"When a person has learned a symbolic system well enough to use it, she has established a portable self-contained world within the mind." (Mihaly Csikszentmihalyi, "Flow", 1990)
"Mathematics […] is mired in a language of symbols foreign to most of us, [it] explores regions of the infinitesimally small and the infinitely large that elude words, much less understanding." (Robert Kanigel, "The Man Who Knew Infinity", 1991)
"Great mathematics seldom comes from idle speculation about abstract spaces and symbols. More often than not it is motivated by definite questions arising in the worlds of nature and humans." (John L Casti, "Reality Rules: Picturing the world in mathematics", 1992)
"Mathematical
modeling is about rules - the rules of reality. What distinguishes a
mathematical model from, say, a poem, a song, a portrait or any other kind of
‘model’, is that the mathematical model is an image or picture of reality
painted with logical symbols instead of with words, sounds or
watercolors." (John Casti, "Reality Rules", 1992)
"Scientific claims or statements are inexact and provisional. They depend on dozens of simplifying assumptions and on a particular choice of words and symbols and on 'all other things being equal'." (Bart Kosko, "Fuzzy Thinking: The new science of fuzzy logic", 1993)
"The insight at the root of artificial intelligence was that these 'bits' (manipulated by computers) could just as well stand as symbols for concepts that the machine would combine by the strict rules of logic or the looser associations of psychology." (Daniel Crevier, "AI: The tumultuous history of the search for artificial intelligence", 1993)
"Above all, words must be recognized as symbolic pointers to truth, not objective containers of truth." (John S Spong, "Resurrection: Myth or Reality?", 1994)
"[...]
images are probably the main content of our thoughts, regardless of the sensory
modality in which they are generated and regardless of whether they are about a
thing or a process involving things; or about words or other symbols, in a
given language, which correspond to a thing or process. Hidden behind those
images, never or rarely knowable by us, there are indeed numerous processes
that guide the generation and deployment of those images in space and time.
Those processes utilize rules and strategies embodied in dispositional
representations. They are essential for our thinking but are not a content of
our thoughts.” (Antonio R Damasio, “Descartes' Error. Emotion, Reason, and the
Human Brain”, 1994)
"Just as music comes alive in the performance of it, the same is true of mathematics. The symbols on the page have no more to do with mathematics than the notes on a page of music. They simply represent the experience." (Keith Devlin, "Mathematics: The Science of Patterns", 1994)
"Every phenomenon on earth is symbolic, and each symbol is an open gate through which the soul, if it is ready, can enter into the inner part of the world, where you and I and day and night are all one." (Hermann Hesse, "The Fairy Tales of Hermann Hesse", 1995)
“Mathematics
is not the study of an ideal, preexisting nontemporal reality. Neither is it a
chess-like game with made-up symbols and formulas. Rather, it is the part of
human studies which is capable of achieving a science-like consensus, capable
of establishing reproducible results. The existence of the subject called
mathematics is a fact, not a question. This fact means no more and no less than
the existence of modes of reasoning and argument about ideas which are
compelling an conclusive, ‘noncontroversial when once understood’."
(Philip J Davis & Rueben Hersh, “The Mathematical Experience”, 1995)
"Schematic
diagrams are more abstract than pictorial drawings, showing symbolic elements
and their interconnection to make clear the configuration and/or operation of a
system." (Ernest O Doebelin, "Engineering experimentation: planning,
execution, reporting", 1995)
"The
logic of the emotional mind is associative; it takes elements that symbolize a
reality, or trigger a memory of it, to be the same as that reality. That is why
similes, metaphors and images speak directly to the emotional mind." (Daniel
Goleman, "Emotional Intelligence", 1996)
“In many
ways, the mathematical quest to understand infinity parallels mystical attempts
to understand God. Both religions and mathematics attempt to express the
relationships between humans, the universe, and infinity. Both have arcane
symbols and rituals, and impenetrable language. Both exercise the deep recesses
of our mind and stimulate our imagination. Mathematicians, like priests, seek
‘ideal’, immutable, nonmaterial truths and then often try to apply theses truth
in the real world.” (Clifford A Pickover, "The Loom of God: Mathematical
Tapestries at the Edge of Time", 1997)
"Reality contains not only evidence, but also the means (such as our minds, and our artefacts) of understanding it. There are mathematical symbols in physical reality. The fact that it is we who put them there does not make them any less physical." (David Deutsch, "The Fabric of Reality", 1997)
"Meaning is conferred not by a one-to-one correspondence of a
symbol with some external concept or object, but by the relationships between
the structural components of the system itself."
"A formal system consists of a number of tokens or symbols, like pieces in a game. These symbols can be combined into patterns by means of a set of rules which defines what is or is not permissible (e.g. the rules of chess). These rules are strictly formal, i.e. they conform to a precise logic. The configuration of the symbols at any specific moment constitutes a ‘state’ of the system. A specific state will activate the applicable rules which then transform the system from one state to another. If the set of rules governing the behaviour of the system are exact and complete, one could test whether various possible states of the system are or are not permissible."
“Cultural
archetypes are the unconscious models that help us make sense of the world:
they are the myths, narratives, images, symbols, and files into which we
organize the data of our life experience” (Clotaire Rapaille, “Cultural
Imprints”, Executive Excellence Vol. 16 (10), 1999)
"In
broad terms, a mental model is to be understood as a dynamic symbolic
representation of external objects or events on the part of some natural or
artificial cognitive system. Mental models are thought to have certain
properties which make them stand out against other forms of symbolic
representations." (Gert Rickheit & Lorenz Sichelschmidt, "Mental
Models: Some Answers, Some Questions, Some Suggestions", 1999)
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