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He published an op-ed titled "Is Algebra Necessary" in the New York Times on July 29, 2012, arguing that while arithmetic and mathematical literacy should be universally taught, higher math concepts like algebra and trigonometry are not useful for the vast majority of students. [3]
[1] His recent book, Higher Education? was written in collaboration with Claudia Dreifus, his wife, a New York Times science writer and Columbia University professor. Professor Hacker is a frequent contributor to the New York Review of Books. In his articles he has questioned whether mathematics is necessary, claiming "Making mathematics ...
Algebra is the branch of mathematics that studies certain abstract systems, known as algebraic structures, and the manipulation of expressions within those systems. It is a generalization of arithmetic that introduces variables and algebraic operations other than the standard arithmetic operations, such as addition and multiplication.
The assertion that Q is necessary for P is colloquially equivalent to "P cannot be true unless Q is true" or "if Q is false, then P is false". [9] [1] By contraposition, this is the same thing as "whenever P is true, so is Q". The logical relation between P and Q is expressed as "if P, then Q" and denoted "P ⇒ Q" (P implies Q).
The scope of algebra thus grew to include the study of algebraic structures. This object of algebra was called modern algebra or abstract algebra, as established by the influence and works of Emmy Noether. [36] Some types of algebraic structures have useful and often fundamental properties, in many areas of mathematics.
In this view, logic is the proper foundation of mathematics, and all mathematical statements are necessary logical truths. Rudolf Carnap (1931) presents the logicist thesis in two parts: [ 39 ] The concepts of mathematics can be derived from logical concepts through explicit definitions.
In 1637, René Descartes published La Géométrie, in which he showed that geometry can be reduced to algebra by means coordinates, which are numbers determining the position of a point. This gives to the numbers that he called real numbers a more foundational role (before him, numbers were defined as the ratio of two lengths).
Rhetorical algebra, in which equations are written in full sentences. For example, the rhetorical form of + = is "The thing plus one equals two" or possibly "The thing plus 1 equals 2". Rhetorical algebra was first developed by the ancient Babylonians and remained dominant up to the 16th century.