"All of this could have been said using notation that kept √-1 instead of the new representative i, which has the same virtual meaning. But i isolates the concept of rotation from the perception of root extraction, offering the mind a distinction between an algebraic result and an extension of the idea of number." (Joseph Mazur, "Enlightening Symbols: A Short History of Mathematical Notation and Its Hidden Powers", 2014)
"Geometry had its origins in the interest of working with lines, figures, and solids that could be imagined in the mind. Algebra had its origins in problems involving number - number hinged by geometric conceptions of iconic figures." (Joseph Mazur, "Enlightening Symbols: A Short History of Mathematical Notation and Its Hidden Powers", 2014)
"In mathematics, the symbolic form of a rhetorical statement is more than just convenient shorthand. First, it is not specific to any particular language; almost all languages of the world use the same notation, though possibly in different scriptory forms. Second, and perhaps most importantly, it helps the mind to transcend the ambiguities and misinterpretations dragged along by written words in natural language. It permits the mind to lift particular statements to their general form." (Joseph Mazur, "Enlightening Symbols: A Short History of Mathematical Notation and Its Hidden Powers", 2014)
"In natural language, even the most carefully chosen words drag along concealed meanings that have the power to manipulate reasoning. [...] Symbols of mathematics too sometimes have concealed meanings, but their purpose is to bring along pure thought. It is possible to learn what a mathematical symbol stands for by context. We learn the meanings of mathematical symbols mostly from their definitions: Mostly, because in formal mathematics not everyone easily grasps definitions that are not linked to the familiar properties of experience." (Joseph Mazur, "Enlightening Symbols: A Short History of Mathematical Notation and Its Hidden Powers", 2014)
"It may come as a surprise that the symbol i (even though it is just an abbreviation of the word 'imaginary') has a marked advantage over √-1. In reading mathematics, the difference between a + b√-1 and a + bi is the difference between eating a strawberry while holding your nose, missing the luscious taste, and eating a strawberry while breathing normally." (Joseph Mazur, "Enlightening Symbols: A Short History of Mathematical Notation and Its Hidden Powers", 2014)
"On a deeper level, the word 'symbol' suggests that, when the familiar is thrown together with the unfamiliar, something new is created. Or, to put it another way, when an unconscious idea fits a conscious one, a new meaning emerges. The symbol is exactly that: meaning derived from connections of conscious and unconscious thoughts." (Joseph Mazur, "Enlightening Symbols: A Short History of Mathematical Notation and Its Hidden Powers", 2014)
"One of the wonderful things about mathematics is that - by its best symbols - its progression expands its vision." (Joseph Mazur, "Enlightening Symbols: A Short History of Mathematical Notation and Its Hidden Powers", 2014)
"Symbolic language surely promotes its own concealed meanings that come from imaginative glimpses into the subconscious, but the best symbols are those that pinpoint meaning and yet permit the mind to quickly roam its databank of similar contextual patterns to compare, to transmit, and to creatively link what is unknown with what is known." (Joseph Mazur, "Enlightening Symbols: A Short History of Mathematical Notation and Its Hidden Powers", 2014)
"Symbols transcend the medium of communication. They are ubiquitous in our language, and play a sizable role (though perhaps not a central one) in mathematical imagery linking the conscious and subconscious, the familiar and unknown, to give us cultural/emotional predispositions to meaning, all to enhance the creative process."(Joseph Mazur, "Enlightening Symbols: A Short History of Mathematical Notation and Its Hidden Powers", 2014)
"Unlike symbols in poetry, mathematical symbols begin as deliberate designs created by mathematicians. That does not stop symbols from performing the same function that a poem would: to make connections between experience and the unknown and to transfer metaphorical thoughts capable of conveying meaning. As in poetry, there are archetypes in mathematics. If there are such things as self-evident truths, then there probably are things we know about the world that come with the human package at birth." (Joseph Mazur, "Enlightening Symbols: A Short History of Mathematical Notation and Its Hidden Powers", 2014)
"What is good mathematical notation? As it is with most excellent questions, the answer is not so simple. Whatever a symbol is, it must function as a revealer of patterns, a pointer to generalizations. It must have an intelligence of its own, or at least it must support our own intelligence and help us think for ourselves. It must be an indicator of things to come, a signaler of fresh thoughts, a clarifier of puzzling concepts, a help to overcome the mental fatigues of confusion that would otherwise come from rhetoric or shorthand." (Joseph Mazur, "Enlightening Symbols: A Short History of Mathematical Notation and Its Hidden Powers", 2014)
"When it comes to algebra, visual conception is beyond any similarities in the physical world. That’s okay; as we’ve noted, it’s not the job of mathematics to be concerned with the physical world, nor with what we call 'reality'. Symbolic consistency and meaning are essentials of mathematics. So is certainty. So is imagination. So is the creative process. So is hypothesis. So is belief beyond experience. So is adventure of knowledge. And, in today’s complexity, there is no better way to do the job of mathematics than by symbolic envisagement." (Joseph Mazur, "Enlightening Symbols: A Short History of Mathematical Notation and Its Hidden Powers", 2014)
"Yet there is a distinct difference between the writer’s art and the mathematician’s. Whereas the writer is at liberty to use symbols in ways that contradict experience in order to jolt emotions or to create states of mind with deep-rooted meanings from a personal life’s journey, the mathematician cannot compose contradictions, aside from the standard argument that establishes a proof by contradiction. Mathematical symbols have a definite initial purpose: to tidily package complex information in order to facilitate understanding." (Joseph Mazur, "Enlightening Symbols: A Short History of Mathematical Notation and Its Hidden Powers", 2014)
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