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What is a Number

  1. Jul 24, 2006 #1
    Is there any consensus among mathematicians (or rather philosophers of mathematics) on the concept of a number - what is really a number?

    This question has quite bothered me recently. Do we left the concept of a number undefined and take it as an intuitive one with no need of definition?

    Leopold Kronecker wrote "God made the integers; all else is the work of man." (article about Kronecker at Wikipedia). Is this idea prevalent?

    Are there any recent books that deal with the concept of a number in modern mathematics?
     
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  3. Jul 24, 2006 #2

    arildno

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    From what I know, a "number" is just the term we use for an ELEMENT of a SET called a "number system". In addition, certain OPERATIONS (like addition or multiplication) must have been defined in order to call our set a "number system".

    So, what you have fractured the number concept into, are the concepts of
    1. A set
    2. Elements of a set
    3.Operations we can do on such elements.

    The concepts of a "set" and "element" are so primitive that they are not defined in terms of other concepts; rather, it is specified what is true (or not true) of elements and sets in a particular set THEORY (for example Zermelo-Franckl).
     
  4. Jul 24, 2006 #3
    A number is an element in a language, much like the english language, that we created to describe what we see in the world.
    The human brain quantifies everything, so it makes sense that we made something to quantify and do other mathematical operations with.

    So imo a number is the way we interpret things.
     
  5. Jul 24, 2006 #4
    a number(Natural number N) is built from the succ() and pred() operators from the set {1}. An integer(Z) is built much teh same way...from there you can build the set(Q-Rational,R-Real,C-Complex,Pn-Polynomial,Vectors etc).
    Like Arilno said, the terms "element" and "set" are rather primitive though i think element comes from set. If you want to go further than that you jump into psychology on why as intelligent beings we begin to label things...once you label things, you call them objects(hence 1)..from there a group of objects is called the set and and element is one of these objects in the set. From there you define and build your operators like succ() and pred()
     
  6. Jul 24, 2006 #5
    Roger Penrose's book "The Road to Reality" has a discussion on this very topic. He seeks a definition of number that is divorced from the physical world, a strictly mathematical definition. He gets one by using sets as follows:

    [itex]0 \equiv \emptyset[/itex]
    [itex]1 \equiv \{\emptyset\}[/itex]
    [itex]2 \equiv \{\emptyset, \{\emptyset\}\}[/itex]
    [itex]3 \equiv \{\emptyset, \{\emptyset\}, \{\emptyset, \{\emptyset\}\}\}[/itex]
    ...

    to define the natural numbers and uses these to define negative numbers, rational numbers, real numbers, and complex numbers.

    If you are looking for a definition of number that IS rooted in the physical world, then I expect that you run up against certain problems such as extremely large numbers having no physical meaning. That is, there are only so many particles in the universe and numbers larger than that may fail to mean anything.
     
    Last edited: Jul 24, 2006
  7. Jul 24, 2006 #6
    jimmysnyder...given a finite large number of objects..the permutation/combination will be larger and given states or time the sequence of permutation will grow.
     
  8. Jul 24, 2006 #7
    Even then, there is a limit. The number of permutations of a finite number of particles in a finite space for a finite period of time is still finite. If you bring in continuity arguments to get an infinite number of spacial and temporal states, then you lose the ability to count altogether.

    I'm not sure that you can't base number on some physical phenomenon. I just am pointing out a problem that you may need to overcome.
     
  9. Jul 24, 2006 #8

    matt grime

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    Just to clarify, the notion of number as cardinals of sets is not an idea of Penrose. It predates him by many decades (as a formal concept, I mean: it predates him as an intuitive idea by millenia).
     
  10. Jul 24, 2006 #9
    Yes. I wrote that from memory, here is what Penrose says (page 64):
    He raises the issue of finiteness and a second issue about basing the definition of numbers on physical phenomena:
    I don't know if he was refering to the idea in quantum mechanics that numbers of objects do have such a tendency in this universe.

    And a third issue:
    My daughter's bedroom?
     
  11. Jul 24, 2006 #10
    The thing about Cantor's approach is that numbers describe sets that are progressively inclusive of one another. The 3 set includes the 2 set, 1 set, and empty set as its elements. This isn't how other languages work. I'd very much like to see how any language can be constructed from nothingness.

    What neurological literature did you scoop that from?
     
  12. Jul 24, 2006 #11
    Mickey: I don't think he needs a neurological reference. Considering As humans we do like to give things labels, doesn't matter the language. But I'm sure you can find one about how we take a retinal image and store it in memory and with language label it. Therefore as the undamaged brain "sees" an image the string of sounds we use to describe objects in that image begin to fire. I'm sure you can also find literature on the brain quantifying size...large/small more/less etc.

    as for teh cantor set...interesting enough he didnt' defin it as
    1={0}
    2={{0},{0,0}}
    3={{0},{0,0},{0,0,0}}

    N=N-1 U {N-1 U 0}
     
    Last edited: Jul 24, 2006
  13. Jul 25, 2006 #12
    I'm sorry, I was under the impression that memory storage has been a persistently elusive problem in neuroscience.

    If you can help point me to some literature that shows how the brain "quantifies everything," or stores quantities, I'd be very thankful. :shy:
     
  14. Jul 25, 2006 #13

    Pythagorean

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    Conjecture:

    I've always assumed that the origin of numbers was more of a philosophical debate, divided mostly between people who think numbers are an intrinsic part of the universe (like pythagoras) and people who think numbers a human construct used to interpret the universe. To me, these can coexist easily, because we are an intrinsic part of the universe, but I am both pluralist and agnostic towards most ambiguous matters.

    I would think the scientific method somewhat avoids confronting that, since it's such an intangible question. As far as most physical scientists are concerned, it's there, and it's the language that developed from the natural philosophers (old school physicists) as answers to the questions they asked.

    I wouldn't be suprised if other life on earth could count, either. The questions to me, is whether or not they're conscious about it.
     
  15. Jul 25, 2006 #14
    Mickey: it is a localization problem. To my knowledge much of what was taught to me states that its in the areas know as the HC,ERC,PRC,SB. Then again the textbooks that i read could be wrong.
     
  16. Jul 25, 2006 #15
    But 0 is a number. Cantor defined numbers in terms of sets alone. What's more, the set {0, 0} is not considered to be anything different from the set {0}.
     
  17. Jul 25, 2006 #16
    I don't remember where I read this, so I can't provide a citation.

    Apparently, baboons have a behavior that indicates that they may be able to count to 3. That is, there was a farm field where baboons were feeding. Three farmers walked into the field and the baboons ran away. The farmers hid themselves in the field, but the baboons did not return. Two farmers then got up and walked away from the field, but the baboons still did not return. The third farmer got up and walked away and then the baboons returned to the field. When four farmers walked onto the field, hid themselves and then three got up and walked away, the baboons 'lost count' and returned to the field with the one hidden farmer in it.
     
  18. Jul 25, 2006 #17
    my bad i donnt' have the EmptySet symbol
     
  19. Jul 25, 2006 #18
    Quote this message and you will have the empty set symbol.

    The problem with the definition you propose is that the set [itex]\{\emptyset\}[/itex] is not considered to be anything different from the set [itex]\{\emptyset, \emptyset\}[/itex]
     
  20. Jul 25, 2006 #19
    yah i realized that...i think iwas thinking of ordered sets ()...but typed {} then again i shouldn't stay up past 5am
     
  21. Jul 25, 2006 #20
    I'm not 100% sure, but I think your ordered sets would work. Now the problem is that your definition which uses ordered sets is by no means simpler than Cantor's which only uses sets.
     
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