Semantics of Mathematics and Science

[Astronuc]

Definition -

1. A statement conveying fundamental character.
2. A statement, or a concise explanation, of the meaning of a word, phrase, term, object or symbol


Proof -

1. The evidence or argument that compels the mind to accept an assertion as true.

2. a. The validation of a proposition by application of specified rules, as of induction or deduction, to assumptions, axioms, and sequentially derived conclusions.
b. A statement or argument used in such a validation.

3. a. Convincing or persuasive demonstration: was asked for proof of his identity;
an employment history that was proof of her dependability.
b. The state of being convinced or persuaded by consideration of evidence.

4. Determination of the quality of something by testing; trial: put one's beliefs to the proof.

5. Law. The result or effect of evidence; the establishment or denial of a fact by evidence.


Convention -

1. General agreement on or acceptance of certain meaning (see definition), practices or attitudes

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It would seem that definitions are beyond proof.

Does the statement - Prove 1 = 1 - make sense?

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scientific method:

The principles and empirical processes of discovery and demonstration considered characteristic of or necessary for scientific investigation, generally involving the observation of phenomena, the formulation of a hypothesis concerning the phenomena, experimentation to demonstrate the truth or falseness of the hypothesis, and a conclusion that validates or modifies the hypothesis.

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I thought this might be useful after reading some threads on proving things that are or seem to be beyond proof.

 

[matt grime]

Even a trivial proof is still a proof (ie X implies X, where X is some axiomatic statement). (The proof here is the definition of equal)

 

[R0man]

The semantics of mathematics and science are deeply intertwined. In mathematics, definitions provide the foundation for understanding and manipulating abstract concepts and objects. They convey the fundamental character of mathematical ideas and serve as the building blocks for more complex theories and proofs. In science, definitions are used to describe and explain the natural world, providing a concise explanation of the meaning of terms and symbols used to represent observations and theories.

Proof, on the other hand, is a crucial aspect of both mathematics and science. In mathematics, it is the process of logically demonstrating the truth of a statement or proposition using established rules and axioms. In science, proof involves conducting experiments and gathering evidence to support or reject a hypothesis, leading to the establishment of new knowledge and theories.

In both disciplines, there is a strong emphasis on the concept of convention, or general agreement on accepted meanings, practices, and attitudes. This is especially important in mathematics, where conventions for notation and terminology must be consistently followed in order for ideas to be communicated effectively. In science, conventions help ensure that experiments are conducted in a standardized and replicable manner, allowing for the validation of results and the advancement of knowledge.

Finally, it is worth considering the question of whether certain things, such as the statement "1=1", can be proven. In mathematics, this statement is considered an axiom, or a self-evident truth that serves as a starting point for building more complex theories. In science, it may be considered a fundamental principle that is supported by overwhelming evidence and therefore accepted as true. While the idea of proof may differ between disciplines, the concepts of definitions, proof, and convention remain essential components of both mathematics and science.