“Greek thought was essentially non-algebraic, because it was so concrete. The abstract operations of algebra, which deal with objects that have been purposely stripped of their physical content, could not occur to minds which were so intently interested in the objects themselves. The symbol is not a mere formality; it is the very essence of algebra. Without the symbol the object is a human perception and reflects all the phases under which the human senses grasp it; replaced by a symbol the object becomes a complete abstraction, a mere operand subject to certain indicated operations.”

p, 125
Number: The Language of Science (1930)

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Tobias Dantzig25

American mathematician 1884–1956

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“This principle, which is thus made the foundation of the operations and results of Symbolical Algebra, has been called "The principle of the permanence of equivalent forms", and may be stated as follows:
"Whatever algebraical forms are equivalent, when the symbols are general in form but specific in value, will be equivalent likewise when the symbols are general in value as well as in form."”

George Peacock (1791–1858) Scottish mathematician

Vol. II: On Symbolical Algebra and its Applications to the Geometry of Position (1845) Ch. XV, p. 59
A Treatise on Algebra (1842)

“Symbolical algebra adopts the rules of arithmetical algebra, but removes altogether their restrictions: thus symbolical subtraction differs from the same operation in arithmetical algebra in being possible for all relations of value of the symbols or expressions employed… all the results of arithmetical algebra which are deduced by the application of its rules, and which are general in form, though particular in value, are results likewise of symbolical algebra, where they are general in value as well as in form: thus the product of a^{m} and a^{n}, which is a^{m+n} when m and n are whole numbers, and therefore general in form though particular in value, will be their product likewise when m and n are general in value as well as in form: the series for (a+b)^{n}, determined by the principles of arithmetical algebra, when n is any whole number, if it be exhibited in a general form, without reference to a final term, may be shewn, upon the same principle, to the equivalent series for (a+b)^n, when n is general both in form and value.”

George Peacock (1791–1858) Scottish mathematician

Vol. I: Arithmetical Algebra Preface, p. vi-vii
A Treatise on Algebra (1842)

“The unnaturalness of mathematical symbolism is attested to by history. The algebra of the Egyptians, the Babylonians, the Greeks, the Hindus, and the Arabs was what is commonly called rhetorical algebra. …on the whole they used ordinary rhetoric to describe their mathematical work. Symbolism is a relatively modern invention of the sixteenth and seventeenth centuries…”

Morris Kline (1908–1992) American mathematician

Source: Mathematics and the Physical World (1959), p. 59

“[A]ll our symbols have the same purpose; words are merely the symbols we use most commonly. The function of words in human thought is to stand for things which are not present to the senses, and allow the mind to manipulate them—things, concepts, ideas, everything that does not have a physical reality in front of us now.”

Jacob Bronowski (1908–1974) Polish-born British mathematician

"The Imaginative Mind in Art" (1978)

“Rather than a series of levels, we have a distinguished level,, the level at which interpretation of the symbols is in the intentional, or cognitive, domain or in the domain of the objects of thought.”

Zenon Pylyshyn (1937) Canadian philosopher

Source: Computation and cognition, 1984, p. 95

“I have endeavoured… to present the principles and applications of Symbolical, in immediate sequence to those of Arithmetical, Algebra, and at the same time to preserve that strict logical order and simplicity of form and statement which is essential to an elementary work. This is a task of no ordinary difficulty, more particularly when the great generality of the language of Symbolical Algebra and the wide range of its applications are considered, and this difficulty has not been a little increased, in the present instance, by the wide departure of my own views of its principles from those which have been commonly entertained.”

George Peacock (1791–1858) Scottish mathematician

Vol. II: On Symbolical Algebra and its Applications to the Geometry of Position (1845) Preface, p. iii
A Treatise on Algebra (1842)

“By the help of God and with His precious assistance, I say that Algebra is a scientific art. The objects with which it deals are absolute numbers and measurable quantities which, though themselves unknown, are related to "things" which are known, whereby the determination of the unknown quantities is possible.”

Omar Khayyám (1048–1131) Persian poet, philosopher, mathematician, and astronomer

Treatise on Demonstration of Problems of Algebra (1070).
Context: By the help of God and with His precious assistance, I say that Algebra is a scientific art. The objects with which it deals are absolute numbers and measurable quantities which, though themselves unknown, are related to "things" which are known, whereby the determination of the unknown quantities is possible. Such a thing is either a quantity or a unique relation, which is only determined by careful examination. What one searches for in the algebraic art are the relations which lead from the known to the unknown, to discover which is the object of Algebra as stated above. The perfection of this art consists in knowledge of the scientific method by which one determines numerical and geometric unknowns.

“Let us conceive, then, of an algebra in which the symbols x, y z etc. admit indifferently of the values 0 and 1, and of these values alone The laws, the axioms, and the processes, of such an Algebra will be identical in their whole extend with the laws, the axioms, and the processes of an Algebra of Logic. Difference of interpretation will alone divide them. Upon this principle the method of the following work is established.”

George Boole (1815–1864) English mathematician, philosopher and logician

Source: 1850s, An Investigation of the Laws of Thought (1854), p. 37; Cited in: William Torrey Harris (1879) The Journal of Speculative Philosophy, p. 109

“Any impatient student of mathematics or science or engineering who is irked by having algebraic symbolism thrust on him should try to get on without it for a week.”

Eric Temple Bell (1883–1960) mathematician and science fiction author born in Scotland who lived in the United States for most of his li…

Source: Mathematics: Queen and Servant of Science (1938), p. 226
Context: Some of his deepest discoveries were reasoned out verbally with very few if any symbols, and those for the most part mere abbreviations of words. Any impatient student of mathematics or science or engineering who is irked by having algebraic symbolism thrust on him should try to get on without it for a week.

“In mathematics, logic, linguistics, and other abstract disciplines, the systems are not assigned to objects. They are defined by an enumeration of the variables, their admissible values, and their algebraic, topological, grammatical, and other properties which, in the given case, determine the relations between the variables under consideration.”

George Klir (1932–2016) American computer scientist

Source: An approach to general systems theory (1969), p. 40.

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Tobias Dantzig25

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