Hypothesis, Axioms and Models: Types and Examples

Ans: First of all, let us see what axioms and postulates are:

• Axioms: These are assumptions made by Euclid, which were not to be proven, and are used throughout mathematics not specifically related to geometry.
• Postulates: On the other hand, these are conjectures that were unique to geometry only.

This question is about axiom 5 of Euclid. Let us now look at the statement for Euclid's Axiom 5:

It states that 'the whole is always greater than the part.' It defines 'greater than'.

This axiom is known as a universal truth because it is always valid for everything in the universe. We can prove this statement by taking an example. Let's take a mathematical example.

12 + 15 = 27

Here the totality of the numbers 12 and 15 is 27. So, the whole number formed by the addition of two numbers is greater than its individual number or we can say that the parts i.e. 12 and 15 are less than the whole number formed by their sums.

Let us take another example.

Suppose you order a cake on your birthday and you cut a piece of cake with a knife. Then the piece you cut from the whole cake is smaller than the whole cake or only a part of it.

Therefore, we can say that Euclid's axiom 5 is a universal truth.

Ans: The universal law of gravitation proposed by Newton is an assumption or hypothesis, that he proposed with his ingenuity. Before him, there were many observations, experiments, and data on the motion of the planets around the Sun, the motion of the Moon around the Earth, pendulums, bodies falling toward the Earth, etc. Each of these required a different explanation, which was more or less qualitative. What the universal law of gravitation states is that, if we assume that any two bodies in the universe attract each other with a force proportional to the product of their masses and inversely proportionate to the square of the distance between them, then we can define all these statements in one stroke. It not only explains these phenomena but also allows the prediction of the results of future experiments.

A hypothesis is a guess without assuming that it is true. It is not fair to ask someone to prove the universal law of gravitation because it cannot be proven. It can be tested and proven through experiments and observations.

An axiom is a truth while a model is a theory proposed to explain an observed phenomenon. But you need not worry about the nuances of using these words at this stage. For example, next year you will learn about Bohr's model of the hydrogen atom, in which Bohr assumed that the electrons in the hydrogen atom follow certain rules (postulates). Why did he do this? He had before him a vast amount of spectroscopic data which no other theory could explain. So Bohr said that if we assume that atoms behave in this way, we can explain all these things together.

Einstein's special theory of relativity is also based on two assumptions, the constancy of motion of electromagnetic radiation and the validity of physical laws in all inertial frames of reference. It is unwise to ask anyone to prove that the speed of light in a vacuum is constant, independent of source or observer.

Even in mathematics, we need axioms and hypotheses at every stage. Euclid's belief that parallel lines never meet is a hypothesis. This means that if we assume this statement, we can explain many properties of straight lines and two- or three-dimensional figures formed from them. But if you don't assume it, you are free to use a different axiom and gel a new geometry, as has indeed happened in the past few centuries and decades.

Ans: All axioms are theorems. However, theorems include not only axioms but also sentences that can be derived from those axioms by means of inference rules. That is why a formal theory of a language system is a set of its axioms closed under logical consequence.

1. An axiom is a statement that is assumed to be true without any proof, while a theory deserves to be proven before it can be assumed to be true or false.
2. Axioms are often self-evident, while a theory often requires other statements to be valid, such as other theories and axioms.
3. Theorems are inherently more challenging than axioms.
4. Basically, theorems are derived from a set of axioms and logical connectives.
5. Axioms are the fundamental building blocks of logical or mathematical statements, as they act as the initial points of theorems.
6. Axioms can be classified as logical or non-logical.
7. The two components of a proof of a theorem are called the hypothesis and the conclusion.

Ans: Below is the general sequence of steps taken to develop a scientific theory:

• Select and define the natural phenomenon that you need to diagram and describe. Collect data about the phenomenon by investigating the source of the phenomenon and analyzing the observations. We can also replicate this phenomenon by experiment or simulation under a controlled environment (usually inside a laboratory) that removes interference from outside variables.
• After extracting enough data, analyze for recurring patterns in the data. Try to describe this recurring pattern by making a provision explanation (hypothesis).
• Test the hypothesis by getting more information to see if the hypothesis holds true to reveal possible patterns. If the available data does not support the hypothesis, it must be changed or eliminated for the better. During the collection of data, we should not ignore information that contradicts the hypothesis in favor of only supporting information (known as "cherry-picking"). This is often misused by pseudo-scientists who try to deceive people who are not familiar with the scientific method.
• If a concrete hypothesis turns out to be true after all testing and is the most accurate explanation for a phenomenon, it is considered a valid theory. An established theory may undergo modifications and rejection if there is sufficient evidence that contradicts it. Thus, a theory is not an eternal or absolute truth.

Ans: An axiom is a statement, usually considered self-evident, that is assumed to be true without proof. It is used as a starting point in mathematical proofs to deduce other truths.

Classically, axioms were considered separate from laws. An axiom refers to an assumption that is common to many fields of inquiry, while a hypothesis refers to a hypothesis specific to a particular line of inquiry, accepted without evidence. For example, in Euclid's Elements, you can compare "general assumptions" (principles) with conjectures.

In most modern mathematics, however, there is generally no distinction between what is classically known as "axioms" and "postulates". Modern mathematics distinguishes between logical axioms and non-logical axioms, the latter sometimes referred to as postulates.

Postulates are assumptions that are specific to geometry but axioms are assumptions used through mathematics and are not specific to geometry.

Ans: During the development of a hypothesis, the researcher should not currently have a complete bias on the possible outcome of the test or experiment. 

• It should be reasonably within the scope of the investigation. Then only experimentation or testing increases the likelihood of getting the valid, correct side of the hypothesis.
• If the investigator already knows the result, it is simply considered a "result" (the investigator must have already assumed this during the formulation of the hypothesis).
• If the researcher cannot test the hypotheses through experience or observation, the hypothesis must be tested by other qualified researchers who provide observations.

Researchers investigating alternative hypotheses may consider the following:

1. Testability
2. Parsimony - discourages the postulation of an unlimited number of entities.
3. Scope - Clear application of the hypothesis to many scenarios of phenomena.
4. Feasibility – the likelihood that a hypothesis can help explain further events in the future
5. Conservatism – Conformity with existing accepted knowledge systems.

Ans: Hypothesis plays an important role in any research project; It is a step towards proving a theory. 

• A hypothesis serves to establish an underlying theory and connection to a particular research topic.
• It helps in processing the data and assesses the reliability and validity of the study.
• It provides the foundation or supporting evidence to demonstrate the validity of the study. 
• A hypothesis allows researchers not only to predict the relationship between variables but also to predict the relationship based on theoretical guidelines and/or empirical evidence.

Ans: Writing a good hypothesis starts before you start typing. As with many tasks, preparation is very important, so you first start by doing your own analysis and reading everything you can about the topic you decide to research. From there, you will receive the information you need to know, where your focus will be on the subject. Keep in mind that a hypothesis can be a prediction of a relationship that exists between 2 or more variables. The hypothesis should be straightforward and concise, and the conclusion predictable, clear, and without assumptions about the reader's knowledge.

Ans: An example of a simple hypothesis is that taking drugs leads to depression. If a person has a proper diet plan, his skin, body, and mind remain healthy and fresh. This is an example of a directional hypothesis. If you consume drugs it not only induces depression but also impacts your brain, leads to addiction, etc. If you pump petrol in your bike, you can go for longer rides, you also become an expert in riding a bike, you explore more places and encounter new things.

Ans: A working hypothesis is a type of hypothesis that is scientifically accepted as a basis for further study in the hope that a plausible theory will be obtained, even if the hypothesis ultimately fails. As with most hypotheses, a working hypothesis is formulated as a statement of assumptions, which can be linked to an exploratory investigation objective in an empirical analysis. In qualitative research, they are frequently used as an abstract foundation.

Ans: Hypothesis testing is the formal procedure of testing our ideas about the world with the help of statistics. It is used by scientists to test specific predictions, called hypotheses, by calculating whether a pattern or relationship between variables could have arisen by chance.

Null and alternative hypotheses are useful in hypothesis testing in statistics. The null hypothesis of a test always predicts no effect or no relationship between variables, while the alternative hypothesis states your research prediction of an effect or relationship.