Overview of how Science and Mathematics course structure is studied and analysed

1. SCIENTIFIC & TECHNICAL
TERMINOLOGIES
ASIF EQBAL
ELECTRICAL ENGINEER

2. Index & Introduction
 In school, science may sometimes seem like a collection of isolated and
statistical facts listed in a textbook, but that's the tip of iceberg. Just as
importantly, science is also a process of discovery that allows us to link
isolated facts into systematic and comprehensive understandings of
the natural world. This article deals the peculiarities and structure into which
Science and Mathematics is organized.
Contents covered:
 Introduction to Science and Mathematics
 Structural setup of Science and Mathematics
 Scientific terminologies

3. Introduction to Science and Mathematics
Well Science, Mathematics, applied Sciences
and finally Engineering is studied and
practiced by many across the globe. However
we seldom spare time or bother to observe how
these subjects are organized and the very
structure and set up of scientific and
engineering subjects is as exciting as the
subject itself.
In simple terms Science is a systematic and
formulated study of nature. Similarly in simple
terms mathematics is formulation of data
pattern and its order to calculate and predict
the hidden data, its order and pattern. So
Mathematics is one of the most important tools
for study of Science.
Word Science is derived from Latin word
scientia, meaning "knowledge. So Science is a
systematic study that builds facts in the form
of testable explanations and reliable
application. A practitioner of science is known
as a scientist.

4. Structural setup of Science and Mathematics
Subjects of Science and
Mathematics their course
description is expressed
and analyzed through:
Axioms
Postulate
Definition
Theorem
Laws
Hypothesis

5. Scientific terminologies

6. Axioms
 Axioms: Axioms are statement or proposition which is regarded as
being established, accepted, or self-evidently true. As per Wikipedia
the word "axiom" comes from the Greek word (axioma), a verbal
noun from the verb (axioein), meaning "to deem worthy", but also
"to require", which in turn comes from (axios), meaning "being in
balance", and hence "having (the same) value (as)", "worthy",
"proper". Among the ancient Greek philosophers an axiom was a
claim which could be seen to be true without any need for proof.
 Why axioms are required: For doing any type of formal or logical
reasoning, or any kind of formulated study to arrive at any inference,
we need to start with some set of known facts. It is not possible to
perform any systematic or formulated inference starting from
absolutely no knowledge. So axioms are basically required for
forming definitions and deriving further theorems. Axioms are the set
of known facts that are accepted as basic primitive unproven facts.

7. Examplesof Axioms are
 Wind blows from high pressure to low pressure
 Space is three dimensional
 A number is equal to itself. (e.g a = a). This is the first axiom of equality.
"Things equal to the same thing are equal to each other."
 Numbers are symmetric around the equals sign. If a = b then b = a. If a = b
and b = c then a = c. "Things equal to the same thing are equal to each other."
 If a = b and c = d then a + c = b + d. If two quantities are equal and an equal
amount is added to each, they are still equal.
 If a=b and c = d then ac = bd. Since multiplication is just repeated addition,
the multiplicative axiom follows from the additive axiom.
 Given any two points, they can be joined by exactly one line.
 Given any finite, non-zero length line segment, it can be extended infinitely
into exactly one line
 If the position of object changes with respect to time and surrounding then it
is said to be in motion
Axioms play a key role in Mathematics as Theorems of Mathematics
are derived from axioms and thereafter axioms and Theorems of
Mathematics is applied in Physical sciences.

8. Postulates
 In Geometry, "Axiom" and "Postulate" are essentially
interchangeable. So when we suggest or assume the existence,
fact, or truth of (something) as a basis for reasoning, discussion, or
belief then it is called postulate. There is no widely recognized
distinction between axiom and postulates. Only distinction
between Postulate and Axioms is what the ancient Greeks
recognized. Axioms are self-evident assumptions, which are
common to all branches of science, while postulates are related to
the particular science.
 Both axioms and postulates are starting point of some reasoning or
formulated study.
 The term “postulate” is from the Latin “postular”, a verb which
means “to demand”. An axiom generally is true for any field in
science, while a postulate can be specific on a particular field.

9. Examplesof Postulates are
 Speed of light is constant regardless of one's frame of reference.
 Newtonian mechanics are true only in inertial frame of reference
 Physical change involves small energy change whereas chemical
change involves high energy change
It is to be noted that postulate and axioms are physically same that is
building block or starting point of any reasoning. The choice to use one
particular term rather than the other is largely a function of the historical
development of a given branch of Mathematics. E.g., geometry has roots
in ancient Greece, where "postulate" was the word used by the
Pythagoreans. So it's largely a matter of history, and context, and the
word favored by the mathematicians that introduced or made explicit their
"axioms" or "postulates."

10. Definition
 Definition can be defined as any self-evident
expression (mathematical or non-mathematical)
which expresses any
unknown physical (measurable) or non-physical
(non-measurable) quantity in those
terms which are very well known.
 Important point here is like axioms and
postulates, definition is also a self-evident
expression however it is expressed in terms
of known/already heard and understood
terms rather than unknown terms. So we
make use of axiomatic and postulate terms
to form a definitions.

11. Definition Continued……………..
 Definitions are not derived. They are formed by using
set of conditions and axioms to parameterize the
behavior of a system. For example definition of circle is
established as: If the locus of a point is constant from a
fixed point then locus of moving point is called a circle.
Here the terms locus, constant and distance are already
understood terms with the help of geometrical axioms.
 Definitions are special axioms under given set of
condition. Definitions are particular in nature whereas
axioms are generalized in nature. We cannot necessarily
just make a particular definition. There is a concept
called "well defined" that mathematics requires. If your
definition gives you inconsistent results, it is not well
defined.

12. Theorem
 Theorems are the point from where we start our scientific
derivation and test its provability. Best way to understand
the term Theorem is let us look at an example: Factorial
of zero is defined and agreed as zero. There is no
mathematical proof for the same. In case any
mathematical proof existed, then factorial of zero that is
0! = 1 should be a theorem and not definition.
 Here one has to note that any provability feature of
Theorem is totally deductive in nature and not inductive.
Any mathematical or scientific inference can be
concluded or proved either by induction or deduction.
When we study lots of particular cases and generalize
them it is called induction but when we derive a
particular case from general case it is called deduction.
For example:

13. Theorem Continued………………….
 Record of millions of men and women born 6--70 years back were
taken from a municipal corporation and studied. It was found that
most of them have died presently and those still alive do not take
tobacco and Alcohol and were teetotaler. Similar data were collected
from Municipal Corporation of other places and same result was
found. So from study of all these cases can be used to derive a fact
that those who do not take tobacco and Alcohal do live long. This way
of concluding things is called deduction. In mathematics same can be
applied to establish Theorem. Like Pythagoras theorem is established
using definitions and properties of similar triangle.
 In science or mathematics, a theorem is a statement (Not a self-evident)
that has been proven on the basis of previously established
statements, such as other theorems—and generally accepted
statements, such as axioms. The proof of a mathematical theorem is a
logical argument for the theorem statement given in accord with the
rules of a deductive system. For example:

14. Examples of Theorem
 For two triangles to be exactly same that is
superimposable upon each other they need to be similar
with equal area
 Line joining mid point of two adjacent side of any triangle
is parallel and half of third side
 A resistive load in a resistive network will abstract
maximum power when the load resistance is equal to the
resistance viewed by the load as it looks back to the
network that is actually the Thevenin equivalent resistance

15. Laws
 A scientific law is a statement based on repeated experimental observations
that describes some aspect of the world. In science or mathematics, a law is
a statement (Not a self-evident) that has been based on repeated experiments
and practical observation. The proof of any scientific law is a logical
argument for the law statement given in accord with the rules of inductive
system. For example: If we analyze lots of particular cases and generalized
them then this is called principle of Induction. Like record of millions of men
and women born 60-70 years back were taken from a municipal corporation
were taken and studied. It was found that most of them have died presently.
Similar data were collected from Municipal Corporation of other places and
same result was found. So study of all these particular cases can be
generalized to say that one who takes birth eventually dies. Or in other words
Men are mortal. This way of concluding things is called principle of
induction. In mathematics same can be applied to establish any law. Same can
be established in Science and Mathematics.

16. Laws Continued………..
 Force is equal to mass times acceleration is Newtons law and is inductively
proved by studying several particular cases of behavior of objects when
required to be accelerated and after several practical observation same is
generalized as Newtons first law.
 In Mathematics several particular cases can be studied to arrive inductively at:
I. Commutative law of addition: m + n = n + m. A sum isn't changed at
rearrangement of its addends.
Commutative law of multiplication: m · n = n · m. A product isn't changed
at rearrangement of its factors.
II. Associative law of addition: (m + n) + k = m + (n + k) = m + n + k. A sum
doesn't depend on grouping of its addends.
Associative law of multiplication: (m · n) · k = m · (n · k) = m · n · k. A
product doesn't depend on grouping of its factors.
III. Distributive law of multiplication over addition: (m + n) · k = m · k + n · k.
This law expands the rules of operations with brackets.
 In view of the above we can say that because of nature of the provability
Newtons law cannot be called as Newtons Theorem, whereas Pythagoras
theorem can be called as Pythagoras law also.

17. Hypothesis
 Hypothesis is a proposed explanation of any
particular phenomenon. Hypothesis is neither
definition, nor theorem, nor laws nor axioms.
During any scientific investigation or result of any
experiment needs reasoning, this is provided by
Hypothesis.

18. BLOCK DIAGRAM
MATHEMATICAL OR SCIENTIFIC FACT
AXIOMS POSTULATES
DEFINITION
THEOREM
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LAWS
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19. THANK YOU