every discipline has its own standard of justification—that is, its own criteria for deciding when a conclusion is adequately supported.
1. Mathematics requires proof
In mathematics, a statement is accepted only if it is logically deduced from definitions, axioms, and previously proven results.
The emphasis is on logical necessity, not observation.
Example 1
Claim:
The angles of a triangle add up to 180° (in Euclidean geometry).
A mathematician does not measure thousands of triangles.
Instead, they prove the statement using axioms and logical deduction.
Once proved, the conclusion follows necessarily.
Example 2
Claim:
√2 is irrational.
Mathematicians do not test many square roots.
Instead they construct a proof (often by contradiction).
The proof demonstrates that assuming √2 is rational leads to a logical contradiction.
Therefore,
√2 must be irrational.
What counts as justification?
- definitions
- axioms
- logical deduction
- valid proof
Not enough:
"I calculated it many times."
Even one valid proof is stronger than a million observations.
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