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  • JRouche
    replied
    Originally posted by mklotz View Post
    The length of the circumference in the world of math is a concept, not a measurement.
    Well, kinda like being a Construct.

    You are a Mathematician. I like that even though I have no math education.

    I accidentally said I dont like Teachers her before. Its not the teachers, it was my issue with getting booted out of HS the first year because I was in a different space back then. And I have always been horrible at math.

    I love Math. It is the defining method where Theorists can solidify their Theory on paper. Otherwise they are just drawing cartoons and hope the Science Community takes them seriously.

    What I like about the math that I know nuthing about? The Continuity of it. It translates very well within its own "laws" and therefore you can prove one experiment with another. Meaning good math is self proving. Its concrete VS all them other sciences. I still like chem though.

    Oh, on a side note while at work board one day I did some long hand multiplications on paper. Like

    143843x4973429=

    Yes, I wrote the entire long hand thing out in pensile. It was a long string. The visual didnt look good. So I did a faster and much shorter way on paper and was surprised to come up with the same answer. You probably already know these things, short hand math.

    I just cant remember all the formulas for what you do.. JR



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  • mklotz
    replied
    Originally posted by lynnl View Post
    One issue that I don't remember concerning myself about, until the last year or so, is the constant pi and it's relation to a circle.
    We know that pi x diameter = the circumference of a circle. But pi is an irrational number with a value having decimals that go on forever. So the exact product of pi X dia that you get will depend on the number of decimal digits you use. In other words, not a precise value.
    Is that to say the circumference of circles is kind of a nebulous thing? Just an approximation.

    What am I overlooking here?
    All measurements in the real world are approximations. The length of the circumference in the world of math is a concept, not a measurement.

    Irrational numbers occur frequently in math. The ratio of the diagonal of a square to its side is irrational. The golden mean is irrational, as is the base of natural logarithms.

    But the fact that these numbers are irrational doesn't mean they can't be calculated. If you can measure your diameter to 1000 digits of precision, we can supply you with a value of pi accurate to 10,000 digits so that you can accurately calculate circumference to 1000 digits.

    Many of these irrational numbers are defined by infinite series and the use of these series (or calculation series based on them) allow the precision to be carried to whatever level is desired.

    Here's another one to try to wrap your mind around...

    In the world of arithmetic we learn about associativity (numbers can be added in any order) and commutativity (numbers can be multiplied in any order). Yet, once we get to matrix algebra, commutativity goes out the window. If A and B are matrices, it's not necessarily true that A*B = B*A.

    Especially in physics, math is used to predict things about the world. Many elementary particles have been discovered by predicting their properties (their existence, in fact) with mathematical theories and then building equipment to find them based on those properties. This is another fascinating feature of math; we can use it to predict things about the real world and many of those predictions pan out. That's freaking amazing.

    So, what does the failure of commutativity in matrix algebra predict about the real world? Tensors, the mathematical structures used to define aspects of the real world, are matrices so the question is not merely theoretical.



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  • lynnl
    replied
    One issue that I don't remember concerning myself about, until the last year or so, is the constant pi and it's relation to a circle.
    We know that pi x diameter = the circumference of a circle. But pi is an irrational number with a value having decimals that go on forever. So the exact product of pi X dia that you get will depend on the number of decimal digits you use. In other words, not a precise value.
    Is that to say the circumference of circles is kind of a nebulous thing? Just an approximation.

    What am I overlooking here?

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  • Bob La Londe
    replied
    I'm sure by now somebody mentioned it, but Joe Pi just did a video on calculating bolt circles that ya'all might find useful or at least entertaining.

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  • darryl
    replied
    Something anomalous seems to be going on at the very large and the very small scales. Within that range, the laws of physics seem to work well enough. So well in fact that mathematics beautifully describes things, and is seen to be essentially perfect. 1 plus 1 plus 1 = 3, no way around it. For how many orders of magnitude does this apply, and at what point would we have to revise our understanding of it all? And for what reasons?

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  • boslab
    replied
    Originally posted by mklotz View Post

    One of the most intriguing questions mathematicians (and many physicists) proffer goes something like this...

    Is mathematics some sort of fundamental property of nature that we have been gradually discovering or is it the most incredible mental construct mankind has ever assembled?

    Think about it. Newton managed to bundle all of the phenomenon of gravity into a single, simple equation. Was this a discovery on his part or an incredible mental construct on his part? If the exponent on the 1/R^2 term is changed by even a slight amount from two, stable orbits are impossible. Did we just luck out with our creation of algebra so that the square of a number was the perfect fit for the description of gravity or are inverse square forces a natural feature that we happened to discover?

    I don't waste any time thinking about philosophical crap like this; it gets in the way of doing something useful.

    However, if you think you can inject theology into math, you should start by answering this question.
    Whilst newton did, it was already described in words by Robert Hooke ( spring guy), Newton hated Hooke,and was as far as I’ve read the originator of a phenomenon we suffer from today, cancel culture, burning his painting, removing references of him from wherever he could, no doubt Newton was smart, but according to the historian at the royal institution, he was also a jealous petty insecure man
    mark

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  • JRouche
    replied
    Originally posted by mklotz View Post

    One of the most intriguing questions mathematicians (and many physicists) proffer goes something like this...

    Is mathematics some sort of fundamental property of nature that we have been gradually discovering or is it the most incredible mental construct mankind has ever assembled?

    However, if you think you can inject theology into math, you should start by answering this question.
    Definitely a construct. Just like the spoken language. The physics are already there. How we interpret them is our construct. The definition of an apple can easily be reinterpreted to be called a banana. One plus one can easily be called A plus A

    a=1, b=2, c=3 and so on. That doesnt change the fact that A+A=B. We gave the number One its value. It doesnt occur in nature as a One. JR

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  • mklotz
    replied
    Originally posted by JRouche View Post
    You dont think there is some Theology within Math?
    One of the most intriguing questions mathematicians (and many physicists) proffer goes something like this...

    Is mathematics some sort of fundamental property of nature that we have been gradually discovering or is it the most incredible mental construct mankind has ever assembled?

    Think about it. Newton managed to bundle all of the phenomenon of gravity into a single, simple equation. Was this a discovery on his part or an incredible mental construct on his part? If the exponent on the 1/R^2 term is changed by even a slight amount from two, stable orbits are impossible. Did we just luck out with our creation of algebra so that the square of a number was the perfect fit for the description of gravity or are inverse square forces a natural feature that we happened to discover?

    I don't waste any time thinking about philosophical crap like this; it gets in the way of doing something useful.

    However, if you think you can inject theology into math, you should start by answering this question.

    Leave a comment:


  • JRouche
    replied
    Originally posted by Jim Stewart View Post

    Sure. But math isn't.

    -js
    Bet me..

    You dont think there is some Theology within Math?
    Dont get me wrong, I see Math as being perfect. JR

    Edit: Sorry, needed the Q.

    Originally posted by nickel-city-fab View Post
    And isn't engineering just a collection of acceptable compromises?
    Last edited by JRouche; 02-19-2021, 01:40 AM.

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  • Jim Stewart
    replied
    Originally posted by nickel-city-fab View Post
    And isn't engineering just a collection of acceptable compromises?
    Sure. But math isn't.

    -js

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  • nickel-city-fab
    replied
    Paul, you have a wonderful way with words. Very clarifying, thanks. I have always heard that one couldn't divide a circle by 7, but I would have to ask "Under what conditions?" The tools and techniques available to the HSM today, are so refined that error is either immeasurable, or negligible. I am a fan of George Thomas' writings about his "Versatile Dividing Head" and his system takes you down to one hundredth of a degree using extremely basic and simple techniques. Would I be able to measure a hundredth of a degree? probably not. But for my shop and what I do, it is an acceptable compromise. And isn't engineering just a collection of acceptable compromises?

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  • Paul Alciatore
    replied
    This post demonstrates the DIFFERENCE between mathematical theory and practical shop methods.

    In our shops we stop refining things and call it "GOOD ENOUGH" when we reach a point where the accuracy is good enough for whatever we are making or when we exceed the ability of our shop measuring tools to reliably detect any error.

    In mathematical work, there are NO PRACTICAL considerations and they are talking about idealized concepts like points that have no dimensions, lines that have no width and are perfectly straight (whatever that may be), and planes that are perfectly flat. Their compass has ideal points - NO AREA.

    In our shops none of those mathematical perfections exist. They can not exist because we are working with actual. physical objects, not theoretical concepts.

    So, YES we can take a compass and draw a "circle" on something. We can then step off ANY number of divisions, using that same compass by starting with a good or not so good guess and adjusting that guess as needed. And if we are not demanding an excessive amount of accuracy and do not have a means of measuring the divisions to a greater amount of accuracy than the compass guess-and-try method, then we can call it "perfect".

    The mathematician does not, CAN not call such a method precise. They KNOW that no matter how small the remaining error is, it will always still be there. The mere fact that neither he/she nor anyone else can actually observe or measure that residual error is of NO CONSEQUENCE to them.

    When you were told that you can divide a circle into any number of parts with a compass, that was a completely true statement for the shop environment. So go ahead, divide, mark, make chips, and use what you have made. It will probably work just fine, 99% of the time. Just do not ask a mathematicians to accept it as a mathematical method.

    Mathematicians use mathematical methods. And we use practical methods that produce results that are within a small margin of error but are functional for whatever device we are making. And if you have had deep enough training in BOTH of these areas of human pursuit, you would understand the differences and when it is appropriate to use one method or the other. And HOW THOSE DIFFERENCES DO AND DO NOT MATTER.



    Originally posted by darryl View Post
    'A true septagon can NOT be constructed with only dividers and straightedge.' Is this true? I'll admit, I never tested the concept, but if this is true then I have been misled. Perhaps a step was left out of the procedure? Perhaps the result was so close that for all intents and purposes is was good enough? I don't know. Maybe I'll pick up some posterboard, a compass and a straightedge and test it out-

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  • Erich
    replied
    No, Close enough is "true" for engineering.

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  • mklotz
    replied
    [QUOTE=darryl;n1928764]... Perhaps the result was so close that for all intents and purposes is was good enough? ...[QUOTE]

    "Close enough" is not a synonym for "true" in mathematics.

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  • darryl
    replied
    'A true septagon can NOT be constructed with only dividers and straightedge.' Is this true? I'll admit, I never tested the concept, but if this is true then I have been misled. Perhaps a step was left out of the procedure? Perhaps the result was so close that for all intents and purposes is was good enough? I don't know. Maybe I'll pick up some posterboard, a compass and a straightedge and test it out-

    Leave a comment:

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