The Most Advanced Mathematical Proof Used in Reasoning, Ranked

Choose the mathematical proof used you think is the most advanced!

Author: Gregor Krambs
Updated on Jul 4, 2024 06:17
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?

  1. 1
    64
    points
    Fermat's Last Theorem

    Fermat's Last Theorem

    Proven by Andrew Wiles, it states that no three positive integers a, b, and c can satisfy the equation an + bn = cn for any integer value of n greater than 2.
    • Field: Number Theory
    • Proven in: 1994
  2. 2
    1
    points
    The Four Color Theorem

    The Four Color Theorem

    This theorem states that any map in a plane can be colored using four colors in such a way that regions sharing a common boundary (other than a single point) do not share the same color.
    • Field: Graph Theory
    • Proven in: 1976
  3. 3
    0
    points
    The Birch and Swinnerton-Dyer Conjecture

    The Birch and Swinnerton-Dyer Conjecture

    One of the Millennium Prize Problems, it describes the set of rational solutions to equations defining an elliptic curve.
    • Field: Number Theory
    • Conjectured in: 1960s
  4. 4
    0
    points
    The Riemann Hypothesis

    The Riemann Hypothesis

    Another of the Millennium Prize Problems, it hypothesizes about the distribution of the zeros of the Riemann zeta function and has profound implications for the distribution of prime numbers.
    • Field: Number Theory
    • Proposed in: 1859
  5. 5
    0
    points
    The Grothendieck-Riemann-Roch Theorem

    The Grothendieck-Riemann-Roch Theorem

    A major result in algebraic geometry, it generalizes the classical Riemann-Roch theorem for Riemann surfaces to higher-dimensional varieties.
    • Field: Algebraic Geometry
    • Proven in: 1957
  6. 6
    0
    points
    The Classification of Finite Simple Groups

    The Classification of Finite Simple Groups

    A monumental theorem in group theory, stating that every finite simple group is of one of four types: cyclic, alternating, Lie type, or one of 26 sporadic simple groups.
    • Field: Group Theory
    • Completed in: 2004
  7. 7
    0
    points
    The Prime Number Theorem

    The Prime Number Theorem

    Describes the asymptotic distribution of prime numbers among the positive integers. It formalizes the idea that primes become less common as they become larger.
    • Field: Number Theory
    • Introduced in: 19th Century
  8. 8
    0
    points

    ZFC and the Independence of the Continuum Hypothesis

    Proven by Kurt Gödel and Paul Cohen, it shows that the Continuum Hypothesis cannot be proven or disproven from the standard Zermelo–Fraenkel set theory with the axiom of choice.
    • Field: Set Theory
    • Proven in: 1938 and 1963
  9. 9
    0
    points

    The Atiyah-Singer Index Theorem

    Provides a relationship between the geometry of manifolds and the analysis of differential operators, playing a crucial role in theoretical physics.
    • Field: Differential Geometry
    • Proven in: 1963
  10. 10
    0
    points
    Proof of the Poincaré Conjecture

    Proof of the Poincaré Conjecture

    Solved by Grigori Perelman, this proof confirms that every simply connected, closed 3-manifold is homeomorphic to the 3-sphere. It's a landmark proof in the field of topology.
    • Field: Topology
    • Proven in: 2003

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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!

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  • 65 votes
  • 10 ranked items

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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.

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More about the Most Advanced Mathematical Proof Used in Reasoning

Fermat's Last Theorem
Rank #1 for the most advanced mathematical proof used in reasoning: Fermat's Last Theorem (Source)
Mathematics relies on proofs to establish truths. Proofs are logical arguments that show a statement is true. They use a series of steps, each following from the last. The most advanced proofs involve deep understanding and creativity.

Proofs start with axioms. Axioms are basic statements accepted without proof. From these, mathematicians build more complex statements. Each step in a proof must be justified. This ensures the conclusion is reliable.

Advanced proofs often use techniques from different areas of math. They may combine algebra, geometry, and calculus. This makes them powerful but also complex. Understanding these proofs requires knowledge in many fields.

Some proofs are very long. They can take hundreds of pages. This is because they cover many details. Each detail must be checked to ensure the proof is correct. Computers sometimes help with this. They can check many steps quickly. However, humans still need to understand the overall structure.

Proofs may also use abstract concepts. These are ideas that are not easy to visualize. For example, higher dimensions or infinite sets. These concepts require a shift in thinking. They are not part of everyday experience.

Collaboration is common in advanced proofs. Mathematicians work together to solve difficult problems. They share ideas and check each other's work. This helps prevent mistakes. It also brings different perspectives to the problem.

Some proofs have taken years or even decades to complete. Mathematicians may work on them for a long time. They may also build on the work of others. This shows the cumulative nature of math. Each generation builds on the last.

Advanced proofs can have practical applications. They may solve problems in physics, engineering, or computer science. They can also lead to new technologies. For example, cryptography relies on number theory. This is an area of math with many advanced proofs.

Mathematicians value elegance in proofs. An elegant proof is simple and clear. It may use a clever idea to solve a problem. Such proofs are admired for their beauty. They show the creative side of math.

Learning to understand advanced proofs takes time. It requires practice and patience. Mathematicians study for many years to develop this skill. They learn to think logically and abstractly. This allows them to tackle complex problems.

In summary, advanced mathematical proofs are intricate and detailed. They build on basic axioms and use many areas of math. They require deep understanding and collaboration. They can take a long time to complete but can have significant applications. They also show the beauty and creativity of mathematics.

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