bn = cn for any integer value of n greater than 2.
Fermat's Last Theorem is a famous mathematical statement that was first proposed by Pierre de Fermat in 1637. It states that no three positive integers a, b, and c can satisfy the equation a^n + b^n = c^n for any integer value of n greater than 2.

Conjecture: a^n + b^n = c^n has no integer solutions for n > 2

Proof: The proof was completed by Andrew Wiles with assistance from Richard Taylor

Year of Proof: 1994

Field: Number Theory

Significance: One of the most famous and long-standing unsolved problems in mathematics
Fermat's Last Theorem in other rankings
- Position 3 of 10 in Most difficult equation to solve
- Position 4 of 10 in Most difficult equation
- Position 4 of 10 in Most difficult math equation
- Position 5 of 10 in Most difficult math question
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34
votes
Poincaré Conjecture

Henri Poincaré

This proof was a famous problem in topology that was solved by Grigori Perelman in 2003. It states that any closed three-dimensional manifold is topologically equivalent to a three-dimensional sphere.
The Poincaré Conjecture is a famous problem in mathematics that deals with the characterization of the three-dimensional sphere. Proposed by the French mathematician Henri Poincaré in 1904, it states that any simply connected, closed three-dimensional manifold is homeomorphic to a three-dimensional sphere.

Problem type: Conjecture

Field: Topology

Dimension: 3

Manifold characteristic: Simply connected

Closed: Yes
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23
votes
Riemann Hypothesis

Bernhard Riemann

This proof is a famous unsolved problem in number theory, which states that all non-trivial zeros of the Riemann zeta function lie on the critical line.
The Riemann Hypothesis is one of the most significant unsolved problems in mathematics. It is a conjecture made by German mathematician Bernhard Riemann in 1859, regarding the distribution of prime numbers. Essentially, it provides information about the behavior of the Riemann zeta function, which is a complex-valued function that holds great importance in number theory. The Riemann Hypothesis suggests that all non-trivial zeros of the zeta function have a real part of 1/2.

Conjecture Type: Unsolved

Field: Number Theory

Conjectured Date: 1859

Importance: Critical

Significance: Determines prime numbers distribution
Riemann Hypothesis in other rankings
- Position 1 of 10 in Most difficult unsolved mathematical problem
- Position 2 of 10 in Most difficult math equation
- Position 2 of 10 in Most difficult equation
- Position 4 of 10 in Most difficult equation to solve
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23
votes
Gödel's Incompleteness Theorems

Kurt Gödel

This proof is a landmark theorem in mathematical logic, which states that any sufficiently powerful formal system of mathematics is either incomplete or inconsistent.
Gödel's Incompleteness Theorems are a set of two theorems in mathematical logic. They were proven by Kurt Gödel in 1931.

1: The first incompleteness theorem states that in any consistent formal system within which a certain amount of elementary arithmetic can be carried out, there exists statements that cannot be proven or disproven within the system itself.

2: The second incompleteness theorem shows that if a formal system is consistent and sufficiently expressive to represent basic arithmetic, then it cannot prove its own consistency.

3: The theorems deal with the limitations of formal systems, showing that certain questions are undecidable or unprovable within those systems.

4: Gödel's work revolutionized the field of mathematical logic and had significant implications for the philosophy of mathematics.

5: The theorems have implications beyond mathematics, impacting fields such as computer science, artificial intelligence, and philosophy.
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19
votes
Sarasan · CC BY-SA 3.0

Four Color Theorem

This proof is a famous problem in graph theory, which states that any planar graph can be colored with at most four colors in such a way that no two adjacent regions have the same color.
The Four Color Theorem is a mathematical theorem that states that any map on a plane can be colored with at most four colors in such a way that no two adjacent regions have the same color. The concept of adjacent in this context refers to regions that share a common boundary, not just a point.

Proof Type: Existence proof

Year of First Proof: 1976

Complexity: High

Significance: The first major theorem to be proven using extensive computer assistance

Application: Graph theory, map coloring, computer science
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7
votes
Banach-Tarski Paradox

This proof is a famous paradox in geometry, which states that a solid ball can be decomposed into a finite number of non-overlapping pieces, which can be rearranged using only rigid motions to form two solid balls of equal volume as the original.
The Banach-Tarski Paradox is a mathematical paradox in set theory that asserts the counterintuitive result that a solid ball can be decomposed into a finitely partitioned set of solid pieces, which can then be rearranged using only translations and rotations to form two complete solid balls, each the same size as the original.

Axiomatic System: The Banach-Tarski Paradox relies on Zermelo-Fraenkel set theory with the Axiom of Choice (ZFC).

Finite Partition: The solid ball is decomposed into a finite number of disjoint non-empty subsets.

Non-measurable Sets: The subsets formed are non-measurable, meaning they cannot be assigned a well-behaved volume that aligns with our intuitive understanding of volume.

Translation and Rotation: The rearrangement of the subsets is done only through translations and rotations in 3-dimensional Euclidean space.

Size Preservation: The two resulting balls have the same size (volume) as the original ball.
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13
votes
Hodge Conjecture

William Vallance Douglas Hodge

This proof is a famous problem in algebraic geometry, which states that for any algebraic variety, its Hodge numbers are determined by the topology of the variety.
The Hodge Conjecture is a famous conjecture in algebraic geometry. It deals with the relationship between two important mathematical structures, algebraic cycles and cohomology theory. The conjecture states that for a smooth projective algebraic variety, there exists a natural decomposition of the cohomology into pieces called Hodge classes, which are algebraic cycles. These algebraic cycles can be understood in terms of the geometry of the variety. The conjecture further asserts that any Hodge class can be represented by a rational linear combination of algebraic cycles.

Field: Algebraic geometry

Status: Unsolved

Conjectured: During the 1950s

Importance: One of the seven Millennium Prize Problems

Impact: Reveals deep connections between algebraic cycles and cohomology theory
Hodge Conjecture in other rankings
- Position 3 of 10 in Most difficult unsolved mathematical problem
- Position 7 of 10 in Most advanced math problem
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9
votes
Navier-Stokes Equations

This proof is a famous problem in fluid dynamics, which describes the motion of viscous fluids. It is still an unsolved problem in mathematics.
The Navier-Stokes equations are a set of partial differential equations that describe the motion of fluid substances such as liquids and gases.

Equations: The equations consist of the continuity equation and the momentum equations.

Continuity equation: Describes the conservation of mass for an incompressible fluid.

Momentum equations: Describe the conservation of momentum for a fluid, taking into account pressure and viscous forces.

Nonlinear nature: The equations are nonlinear, making them difficult to solve analytically.

Important in fluid dynamics: They provide a mathematical description of the behavior of fluids in various scenarios.
Navier-Stokes Equations in other rankings
- Position 1 of 10 in Most difficult math equation
- Position 1 of 10 in Most difficult equation
- Position 3 of 10 in Most difficult math question
- Position 4 of 10 in Most difficult unsolved mathematical problem
- Position 5 of 10 in Most advanced math problem
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6
votes
Yang-Mills Theory

This proof is a famous problem in theoretical physics, which describes the behavior of subatomic particles. It is still an unsolved problem in mathematics.
The Yang-Mills Theory is a mathematical framework that describes the behavior of elementary particles in the context of Quantum Field Theory. It provides a way to understand the strong nuclear force, electromagnetism, and weak nuclear force as different aspects of a single mathematical structure.

Gauge invariance: The theory is invariant under local gauge transformations.

Field equations: The theory is described by a set of nonlinear differential equations.

Quantum chromodynamics: Describes interactions between quarks and gluons.

Yang-Mills potential: The gauge field is represented by a set of potentials.

Lie algebra: The symmetry group of the theory is described by a Lie algebra.
Yang-Mills Theory in other rankings
- Position 6 of 10 in Most difficult equation
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3
votes

Grothendieck's Theory of Schemes

This proof is a famous theory in algebraic geometry, which revolutionized the subject by introducing the use of abstract algebraic geometry.

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Discussion

Ranking factors for advanced mathematical proof used

Complexity

The level of complexity of a mathematical proof can be determined by the number of steps involved in proving the theorem, the presence of many interwoven ideas, and the difficulty in understanding each step. More complex proofs may be considered more advanced.
Novelty

If a proof introduces new techniques, concepts, or methods for solving a problem, it may be considered more advanced. Novelty can be measured by the extent to which a proof departs from the existing body of knowledge in mathematics.
Significance

The overall importance of the theorem being proved is a key factor. A proof that solves a long-standing problem, answers a fundamental question, or has wide-reaching implications within or beyond mathematics is considered more advanced.
Elegance and efficiency

An elegant proof is one that is simple, concise, and aesthetically pleasing. Advanced proofs often have a sense of elegance to them, revealing deep connections between seemingly unrelated areas of mathematics or providing surprising insights into a problem.
Degree of abstraction

Advanced mathematical proofs may require a high degree of abstraction, relying on sophisticated concepts from one or more fields of mathematics. This can make the proof challenging for even experts in the field to understand.
Scope

The applicability of the proof – whether it serves as a cornerstone for a large body of knowledge, connects several domains of mathematics, or opens up new areas of research – is another significant factor that indicates the level of advancement.
Rigor

Advanced proofs should be thoroughly and rigorously constructed, leaving little room for doubt in their conclusions.
Originality

The extent to which a proof is based on the work of others can play a role in determining its level of advancement. Proofs that rely more on fresh ideas or make unexpected connections can be considered more advanced.
Accessibility

While not a direct measure of advancement, the level of difficulty in understanding a mathematical proof can influence how advanced it is considered. This can be due to the complexity of the underlying ideas, the mathematical language used, or the way the proof is presented.
Contribution to the field

A proof's impact on the development of mathematics is an important factor to consider. Advanced proofs often lead to new insights, theories, or advancements in the field.

About this ranking

This is a community-based ranking of the most advanced mathematical proof used in reasoning. We do our best to provide fair voting, but it is not intended to be exhaustive. So if you notice something or proof is missing, feel free to help improve the ranking!

Statistics

1582 views
201 votes
10 ranked items

Voting Rules

A participant may cast an up or down vote for each proof once every 24 hours. The rank of each proof is then calculated from the weighted sum of all up and down votes.

More information on most advanced mathematical proof used in reasoning

Mathematics is an ever-evolving field that has been a subject of great interest to scholars and researchers for centuries. Some of the most advanced mathematical proofs are used in reasoning to solve complex problems in various disciplines, including physics, engineering, and computer science. One of the most famous proofs is the Fermat's Last Theorem, which remained unsolved for over 350 years until it was finally proved by Andrew Wiles in 1994. This theorem states that there are no whole number solutions to the equation x^n + y^n = z^n, where n is an integer greater than 2. Another famous proof is the Four Color Theorem, which states that any map can be colored using only four colors in such a way that no two adjacent regions have the same color. This theorem was first proposed in the 1850s and was finally proved in 1976 by Kenneth Appel and Wolfgang Haken using a computer program. In recent years, the Poincaré Conjecture has gained a lot of attention as one of the most complex and advanced mathematical proofs. This conjecture, which was first proposed by Henri Poincaré in 1904, states that any three-dimensional shape without holes is equivalent to a three-dimensional sphere. It was finally proved in 2003 by Grigori Perelman, who was awarded the Fields Medal for his work. Overall, advanced mathematical proofs play a crucial role in solving complex problems and advancing our understanding of various fields. Their significance cannot be overstated, and they continue to be