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May 9, 2023 · The essay describes the developments in mathematics over human history. It is a short account of important steps taken by humanity in the development of mathematics, its methods and...
Where Mathematics Comes From: How the Embodied Mind Brings Mathematics into Being (hereinafter WMCF) is a book by George Lakoff, a cognitive linguist, and Rafael E. Núñez, a psychologist. Published in 2000, WMCF seeks to found a cognitive science of mathematics, a theory of embodied mathematics based on conceptual metaphor.
- George Lakoff, Rafael E. Núñez
- 2000
- Overview
- Mathematics in ancient Egypt
- The numeral system and arithmetic operations
- Geometry
The introduction of writing in Egypt in the predynastic period (c. 3000 bce) brought with it the formation of a special class of literate professionals, the scribes. By virtue of their writing skills, the scribes took on all the duties of a civil service: record keeping, tax accounting, the management of public works (building projects and the like), even the prosecution of war through overseeing military supplies and payrolls. Young men enrolled in scribal schools to learn the essentials of the trade, which included not only reading and writing but also the basics of mathematics.
One of the texts popular as a copy exercise in the schools of the New Kingdom (13th century bce) was a satiric letter in which one scribe, Hori, taunts his rival, Amen-em-opet, for his incompetence as an adviser and manager. “You are the clever scribe at the head of the troops,” Hori chides at one point,
a ramp is to be built, 730 cubits long, 55 cubits wide, with 120 compartments—it is 60 cubits high, 30 cubits in the middle…and the generals and the scribes turn to you and say, “You are a clever scribe, your name is famous. Is there anything you don’t know? Answer us, how many bricks are needed?” Let each compartment be 30 cubits by 7 cubits.
This problem, and three others like it in the same letter, cannot be solved without further data. But the point of the humour is clear, as Hori challenges his rival with these hard, but typical, tasks.
The introduction of writing in Egypt in the predynastic period (c. 3000 bce) brought with it the formation of a special class of literate professionals, the scribes. By virtue of their writing skills, the scribes took on all the duties of a civil service: record keeping, tax accounting, the management of public works (building projects and the like), even the prosecution of war through overseeing military supplies and payrolls. Young men enrolled in scribal schools to learn the essentials of the trade, which included not only reading and writing but also the basics of mathematics.
One of the texts popular as a copy exercise in the schools of the New Kingdom (13th century bce) was a satiric letter in which one scribe, Hori, taunts his rival, Amen-em-opet, for his incompetence as an adviser and manager. “You are the clever scribe at the head of the troops,” Hori chides at one point,
a ramp is to be built, 730 cubits long, 55 cubits wide, with 120 compartments—it is 60 cubits high, 30 cubits in the middle…and the generals and the scribes turn to you and say, “You are a clever scribe, your name is famous. Is there anything you don’t know? Answer us, how many bricks are needed?” Let each compartment be 30 cubits by 7 cubits.
This problem, and three others like it in the same letter, cannot be solved without further data. But the point of the humour is clear, as Hori challenges his rival with these hard, but typical, tasks.
The Egyptians, like the Romans after them, expressed numbers according to a decimal scheme, using separate symbols for 1, 10, 100, 1,000, and so on; each symbol appeared in the expression for a number as many times as the value it represented occurred in the number itself. For example, stood for 24. This rather cumbersome notation was used within the hieroglyphic writing found in stone inscriptions and other formal texts, but in the papyrus documents the scribes employed a more convenient abbreviated script, called hieratic writing, where, for example, 24 was written .
In such a system, addition and subtraction amount to counting how many symbols of each kind there are in the numerical expressions and then rewriting with the resulting number of symbols. The texts that survive do not reveal what, if any, special procedures the scribes used to assist in this. But for multiplication they introduced a method of successive doubling. For example, to multiply 28 by 11, one constructs a table of multiples of 28 like the following:
The several entries in the first column that together sum to 11 (i.e., 8, 2, and 1) are checked off. The product is then found by adding up the multiples corresponding to these entries; thus, 224 + 56 + 28 = 308, the desired product.
To divide 308 by 28, the Egyptians applied the same procedure in reverse. Using the same table as in the multiplication problem, one can see that 8 produces the largest multiple of 28 that is less then 308 (for the entry at 16 is already 448), and 8 is checked off. The process is then repeated, this time for the remainder (84) obtained by subtracting the entry at 8 (224) from the original number (308). This, however, is already smaller than the entry at 4, which consequently is ignored, but it is greater than the entry at 2 (56), which is then checked off. The process is repeated again for the remainder obtained by subtracting 56 from the previous remainder of 84, or 28, which also happens to exactly equal the entry at 1 and which is then checked off. The entries that have been checked off are added up, yielding the quotient: 8 + 2 + 1 = 11. (In most cases, of course, there is a remainder that is less than the divisor.)
For larger numbers this procedure can be improved by considering multiples of one of the factors by 10, 20,…or even by higher orders of magnitude (100, 1,000,…), as necessary (in the Egyptian decimal notation, these multiples are easy to work out). Thus, one can find the product of 28 by 27 by setting out the multiples of 28 by 1, 2, 4, 8, 10, and 20. Since the entries 1, 2, 4, and 20 add up to 27, one has only to add up the corresponding multiples to find the answer.
Computations involving fractions are carried out under the restriction to unit parts (that is, fractions that in modern notation are written with 1 as the numerator). To express the result of dividing 4 by 7, for instance, which in modern notation is simply 4/7, the scribe wrote 1/2 + 1/14. The procedure for finding quotients in this form merely extends the usual method for the division of integers, where one now inspects the entries for 2/3, 1/3, 1/6, etc., and 1/2, 1/4, 1/8, etc., until the corresponding multiples of the divisor sum to the dividend. (The scribes included 2/3, one may observe, even though it is not a unit fraction.) In practice the procedure can sometimes become quite complicated (for example, the value for 2/29 is given in the Rhind papyrus as 1/24 + 1/58 + 1/174 + 1/232) and can be worked out in different ways (for example, the same 2/29 might be found as 1/15 + 1/435 or as 1/16 + 1/232 + 1/464, etc.). A considerable portion of the papyrus texts is devoted to tables to facilitate the finding of such unit-fraction values.
The geometric problems in the papyri seek measurements of figures, like rectangles and triangles of given base and height, by means of suitable arithmetic operations. In a more complicated problem, a rectangle is sought whose area is 12 and whose height is 1/2 + 1/4 times its base (Golenishchev papyrus, problem 6). To solve the problem, the ratio is inverted and multiplied by the area, yielding 16; the square root of the result (4) is the base of the rectangle, and 1/2 + 1/4 times 4, or 3, is the height. The entire process is analogous to the process of solving the algebraic equation for the problem (x × 3/4x = 12), though without the use of a letter for the unknown. An interesting procedure is used to find the area of the circle (Rhind papyrus, problem 50): 1/9 of the diameter is discarded, and the result is squared. For example, if the diameter is 9, the area is set equal to 64. The scribe recognized that the area of a circle is proportional to the square of the diameter and assumed for the constant of proportionality (that is, π/4) the value 64/81. This is a rather good estimate, being about 0.6 percent too large. (It is not as close, however, as the now common estimate of 31/7, first proposed by Archimedes, which is only about 0.04 percent too large.) But there is nothing in the papyri indicating that the scribes were aware that this rule was only approximate rather than exact.
A remarkable result is the rule for the volume of the truncated pyramid (Golenishchev papyrus, problem 14). The scribe assumes the height to be 6, the base to be a square of side 4, and the top a square of side 2. He multiplies one-third the height times 28, finding the volume to be 56; here 28 is computed from 2 × 2 + 2 × 4 + 4 × 4. Since this is correct, it can be assumed that the scribe also knew the general rule: A = (h/3)(a2 + ab + b2). How the scribes actually derived the rule is a matter for debate, but it is reasonable to suppose that they were aware of related rules, such as that for the volume of a pyramid: one-third the height times the area of the base.
foundations of mathematics, the study of the logical and philosophical basis of mathematics, including whether the axioms of a given system ensure its completeness and its consistency.
Jan 26, 2024 · We don’t know exactly how prehistoric humans dealt with math problems (like counting how many berries they picked, or figuring out the distance between two places), but researchers believe that people were using addition, multiplication, and other math concepts in early China, India, and Mesopotamia.
Jan 13, 2020 · Starting from A to Z: Learn about the history of math and the inventions associated with mathematics.
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Oct 4, 2021 · Mathematics today profoundly affects our societies and has had a lasting impact on science and technology. However, before Greek mathematicians developed many wonderful achievements in this area, what are the origins of mathematics and how did this field evolve in its early history? [1]
