Scientific terminology

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

1. Asif Eqbal Electrical Engineer SCIENTIFIC & TECHNICAL TERMINOLOGIES

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. Examples of 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. 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." Examples of Postulates are

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. 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. Definition Continued……………..

12. 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 : Theorem

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. 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 Examples of Theorem

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: 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.  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.  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   POSTULATES AXIOMS DEFINITION THEOREM LAWS Through Induction Deduction

19. THANK YOU