I always thought the derivatives (or generally any operation with infinitesimal (infinitely large) values) belongs to the different infinity (= it is different variable). And obviously, there is no transition between two infinities (that is why is it different variable - e.g. big bang singularity problem). I thought every time only one infinity can be intended as real, in that case other infinities are just inf. smaller (i.e. [itex]0[/itex] is expressing all values of smaller infinity) respectively inf. larger (i.e. every real value can be expressed only as [itex]0[/itex] in larger infinity). But with this approach I stumbled. As I think about it, I realized that something (let's call it a reality) is always overstated to all infinities, making every single infinity REAL (simultaneously, no matter those are different infinities). Thus I can write real values ([itex]5,26,356,-14...[/itex]) of different derivatives (i.e. different infinities) side by side, although they are infinitely larger each other. Or I can solve differential equations. This illustrates the image: So I want to ask: is it true consideration? Because I doubt it a little bit. For example in the picture: [itex]dx_{1}[/itex] should be infinitely smaller than the rest of real graph, as I previously thought. Real ratio of those two variables evokes for me that they belongs to the same infinity.
It seems that you are very vaguely talking about the hyperreal (or related) umber system. Try reading this (free) calculus book http://www.math.wisc.edu/~keisler/calc.html Also, check out our FAQ: https://www.physicsforums.com/showthread.php?t=507003
If you want to discuss non-standard analysis you need to say so. The standard definition of derivative does not use infinity anywhere. I have no idea what this sentence means. Infinity is never a real number. Again, if you wan to talk about hyperreals, you need to be explicit. Note that even in the hyperreals "infiniticimals are real" is still a false statement. I don't know if you are talking about "real" as in real numbers, or "real" as in real life. The latter is a philosophy topic and not for this forum. Again, derivatives do not use infinity.
Some down-to-earth points that I think may allow the OP to be more concrete and more Mathematical: One of the main differences between the Hyperreals H and Reals R is that in H you can have indefinitely-small numbers that are not 0, but this is not so in R. A similar result is true for indefinitely-large numbers, but I cannot think of a clear way of stating this. There are many different constructions of H; one I think most are familiar with is the one in model theory , building new models using ultraproducts and filters, where a hyperreal number is an equivalence class of sequences, and it is a model for the standard axioms of the Reals by the Compactness theorem in logic. For more on ultraproducts , watch the TV show: "ultraproducts: America's new supermodel"(kidding.)
Sorry for everyone, I'm perhaps creating my own definitions. pwsnafu: I don't know what do you mean. A derivative is a ratio of two infinitely small values. If it wasn't defined as infinite values, it wouldn't (reflect, express) - let's say a slope of the tangent - precisely. Why couldn't I consider any number axis as real? With real number axis, there exists infinitely small number axes everywhere on the real axis (in fact those are points on the real number axis). And the whole real number axis is a single point on the infinitely large number axis. Like this, I can consider arbitrary number axis as real (then the others are inf. small/large). That's what I meant by "I thought every time only one infinity can be intended as real". Infinity = number axis. That's simple. Let's take an amount of volume. Real degree of volume (i.e. belongs to certain infinity) cannot become infinitely small (i.e. cannot move into another infinity), nor conversely (I'm refering to the beginning of big bang). As the volume is decreasing, it obviously keeps always in same infinity. If you now consider infinitely small number axis as real, that volume again cannot become infinitely large. I meant of course real numbers. The physical reality (that is the only one for me) "behaves" mathematically, all mathematical notions were observed from reality (let's say from behavior of matter in timespace). I'm thinking about the concept of reality commonly. For me there is nothing philosophical about reality. I'm thinking about this: [itex]\frac{df(x)}{dx}=[/itex] infinitesimal result? Or real result? I don't understand how INFINITESIMAL result could "be equal" to REAL slope of tangent :( And finally, a lot of patience with me, please...
No, it isn't. In standard calculus, the derivative is a limit, not a ratio. As others have mentioned in "non-standard analysis" we extend to the "hyper-reals" an have "infinitesimal" and "infinite" quantities but you don't seem to be talking about that. Then, even in non-standard analysis, you are talking about a real number. It does not belong to any kind of "infinity". Are you talking about mathematics or physics? Now I have no idea what you mean by "axis". You cannot "infinitely small number axes" on a single axis using any reasonable definition of "axis". Again, you seem confused about the difference between mathematics and physics. Then I strongly recommend that you take a Calculus course. There are Calculus texts (and, I imagine, classes) that deal with Calculus from a "non-standard", "hyper-reals" point of view but I would not recommend one of those until after you have taken a traditional "limits" based Calculus class which uses only real numbers. Just developing the hyper-reals takes a heck of a lot of deep symbolic logic.
Please note that, as many have said, both Mathematics and Physics use precise definitions. There is room for discussion, and it may take years before the right definition is found, but once it is found, it is made precise. If you are not dealing here with precise definitions, you are not dealing with neither Mathematics nor Physics; nothing wrong with that, but it is difficult to have a discussion with someone if that someone is using definitions other than those usually used. Maybe you want to ask why some terms are defined the way they are defined? This is not strictly Mathematics ( it is Meta- Mathematics), but still seems within the scope of this forum.
That is true. I never understood what math really is. I'm mainly trying to avoid an approach that math is something unreal. I'm trying to unify my view to "math" and "physics" to the one and only reality. Ultimately, I have no problem with this. As every infinitesimal value can be expressed only as [itex]0[/itex] in [itex]\mathbb{R}[/itex], the definition with [itex]0[/itex] is correct. Afterwards I don't see the meaning of limit in that definition. Why isn't there just written zero changes, but is there used limit? After all, the subtraction of two, above all limits near values is always [itex]0[/itex] in [itex]\mathbb{R}[/itex]. The definition with limit is correct, but why cannot be the definition simply with [itex]0[/itex] changes?
Think of it in this way, there are no infinitesimal values, every time you think you have an infinitesimal value, you have something different. This should start to make things clearer. If you have a sequence that decreases all the way to 0 but never gets there, the limit is 0, it's not some other type of value. Then you should find that it is less confusing.
I need some proof that division of two infinitesimal values is a real value. That's what I can't see intuitively, but my theory is based on the opposite.
You have not been listening to us. In the real numbers, there are no infinitesimals! They don't exist. If you want a theory with infinitesimals, then you will need some other number system such as the hyperreal numbers. Relevant theorems can be found in the calculus book I linked in Post 2.
I don't understand - what means something different? If there are no infinitesimal values, I would answer that the number axis is not differentiable. Or, if I consider only [itex]\mathbb{R}[/itex] number axis and there are infinitesimal values, then I may be only [itex]0[/itex].
I need some proof that division of two infinitesimal values in hyperreal numbers is a real value. That's what I can't see intuitively, but my theory is based on the opposite.
There is no such proof since it is not true. The division of two infinitesimals can be infinitesimal, finite or infinite. See the book I linked in Post 2 page 31.
Then you need to take a course in, or read a book on, "non-standard analysis". You will NOT find "infinitesimal" in any standard text on Calculus.
What does it mean that the number axis is continuous? Also, you don't need any infinitesimals in order for the limit definition of derivatives to work.
However, I thought there is such proof in [itex]\mathbb{R}[/itex] - that is probably false, just because there are no infinitesimal values in [itex]\mathbb{R}[/itex]. Also such operations like: [itex]\frac{\varepsilon^{2}}{\varepsilon}[/itex] = infinitesimal, because = [itex]\varepsilon[/itex] are only predicted from the behavior of real values (basically we don't know whether this relation works with infinitesimal values (and it's irrelevant) - those relations are postulated from [itex]\mathbb{R}[/itex]). No problem with that [itex]\mathbb{R}^{*}[/itex] definition, it is probably the easiest way. I can just define infinitesimals another way. I can probably every value from different infinity ([itex]f(x)[/itex],[itex]\frac{\mathrm{d} f(x)}{\mathrm{d} x}[/itex],[itex]\frac{\mathrm{d^{2}} f(x)}{\mathrm{d} x^{2}}[/itex]...) consider on different [itex]\mathbb{R}[/itex] number axis. I just cannot make any comparisons of those axes (I cannot compare them as two real, equal axes). The assumption is that all values from one axis are just [itex]0[/itex] in higher axis. As for [itex]\frac{\varepsilon^{2}}{\varepsilon}=\varepsilon[/itex], I'm doing exactly the same. I just postulate "miraculous transition" of [itex]\varepsilon[/itex] value to another number axis (let's say it's decreasing from higher axis). Then I divide real numbers. Also this theory is quite similar to [itex]\mathbb{R}^{*}[/itex] (if it is not the same). The entire graph (like in my first post) corresponds to one real axis. All single values from other lower axes are then [itex]0[/itex] in that graph. The advantage of this theory for me is, that I can nicely imagine the impossibility of transition from one number axis (from one infinity) to another number axis (to another infinity). As for the slope of tangent in the graph - the problem would be solved if I defined the slope in the graph as [itex]\int (\frac{\mathrm{d} f(x_{1})}{\mathrm{d} x_{1}}=0)dx[/itex]. This integral expresses every tangent slope ([itex]C(int)\in \mathbb{R}[/itex] - some solution of integral expresses every tangent). I haven't redefined much, much other stuff so far. I shall see later whether this theory would be sufficiently consistent and transparent for me.
Use the sequence definition of hyperreals, and the definition of division in it, applied to infinitesimals.