9th

August, 2013

How Mathematics Affects Material Culture

An essay by Akshay Verma, 7th Semester, FVC

Introduction

Mathematics is the principle that governs our universe. It is the underlying ‘logic’

of things. The systems of mathematics developed by humans are a most

fascinating and unique reflection of the species’ understanding of the cosmos. It

is a very abstract and highly conceptual system that allows us to manipulate

matter using the laws that govern it. This essay will explore how it has affected

material culture by primarily focusing on the macroscopic picture and delving

into a little bit of history.

Numbers Versus Mathematics

One of the cornerstones of human intelligence is the sense of numbers.

Interestingly, most animals and birds have it too. Although they may not be seen

counting, they know if one of their offspring goes missing. But, there are very few

animals that can apply this sense into mathematics. Human beings not only sense

numbers and quantities, they perform complex operations on them.

The sense of numbers is not one of the five ‘pure’ senses. The perception of

numbers is not possible without the five senses. Most people perceive numbers

through the sense of sight. Then those fundamental visual ideas are given names

and corresponding sounds. Unlike numbers, generally, mathematics doesn’t end

up being the final product; it is mostly an intermediate. Numbers are the

outcome of things, while mathematics is the operator that makes other numbers.

Therefore, a sense of numbers can be called a ‘Level 0’ faculty, while

mathematics, a ‘Level 1’ faculty.

The sense of numbers is closely related to linguistics. There are certain tribes

that do not have words for numbers. One such tribe is the Piraha tribe of the

Amazon.1 They have only three words to denote quantity: Hòi means "small size

or amount," hoì, means "somewhat larger amount," and baàgiso indicates to

"cause to come together, or many." The following is from a study [1]:

… one test involved 14 adults in one village that were presented with lines of spools of

thread and were asked to create a matching line of empty rubber balloons. The people

were not able to do the one-to-one correspondence, when the numbers were greater

than two or three.

1 Science News (2012): You Can't Do the Math Without the Words: Amazonian Tribe Lacks Words for Numbers

The oldest known possibly mathematical object is the Lebombo bone and is

dated to approximately 35,000 BC and has what are believed to be tally marks.

Also, prehistoric artifacts discovered in Africa and France, dated

between 35,000 and 20,000 years old, suggest early attempts to quantify time.2

A sense of numbers doesn’t guarantee the development of the faculty of

mathematics. It’s when numbers are dematerialized and values attached to them,

that mathematics starts to evolve.

2 + 2 = 4

Basic mathematical operations could have evolved as the visual ideas of adding

(or collecting) similar objects together, subtracting from them, dividing them

into equal or proportional parts, or multiplying them with small numbers. Small

quantities can be represented or counted on fingers, but the representation of

larger quantities could not have evolved without a proper system of denoting

them. Many civilizations evolved tools like tally marks, abaci and written

symbols. These principles were applied to geometric shapes by exploring the

relationships between the lengths, angles and areas. Systems of weights and

measures were also developed.3 Further, place value systems, decimal and zero

were invented.

Although measurement was an inaccurate science then, it lead to great advances

in material technology. Chemical processes became more accurate. Architecture

is the most obvious field that benefitted from this. The great pyramids of Egypt

are a striking example. Astronomy began to be observed and understood, and

calendars were invented.4 Technology improved in general with the

development of number systems and mathematics.

Gradually, mathematics moved from a direct application of numbers to objects,

to higher levels of abstraction. Concepts like Pi (π), negative numbers, recurring

decimals, irrational numbers, square roots, quadratic equations, etc. were

discovered.5 In 820 AD, Al-Khwarizmi wrote Al-Jabr, later transliterated as

Algebra. It was a milestone in mathematics, as it made numbers variable and

realized the concept of dematerialized quantities. This can be called as a ‘Level 2’

faculty.

2 Wikipedia: History of Mathematics

3 California State University, Northridge: The evolution of notation for decimal fractions

4 Wikipedia: Egyptian Calendar

5 Wikipedia: Timeline of Mathematics

The Concept of Value

The concept of quantitative value is a limited one. It made possible for objects to

be measured on various parameters, but could not represent the relative

qualitative values of things. Thus, the concept of currency was evolved. The

values of different objects were in direct proportion to the unit of the currency.

These values were variable, depending on subjective terms of demand and

supply. Selling and buying became easier and no longer depended on

coincidence of wants. Though banking systems had already existed before the

development of mathematics, they greatly improved and now made use of

compound interest and collaterals. Foreign exchange and trade also started to

take place.

With global exchange of information and the establishment of systems of

learning, the concept of value reached greater heights. Calculus, logarithms,

statistics and graphing were discovered. In 1847, George Boole formalized

symbolic logic in The Mathematical Analysis of Logic, defining what is now called

Boolean algebra.6 This was a huge breakthrough that paved way for computer

science. It added another layer of abstraction to mathematics. Earlier, numbers

used to denote quantity, but now numbers were realized as ‘contrast,’ relative

difference in state. Anything that differs from another has a mathematical

relationship to it. For example, if an apple is red and a banana is yellow, I may

call an apple ‘0’ and a banana ‘1.’ Now, I can define a mathematical set of logic to

these numbers and perform logical computations on them. This can be called as a

‘Level 3’ faculty.

The expansion of the domain of mathematics from simple quantitative

expressions and operators, to the something beyond addition and subtraction —

logic, the state of being true or false — created the world we are living in today.

From a philosophical perspective, mathematics is now starting to be applied to

the duality principle.

As we are going higher up the level of abstraction in mathematics, we are moving

away from a system of constant measures to one that accounts for variability.

The only constants are the concepts of the numbers, not matter itself. Our

material culture is gradually shifting to a relative value based system from an

absolute value system. This is also affecting the importance of things, which can

be seen in the marked changes that are taking place in people, both at an

individual level as well as at the social level. It is a change in human intelligence,

the ripples of which are reaching the surface in the form of behavioral changes.

6 Wikipedia: Boolean Algebra

The Future

As we have seen, our understanding of mathematics places the brick-wall of the

limit of material advances. In a universe where everything is in mathematical

relationship with everything else, we achieve higher levels of manipulability with

our increasing abstraction of ideas into mathematics. The current level of

understanding has allowed us to transfer our intelligence to machines in the

form of computers and artificial intelligence. In the future we may also be able to

find mathematical parallels to our emotions, intuition and ego. Our universe is

expanding both outwards and inwards, and mathematics is the invisible force

that is driving it.