What is Mathematics? 

Some Quotes

 

 

            Mathematics has traditionally been described as the science of number and shape.  The school emphasis on arithmetic and geometry is deeply rooted in this centuries-old perspective.  But as the territory explored by mathematicians has expanded … the historic boundaries of mathematics have all but disappeared.  So have the boundaries of its application: no longer just the language of physics and engineering, mathematics is now an essential tool for banking, manufacturing, social science, and medicine.  When viewed in this broader context, we see that mathematics is not just about number and shape but about pattern and order of all sorts.  Number and shape  arithmetic and geometry  are but two of many media in which mathematicians work.  Active mathematicians seek patterns wherever they arise.

 

Lynn Arthur Steen, On The Shoulder’s of Giants: New Approaches to Numeracy, 1990.

 

 

            Most cultures do not set mathematics apart as a distinct, explicit category.  But with or without that category, mathematical ideas, nonetheless, do exist.  The ideas, however, are more often to be found elsewhere in the culture, namely, integrated into the contexts in which they arise, as part of the complex of ideas that surround them.  The contexts for the ideas might be, for example, what we categorize as navigation, calendrics, divination, religion, social relations, or decoration…Were we to present the ideas divorced from their contexts, they might look more like our own modern mathematics.  This approach, however, would distort a major difference  most practitioners of modern mathematics value their ideas because they believe them to be context-free; others value their ideas as inseparable from the cultural milieu that gives them meaning.

Marcia Ascher, Mathematics Elsewhere: An Exploration of Ideas Across Cultures, 2002.

 

 

            Mathematics is a vast adventure in ideas; its history reflects some of the noblest thoughts of countless generations.

Dirk Struik, 1948 (quoted in Schmalz, Rosemary, Out of the Mouths of Mathematicians: A Quotation Book for Philomaths) 

 

 

Creativity is the heart and soul of mathematics at all levels.  The collection of special skills and techniques is only the raw material out of which the subject itself grows.  To look at mathematics without the creative side of it, is to look at a black-and-white photograph of a Cezanne; outlines may be there, but everything that matters is missing. 

R. C. Buck, 1962 (quoted in Schmalz)

Mathematics in general is fundamentally the science of self-evident things. 

Felix Klein, 1902 (quoted in Moritz, Memorabilia Mathematica: The

            Philomath’s Quotation Book).

 

 

            Pure mathematics consists entirely of such asseverations as that, if such and such a proposition is true of anything, then such and such another proposition is true of that thing.  It is essential not to discuss whether the first proposition is really true, and not to mention what the anything is of which it is supposed to be true…. If our hypothesis is about anything and not about some one or more particular things, then our deductions constitute mathematics.  Thus mathematics may be defined as the subject in which we never know what we are talking about, nor whether what we are saying is true.

                        Bertrand Russell, 1901 (quoted in Moritz)

 

 

            The object of pure Physic is the unfolding of the laws of the intelligible world; the object of pure Mathematic that of unfolding the laws of human intelligence.

                        J.J. Sylvester, approx 1870 (quoted in Moritz)

 

 

I propose the following, if not as a definition, then at least as a partial description: mathematics is persistent intellectual honesty.

Moses Richardson, 1952 (quoted in Schmalz)

 

 

Mathematics is a human activity almost as diverse as the human mind itself.

Gábor Szegö, 1963 (quoted in Schmalz)

 

 

The word “mathematics” is a Greek word, and, by origin, it means “something that has been learned or understood,” or perhaps “acquired knowledge,” and perhaps even, somewhat against grammar, “acquirable knowledge,” that is, “learnable knowledge,” that is, “knowledge acquirable by learning.”

Salomon Bochner, 1966 (quoted in Schmalz)

 

 

To find the simple in the complex, the finite in the infinite  that is not a bad description of the aim and essence of mathematics.

Jacob Schwartz, 1986 (quoted in Schmalz)

 

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