Mathematics as a Language

Blake wrote: "I have heard many People say, 'Give me the Ideas. It is no matter what Words you put them into.'"
To this he replies, "Ideas cannot be Given but in their minutely Appropriate Words."

- William Blake
(quoted by J. Newman, The World of Mathematics, 1956)

This opinion is seconded by Bertrand Russell. Here is what he says in his Autobiography about meeting G.Peano at an International Congress on Philosophy in 1900:

The Congress was a turning point in my intellectual life, because I met there Peano. I already knew him by name and had seen some of his work, but had not taken the trouble to master his notation. In discussions at the Congress I observed that he was always more precise than anyone else, and that he invariably got the better of any argument upon which he embarked. As the days went by, I decided that this must be owing to his mathematical logic. I therefore got him give me all his works, and as soon as the Congress was over I retired to Fernhurst to study quietly every word written by him and his disciples. It became clear to me that his notation afforded an instrument of logical analysis such as I had been seeking for years, and that by studying him I was acquiring a new and powerful technique for the work that I had long wanted to do.

When I think of the development of Mathematics over the last 2500 years, I am less surprised that early mathematicians left lasting results than that, given the tools they possessed, they achieved anything at all that could have lived through centuries. Just think of it. Zero gained widespread use only in the last millennium. Systematic introduction of modern algebraic notations began only in the sixteenth century and is most often associated with the French mathematician François Viète (1540-1603). René Descartes (1596-1650) was first to use letters at the end of the alphabet for unknowns. He also introduced the power notations: x2, x3. The sign of equality (two equal parallel strokes) has been invented by Robert Recorde (c. 1510-1558) in his The whetstone of witte (London, 1557):

I will sette as I doe often in worke use, a paire of paralleles, or Gemowe lines of one lengthe, thus: =, bicause noe.2. thynges, can be moare equalle.

To help you appreciate the expressive power of the modern mathematical language, and as a tribute to those who achieved so much without it, I collected a few samples of (original but translated) formulation of theorems and their equivalents in modern math language.

If a straight line be cut at random, the square on the whole is equal to the squares on the segments and twice the rectangle contained by the segments. (Euclid, Elements, II.4, 300 B.C.) (a + b)2 = a2 + b2 + 2ab
If as many numbers as we please beginning from a unit be set out continuously in double proportion, until the sum of all becomes prime, and if the sum multiplied into the last make some number, the product will be perfect. (Euclid, Elements, IX.36, 300 B.C.) If 1 + 2 + ... + 2n is prime, then
2n(1 + 2 + ... + 2n) is perfect
The area of any circle is equal to a right-angled triangle in which one of the sides about the right angle is equal to the radius, and the other to the circumference, of the circle. (Archimedes, Measurement of a Circle, 225 B.C.) A = 2pr·r/2 = pr 2
The surface of any sphere is equal four times the greatest circle in it. (Archimedes, On the Sphere and the Cylinder, 220B.C.) S = 4pr 2
Rule to solve x3 + mx = n
Cube one-third the coefficient of x; add to it the square of one-half the constant of the equation; and take the square root of the whole. You will duplicate this, and to one of the two you add one-half the number you have already squared and from the other you subtract one-half the same... Then, subtracting the cube root of the first from the cube root of the second, the remainder which is left is the value of x (Gerolamo Cardano, Ars Magna, 1545).
x =
D in R - D in E aequabitur A quad. (Francois Viete, In artem analytican isagoge, 1590) DR - DE = A2

Another example shows how different mathematicians might have expressed the modern equation 4x2 + 3x = 10.

However, the language of Mathematics does not consist of formulas alone. The definitions and terms are verbalized often acquiring a meaning different from the customary one. Many students are inclined to hold this against mathematics. For example, one may wonder whether 0 is a number. As the argument goes, it is not, because when one says, I watched a number of movies, one does not mean 0 as a possibility. 1 is an unlikely candidate either. But do not forget that ambiguities exist in plain English (the number's number is just one of them) and in other sciences as well. As a matter of fact, mathematical language is by far more accurate than any other one may think of. Do not forget also that every science and a human activity field has its own lingo and a word usage in many instances much different from that one may be more comfortable with.

Lest you think that my defense of the mathematical language has no solid basis, I began collecting word usage surprises from non-mathematical fields of activity and sciences. I welcome any examples of language misuse or inherent ambiguity you may want to send me.

Language of Mathematics, Language of Science and Plain Language


  1. W. Dunham, Journey through Genius, Penguin Books, 1991
  2. H. Eves, Great Moments in Mathematics Before 1650, MAA, 1983
  3. R. Hersh, Math Lingo vs. Plain English: Double Entendre, The Amer Math Monthly, V104, N1, 1997, pp 48-51.
  4. My Letter to the Monthly Editor
  5. R. Hersh, "Math Lingo": a follow-up article

On Internet

  1. Earliest Uses of Various Mathematical Symbols
  2. Earliest Known Uses of Some of the Words of Mathematics

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