Mathematics in the Modern World, Nature of Mathematics, Definition of Mathematics. This slide is patterned after the book Mathematics in the Modern World by Juan Apolinario C. Reyes, published by Unlimited Books Library Services and Publishing Inc, 2018.

1. 1.11 Mathematics
in Our World
Nature of Mathematics, Definition of Mathematics, Patterns in Nature
and Society,

2. Nature of Mathematics
a. It is partly invented.
b. Much of it is discovered.
c. It is a system. It is a collection of objects and
relations among objects whose characteristics are
discovered through accepted rules of reasoning.

3. Mathematics is a human invention
 Mathematics is a form of language. It is a language to
describe quantity, form, patter, and change.
 Language, in general, is a human invention. This is a
fundamental proposition about language agreed upon by
anthropologists, archeologists, and social scientists in
general.
 Therefore, mathematics is a human invention.

4. Mathematics as an invention
 Early mathematics was invented to represent the
idea of QUANTITY.
Number of cattle
Quantity of grains
Length of time
Size of armies, etc.

5. Earliest artifact in mathematics
 Ishango bone
a fibula of a baboon discovered in the
area of Ishango near Semliki River in
Congo
At least 20,000 years old
It contained notches to track the
menstruation cycle of women

6. Why was mathematics invented?
 It was invented to help people act on things,
events, other people, etc.
 To act on things, events, etc., means to act on
them with purpose.
 Mathematics told them when to act (time).
 It helped them answer questions like “how much”
and “how often”?

7. Why was mathematics invented?
 It was invented to guide human behavior on issues
that have to do with “when”, “how much”, “how
often”, etc.

8. Why was mathematics invented?
 What was the ishango bone invented for?
Judging from the superstitions people in primitive
cultures, and even in modern cultures, have about
menstruation, people in Ishango wanted to know
when will the period come and leave.
They wanted to know when so that they can act to it
in ways that fit their beliefs about menstruation.

9. Why was mathematics invented?
 It was invented to manage human behavior at the
person-to-person level.
 It was also invented to manage collective human
behavior.

10. Why was mathematics invented?
 “What cannot be measured cannot be managed.”
The institutions of power and authority in society
needed mathematics to effectively manage people
and society in general.
For example, in apportioning lands to people, in
assigning and collecting taxes, to count the sacks of
grains in a granary, to track and facilitate commerce
and trade, to manage armies, etc.

11. Why was mathematics invented?
 Sir Isaac Newton and Gottfried Leibnitz invented
calculus
It was invented to measure and reveal the properties
of rates of change that take place in infinitesimally
small changes in time.

12. Why was mathematics invented?
 George Boole invented the modern binary system.
It was invented so that people can apply
mathematics to solving problems in logic and
philosophy.

13. Mathematics as an invention
 As an invention, it helps guide individual human
behavior.
 At the society level, it helps organize society.
 It contributes to an orderly social interactions in a
society.

14. Mathematics is also discovered.
 Much of mathematics is discovered.
The foundation of mathematics was an act of human
invention.
But what came after it were discoveries.

15. Euclidean Geometry as Invention and
Discovery
 The five postulates (Euclid’s postulates) of plane
geometry were axioms.
They were taken to be true without proof.
The decision to make these the foundation of plane
geometry is an act of human invention.

16. Euclidean Geometry as Invention and
Discovery
 Propositions or theorems that came after the five
postulates were discoveries.
 They were discovered through logical reasoning.

17. Euclidean Geometry and Non-Euclidean
Geometry
 Different geometries arise from different sets of
assumptions.
Removing the fifth postulate gives rise to non-Euclidean
Geometries.
Non-Euclidean geometries have their own structure and
properties which can be discovered through logical
reasoning.
Validity of this structure and truthfulness of properties are
demonstrated by methods of proof.

18. Definition of Mathematics

19. Definition of Mathematics
 Mathematics is a form of language. It is also a
body of knowledge.

20. Mathematics as language and a body of
knowledge
 Mathematic is a tool to name and describe
objects or events that reveal quantity, form,
pattern, and change.
 It is a set of concepts or ideas and relations
among them whose properties and characteristics
are obtained through logical reasoning.

21. Mathematics as language
 Numeration systems
 Symbols and notations to describe quantity, time,
change, patterns.

22. Mathematics as a body of knowledge
 Geometry
 Algebra
 Statistics
 Calculus
 Combinatorics
 Graph Theory
 Number Theory
 Optimization
 And many more