
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)(xa) + f(a) [2] g(x) ≈ g`(a)(xa) + g(a)
III step. Given that: [3] h(x) ≈ k(xa) + l
IV step. Because: h(x) = f(x) + g(x), puttin [3] to [1] and [2], [4] h(x) ≈ f`(a)(xa) + f(a) + g`(a)(xa) + g(a)
V step. Rechanging elements in [4] allows us to get that all elements before "(xa)" is equal to k, then:
[5] h(x) ≈ (f`(a) + g`(a)) × (xa) + f(a) + g(a)
VI step. Because h(x) = k(xa), 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.


sai123
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Post by sai123 on Nov 10, 2020 2:16:41 GMT
Where do you get this things? Are you taking some course? Or reading some textbooks? I'm curious to know about it!



Post by Eugene 2.0 on Nov 10, 2020 20:29:49 GMT
Where do you get this things? Are you taking some course? Or reading some textbooks? I'm curious to know about it! Well, usually from different sources of literature. On one hand, I've got a not bad library at home, but most often I don't have much time to read it. On the other, I just tried not to be afraid of posting mathlike posts – sometimes people all around us make us br like this or like that, and I always want to let myself out of those social prisons. Exactly this rule of addition is very simple, I think it's not so hard to find it. Because I took it from the series called "Manga Guide To Calculus". This book is about differentials and integrals, but I've heard there's another one about derivatives. About the logic: there are two books which I find to be very good (i.e. "good" as far as I, personally, know about it): B. Russell's "My Philosophical Education" and "Introduction to Philosophy of Mathematics', and E. Landau "Introduction to Integrals and Differentials". Last two books are painly hard even for skilled mathematicians (I asked about it some mathematicians). I took it without deeply studying it, because my sphere of interest is beyond of it. However, I'm really appreciate your questioning!



Post by archlogician on Dec 15, 2020 15:31:13 GMT
This certainly only qualifies as an informal proof, to really prove the linearity of differentiation one would want to work with the limit definition
f'(x) := lim_{u \to 0} (f(x + u)  f(x))/u
Which would be what a mathematician would consider the "official" definition of the derivative. From here, one can proceed without any dubious inferences to prove the standard limit properties.

