Can Any Statement Be Conclusively Proven Or Do We Have To Fall Back On "impossible Things"?

Introduction

The concept of proof is a fundamental aspect of mathematics, logic, and philosophy. It is the process of establishing the truth or validity of a statement through a series of logical steps. However, the question of whether any statement can be conclusively proven has been a subject of debate among philosophers and mathematicians for centuries. In this article, we will explore the concept of proof, the limitations of proof, and the implications of the statement "impossible things."

The Nature of Proof

Proof is a process of establishing the truth or validity of a statement through a series of logical steps. It involves using axioms, definitions, and previously proven statements to derive a conclusion. The goal of proof is to provide a convincing argument that a statement is true, and to demonstrate that it is not possible to contradict the statement without violating the axioms or previously proven statements.

Types of Proof

There are several types of proof, including:

• Direct Proof: A direct proof is a straightforward argument that establishes the truth of a statement. It involves using axioms, definitions, and previously proven statements to derive a conclusion.
• Indirect Proof: An indirect proof is an argument that establishes the truth of a statement by showing that its negation is false. It involves using axioms, definitions, and previously proven statements to derive a conclusion.
• Proof by Contradiction: A proof by contradiction is an argument that establishes the truth of a statement by showing that its negation leads to a contradiction. It involves using axioms, definitions, and previously proven statements to derive a conclusion.

The Limitations of Proof

While proof is a powerful tool for establishing the truth or validity of a statement, it is not without its limitations. There are several reasons why proof may not be able to conclusively establish the truth of a statement:

• Axioms: Axioms are statements that are assumed to be true without proof. They are the foundation of a mathematical or logical system, and they are used to derive conclusions. However, axioms can be arbitrary, and they may not be universally accepted.
• Definitions: Definitions are statements that define the meaning of a term or concept. They are used to establish the truth or validity of a statement. However, definitions can be ambiguous, and they may not be universally accepted.
• Previously Proven Statements: Previously proven statements are statements that have been established as true through a series of logical steps. However, previously proven statements can be incorrect, and they may not be universally accepted.

The Implications of "Impossible Things"

The concept of "impossible things" refers to statements that cannot be proven or disproven. These statements are often considered to be true or false, but they cannot be established through a series of logical steps. The implications of "impossible things" are far-reaching, and they have significant consequences for mathematics, logic, and philosophy.

The Liar Paradox

The liar paradox is a classic example of an "impossible thing." It states that a sentence that says "this sentence is false" is either true or false. However, if the sentence is true, then it must be false, and if it is false, then it must be true. This creates an infinite loop, and it is impossible to establish the truth or validity of the sentence.

The Halting Problem

The halting problem is another example of an "impossible thing." It states that there is no algorithm that can determine whether a given program will halt or run indefinitely. This problem has significant implications for computer science, and it has been used to establish the limits of computation.

Conclusion

In conclusion, the concept of proof is a fundamental aspect of mathematics, logic, and philosophy. However, the question of whether any statement can be conclusively proven has been a subject of debate among philosophers and mathematicians for centuries. While proof is a powerful tool for establishing the truth or validity of a statement, it is not without its limitations. The concept of "impossible things" refers to statements that cannot be proven or disproven, and it has significant implications for mathematics, logic, and philosophy.

References

• Gödel, K. (1931). "On Formally Undecidable Propositions of Principia Mathematica and Related Systems." In M. Davis (Ed.), The Undecidable: Basic Papers on Undecidable Propositions, Unsolvable Problems and Computable Functions (pp. 5-38).
• Turing, A. M. (1936). "On Computable Numbers, with an Application to the Entscheidungsproblem." Proceedings of the London Mathematical Society, 42, 230-265.
• Kleene, S. C. (1936). "General Recursive Functions of Natural Numbers." Mathematische Annalen, 112, 727-742.

Further Reading

• Gödel, K. (1947). "What is Cantor's Continuum Problem?" The American Mathematical Monthly, 54(9), 515-525.
• Turing, A. M. (1950). "Computing Machinery and Intelligence." Mind, 59(236), 433-460.
• Kleene, S. C. (1952). "Introduction to Metamathematics." North-Holland Publishing Company.
  

Q&A

Q: What is the difference between a proof and a demonstration?

A: A proof is a formal argument that establishes the truth or validity of a statement, while a demonstration is a more general term that refers to any argument or evidence that supports a claim.

Q: Can we prove that a statement is true, but not necessarily that it is the only true statement?

A: Yes, this is a common situation in mathematics and logic. For example, we can prove that a particular number is a solution to a given equation, but we may not be able to prove that it is the only solution.

Q: What is the relationship between proof and certainty?

A: Proof and certainty are related but distinct concepts. A proof can establish the truth or validity of a statement with a high degree of certainty, but it is not necessarily a guarantee of absolute certainty.

Q: Can we prove that a statement is false, but not necessarily that it is the only false statement?

A: Yes, this is also a common situation in mathematics and logic. For example, we can prove that a particular number is not a solution to a given equation, but we may not be able to prove that it is the only number that is not a solution.

Q: What is the role of axioms in proof?

A: Axioms are statements that are assumed to be true without proof. They are the foundation of a mathematical or logical system, and they are used to derive conclusions.

Q: Can we prove that a statement is independent of a particular axiom or set of axioms?

A: Yes, this is a common situation in mathematics and logic. For example, we can prove that a particular statement is independent of a particular axiom or set of axioms, meaning that it cannot be derived from those axioms.

Q: What is the relationship between proof and incompleteness?

A: Proof and incompleteness are related but distinct concepts. A proof can establish the truth or validity of a statement, but it may not be able to prove that the statement is true in all possible cases.

Q: Can we prove that a statement is true in all possible cases, but not necessarily that it is the only true statement?

A: Yes, this is a common situation in mathematics and logic. For example, we can prove that a particular statement is true in all possible cases, but we may not be able to prove that it is the only statement that is true in all possible cases.

Q: What is the role of Gödel's incompleteness theorems in proof?

A: Gödel's incompleteness theorems establish that any formal system that is powerful enough to describe basic arithmetic is either incomplete or inconsistent. This has significant implications for the nature of proof and the limits of formal systems.

Q: Can we prove that a statement is true in a particular formal system, but not necessarily that it is true in all possible formal systems?

A: Yes, this is a common situation in mathematics and logic. For example, we can prove that a particular statement is true in a particular formal system, but we may not be able to prove that it is true in all possible formal systems.

Q: What is the relationship between proof and computability?

A: Proof and computability are related but distinct concepts. A proof can establish the truth or validity of a statement, but it may not be able to prove that the statement is computable.

Q: Can we prove that a statement is computable, but not necessarily that it is the only computable statement?

A: Yes, this is a common situation in mathematics and logic. For example, we can prove that a particular statement is computable, but we may not be able to prove that it is the only computable statement.

Q: What is the role of Turing's halting problem in proof?

A: Turing's halting problem establishes that there is no algorithm that can determine whether a given program will halt or run indefinitely. This has significant implications for the nature of proof and the limits of computation.

Q: Can we prove that a statement is true in a particular computational model, but not necessarily that it is true in all possible computational models?

A: Yes, this is a common situation in mathematics and logic. For example, we can prove that a particular statement is true in a particular computational model, but we may not be able to prove that it is true in all possible computational models.

Further Reading

• Gödel, K. (1931). "On Formally Undecidable Propositions of Principia Mathematica and Related Systems." In M. Davis (Ed.), The Undecidable: Basic Papers on Undecidable Propositions, Unsolvable Problems and Computable Functions (pp. 5-38).
• Turing, A. M. (1936). "On Computable Numbers, with an Application to the Entscheidungsproblem." Proceedings of the London Mathematical Society, 42, 230-265.
• Kleene, S. C. (1936). "General Recursive Functions of Natural Numbers." Mathematische Annalen, 112, 727-742.

References

• Gödel, K. (1947). "What is Cantor's Continuum Problem?" The American Mathematical Monthly, 54(9), 515-525.
• Turing, A. M. (1950). "Computing Machinery and Intelligence." Mind, 59(236), 433-460.
• Kleene, S. C. (1952). "Introduction to Metamathematics." North-Holland Publishing Company.