Discussion Overview
The discussion revolves around the distinction between the phrases "holds" and "holds true" in the context of mathematical statements and proofs. Participants explore the implications of these terms regarding the truth of statements for arbitrary numbers in the real numbers, R.
Discussion Character
- Conceptual clarification
- Debate/contested
Main Points Raised
- One participant questions whether claiming that a statement P holds for some x in R implies that P holds true for arbitrary numbers in R but not necessarily for all numbers in R.
- Another participant asserts that they have not encountered any distinction between "holds" and "holds true," suggesting both phrases convey the same meaning of "it is the case."
- A participant seeks clarification on whether "theorem holds for all x in R" and "theorem holds true for all x in R" are equivalent, pondering if "hold" allows for the possibility of a theorem being false for some x in R, while "holds true" implies certainty.
- Responses indicate that the two phrases are considered to mean exactly the same thing, with no difference in implication regarding the truth of the theorem.
- Another participant emphasizes that the expression of which members of a set satisfy a property should be made clear through quantifiers or proper phrasing, reinforcing the idea that the terms are interchangeable.
Areas of Agreement / Disagreement
Participants generally agree that "holds" and "holds true" are interchangeable in meaning, although some express uncertainty about the implications of each term in specific contexts.
Contextual Notes
Some participants express confusion regarding the subtleties of grammar and terminology in mathematical discourse, indicating a potential limitation in understanding the implications of these terms.