What do variables represent, anyway?

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SUMMARY

The discussion centers on the interpretation of mathematical statements involving variables and quantifiers, specifically comparing "let x be a real number" with "for all x in ℝ." Both expressions convey the same mathematical truth that \(x^2 + 1 \neq 0\) for real numbers. The conversation highlights a distinction between singular and plural usage in mathematical language, emphasizing that the second statement is more general. Participants recommend studying quantifiers in propositional calculus or mathematical logic for deeper understanding.

PREREQUISITES
  • Understanding of basic mathematical concepts, including variables and equations.
  • Familiarity with quantifiers in logic, specifically universal quantification.
  • Knowledge of propositional calculus and its principles.
  • Basic understanding of mathematical proofs and their structure.
NEXT STEPS
  • Study the principles of quantifiers in mathematical logic.
  • Read a textbook on propositional calculus to grasp foundational concepts.
  • Explore mathematical proof techniques to enhance logical reasoning skills.
  • Investigate the differences between singular and plural expressions in mathematical language.
USEFUL FOR

Students of mathematics, educators teaching mathematical logic, and anyone interested in the formal language of mathematics and its implications for understanding variables and quantifiers.

HyperbolicMan
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"let x be a real number. Then x^2+1 does not equal 0."

"For all x in lR, x^2+1 does not equal 0"

As far as I know, both of these statements mean exactly the same thing. From a grammatical perspective, in the first statement, x is singular ("a real number"), while in the second, x appears to be plural (preceded by "all" which implies more than one). Is one way 'more correct' than the other, or is there some principle in mathematics that equates the two? Instinctively, I feel like the second is better because it sounds more general, but the first one is easier to visualize.
 
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i think both usages are singular. i.e. "everyone here is a nerd." every is singular. it the same as for all x, to me at least.
 
your question is not about variables, but about quantifiers. get a book on quantifiers. i.e. propositional calculus, or mathematical logic. or just a book on proofs and read the part on quantifiers. you will never regret it.
 

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