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Does the number π really exist? watch

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    (Original post by majmuh24)
    Isn't mathematical logic a whole different thing from philosophical logic though :holmes:
    It seems to me, and i might be wrong, that the main reason for the divide is that philosophers get off on modal logic, whereas mathematicians don't really care.

    Take the axiom: <>[]P--> []P

    Mathematicians are like "meh"

    Philosophers be like " dem lines"
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    (Original post by KingStannis)
    It seems to me, and i might be wrong, that the main reason for the divide is that philosophers get off on modal logic, whereas mathematicians don't really care.

    Take the axiom: <>[]P--> []P

    Mathematicians are like "meh"

    Philosophers be like " dem lines"
    Logic is a branch of methamatics though. The same would be true of you showed a topologist a theorem from multvariate analysis.
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    (Original post by james22)
    Logic is a branch of methamatics though. The same would be true of you showed a topologist a theorem from multvariate analysis.

    I wouldn't say logic is a branch of mathematics, mathematical logic certainly is but logic itself does not appear mathematical.
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    (Original post by KingStannis)
    Wow, touching upon a lot of huge philosophical points here. I don't want to get into a long discussion about all of them tbh
    Noted. I'll try and be short and sweet in any responses


    (Original post by KingStannis)
    You're correct that numbers have no physical existence, but them being abstracts doesn't prove they don't exist.
    That's precisely my point - numbers don't exist physically and they do exist as abstract concepts. And that, I contend, is a complete and trivial solution to this thread (its original direction anyway!)


    (Original post by KingStannis)
    Many mathematicians and philosophers would say that if all sentient beings died then the rules of mathematics and valid inference (which includes numbers), exist as a fundamental part of the universe. Others would argue that they're a purely human construct that happens describe the way the universe is, but are not an actual part of the way the universe is structured. Big philosophical issue there.
    I don't think this is a big philosophical issue (or, if it is, it says a lot about the scale of other philosophical issues ) I think whether or not logical axioms and rules of inference are objective is either an answerable or a meaningless question. The rules of mathematics (i.e. non-logical axioms are the conclusions derived from these using rules of inference), however, are fairly evidently a human construct since we could quite happily pick completely different defining axioms if we wanted to.

    First consider the possibility that rules of inference are objective. Then clearly modern mathematics is also objective to anyone taking the same set of axioms, since it is derived purely from those axioms using rules of inference.

    Next, consider the possibility that rules of inference are not objective. Then we cannot deduce anything about mathematics.

    Whether or not abstract concepts exist in the absence of sentient creatures to hypothetically construct them is meaningless without a better definition of existence and trivial with one.


    (Original post by KingStannis)
    Arguing over how to define things is far more important then you make out; logic, in practical terms, is useless without definitions that comply with reality.
    But definitions are arbitrary. If a question hinges only on the definition of a concept that is yet undefined in the context, then it is a trivial question. It is a trivial question masquerading as a "deep" and "philosophical" question by employing stupid linguistic techniques (whether intentional or not). Namely: ****ty, vague definitions.

    Do slithy toves gyre in the wabe? Or do they gimble? Or both?


    (Original post by KingStannis)
    You're making an empiricist epistemological assumption; which is by no means a closed debate in philosophy.
    I don't know what this means
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    I would not say logic is a branch of mathematics at all. It was part of my mathematics syllabus but I don't class it as mathematics
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    (Original post by Implication)
    Noted. I'll try and be short and sweet in any responses




    That's precisely my point - numbers don't exist physically and they do exist as abstract concepts. And that, I contend, is a complete and trivial solution to this thread (its original direction anyway!)




    I don't think this is a big philosophical issue (or, if it is, it says a lot about the scale of other philosophical issues ) I think whether or not logical axioms and rules of inference are objective is either an answerable or a meaningless question. The rules of mathematics (i.e. non-logical axioms are the conclusions derived from these using rules of inference), however, are fairly evidently a human construct since we could quite happily pick completely different defining axioms if we wanted to.

    First consider the possibility that rules of inference are objective. Then clearly modern mathematics is also objective to anyone taking the same set of axioms, since it is derived purely from those axioms using rules of inference.

    Next, consider the possibility that rules of inference are not objective. Then we cannot deduce anything about mathematics.

    Whether or not abstract concepts exist in the absence of sentient creatures to hypothetically construct them is meaningless without a better definition of existence and trivial with one.



    But definitions are arbitrary. If a question hinges only on the definition of a concept that is yet undefined in the context, then it is a trivial question. It is a trivial question masquerading as a "deep" and "philosophical" question by employing stupid linguistic techniques (whether intentional or not). Namely: ****ty, vague definitions.

    Do slithy toves gyre in the wabe? Or do they gimble? Or both?



    I don't know what this means
    empirical = derived from sensory experience
    epistemological = of or relating to epistemology, i.e the theory of knowledge.

    It appears you hold a constructivst view of mathematics, which is actually perfectly fine and logical, the issue is much, much more complicated than you and me would like however: http://plato.stanford.edu/entries/ph...y-mathematics/
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    (Original post by ClickItBack)
    When you say that there will be statements which are true but we cannot prove to be true, are you referring to non-arithmetical statements? Why does this follow from Godel's theorem and the assumption that humans are Turing machines?
    Assuming humans are emulatable by Turing machines, then Gödel's incompleteness theorem applies to our systems of reasoning. (If we are not emulatable by Turing machines, then we could be doing all kinds of weird reasoning that somehow escapes Gödel.) The reason we know that some things are true but unprovable is because we can "jump out of the system": we can reason about first-order mathematics without being subject to the same rules as first-order mathematics is, so we are not bound by the fact that "there are statements of first-order mathematics which are true but cannot be proven by first-order mathematics". We can use second-order mathematics, so we can recognise that kind of fact.
    However, if we are emulatable by Turing machines, then there is a maximum level of meta for us: there is a point beyond which we can simply jump no further (if nothing else, because we have only a finite time to think and we think at a finite speed). Gödel's theorem then says that in that maximum-level-of-meta system, there are statements which are true but cannot be proven in that system. Ergo, there are statements which are true but which cannot be proved by us. [Someone who knows more about this, please feel free to correct me.]

    I'm talking about arithmetical statements. However, all statements in the real world can be translated into (particularly large) arithmetical statements (along the lines of "state_of_universe_at_time_t bears this-that-and-the-other relation to state_of_universe_at_time_t+1).

    (Original post by KeepYourChinUp)
    Anyway this discussion has exhausted me so I think I'm going to pull out lol.
    By all means feel free not to reply to this!

    I would say that anyone who deemed it acceptable to kill an innocent person to save 6 ill people belongs in a mental home. It isn't about what "I" think is correct. It's about what 99% of the world view as correct and sure there might be that 1% who think it's ok but when we have 99% of the population deeming it unacceptable... it's probably because it's morally wrong. Morals do change with time and this is because we learn things about the world. We don't all have the exact same morals but the 21st century world generally has similar shared morals although there are some countries which still behave like Ancient Egypt.
    A thousand years ago (or even a hundred years ago), the same statement would be true if "kill one to save six" were replaced by "white person to marry black person". Your argument Proves Too Much: it cannot be correct, because it produces a clearly-wrong answer in a situation in which it is clearly still applicable. When you say morals change, why do you assert that this particular moral (that you will not murder six patients to save one healthy person) will not?

    If you were to ask me how many cats would I sacrifice to save 1 child? I would sacrifice as many as needed, but not so many that they would go extinct. We could sit here all day and ask thousands of pointless questions about different things but why not focus on questions which are worth debating and thinking about? We're never going to find ourselves in a situation where we are killing cats to save 1 person so the question isn't worth discussing at any meaningful length.
    You've missed my point - my point was that "your argument doesn't stand up well to consideration under a continuum". I was setting up a continuum where most of the answers were obvious, then dragging the continuum towards where it was actually useful. I was attempting to ascertain the nature of the dividing line you have between "I will sacrifice all the nematode worms to save myself" and "I will not sacrifice one person to save another six people". Easy questions aren't necessarily pointless - there was an aim to mine.

    By the way, the statement "I will not sacrifice one person to save six" is functionally identical to the statement "I will sacrifice six people to save one". That's clearly wrong, in exactly the same way as you assert the original is - unless it's different in some way I've missed.

    An example of a good question: - A 6 year old and a 45 year old are both on the donar list for a heart transplant. The 45 year old is first on the list and the child is second on the list. Does the doctor give the heart to the child or the 45 year old? Every fibre of your being would want to give the heart to the child but this is morally wrong as the 45 year old is before the child on the list.
    Ah, I understand - you believe your morals are universal. What you meant to say was that "every fibre of *my* being…". To me, it's not clear-cut like that. Imagine for some reason that the child had a 10% chance of survival after the operation, but the 45-year-old a 90% chance. Then to me it's obvious that the 45-year-old should get it - I don't know about you. (If all other things are equal - the 45-year-old is not some kind of scientist on whom the fate of the world depends, etc - then I agree with you that the child should get the heart, so as to maximise the number of life-years output from the operation.)

    An example of a bad question: - How many Dolphins would you kill to save a baby? While this question can be discussed and opinions shared it's just a pointless question. If the baby is in a pool of Dolphins then obviously you're going to kill them all to save the baby if they're attacking the baby.
    Don't avoid the question - I meant "kill the dolphins, which are doing no harm to the baby…" - it's a question designed to determine how you measure one life against another. And what if the pool of dolphins is every dolphin in existence? Your previous response with the cats suggests that you'd let the baby die.

    Anyway, you've tapped out, so I will too now unsubscribed! (unless someone quotes me)
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    (Original post by John Stuart Mill)
    empirical = derived from sensory experience
    epistemological = of or relating to epistemology, i.e the theory of knowledge.

    It appears you hold a constructivst view of mathematics, which is actually perfectly fine and logical, the issue is much, much more complicated than you and me would like however: http://plato.stanford.edu/entries/ph...y-mathematics/
    I'm digging in :cool:
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    (Original post by james22)
    Logic is a branch of methamatics though. The same would be true of you showed a topologist a theorem from multvariate analysis.
    Well, you can say that (or that mathematics is a branch of logic. I find the divide to be arbitrary), but that was just a silly post showing that modal logic is central to philosophy but not mathematics.

    I think that mathematics and mathematical logic end where dealing with language and statements begins, tbh.
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    (Original post by Implication)
    Noted. I'll try and be short and sweet in any responses




    That's precisely my point - numbers don't exist physically and they do exist as abstract concepts. And that, I contend, is a complete and trivial solution to this thread (its original direction anyway!)

    Do abstract concepts have an existence? Are they purely a construction of the human mind, or is the universe built on these rules?


    I don't think this is a big philosophical issue (or, if it is, it says a lot about the scale of other philosophical issues ) I think whether or not logical axioms and rules of inference are objective is either an answerable or a meaningless question. The rules of mathematics (i.e. non-logical axioms are the conclusions derived from these using rules of inference), however, are fairly evidently a human construct since we could quite happily pick completely different defining axioms if we wanted to.

    First consider the possibility that rules of inference are objective. Then clearly modern mathematics is also objective to anyone taking the same set of axioms, since it is derived purely from those axioms using rules of inference.

    Next, consider the possibility that rules of inference are not objective. Then we cannot deduce anything about mathematics.

    Whether or not abstract concepts exist in the absence of sentient creatures to hypothetically construct them is meaningless without a better definition of existence and trivial with one.
    Why would the rules of valid inference be wrong, when using anything other than them creates contradictions? By extension then, mathematics must be objective too, correct? But, are they objective because they correctly reflect the universe, or does the universe reflect mathematics ? That, for me, is an interesting question.

    But definitions are arbitrary. If a question hinges only on the definition of a concept that is yet undefined in the context, then it is a trivial question. It is a trivial question masquerading as a "deep" and "philosophical" question by employing stupid linguistic techniques (whether intentional or not). Namely: ****ty, vague definitions.

    Vagueness is precisely what philosophers try to avoid by using logical symbolism. Language leads to ambiguity and mistakes. Philosophers simply try to find definitions that are true, for instance, we know the inference A --> B must be unsound if the definition of A is inconsistent with the existence of B.


    Do slithy toves gyre in the wabe? Or do they gimble? Or both?
    Both, and the moongrath outgrabe

    I don't know what this means

    The is a debate between rationalists and empiricists, empiricists believing that all knowledge comes from experience, rationalists believing knowledge can also come from our reason.
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    (Original post by KingStannis)
    I still think many of the questions you pose are resolveable by clarify of definitions... but after reading the link posted by John Stuart Mill (presumably not the real one!) I'm starting to think I might be wading towards the deep end here. I'm taking a module or two on formal or mathematical logic in September if my course organiser and home School allow it, so expect me back with revised opinions next year
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    (Original post by StrangeBanana)
    Circles exist; they can be manufactured by humans, or they can be found in nature. pi is defined to be the ratio of the circumference of a circle to the diameter of that circle, so, if circles exist, then pi exists.

    The question is meaningless until you define existence, though. Is existence being able to be expressed as a decimal with a finite number of digits? Then, from the manner in which we have defined existence, pi, the square root of 2, and all other irrational numbers do not "exist". Is existence not being able to be written down accurately on paper with simple instruments? Then pi, e, and the transcendental numbers do not "exist".

    Ultimately, numbers can't really exist on their own - we can use them to count things, and the we count exist, but the numbers with which we count do not. Mathematics is an abstract field of study, after all.
    I do like this reply.

    If human beings don't exist, circles can exist? Circles still exist if no one finds them? Stupid questions? Whatever. I can't help it.



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    (Original post by StarvingAutist)
    So? Pi is clearly a number, and it is very real and very important.

    If it wasn't a number, how could it be in any way useful? Sure, it's not algebraic, but it has many applications. I'll just remind you:

    Attachment 264151
    Okay, thanks for reminding me.

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    (Original post by Implication)
    I'm just saying that the question is meaningless until you define "existence". Abstract concepts don't have physical existence by definition.

    Can you see a pi in real life (please no lame jokes )? No. But can you see a 2 in real life either? No. Can you apply pi to concepts in real life? Yes. Can you apply 2 to concepts in real life? Yes. Can you apply them both to the same concepts? Not necessarily.
    Cut-and-dried answer. Thanks.

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    (Original post by Obiejess)
    It exists as much as X does.

    E.g. Let C be exact length of circumference and D diameter:

    C × X = D therefore D ÷ C = X

    X = Pi

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    Hahahahaha.... Pi really exists in your mind while you are thinking about it. Agree? But where is it while you are thinking about something else?

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    (Original post by cuckoo99)
    Do numbers really exist at all?
    It depends.

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    (Original post by ockhamsshotgun)
    Define exist. How does a number exist?

    Why is the idea of the ratio between the length of two points and the locus of a single point, that point located at the cenre of that line, beyond existence in a way that say, 2, is not?

    The question implies the OP really needs to improve his/her knowledge of mathematics before trying to wade into more complicated question.

    A better question might have been, "do numbers exist if human beings don't?" That is a real epistemological issue, in my opinion.

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    Are you a psychic? You are absolutely right! You can read my mind!

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    (Original post by Liamnut)
    You know the first link says "But John Adam, a mathematics professor at Old Dominion University and the author of Mathematics in Nature: Modeling Patterns in the Natural World, said that no perfect circle can occur in nature "since a perfect circle is a geometrical idealization.""

    Also there are no perfect circles in nature and thence, the points about pi being related to the structure of DNA are wrong.
    This is where the physicists of the world would get up and leave... a field from a stationary point object MUST have spherical symmetry and a frame exists in which any given point object is stationary so therefore it follows that circles do exist in nature. Though not so much on a macroscopic scale.

    Pi being related to the structure of DNA is not strictly true BUT it is meant to be only an approximation of reality and happens to be a very good one.

    In fact, pi is so pervasive in physics that the magnetic permeability of free space was chosen to be equal to 4piE-7.
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    (Original post by miser)
    There are three major philosophical stances regarding the question of whether numbers exist. The first is that they do exist; the second is that they don't exist; the third is that they are a description of what exists.

    The first view - that they exist - is called mathematical platonism and holds that numbers are definite objects, but they are abstract objects, which means they don't interact with physical reality or hold any location in it. The second view - that they don't exist - is called mathematical fictionalism and holds that we invented numbers merely to help us make sense of the world. The third view is called mathematical nominalism and holds that mathematical claims are true, but they are really claims about the world (e.g. the claim that 2+2=4 is simply a description that if I had 2 objects and then received 2 more, I would have a quantity of objects we choose to represent with the symbol '4').

    Whether pi exists depends on which of these schools of thought is true. Pi is not a special number in respect to existing or not - it is not more or less of a candidate for existence than any other number (even so-called 'imaginary' numbers). It is an irrational number, sure, but all this means is that it cannot be expressed as a ratio of two integers. It exists as much as the number 1, which may or may not exist.
    Thanks for your help, mod. I prefer Buddhism...hehe...

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    (Original post by felamaslen)
    The answer would be similar to the answer to the question: do perfect circles really exist?

    Both exist in the mind of a mathematician.
    Always exist in the mind?

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