The Squared Circle

Discussion in 'Physics & Math' started by Motor Daddy, May 7, 2022.

  1. exchemist Valued Senior Member

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    The last sentence is not relevant. This discussion is about pure mathematics, not natural science.

    And for what it is worth, quantum theory and chemistry make extensive use of calculus and other mathematical concepts that rely on limits and converging infinite series. So to suggest they are some way ruled out by modern science is just bullshit.
     
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  3. Sarkus Hippomonstrosesquippedalo phobe Valued Senior Member

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    It seems odd that phyti seems only to be considering real-world application, as if maths is only applicable to the real-world, rather than consider it in the abstract.
     
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  5. exchemist Valued Senior Member

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    Indeed. Next he’ll be telling us mathematics can’t deal with more than 3 dimensions, because that is all that physical space requires. There’s a basic failure to understand what pure mathematics is, by the look of it.
     
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  7. phyti Registered Senior Member

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    Sarkus;

    How can you describe something you haven't experienced, like an alien abduction?
    Measurement was a necessary practice in early history. Tracking time, recording inventory, business transactions, building structures, etc. That is human experience.
    Math as a language was developed over centuries of experience. Today it is the verification tool for science.

    The set of integers is 'infinite', is a figure of speech for 'has no limit' or 'has no greatest element'. If it is without a boundary/limit it cannot be measured/counted. Boundaries are the requirements that enable measurement in a world of finite entities. Math as a tool is useful when its calculations match experience. I prefer to integrate a spherical volume and apply a density factor vs counting the number of particles of mass in the sphere.

    The algorithm for that series is to add 1/2 of the current term as the increment to the sum S. That increment also equals the difference |Limit-S|. Thus S never terminates.
    That is what you inherit with the concept of continuous division (a continuum).
    Being a geometric progression, the limit will be 2. You round up the current value which approximates the limit and claim it equals the limit of 2, because no one can imagine an infinite number. Anyone can say both a 1 meter stick and a 2 meter stick have an
    'infinite' number of points, but that doesn't determine which one is longer.

    Except integers. There is no number between 4 and 5.
    The sequence .9R has no last decimal position, (because it has been defined as such), so it never terminates.

    I mean using 'infinity' for measurement AS IF it's a number, which are abstractions.
    Number is a concept. So are all the means of representation originating in the mind.
    Mental images are real even if they only exist in the mind.

    You stated:
    "Maths doesn't require "time"."
    "One can sum an infinite sequence in seconds."
    Sure looks like it.

    Then it it's fiction.

    Then your math is for entertainment and is classified as 'recreational math'.
     
  8. exchemist Valued Senior Member

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    11,466
    No, it is pure mathematics, that’s all.
     
    Sarkus likes this.
  9. Neddy Bate Valued Senior Member

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    phyti,

    All we are saying is that when we divide 1 by 3 it produces the decimal 0.333R as the result. So that answer, with its non-terminating decimal, is equal to 1/3. You can prove this to yourself by doing the long division and dividing 1 by 3. You will see that it produces exactly the non-terminating decimal 0.333R and no other decimal but that one. So 1/3 = 0.333R in mathematics. Easy right?

    You are making things far more complicated by trying to invent reasons why 0.333R might actually be less than 1/3. In the process of doing this, you reveal that you yourself are not using the non-terminating decimal 0.333R, but rather you are using a decimal that has a finite number of decimal places instead. You say things like humans don't experience infinity, and there is not enough time for us to write out the entire non-terminating decimal. These hand wavy statements just further reveal that you are not using 0.333R but something like 0.3333333333333333333333 instead. We would all agree with you that number is less than 1/3, but that number is not 0.333R. Do you see the difference?
     
  10. Sarkus Hippomonstrosesquippedalo phobe Valued Senior Member

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    9,811
    I guess if people want to remain ignorant then it's their prerogative. I'm done here, so I'll leave you to trying to get through. Good luck.

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  11. Neddy Bate Valued Senior Member

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    Has phyti ever answered this? If not, it speaks volumes.
     
  12. James R Just this guy, you know? Staff Member

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    37,550
    phyti:

    The series represented by 0.9999... converges to a sum of 1. Agreed?
    An infinite string of 1s.
    The point is that 0.999... = 1. The two notations refer to the same number, just as in base 3, 0.2222.... = 1.

    If you disagree, then tell me what number falls between 0.999.... and 1. There must be at least one, if 0.999... < 1.
    Yes, and there's a countable infinity of natural integers.
    I think Mr Cantor wants a word with you.
    I agree. In other words, the cardinality of the set of integers is an infinite number. Mr Cantor called it "Aleph zero".
    There's a whole subfield of mathematics that deals with well-defined (i.e. not abstract) infinite numbers. There is even a "continuum hypothesis". Look it up!
    I don't know why you imagine that chemistry and physics determine mathematics.
     
  13. phyti Registered Senior Member

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    680
    forum;

    Here is a modern day Zeno scenario of the chase.
    It is a product of imagination based on real world events.
    The language of math works the same regardless of the source.

    A person robs a bank close to an interstate highway. Police respond quickly, issue a description of the robbers car. A person with a scanner reports sighting the car 12 miles west on the interstate. If the robber's speed is 60 mph (the speed limit) to avoid getting stopped, and the police car pursues him at 90 mph, how long before they overtake him?

    Using basic math with t in hrs.,
    for the robber, distance r=12+60t.
    for the police, distance p=90t.
    If r=p then t(90-60)=12.
    t=12/30=.4 hr or 24 min.
    The distance is a finite 24 miles and the time is a finite .4 hr.

    Where is anything that is 'infinite'?
     
  14. Sarkus Hippomonstrosesquippedalo phobe Valued Senior Member

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    Same place as the price of eggs, probably.
     
  15. James R Just this guy, you know? Staff Member

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    37,550
    To drive 24 miles, one first has to drive 12 miles. Then another 6 miles. Then another 3 miles. Then another 1.5 miles. And so on. As a series we get:

    24 = 12 + 6 + 3 + 1.5 + 0.75 + ...

    which has an infinite number of terms. This is what Mr Zeno was discussing with his paradox of the arrow.
     
  16. phyti Registered Senior Member

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    680
    What was Zeno's point in the runner and the tortoise?
     
  17. Neddy Bate Valued Senior Member

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    2,411
    So, you don't see the point James made in post #212? Seriously?!?

    All of your objections to 1/3=0.333... would also apply to the series James wrote, if they were valid.

    There is not enough time to write out all of the terms, so you claim it is invalid, right?

    It is an infinite series, and humans do not deal in infinities, so you claim it is invalid, correct?
     
  18. phyti Registered Senior Member

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    680
    My example was to show the time and distance can be determined without using 'infinite' series.

    Zeno was not promoting 'infinite'series, nor was he denying motion, since it was a common occurrence.
    He was demonstrating the nonsensical results from assuming a continuum with its perpetual division.

    This subject is another instance of 'truth will never be decided by an opinion poll'.
     
  19. Neddy Bate Valued Senior Member

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    2,411
    So, to you, it is a "nonsensical result" that the infinite series 12 + 6 + 3 + 1.5 + 0.75 + ... equals 24.

    It is really not so difficult for the rest of us to understand. What decimal result do you get when you divide 1 by 3? Can you write it down? Can you say that it equals 1/3? The rest of us can do it, but you can not. So what advantage does your opinion offer?
     
    Last edited: Jun 11, 2022
  20. James R Just this guy, you know? Staff Member

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    37,550
    Fine, but Zeno's point was different.
    He thought that his reasoning was paradoxical. Hence the label "Zeno's paradox" is often attached to these kinds of examples.

    I don't know how good the historical record is, for us to know what Zeno himself thought about any possible resolution of his paradox. But modern physics/mathematics certainly has a solution: the infinite series that sums to a finite number. Zeno recognised the infinite series in terms of distance travelled, but of course there's also a corresponding infinite series of times that it takes to travel the various distances. Divide one infinite series by the other, in effect, and we end up with a finite velocity. Moreover, it turns out that motion is possible after all, despite Zeno's "paradox". We can all breathe a sigh of relief and get on with our lives, comforted by the knowledge that infinite series solve the problem.
     

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