When1: 1879
When2: 1892
Who: Gottlöb Frege [Frege, Gottlöb]
What: mathematician
Where: Jena, Germany
works\ Calculus of Concepts or Concept Script [1879]; Fundamental Laws of Arithmetic [1884]; Function and Concept [1891]; Concept and Object [1892]; Sense and Reference [1892]
Detail: He lived 1848 to 1925, axiomatized counting numbers using equivalence and symbolic logic, and axiomatized arithmetic. He founded axiomatic logic, using sets and propositions with quantifiers, to make the first propositional calculus.
Epistemology
All mathematics is formal {logicism, Frege}. Numbers and arithmetic form logical systems {analytic system}. They are not about intuition or empirical fact {synthetic system}.
Number is not an object property or subjective idea. Numbers are objective objects, and statements about numbers are objective.
Number {number, Frege} is set of elements whose quantity is the number. For example, two is set of all pairs. Zero is set of all sets having same number of elements as set of elements not identical to themselves. Classes have an element, number of elements, and number of elements {successor, Frege} not identical to element.
Higher object or set category {ancestor, Frege} includes lower category {ancestral relation}.
Second-order logic needs this concept.
Symbol systems {propositional calculus, Frege} can show truth or falsehood of logical statements containing IF ... THEN ..., AND, OR, and NOT, depending on clause truth. In particular, symbol system can express ideas of ALL, SOME, ANY, EVERY, and NONE {quantification theory}. First-order predicate calculus, second-order predicate calculus, and set theory can develop from propositional calculus.
Expressions {saturated expression} can be about objects and have completed senses. Expressions {unsaturated expression} can be functions and need objects to complete them.
Language objects, concepts, features, phrases, or sentences {reference, Frege} can denote {bedeutung}. Logical statement terms should have references.
Objects, concepts, or sentences can connote {sense, idea} {sinn}. Word sense is reference method used, so all words, even proper names, have sense. Word sense is constant objective fact, not subjective idea.
Logic laws are not laws of thought.
Sentence is function with arguments and should be either true or false. Declarative sentence represents situation. Word meanings and sentence structures supply conditions for understanding sentences {truth-condition, Frege}. Sentence meaning is conditions that make sentence true {truth-conditional semantics, Frege} {model-theoretic semantics, Frege} {Situation Semantics, Frege}. Only whole sentences have meaning.
Truth depends on objective-reality state that sentence depicts, not on mental judgments or ideas. Sentences with same meaning can be in different forms.
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Date Modified: 2022.0224