source:https://www.bbc.co.uk/programmes/articles/2hyty1swxN9b0sBfc5nfjC6/13-juicy-facts-about-mangoes

 

Turn your sight to a box of mangoes. What did you see as soon as you looked? What was the initial cognition that came in a flash as soon as you looked, and lasted for a fraction of a second? — that ‘there is something there!’ This primary stage of cognition was followed by the appreciation of some mangoes, a collection of mangoes. This collection, in mathematical term, is called set.

Now, this sight of a set of a mangoes has, or rather may be seen to be having, a number of features or characteristics. Just by looking at the sight, you can understand the following features.

  1. You may see that some mangoes got pressed due to some kind of pressure and are deformed. These deformed mangoes do not look like other mangoes in shape or size. But, if these deformed mangoes are not split open, they can be brought back to original size by properly pressing from different sides. So, there is some characteristic common between a deformed mango and a non-deformed mango. In other words, there is an identity of the mango that is not affected by deformation-without-split. Mathematically speaking, this deformation-without-split is called a continuous transformation of the mango. And the aforementioned identity or property of a mango, or anything, which is not changed by continuous transformation is called topological property. A set will have the topological property when it can be in some way associated with a mathematical object called topology. A set equipped with a topology is called topological structure or topological space. The branch of mathematics that studies topological spaces is called topology as well.

  2. Note that size and shape of a mango themselves are not topological properties because they change by continuous deformation. What also change by continuous deformation are the relative position of the top and the bottom of the mangoes, or the curve around the stem on the top. All these properties — size, shape, relative positions of parts and curvature — are called geometric properties. A set will have geometric properties when it can be associated with a mathematical object called metric, which is a generalized concept of distance. A set equipped with a metric is called metric structure or metric space. The branch of mathematics that considers the study of metric spaces is called geometry.

  3. After checking the deformation, size, shape, relative positions of parts and curvature of the mangoes, what else may we be interested to know next? We may like to count how many mangoes are there. Further, in addition to knowing whether the mangoes are relatively small, medium or big in size, we may like to know the exact surface area or volume of each mango as numbers. We may also want to know the chance or probability of blindfoldedly picking up one of the largest three mangoes, or chances of various such outcomes. In general, we may want to know certain properties about these set of mangoes as numbers, which is to say, we want some kind of measurement about these mangoes. Any property of the mangoes which can be measured numerically is called a measurable property. There can be numerous properties which can be expressed as a number. All these are measurable properties. A set will have measurable properties when it can be associated with a mathematical object called measure. A set equipped with a measure is called measure structure or measure space. The mathematical branch that considers the study of measure spaces is called measure theory.

  4. Is there anything else we may like to know about this collection of mangoes? Let your imagination run wild and free. Well, what about this one? We may like to know, if we tilt the mango box how fast the position of the mangoes will change. Or if we drop a heavy slab of stone on the box from top how fast the shape or size of the mangoes will change. The softer mangoes should be crushed faster than the harder ones. Phrases such as ‘how fast’ or ‘how slow’ signify change. In general, we may want to know the consequence of any kind of change — its rate or its outcome. The mathematical concept used to analyze change is called limit. The branch of mathematics that considers the study of limit is called mathematical analysis, which is an outgrowth of calculus. In particular, rate of change is measured using derivatives, obtained by differentiation. A set on which differentiation can be performed forms a differential structure.

All these above properties about the box of mangoes are not opposed to each other. Each property is the formalization of one assumed perspective of knowing the box of mangoes. Some of them or all of them may be present simultaneously. Also, one perspective can be used in the understanding of another perspective. In general, we may say, each perspective imposes a mathematical structure on the set of mangoes — which itself can be seen as a common perspective of all perspectives. If a mathematical structure on a set arises out of some operation between elements of the set that structure is called an algebraic structure, else it is a non-algebraic structure. The study of algebraic structures is algebra. Mathematical structure, algebraic or not, is studied in general in the branch called category theory. Thus, mathematics can largely be thought of as the science (or art!) of structures. Any category of structure is by definition a set of statements called axioms. These axioms are taken to be valid, and based on them the whole development of that mathematical subfield proceeds. If you reduce the number of axioms, you will get a more general structure. Now, whether these statements are true or false is a question whose answer demands a bridge between the mathematical world of symbols and the world of our perception—a topic dealt in the philosophy of mathematics. Without recourse to the philosophy, mathematics does not present an ontological position, which means it does not say whether such things are true but only presents a conditional position saying that if {such axioms hold} then {such happens}. In other words, the conditional position explores possibilities…(may be continued later)



source:https://www.deviantart.com/l33tm0b1l3/art/Disco-Superclub-32662937

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