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What's a number?

Discussion in 'Off-Topic' started by Souron, Apr 10, 2010.

  1. Birdjaguar

    Birdjaguar Hanafubuki Retired Moderator Supporter

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    No, since we created the notion of the relationships to begin with, based on our conventions. Mathematics is a system we created to organize things and those relationships only exist within that context.
     
  2. Souron

    Souron The Dark Lord

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    Bees see honey combs, we see hexes, but whatever it's called, it's still one of 3 regular polygons that can be arranged to form a grid on a plane. That's meaningful to bees as well as to us. The bees may not be aware, but they wouldn't be able to build hives if it were not true.
     
  3. Birdjaguar

    Birdjaguar Hanafubuki Retired Moderator Supporter

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    Honey combs are real, but the mathematics of regular polygons is a product of our imaginations. Math (and science) are tools we use to impose orderliness upon the physical world.
     
  4. Perfection

    Perfection The Great Head.

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    Couple things to think about here:

    1. Let's imagine we encounter an alien race, it stands to reason that they should have mathematics in a form recognizable to ours. We might call that relationship between the two something that exists, might we not? If so what is the nature of that relationship?

    2. consider this:
    n=34238497523094875203984752309487528930475^23
    m=34238497523094875203984752309487528930475^23-25
    p = the lowest prime number between m and n
    does p exist?

    I have this concept of what p is. And that concept exists, however p might or might not exist! It seems to me that p either does exist or doesn't exist and it has nothing to do with any of our mental states. It only has to do with if there is some number between m and n that happens to be prime. Something I have no clue about!

    How can you account for that?
     
  5. Birdjaguar

    Birdjaguar Hanafubuki Retired Moderator Supporter

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    Within the context of mathematics all of those things are true. Just like in the context of chemistry H+H+O = water. The relationships are defined by the system we created. Your aliens may see similar relationships (or not) depending upon their sensory inputs and tools.

    When you take away our way of looking at things through mathematical systems and imagine that they were never created, the world doesn't change a bit. Without math, people will find a different way of organizing things to improve their understanding; In most cases our imposed systems change as we learn more and the old system is discarded as too far from the new truth. Part of the elegance of math is that appears to be so internally consistent and we keep refining things to make it more so. None the less it is still just a product of our imagination.

    EDIT: "What is the nature of relationships? is an interesting question.
     
  6. Souron

    Souron The Dark Lord

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    So you are saying that it is a product of our imaginations that it is impossible to arrange regular pentagon shaped honey combs to form a flat grid?

    I certainly don't agree.
     
  7. Birdjaguar

    Birdjaguar Hanafubuki Retired Moderator Supporter

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    I am saying that mathematics as a system is a product of our imaginations. We observe the world and then created a system to organize it using numbers and relationships. For the most part those relationships are true within its mathematical context. The fact that mathematics can model the real world (it is based on our observations of that world after all) doesn't mean that it exists outside of the minds that created it or use it.

    Language is similar if less precise. We use it to organize our thoughts and communicate them to others. It demonstrates relationships and can be used to explain all kinds of useful things (like mathematics). Does language also exist whether or not people ever existed?
     
  8. warpus

    warpus In pork I trust

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    Something you imagine.. with your mind!

    Even prime numbers don't exist by definition. You might as well wonder why single married men don't exist!

    What is blue? How do you touch it? How do you get to know it?

    I don't find them interesting.. To me it's all clear cut and makes perfect sense.

    Well, just because we can use numbers to do those things, doesn't mean that other things can't!
     
  9. Souron

    Souron The Dark Lord

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    I'm not sure what you mean by "mathematics as a system", but mathematics generally does not stem from the physical world, but rather from basic axioms. This is what separates from science, which is based on observation.

    For instance, all of euclidean geometry can be derived from [wiki]Euclid's postulates[/wiki], of which there are only 5. This includes Pathogen's theorem, tessellations mentioned above, and more. By contrast, Newton's laws of motion, were accepted because they agree with experiment. In 1915, Albert Einstein Published the [wiki]Theory of General Relativity[/wiki], which describes a world that does not follow euclidean geometry or newton's laws of motion. Newton's laws are now considered incorrect (though very good approximations for most applications). However, euclidean geometry is still considered true, and not dispoven by Einstein. You can no longer expect the world to obey euclidean geometry, but the principles that are derived from Euclid's postulates still follow from Euclid's postulates.

    Language is not build off axioms.
     
  10. Souron

    Souron The Dark Lord

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    Blue is a physical property, or alternately the mental sensation associated with seeing blue things*.

    Saying numbers are like the color blue would contradict your claim that numbers are purely abstract.

    Now Duo did earlier claim that numbers are like the color blue. And like the relation "above". Basically that they are a physical property. Do you agree or disagree?

    *color is kinda special, because unlike other physical quantities like weight, it is really a sensation that doesn't map directly to a physical trait; color is not a measure of wavelength. So for the purposes of this thread it may be better to avoid talking about color when the example of mass would serve just as well. Unless of course one means to say that numbers are a sensation.
     
  11. Perfection

    Perfection The Great Head.

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    I think Souron addresses a lot of important misconceptions about the nature of mathematics to you. It's not so much that we refine math rather, we discover new facts about it!

    And you still haven't explained to me why my prime number p appears either already exist or can never exist, quite independent of my imaginings about it.

    Well, I think what needs to be addressed is the distinguishing between our body of mathematical knowledge, and mathematics itself. I don't think they are the same. Our body of knowledge doesn't know whether or not P exists, but that's different from if P does exist. If I knew everything everyone knows about mathematics, I still wouldn't know if P exists, but it could very well exist!
     
  12. warpus

    warpus In pork I trust

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    I'm not saying it's the same, but it's similar in the sense that "blue" is a concept that you can't touch, and yet.. things in the real world are blue.
     
  13. Birdjaguar

    Birdjaguar Hanafubuki Retired Moderator Supporter

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    If you want to root mathematics in axioms rather than counting that's fine. I tend to think of numbers and such from an application standpoint.
    Such a basis seems to make the existence of numbers as entities independent of humans less likely. My position is that mathematics (and numbers) are constructs of people (our minds) and do not exist without us. Is there evidence that they do? I see them as tools of our convenience and nothing more. The fact that we tend to discard the math that no longer suites our needs or that has been supplanted by newer approaches further supports the idea that such systems do not exist outside our minds.

    While language may not be rooted in axioms, it is a similar system used to organize our place in the world and communicate ideas. Our brains are well be suited to create such useful systems along with many others.
     
  14. Souron

    Souron The Dark Lord

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    Things in the real world can also be beautiful, dark matter, enchanting, positively charged, or finite. None of these concepts represent things that can be touched, yet all for a different reason. Saying that like the color blue, numbers can't be touched isn't saying much at all, unless you mean to imply further similarity between being blue, and being/being associated with a number.
     
  15. Perfection

    Perfection The Great Head.

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    I can't imagine every single natural number, do they not exist?

    Actually, an even prime number does exist...
    There's an issue here these sorts of contradictions can be insanely hard to tease out for instance, take Fermat's last theorem. It took us hundreds of years to realize there are positive integers a, b, and c such that a^n+b^n=c^n for integer n greater than 2. There's something very special going on here.

    See Sauron

    Then you haven't thought it through enough.

    Earlier you said
    If there's nothing else to it, then your description should not only be what is necessary for something to be a number but also what is sufficient to be a number. You have failed to do that, and thus your description is incomplete and your statement of "Nothin else to it." is wrong.
     
  16. Birdjaguar

    Birdjaguar Hanafubuki Retired Moderator Supporter

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    I have no idea about "p" or its nature. Within the framework of mathematics it has its place, but outside of that framework, it has no existence. Whether P exists within our body knowledge of outside of it or if it exists at all, all of that discussion takes place within mathematics.

    I do not dispute that numbers and mathematics have value and have proven themselves to be true at both the detail level or the more global perspective. What I am saying is that the entire system of numbers and math that we rely on every day is a product of our minds (and I would add, its interaction with the world).Given that, they do not exist outside of our minds.
     
  17. Perfection

    Perfection The Great Head.

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    :confused: What math have we discarded?
     
  18. Birdjaguar

    Birdjaguar Hanafubuki Retired Moderator Supporter

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    Of course blue is a number...
     

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  19. Birdjaguar

    Birdjaguar Hanafubuki Retired Moderator Supporter

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    Perhaps "discarded" is too strong a word, but we do keep moving on to more precise ways of calculating.
     
  20. Perfection

    Perfection The Great Head.

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    I don't think so. P may exist mathematics (and if not we could define a Q,R, and S at least one of which does, but none of which we know for sure), but it isn't within our knowledge of mathematics. This is something very bizzare for something that is supposedly within our heads.

    The other thing that needs addressing is to note that there are relationships that exist in the real world that are quite math-like. An explanation of the nature of mathematics would be incomplete if one doesn't tell how this fits into the picture. Why do the patterns we see in mathematics keep showing up in the real world?
     

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