Are Infinitesimals Really Numbers?

Discussion in 'Physics & Math' started by TruthSeeker, Mar 29, 2017.

  1. Write4U Valued Senior Member

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    Fractals. See Causal Dynamical Triangulation (CDT)by Renate Loll.
    https://en.wikipedia.org/wiki/Causal_dynamical_triangulation

    As layman it occurs to me that unless something has a value it cannot be assigned a numerical symbol.
     
    Last edited: Apr 19, 2017
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  3. Write4U Valued Senior Member

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    According to wiki, they approach zero (value).
     
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  5. Write4U Valued Senior Member

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    uestion:
    Do infinitesimals necessarily exist?
     
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  7. geordief Registered Senior Member

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    Can anything that successfully predicts anything be said to "not exist"? Does function trump form?
     
  8. origin Trump is the best argument against a democracy. Valued Senior Member

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    The 'size' of the infinitesimals approaches zero as the 'number' of infinitesimals approach infinity.

    \(\int \limits_a^b f(x)~dx = \lim_{n \rightarrow \infty }\sum \limits_{i\rightarrow }^n f(x_i)\Delta x\)
     
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  9. origin Trump is the best argument against a democracy. Valued Senior Member

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    To integrate the area under a curve you would use infinitesimals. Imagine some sort of curve on a graph and you want to find the area under the curve. You can place a square under the curve to approximate it but that is not very good. It would be better to put a series of rectangles under the curve and add them up to approximate the area under the curve. The narrower the rectangles are the closer they will approximate the area under the curve. As the number of rectangles increases; the closer you get to to the true area. As the rectangles become infinitesimally small and the number of the rectangles approaches infinity the area calculated becomes an accurate assessment of the area under the curve.

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    Do infinitesimally small rectangles exist. I guess not - but so what?
     
  10. Write4U Valued Senior Member

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    According to Loll, infinitesimal fractals (triangles) do exist, perhaps even in the abstract.
     
  11. Write4U Valued Senior Member

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    IMO, functions trumps form. A function can produce many .forms , but a form can produce only a limited number of functions?
    Perhaps an example might be the fractal function, which can produce an unlimited number of forms
    .
     
    Last edited: Apr 19, 2017
  12. QuarkHead Remedial Math Student Valued Senior Member

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    Well, I am not sure I understand what is meant by "produces" or "trumps", but you are correct is a sense (though I suspect inadvertently).

    Suppose the domain of polynomial forms over the integers mod any prime \(\mathbb{Z}_p\). Then the polynomial forms \(f_1(x)= x^p-x\) and \(f_2(x)=0\) determine the same function, namely \(f(x)=0\)
     
  13. Write4U Valued Senior Member

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    Thank you, if I understand an aspect of a priori causality.I'm a happy guy...

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    p.s. is *function* possibly tangently related to the fundamental law of *necessity and sufficiency*?
     
    Last edited: Apr 19, 2017
  14. Write4U Valued Senior Member

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    I cannot comment on the symbolic maths, but IMO, forms are expressions of mathematical functional patterns, as Roger Antonsen demonstrated with his example of two connecting rotating arms turning at a 4/3 rate, creates a remarkably beautiful form, the image of the number 4/3
     
  15. Write4U Valued Senior Member

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    Can one drawa comparison to the law of *neccessity and sufficiency*?
     
  16. someguy1 Registered Senior Member

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    I know what necessary and sufficient conditions are, but I couldn't Google any such law. Can you say what it means?

    I just think it's a little ironic/odd/interesting that the standard reals are complete, while the two common alternative models of the reals are incomplete by virtue of having too few or too many points. I wonder if that has some philosophical significance.
     
  17. someguy1 Registered Senior Member

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    No this is not true. Each rectangle in a Riemann sum has nonzero base. There are no infinitesimals in the real numbers.

    Perhaps I should start by defining infinitesimal. An infinitesimal is a number that is greater than \(0\) but less than \(\frac{1}{n}\) for any positive integer \(n\).

    It's clear that there are no infinitesimals in the real numbers. If you claim \(\epsilon\) is infinitesimal, I'll just keep going \(.1, .01, .001, .0001, \dots\) till I find an \(n\) such that \(0 < \frac{1}{n} < \epsilon\), falsifying the claim that \(\epsilon\) is infinitimal.

    When we form a Riemann sum, we partition the \(x\)-axis into a finite collection of subintervals, each one finite, and form the sum. If the collection of all such sums has a limit, we call that sum the value of the integral.

    No infinitesimals are involved in this process, nor could they be, since there are no infinitesimals in the real numbers.

    The notation \(dx\) is not an infinitesimal. It is simply a notation telling you which variable you're integrating with respect to. In more advanced math such as differential geometry, they give formal meaning to \(dx\) and other differential forms, as they're called, but they are not infinitesimals.

    It's true that many students come away from calculus class thinking that \(dx\) is an infinitesimal, but it is not. The historical achievement of making calculus logically rigorous involved banishing infinitesimals from math and replacing them with limits. Instead of talking about "infinitely small" we talk about "arbitrarily small" and this makes all the difference.

    It's true that there are alternative number systems such as the hyperreals and the surreals that contain infinitesimals, but these systems add no clarity to calculus and have their own logical difficulties, such as being incomplete and requiring more set theoretic assumptions than do the standard reals. They're interesting to study in their own right, but they do not bear on the question of whether there are infinitesimals in the real numbers.

    There are no infinitesimals in the real numbers.
     
    Last edited: Apr 20, 2017
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  18. someguy1 Registered Senior Member

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    Infinitesimals do not exist in the real number system.

    Whether any mathematical entities at all may be said to exist depends on what you mean by exist. Does the number \(3\) exist? Does \(-\sqrt{\pi}\) exist? How about a noncomputable number that represents a point on the number line but that has no finite-length description? Does it exist? These are questions of philosophy.

    But it is a mathematical truth that there are no infinitesimals in the real number system.
     
  19. someguy1 Registered Senior Member

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    No this is mathematically wrong. There are no infinitesimals involved. Please consult any book on real analysis and don't rely on vague misunderstandings picked up in calculus class. Not your fault personally, the teaching of calculus is geared toward practical applications and not to theory, so many students come away with these misconceptions.
     
  20. origin Trump is the best argument against a democracy. Valued Senior Member

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    I disagree.
     
  21. someguy1 Registered Senior Member

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    With the fact that calculus doesn't teach theory? Or that students come out of calculus class confused? Or that there are no infinitesimals in the reals? I think if you supply some more details, there might be an interesting discussion. Of course my opinions about math pedagogy are my own; but my statements about Riemann integrals and infinitesimals are mathematically correct.
     
  22. Write4U Valued Senior Member

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    Perhaps my use of the term "law" is misleading.
    (Highlght mine).
    IMO, this relational statement holds true in mathematics also.
    Perhaps we have no numerical symbols for values which are unmeasurable to us. Numbers are invented by humans from observation of natural mathematical functions. and values (the universe uses no numbers, it uses latent and dynamic values, potentials). Sometimes we can infer the truth of an *equation* by applying the logic of *necessity and sufficiency*.
    Consider the number 4/3 = 1.333333333......in decimal math can also be wxpressed as 1.010101010101,,,, in binary form, yet the value remains the samefrom this *perspective*
    Is that not the definition of *axioms*?

    The interesting part is that an open number (value) as 1.3333..........can yield a *closed form*


    But IMO, logic can resolve a lot of mathematical and observational problems/ Philosopjy gives logical (abstract) perspectives and implicatons to our understanding of the physical world.
     
    Last edited: Apr 21, 2017
  23. someguy1 Registered Senior Member

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    Yes that's true. There are real numbers for which we can not have any symbols or descriptions or ways of specifying them.

    This is a bit off-topic from infinitesimals. There are the non-computable numbers. These are real numbers that can not be described by any algorithm or finite-length string of symbols. There are many noncomputable real numbers. If you pick a random real, it will almost certainly be noncomputable. It's a little mysterious.

    By the way "measurable" is a technical term in math that means something else, so best to avoid it.


    If I'm understanding you, you're making the distinction between a number and any of its many possible representations. For example 2 + 2, 4, and 10 (base 4) are different expressions that both point to the same number. The number itself is an abstraction.

    Yes in this case that's true. But most real numbers don't have closed-form representations. All the familiar numbers we use do have closed-form representations though. For example the digits of pi or sqrt(2) can be cranked out by a computer program, and a program is a finite string of symbols. We can think of a program as a closed form for the number whose digits it cranks out. But since there are fewer programs than real numbers, there are real numbers without closed forms.

    To be fair, nothing in this thread is about the physical world. We're talking about the mathematical abstraction of the real numbers. As far as we know, real numbers can't be instantiated in the physical world because it takes an infinite amount of information to specify most real numbers.

    It's always important to distinguish math from physics.
     
    Last edited: Apr 21, 2017
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