Originally Posted by

**ThePerfectHacker** I am not exactly sure what you are asking thus I will as much as a can even if you were not asking that.

The word **rigorous** is a English word that mathematicians love to use. Another word mathematicians love to use is **intuitive**. For example, many many concepts from Calculus, are intuitively understood. Concepts such as: area, volume, length, tangent lines. And for the most part physicists/engineers only need the intuiteve concepts, because they are based on the physical applications of the subject matter. Mathematicians, however, are strange. They cannot, at all, never, ever, woe on their soul, accept anything intuitive. Meaning, if f(x)>g(x) on interval [a,b] then INT(a,b) f(x) > INT (a,b) g(x). The intuition says that since integration is area it makes sense to says that this statement is true. However, a mathemation does not accept such things, he will spend hours and days proving even the most simple of concepts. The reason why is that mathemations do not make proofs based on their intuition (they can use it to construct idethinks but not proofs). And in fact, those who study math at a serious level will agree that intution is a dangerous thing. For example, the thing I said about f(x) >g(x) is not acutally true when f and g are discontinous at too many points! Now there is no way human intutition can see something like that. And the beauty is when you prove something rigorously then there is absolutely no way that you are wrong. Another example, everybody uses Euclidean geometry, everybody knows it nobody denys it. However, mathemations found it hard to accept because Euclid does not define *rigorously* the meaning of a point, a line, a plane,..... And about 2000 years later a mathemation comes along, Hilbert, and he turns it into a rigorous acceptable subject, so rigorous that you do not even need diagrams/pictures to make geometric proofs! However, his geometry is extremely complicated and is at all not necessary for people to learn if they just want to solve physical problems. Mathemations have a thing in them that they need to see a rigorous developement of the subject matter. That is what makes them so different from every single disciple that exists. Now that is what I mean by rigorous. The transitive property is a **relation**, something that relates two elements/objects to each other. Thus we say if a related to b and b related to c then a related to c. That is a rigorous statement. However, the substitution property states you can replace similar things into each other. The difficutly is what does it mean "similar"? Mathemations therefore disfavor it. One last example, when Calculus was first introduced by Newton and Leibniz mathemations had difficutly accepting it because certain concepts were never defined. Like what does it mean a limit "approaches", what does "approaches" mean. Only later on did mathemations improve Calculus so it is acceptable among them. Many people (non-math) think proving these simple looking things is stupid but I disagree not only did they make Calculus beautiful they discovered several shattering statements.