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Different infinities problem

  1. Jun 8, 2014 #1
    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.
    Last edited: Jun 8, 2014
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  3. Jun 8, 2014 #2
    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 [Broken]
    Last edited by a moderator: May 6, 2017
  4. Jun 8, 2014 #3


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    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.
  5. Jun 8, 2014 #4


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    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.)
  6. Jun 9, 2014 #5
    Sorry for everyone, I'm perhaps creating my own definitions.

    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...
    Last edited: Jun 9, 2014
  7. Jun 9, 2014 #6


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    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.

  8. Jun 9, 2014 #7


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    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.
    Last edited: Jun 9, 2014
  9. Jun 9, 2014 #8
    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?
  10. Jun 9, 2014 #9


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    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.
  11. Jun 10, 2014 #10
    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.
  12. Jun 10, 2014 #11
    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.
  13. Jun 10, 2014 #12
    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].
  14. Jun 10, 2014 #13
    What does it even mean for a number axis to be differentiable?
  15. Jun 10, 2014 #14
    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.
  16. Jun 10, 2014 #15
    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.
  17. Jun 10, 2014 #16


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    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.
  18. Jun 10, 2014 #17
    That the number axis is continuous - so the limit derivation definition can be applied.
  19. Jun 10, 2014 #18
    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.
  20. Jun 10, 2014 #19
    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.
  21. Jun 10, 2014 #20


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    Use the sequence definition of hyperreals, and the definition of division in it, applied to infinitesimals.
    Last edited: Jun 10, 2014
  22. Jun 10, 2014 #21


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    There are different ways in which something can be "real" in reality. A unicorn is not real because we don't have a living specimen. The concept of unicorns is real because we can define it with words. Mathematics, like "honor" or "congress", falls in the latter.

    Infinitesimals are defined as numbers which satisfy ##0<x<r## for every real number ##r>0##. You can't express (or represent) infinitesimals with 0 because it means ##0\neq0##.

    There are two ways to obtain infinitesimals
    1. Add elements to the reals (hyperreal method)
    2. Strengthen the axioms of the reals in order to distinguish infinitesimals (Internal Set Theory method)
    In neither case does 0 represent an infinitesimal.

    Do you even know the epsilon-delta definition of limit?

    Because they are different.

    If you mean the numerator in the definition of derivative, that is always non-zero for non-constant functions.

    Serious question, when your teacher showed you the definition, didn't you do any exercises where you calculate the derivative using first principles?

    Taking the limit to zero is not the same thing as substituting in zero.

    This forum is not the place for original research.
    More to the point, you have severe misunderstandings about what is going on.

    An axis is a line which we use to orient the Cartesian coordinate system. It has nothing to do with calculus.

    If you want to know why we use the reals to do calculus with, it is because the reals are Dedekind-complete (that means: any subset of the reals which is bounded above has a least upper bound).
  23. Jun 11, 2014 #22
    Because there exists such an arrangement of matter that creates a statue of unicorn, or a movie with moving unicorn, or a brain structure that can somehow imagine unicorn. The concept of unicorn is real because there exists an arrangement of matter that we can define as unicorn.

    Yes, I know that. I have no problem with limit derivation definition.

    Because it's defined so. I'm suggesting that in physical reality there is no meaning that e.g.:
    ##\varepsilon =infinitesimal,\frac{\varepsilon^{2} }{\varepsilon}=\varepsilon##

    What definition do you mean? You mean the limit derivation definition? What exercises do you mean?

    I thought that we use reals because real numbers have meaning in physical reality.
  24. Jun 11, 2014 #23


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    It's just a name. The "real" numbers are not real in the ordinary sense of the word. The "natural" numbers are not natural in the ordinary sense of the word. The "imaginary" numbers are not any more imaginary than any other class of numbers. The "rational" numbers are not rational in the sense of rational thought. The "irrational" numbers are not irrational in the sense of rational thought.

    The real numbers are convenient to use in our models because they have enough range and precision to match any conceivable measurement and because as a number system they are closed under the kinds of calculations that we want to perform with them.

    [In practice, we may actually use IEEE floating point because we can then fit descriptions of the model numbers into a computer and because we have an entire discipline within mathematics that can deal with the quantization errors that result from this departure from the ideal models]
    Last edited: Jun 11, 2014
  25. Jun 11, 2014 #24
    There is an abstract meaning (either created in my brain or even "deeper" existing in physical reality) that those apples are - what we call - two. I consider that meaning as some feeling, or intuition in my brain.
  26. Jun 11, 2014 #25
    First of all, "two" is a natural number. One could indeed argue that those have a special significance. But the real numbers consist out of way more numbers than the natural numbers. In fact, most real numbers can't even be defined or described. They just exist to make our mathematical theory more beautiful. There is no reason to expect that those numbers have a special significance in the real world, much unlike numbers like two which are special.

    Second, one could even argue that the natural numbers aren't so special. There are some "primitive" tribes which only recognize the numbers 1,2,3, many. Something like 4 and 6 are not distinguished. This is very much like animals see numbers.

    Also, this is getting dangerously in the realm of philosophy, something that this not permitted in the math sections. But I will allow the discussion for now.
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