Post by Eugene 2.0 on Nov 2, 2020 12:10:14 GMT
I step.
(2.1) If h(x) = f(x) + g(x), then h`(x) = f`(x) + g`(x)
II step.
[1] f(x) ≈ f`(a)(x-a) + f(a)
[2] g(x) ≈ g`(a)(x-a) + g(a)
III step.
Given that:
[3] h(x) ≈ k(x-a) + l
IV step.
Because: h(x) = f(x) + g(x), puttin [3] to [1] and [2],
[4] h(x) ≈ f`(a)(x-a) + f(a) + g`(a)(x-a) + g(a)
V step.
Rechanging elements in [4] allows us to get that all elements before "(x-a)" is equal to k, then:
[5] h(x) ≈ (f`(a) + g`(a)) × (x-a) + f(a) + g(a)
VI step.
Because h(x) = k(x-a), then:
[6] k = f`(a) + g`(a).
VII step.
Then:
[7] k = h`(a) = f`(a) + g`(a)
Except for V, everything seems to be ok. The fifth step doesn't seem to be well, because we use analogy, while all of the time we use only deductive methods. As long as this operation is approximation, not a strict deductive proof, it's possible to achieve such a logic component here. For honestly, it's not really okay. If to use approximation only, like presenting 1/3, 1/6, 2/3, etc in short real numbers i.e. 3.333 instead of 3.333...., 0.166, instead of 0.1666..., etc. We can say that 1/3 ≈ 3.333, but it's not the same as to choose which coeficient belongs to which argument.
However, this logic doesn't seem to be so abnormal to refuse it. What makes it be so clear and accaptable then? Probatly, it's nature of numbers. (So, some philosophical questions might be arisen here.) Either numbers' nature is one and the same, or we have no real chances to confuse when differ one number from another. Briefly, numbers' metaphycisal form is one and the same for all the numbers.
For Cantor's or Peano representation of numbers it still cannot be absolute seen. Why? Because numbers in their system require extra formalization, or in other words a number is a certain set representation, e.g. {∅,{∅}} can be maintained as "2" (two). Another ways to represent it are:
{A}, {A,{A,B}}... = <A>, <A,B>... = 1, 2...;
{}, {{},{{}}}, {{},{{,{{}}}}... = 1, 2...;
a = 0, a` = 0+1, (a+b)` = (0+1)+1 = 2...
etc.
As you can see many obstacles raise when we try to understand the nature of numbers and how to represent them corretly.
I could provide more examples in Calculus, but I think that this is enough representable to meet with those problems closely.
(2.1) If h(x) = f(x) + g(x), then h`(x) = f`(x) + g`(x)
II step.
[1] f(x) ≈ f`(a)(x-a) + f(a)
[2] g(x) ≈ g`(a)(x-a) + g(a)
III step.
Given that:
[3] h(x) ≈ k(x-a) + l
IV step.
Because: h(x) = f(x) + g(x), puttin [3] to [1] and [2],
[4] h(x) ≈ f`(a)(x-a) + f(a) + g`(a)(x-a) + g(a)
V step.
Rechanging elements in [4] allows us to get that all elements before "(x-a)" is equal to k, then:
[5] h(x) ≈ (f`(a) + g`(a)) × (x-a) + f(a) + g(a)
VI step.
Because h(x) = k(x-a), then:
[6] k = f`(a) + g`(a).
VII step.
Then:
[7] k = h`(a) = f`(a) + g`(a)
Except for V, everything seems to be ok. The fifth step doesn't seem to be well, because we use analogy, while all of the time we use only deductive methods. As long as this operation is approximation, not a strict deductive proof, it's possible to achieve such a logic component here. For honestly, it's not really okay. If to use approximation only, like presenting 1/3, 1/6, 2/3, etc in short real numbers i.e. 3.333 instead of 3.333...., 0.166, instead of 0.1666..., etc. We can say that 1/3 ≈ 3.333, but it's not the same as to choose which coeficient belongs to which argument.
However, this logic doesn't seem to be so abnormal to refuse it. What makes it be so clear and accaptable then? Probatly, it's nature of numbers. (So, some philosophical questions might be arisen here.) Either numbers' nature is one and the same, or we have no real chances to confuse when differ one number from another. Briefly, numbers' metaphycisal form is one and the same for all the numbers.
For Cantor's or Peano representation of numbers it still cannot be absolute seen. Why? Because numbers in their system require extra formalization, or in other words a number is a certain set representation, e.g. {∅,{∅}} can be maintained as "2" (two). Another ways to represent it are:
{A}, {A,{A,B}}... = <A>, <A,B>... = 1, 2...;
{}, {{},{{}}}, {{},{{,{{}}}}... = 1, 2...;
a = 0, a` = 0+1, (a+b)` = (0+1)+1 = 2...
etc.
As you can see many obstacles raise when we try to understand the nature of numbers and how to represent them corretly.
I could provide more examples in Calculus, but I think that this is enough representable to meet with those problems closely.
