A sentence is considered to be true if it includes a statement that is true. The statement "2 is greater then 1" is true, therefore the sentence containing this statement is true. There is no difference in the notation of the sentence and its statement. One can also say the sentence represents its statement.
What can we tell about the barber who is shaving all the people into a village who don't shave themselves? If we count him to the population of the village we can not say if he is shaving himself or not.
If he is shaving himself so he can not be shaved by the barber, because this human being is shaving all the people who don't shave themselves. If he don't shaves himself he is shaved by the barber.
How simple is the statement "I don't like mathematics".
Consider the sentence "god is almighty". If god is almighty then he can create a stone which is so big that he can not lift it. But can we say then that he is still almighty? I can not say anything more to this.
What are the problems mathematics can formulate? Simple statement are "the square over a hypothenuse is the sum of the squares over the kathetes" (consider a triangle with one angle of 90 degrees). It is much more complicate to say "this problem can not be solved with a computer". There exist case studies where such problems are treated.
In the everydays life this problems do not occur.
I don't know it this discussion is important or not but sometimes I hear about it.
The thing that fascinates me on mathematics is the term "infinity" and what can be done with it, probably as a result from the insigth that my life is limited. With the term "infinity" it is possible to create numbers. Natural numbers (1,2,3, ...), rational numbers (fractions, like 2/3, -4/5 ...), real numbers like 1.3245466543657376... with a limited or unlimited number of decimal places. If someone writes down a decimal number between 0 and 9 everyone else can add a new one. It is this dynamics that characterizes the real numbers. One can not grasp them completely, but one can believe that they exist, and noone is able to disprove it.
It is possible to expand the number system to complex numbers and hypercomplex numbers. With complex numbers one can describe current-voltage relations in Electrodynamics. They allow a representation in the plane.
The hypercomplex numbers have properties which are important for formulations in Quantum Mechanics (Dirac equation).
Real numbers are used for integration and differentiation. They allow the presentation of a continuum.
My personal interests in mathematics belong to the calculus of differentials. With this theory it is possible to describe the foundations of General Relativity. There is no need of complex or hypercomplex numbers.