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Thread: Frege's 0 Lemma is not a definition of zero

  1. #1 Frege's 0 Lemma is not a definition of zero 
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    Gottlob Frege's definition of 0 in In λ-notation is 0 = #[λx x≠x].

    In words it means 0 is the number of the property of x such that x is not self identical (# signifies 'number', [] 'property', λx 'x such that', and x≠x is the condition).

    Please see Proof of Lemma Concerning Zero at plato.stanford.edu/entries/frege-theorem/proof4.html

    The theory relies on the belief that there are no instances of x such that x≠x. Hence Frege claims 0 represents zero, i.e. nothingness.

    First, examine what x is with fine scrutiny.

    In logic there are various types of x. I attempt to list them all in the table below. The degree of binding increases downwards:-

    Code:
    TYPE of x           VALUES(s) of x    CONDITION
    
    free variable       undefined         value definable but not defined yet
    bound variable      uncountable       e.g. natural numbers.  1,2,3,4,....
    bound variable      multi-value       e.g. value of x is 3 or 7.  3,7. 
    bound constant      single value      e.g. value of x is 5.
    bound symbol        undefinable       x cannot have a value


    The least bound case is with undefined value and the most bound case is with undefinable value.

    One way to understand this is using a computational view. A computer variable has a name (x say) which is stored as a character symbol in a lookup table with its associated value location. This is called a 'name value pair'. On declaration of x the value can be undefined but the location still exists. In this case x is a free variable. It is perfectly possible to have a lookup table where the value location can be removed completely from association with x. In this case the table contains x as a symbol only, with undefinable value. x is as bound as it can possibly be without the possibility of even having a value.

    It is important to also distinguish between x the symbol alone, x the symbol with value, and the value alone. They are three different meanings. x is not a value, it is a symbol, or a symbol with a value.

    Now take say λx x=3, what is the meaning of the x's in this? You might say x such that x is 3. But the precise meaning is x such that the value of x is 3. If the whole x (symbol and value) IS 3, then λx x=3 would become just '3', and not 'x with value of 3'. So it should be written:-

    λx valueOf(x)=3

    The x in λx refers to the whole x, i.e. symbol x, or symbol x with associated value.

    In the same way λx x≠x really means:-

    λx valueOf(x)≠valueOf(x).

    x cannot have a value and satisfy that inequality. So the only x which might satisfy it is the lowest type from the table, a bound symbol prohibited from having a value.

    Since x cannot have a value, x as a bound symbol must satisfy all conditions using values because the condition test does not relate to x. x is independent of the condition. All that could matter is whether the condition is contradictory or not.

    If contradictory it still doesn't affect x but could render the λ expression invalid. The result of the λ expression is either x or invalid.

    If non-contradictory, the result of the λ expression is just x.

    Therefore there are only two possible conclusions:-

    0 = #[x the symbol alone, which cannot have a value]
    0 = #[invalid]

    Neither of these two cases define 0 as zero (nothingness). Frege's definition of 0 is incorrect.

    This highlights the problem of defining 0 in general. Similar reasoning should also disprove all other attempts, such as the definition of the empty set in set theory.

    An informal proof of the general case is as follows:-

    If nothingness were a definable concept then it would be a concept, and any concept is something, at the very minimum some brain signals. If nothingness were a definable thing, then it would be something.

    Nothingness must therefore be an illusion.

    Having 0 as a placeholder in our decimal system since around 500 AD was the foundation for its general acceptance in mathematics. However, as a placeholder it does not signify nothingness and James E Foster has demonstrated a fully functional bijective decimal number system without a zero symbol. See jstor.org/stable/3029479. This could even be a better system because each number maps 1 to 1 with the unique character string representing it. With 0 this is not the case, e.g. 0023 and 023 both map to 23.

    0 cannot be nothingness. The wording 'there are 0 apples in a box' is misleading because there cannot 'be' 0 apples as there is not anything there to 'be'. 0 cannot be an amount nor a quantity nor a number. If anything it is the failure of detection of something, and that is a logical result, not a number. Maybe 0 could be defined as not(1), with interpretation '1 was undetected' rather than 'anything but 1'.


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    x-posted from here ...

    Frege's zero Lemma is not a definition of 0: Philosophy Forums

    Is this regarded as ok on this forum? Just asking.


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    (Not a moderator but) I think it's generally frowned upon just slightly, unless the poster is just copy/pasting the same thing all over the internet without actually stopping to discuss it.

    As for the content of the post, I agree with Nagase from the other forum. This mostly seems like a "I don't believe it so it can't be true" kind of argument (along with some "using my own definitions" errors). The last paragraph seems to sum all the errors up neatly.
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    Quote Originally Posted by MagiMaster View Post
    (Not a moderator but) I think it's generally frowned upon just slightly, unless the poster is just copy/pasting the same thing all over the internet without actually stopping to discuss it.
    This post mixes maths, logic and philosophy so I posted on a few different forums. It's hard to know which exactly are suitable.

    I guess the post was somewhat bloated. The important bit can be summarised as follows:-

    My communication from an expert on the Frege Lemma page I listed was as follows: "0 could be defined as the number of the *property* of being an x such that x≠x, since nothing falls under that property". In my understanding that translates to 0 = #[x such that x≠x].

    There either is such an x or there is not. The expert claims not. I claim there is one. So my argument hinges on the "x such that x≠x" part only.

    The x that satisfies x≠x is: x such that x cannot have a value.

    x is not its value. x is a name. Any value is a property of x, rather than x itself.

    If the name x cannot have a value then the condition x≠x does not apply to x since conditions compare values rather than names. Therefore x as stated remains as the one solution.

    That's all there is to it.
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    x is not a name, it's a placeholder. x as a name is completely unimportant and one of the axioms of lambda calculus (and an idea applicable to algebra in general) is that you can change a name without changing the meaning of anything as long as there are no conflicts.

    Also, if x-as-a-name cannot have a value, then exactly how many x are there? After all, you're not trying to count how many names x has. You're trying to count how many objects from the universe of discussion can be slotted in to the x placeholder and leave the expression true. "x such that x cannot have a value" is not an object from the universe of discussion. It might be an object in a different universe, but I can't think of many universes that would allow such a thing. (Maybe an empty type in type theory or something. Not really my area of expertise.)

    (Again, not a mod but) Math and formal logic both fit well under the Mathematics section, but save the philosophy for elsewhere.
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    Thanks for your reply which I found interesting and to the point.

    Also, if x-as-a-name cannot have a value, then exactly how many x are there?
    There is one x. It could have a multiple value, e.g. x such that x^2=4 is an x with values 2 or -2, it could have one value, or it could be without a value. I see no problem with the final option. The x is still there in all cases. It references (points to) a value like an object in computing does. Just because the reference is null (without a value) doesn't mean the object doesn't exist.

    If we say x=4 that is not the same as just saying 4. The x is not the 4. An object named x is created which references a value of 4. The object has a name and a value. It is a pair. It is true that you could have the same value for different objects called y or z etc, but you would still have a symbol to reference the value. So in that sense the x doesn't matter as you say, but there being a symbol does matter and so my argument remains as follows:-

    (a symbol) such that valueOf(a symbol) ≠ valueOf(a symbol) is satisfied by:-

    (a symbol) such that (a symbol) cannot have a value.

    Of course assuming that (a symbol) is the same symbol.

    You're trying to count how many objects from the universe of discussion can be slotted in to the x placeholder and leave the expression true
    Yes, and there is one as above.

    "x such that x cannot have a value" is not an object from the universe of discussion.
    It is an object as I described above. An object without a value is still an object. An object is a container. Could be empty. What Frege seems to suggest is that an empty container should vanish. The obvious response is well why use a container in the first place? Perhaps his expression x such that x≠x should have been 3 such that 3≠3 or similar.
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    You're mixing up a lot of different theories here, and in ways that aren't really meaningful.

    If you want to go the programming route, null != null is still false in most languages. Even undefined != undefined isn't usually true (although it might be undefined instead of false).

    If you want to go the predicate logic route, x really isn't a thing. It has no substance and cannot have any properties. It really is just a blank to be filled in. All meaning is attached to the objects you're putting in to the container and none to the container itself. Even higher order logic doesn't actually ascribe any meaning to the containers. It just expands the universe of discussion to allow you to put predicates into the containers.

    I'm not overly familiar with type theory, but I'm pretty sure that even an empty type wouldn't make that expression true.

    If we say x = 4, then what we're saying is go through the expression and fill all the x-shaped blanks with a 4. x is not an object in any of this. It's just an arbitrary label to keep the various different blanks separate, and those blanks are just that, blanks, with no properties of their own. Even when we say x^2 + 2 such that x > 1, that's not a property of the x-shaped-hole. It's just a compound expression with 2 x-shaped-holes to be filled.

    Now, if I'm not mistaken, there are some theoretical systems out there that do try and assign some properties to these blanks, such as specifying what kind of objects can fit in to each one, but since neither Frege's Theorem nor ZMF set theory are built on those axioms, such systems have no bearing on these definitions of zero. (You can't arbitrarily pull bits of system B in to try and prove or disprove part of system A.) And besides that, all these other systems still define a 0 or some equivalent.
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    Quote Originally Posted by MagiMaster View Post
    If you want to go the programming route, null != null is still false in most languages. Even undefined != undefined isn't usually true (although it might be undefined instead of false).
    Let me try the programming route in detail. What's going on should be clear in the physical example of a machine. Frege's 0 Lemma applies to the foundations of arithmetic and therefore to programming too.

    I used null not as the value but the reference. As in x --> reference --> physical address --> value. If the reference is null then there is not a value and not an address for it either. That is how objects and variables can work on computers.

    There is a simpler case x --> physical address --> value is when the physical address is next to where x is found in a linked list for example. In the reference case a variable y could have the same reference as x and so refer to the same physical address. In the simpler case it could not.

    In all cases there must be a physical address (PA) for a value.

    So in the reference case it is easy to ensure x cannot have a value. We can even forget x completely and use the reference (ref) itself as if it was x as follows:-

    ref such that value of PA ≠ value of PA. That is satisfied by ref = null as there is not a physical address for the value of the condition.

    It could also be satisfied by ref = "address greater than the limit of the computer" if there was objection to use of null.

    Now look at the simplest case eliminating both x and the reference, and using the physical address as if it was Frege's variable x we get the expression:-

    PA such that value of PA ≠ value of PA.

    When x is first declared as a variable, PA is set up next to it without a value. So that satisfies the expression. There is not a value, so any condition using value is inapplicable to remove this solution.

    That is all the possibilities I believe. So in all cases there is a solution for the expression.

    The simple case is the most basic interpretation of Frege and could be something interesting. The PA in the solution is initially without a value but it contains space (volume) in which to put one. Imagining this PA as the smallest possible 1 bit physical address, could Frege's zero literally be defined as an indivisible foundational volume of space? Then also could a 1 moving between fixed 0's be like an object moving in space. This may explain a lot more than I care to go on about here for fear of diverting the topic.
    Last edited by JohnMiddlemas; July 26th, 2014 at 01:57 PM.
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    Even in those cases, you don't get x != x. You might get undefined or an error, depending on your programming language, but you won't get true. valueAt(null) still equals valueAt(null) if it returns an answer at all (and no answer is not the same thing as true or false).

    Although, again, Frege's Theorem isn't proven in computer languages, it's proven in second-order logic. Nothing you say about it outside of that context has any bearing on it's validity. In second-order logic, x is not a thing and has no properties, so you can't even say "x cannot have a value." You can't say, for example, "x+4 = 8 but you can't put anything in x."
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    Quote Originally Posted by MagiMaster View Post
    You can't say, for example, "x+4 = 8 but you can't put anything in x."
    I am saying this:-

    Expression: x such that x+4=8
    Solution1: x=4
    Solution2: x without a value

    Expression: x such that x≠x
    Solution1: x without a value

    x without a value is a solution to both for the same reason. The condition cannot apply if there is not a value for it. If the condition does not apply then it can't invalidate the expression. There isn't any condition that can apply without a value.
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    x without a value does not make the expression x + 4 = 8 true; therefore it is not a solution. The condition can't apply so there is no output, especially not a true output.
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    Does this have anything to do with the use of sets to define numbers? In other words, defines the empty set which is mapped to zero?
    There are no paradoxes in relativity, just people's misunderstandings of it.
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    Yes, it's related to that. From Wikipedia:
    Frege's theorem is a metatheorem which states that the Peano axioms of arithmetic can be derived in second-order logic from Hume's principle
    The Peano axioms don't say 0 is equivalent to anything. They just include "0 is a natural number" as one of the axioms. In the set-theoretic model of the Peano axioms, 0 is equated with the empty set.

    I'm not exactly sure where the OP got the lambda definition, but I think most of the same arguments apply either way.
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    Quote Originally Posted by MagiMaster View Post
    x without a value does not make the expression x + 4 = 8 true; therefore it is not a solution. The condition can't apply so there is no output, especially not a true output.
    x without a value is the proposed solution. Since the condition does not falsify the proposal then the proposal stands. (value of x) + 4 = 8 is not false if x has not got a value. To show your view you need to show it is false.

    The condition does not have to apply for there to be an output. There already is an output, namely the proposal. That is how the expression starts, i.e. "x such that". Any x is assumed and one of the possible assumed x's is an x empty of value. It is for the condition to deny those assumptions.
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    Quote Originally Posted by KJW View Post
    Does this have anything to do with the use of sets to define numbers? In other words, defines the empty set which is mapped to zero?
    I think I agree with that. This seems similar to what I am saying that x without a value is a solution of x≠x, namely x empty of value.
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    Quote Originally Posted by JohnMiddlemas View Post
    Quote Originally Posted by MagiMaster View Post
    x without a value does not make the expression x + 4 = 8 true; therefore it is not a solution. The condition can't apply so there is no output, especially not a true output.
    x without a value is the proposed solution. Since the condition does not falsify the proposal then the proposal stands. (value of x) + 4 = 8 is not false if x has not got a value. To show your view you need to show it is false.

    The condition does not have to apply for there to be an output. There already is an output, namely the proposal. That is how the expression starts, i.e. "x such that". Any x is assumed and one of the possible assumed x's is an x empty of value. It is for the condition to deny those assumptions.
    Two problems. "x such that" does not mean any x, it means any x from the current universe of discussion. The last bit is usually left off as implied.

    And, (value of x) + 4 = 8 is not true if x has not got a value. To show your view you need to show it is true. (See what I did there?) No value is not the same as either true or false and cannot be used the way you're trying to use it.
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    Quote Originally Posted by MagiMaster View Post
    Two problems. "x such that" does not mean any x, it means any x from the current universe of discussion. The last bit is usually left off as implied.

    And, (value of x) + 4 = 8 is not true if x has not got a value. To show your view you need to show it is true.
    Thanks for your replies.

    I read "value of x" in this case as "value of that which has not got a value". That is jibberish and so (value of that which has not got a value) + 4 = 8 is not a condition anymore, just jibberish and something else. Since there now is no condition then what is there to be shown true as you say? This x without value modifies the condition to not(a condition).

    x such that condition undetected, or x such that no condition, or x such that not(a condition), are all satisfied by x without value. If an x with value is tried then the condition remains a condition.

    I guess we won't agree, but I would like your interpretation of Frege's Lemma.

    Going back to "x such that x≠x". For me, this gives:-

    0 = #[x without value].

    The definition of 0 = #[λx x≠x] in words is:-

    "0 could be defined as the number of the *property* of being an x such that x≠x, since nothing falls under that property"

    Where number refers to #, and property refers to the [].

    What is your interpretation then of 0 = #[being an x such that x≠x]? Maybe one of these.

    0 = #[nothing]
    0 = #[no instances of x]
    0 = #[]
    0 = #[non existance of x]
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    If it's not even a condition, then how can it satisfy anything?

    Can you link to the original? It would be much easier to argue about this if I had the full context.

    As for my interpretation, it looks like #[λx p(x)] could be read as the number of objects that satisfy p(x). That is, how many ways can you fill in x so that p(x) is true, or more verbosely, how many elements of the universe of discourse make this expression evaluate to true. For example, #[λx x=1] would be 1. There's one way to fill in x to make that expression return true. #[λx x∈(1,5)] would be 3 if we're talking about the integers, or if we're talking about the real numbers. ((1,5) is the set of numbers between 1 and 5, not including 1 or 5. is the cardinality of the real numbers.)

    In that case, #[λx x≠x] would be the number of ways of assigning x so that whatever object you chose doesn't equal itself. This would be closest to 0 = #[nothing], but might be better written as 0 = #[false] since the #[] notation seems to be set up to count truths. (By itself, λx x≠x is a valid lambda expression that always evaluates to false.)
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    Sorry about the posting delay it's a tricky topic.

    Quote Originally Posted by MagiMaster View Post
    If it's not even a condition, then how can it satisfy anything?
    I guess I'm saying (x without value) : no condition --> (x without value) & no condition --> (x without value) hence the expression is satisfied. However, since I did more on this (please see below) I also now think x without value may represent a location without value.

    Can you link to the original? It would be much easier to argue about this if I had the full context.
    The original place I investigated is the Stanford page. There is a book page here which is more directly about Frege.

    On the book page it says "Frege defines zero as the number of objects which are not self identical. 0 = Nx : x≠x".

    As for my interpretation, it looks like #[λx p(x)] could be read as the number of objects that satisfy p(x). That is, how many ways can you fill in x so that p(x) is true, or more verbosely, how many elements of the universe of discourse make this expression evaluate to true. For example, #[λx x=1] would be 1. There's one way to fill in x to make that expression return true. #[λx x∈(1,5)] would be 3 if we're talking about the integers, or if we're talking about the real numbers. ((1,5) is the set of numbers between 1 and 5, not including 1 or 5. is the cardinality of the real numbers.)

    In that case, #[λx x≠x] would be the number of ways of assigning x so that whatever object you chose doesn't equal itself. This would be closest to 0 = #[nothing], but might be better written as 0 = #[false] since the #[] notation seems to be set up to count truths. (By itself, λx x≠x is a valid lambda expression that always evaluates to false.)
    How can you define the number zero as the number of ways of assigning x? It's a circular argument from the start. It uses 'number' in the definition of a number. However this is a side issue.

    Another side issue: If we say 0=#[false] is the interpretation of #[λx x≠x] then λx x≠x is false under all possibilities and isn't λx x≠x a false expression itself, namely a contradiction? It's not just that x≠x is a contradiction, which it is if x is taken to be any same number on both sides of the inequality, e.g. 2≠2, but also that x≠x makes the whole lambda expression a contradiction too in this case, i.e. 2 such that 2≠2 --> (2 & 2≠2) --> (2 & !2). So depending on what 'x' means, in some meanings of 'x', the interpretation of 0=#[λx x≠x] is definitely 0=#[contradiction]. It is the number of objects in a contradiction. It is like saying the number of objects in (A & !A) is 0, therefore 0=#[0]. That is completely circular. Not a definition. Also we cannot use "number of" in a definition of a number, nor even the [] either has any meaning here. Frege's definition reduces to 0=0. Circular.

    "no objects" or "nothing" cannot be located with a symbol (nor by {}) because they haven't got any location. All we are talking about with 0 and {} are different ways of writing the same contradiction. This can only confuse the system. A contradiction cannot be a number, and the number of a contradiction is a contradiction. Also the number of the property of a contradiction (#[contradiction]) is also a contradiction.

    I see the main issue as being what is the meaning of 'x' in:-

    A) x such that x≠x

    This appears to be the source of the problem. It can be taken to have multiple meanings.

    The meaning of x could be:-

    1) x without value
    2) varying value replacing x
    3) x with varying value

    These meanings could be mixed across the three occurrences of x in A) so there would be 3 x 3 x 3 = 27 possible views! Below, I only consider all occurrences having the same meaning. The usual view is all occurrences with meaning 3). Above here we have been considering the mixed views (x without value) such that (x with value≠x with value) or (x without value) such that (value replacing x≠value replacing x).

    Let the value be any number n. I don't think we lose generality by doing that and it makes things clearer.

    Meaning 1). x is just a symbol, 'x' : 'x'≠'x'. Variability is not possible. --> 'x' & ('x'≠'x') --> 'x' & !'x'. The meaning is contradiction.

    Meaning 2). x is taken to be replaced by n, e.g. n : n≠n. Now this could mean n in three different locations in which case say 3 : 4≠7 will satisfy the expression. The numbers are not tied to one location and can vary independently. In the case of say 3 : 3≠3 we have the same contradiction as for meaning 1). So all cases either satisfy the expression or produce a contradiction. In no case do you get nothing or non-existence.

    Meaning 3).
    This is the majority view when applied to all occurrences. In this view the expression A) becomes:-

    B) x,n such that x,n ≠ x,n

    where x,n is substituted for x in A) "x = x,n". The "variable" is a pair x,n. It is clearly confusing since the x is referred to by itself. This leads to continual regress, i.e. x = x,n,n,n,n,n,n,n....... or x = .......n,n,n,n,n,n,n,x if you write it x = n,x.

    So the "x = x,n" must be rethought. To fix this the right hand side x can be thought of as a fixed location, say L, and the left hand side x as an object. So B) becomes:-

    C) L,n such that L,n ≠ L,n

    L being the same fixed location ensures all three varying n's are the same varying n's. In this view, x is thought of as a variable number at a fixed location. If we used y instead of x then C) would still be the same format. It is the fixing by the location L which is important, not the x or y or even the L, just the fixing. All possibilities here are like these:-

    L,2 such that L,2 ≠ L,2
    L,3 such that L,3 ≠ L,3
    .
    .

    where the 2's are the same 2 since they are fixed at the same location. So the first reduces to "2 such that 2 ≠ 2" which is the same contradiction as for meaning 1) above.

    In all cases this meaning 3) therefore resolves to a contradiction and this meaning is what the majority associate with an object or variable in logic or mathematics I believe.

    In none of the meanings 1) to 3) does the expression resolve to "nothing" or "non-existence", only to either a contradiction or a satisfaction of the expression.

    I have checked many of the other 26 views and found the same. It's clear they all will be.

    Frege's Lemma resolves to one of:-

    0=#[contradiction] which just says 0 is a contradiction, which is exactly what I am saying. You can't define 0 unless you define something.
    0=#[satisfied]

    It may be that a nice definition of zero is among those satisfied ones since x≠x does seem to give an expression with maybe the highest possible restriction. I suspect 0 may best be defined as an unoccupied location similar to my x without value case. A location not required to exist in our realm but which could exist if occupied at a later time by a value. The location would be number(s), coordinates maybe, but without further properties. The consequences are drastic however for it means that there are different 0's for every such different location. 0 + 0 ≠ 0 anymore. Instead it means 0L1 + 0L2 which means (found L1 unoccupied) + (found L2 unoccupied). Not sure if that could have a meaning under addition.
    Last edited by JohnMiddlemas; August 5th, 2014 at 02:24 AM.
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    There are a couple problems here. (λz z≠z) is not a contradiction in and of itself. You can equate "false" and "contradiction" (I think at least some systems do use contradiction as the definition of falsehood) but (λz false) is still perfectly valid. Lambdas are formalizations of functions, so in C++, this would look like:
    Code:
    bool someFunction(object x) { return false; }
    Or if you consider a contradiction as an error:
    Code:
    bool someFunction(object x) { throw ContradictionError; }
    There's nothing at all wrong with either function. And if you equate falsehood and contradictions, then truth would equate with satisfyability and you're back at the definition that #[λz ...] is the number of objects in the universe of discourse that satisfies the expression.

    Another thing, in your meaning 2, all you've done is renamed x to n. Pretending n is some specific number is exactly the same as pretending x is some specific number. That's pretty much the definition of a variable. (Although pretending it's a number does restrict the universe of discourse to numbers.)

    Also, I still don't agree that "no value" satisfies anything. "Satisfied" means that you fully reduce the expression and don't get a contradiction in the end. If you don't give a value to a variable, it's an expression and isn't yet satisfied. You could even take #[λx p(x)] to mean "how many ways can you reduce this expression without generating a contradiction," in which case the obvious result of #[λx x≠x] would be "none."

    However, I do agree that "the number of these things is the definition of the number 0" would be a little bit circular. I doubt the full, rigorous definition used is though. (You tend to lose a lot of subtlety when you convert mathematical expressions to natural languages.) Besides that, most numerical systems include something like "0 is a number" as an axiom, which means that it does not follow from anything and is taken as a given, which would break any circularity.
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    Quote Originally Posted by MagiMaster View Post
    There are a couple problems here. (λz z≠z) is not a contradiction in and of itself. You can equate "false" and "contradiction" (I think at least some systems do use contradiction as the definition of falsehood) but (λz false) is still perfectly valid. Lambdas are formalizations of functions, so in C++, this would look like:
    Code:
    bool someFunction(object x) { return false; }
    An expression is a contradiction is when all possibilities for the expression are individually contradictions.

    I'm not saying (λz z≠z) is a contradiction but that it is either a contradiction or can be satisfied. It depends on your interpretation of the three z's.

    In the common interpretation though it is a contradiction, because all possibilities are individually contradictions. In the common interpretation the three z's are considered to be in the same location at the same time, thereby supposedly eliminating location and time from consideration. However, this means that each of the three z's is exactly the same z in the same location at the same time. This means all possibilities for the expression are of the form λ2 2≠2 or 2 such that 2≠2 where the 2's are exactly the same identical 2. This implies 2 & 2≠2 which implies 2 & !2 which is a contradiction.

    If you want to interpret the 2's as being in different locations then one possibility is 2L1 such that 2L2 ≠ 2L3 which is satisfied by 2L1 because 2L2 is not self identical to 2L3 having a different location. So if location is considered you can get a solution but if it is ignored you always get a contradiction.

    I would be happy if you could address these points about location since it is the crux of the matter. Aristotle in his definition of contradiction was forced to involve time and so logic is not separate from considerations of reality. Even location and time are required in all abstract thinking because thinking itself requires them.

    There is no such concept as "nothing", only the finding of an unoccupied location.

    Or if you consider a contradiction as an error:
    Code:
    bool someFunction(object x) { throw ContradictionError; }
    There's nothing at all wrong with either function. And if you equate falsehood and contradictions, then truth would equate with satisfyability and you're back at the definition that #[λz ...] is the number of objects in the universe of discourse that satisfies the expression.

    Another thing, in your meaning 2, all you've done is renamed x to n. Pretending n is some specific number is exactly the same as pretending x is some specific number. That's pretty much the definition of a variable. (Although pretending it's a number does restrict the universe of discourse to numbers.)

    However, I do agree that "the number of these things is the definition of the number 0" would be a little bit circular. I doubt the full, rigorous definition used is though. (You tend to lose a lot of subtlety when you convert mathematical expressions to natural languages.) Besides that, most numerical systems include something like "0 is a number" as an axiom, which means that it does not follow from anything and is taken as a given, which would break any circularity.
    In the meaning 2) I could have used v for value rather than n for number. Using numbers instead of values seems sharper. Couldn't any value be represented by a number anyway? I am examining the case of values without an x as compared to the case 1) of x without value. Seems a valid approach. Like 1) x alone, 2) v alone, 3) both x and v.

    Also, I still don't agree that "no value" satisfies anything. "Satisfied" means that you fully reduce the expression and don't get a contradiction in the end. If you don't give a value to a variable, it's an expression and isn't yet satisfied. You could even take #[λx p(x)] to mean "how many ways can you reduce this expression without generating a contradiction," in which case the obvious result of #[λx x≠x] would be "none."
    But it isn't "none", it's all reductions of #[λx x≠x] (in the common view) generate contradictions therefore the expression is a contradiction (under that view). There is no such answer as "none".

    Either you are saying that all the x's are the same identical x or you are not. If I say all the x's refer to different locations I can get solutions like 3 such that 4≠5. Just because the x's have the same symbol doesn't necessarily mean they refer to the same location. Each x can have a different varying value at its different location. That is easy to imagine. On the other hand, if the x's do refer to the same location, i.e. supposedly ignoring location (as you assume), then you get a contradiction in all cases.

    If there is a value then you need somewhere to put it. Either in your mind as a concept, on the paper as a symbol, in a computer, or as an object in reality. All of these examples of a value have location. You can't have a value without a location for it. Actually you can't have anything at all without a location for it. If there was a value without location where would it be?
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    Lambda expressions have rigid rules, one of which is that all x's get replaced together. If you're going to ignore that rule, then you're working in a different system than the proof we're discussing and none of your conclusions will really apply.

    So, if we ignore that possibility, (λz z≠z) is still not a contradiction in and of itself. It is a lambda expression, which is a specific thing, again, with well-defined rules. Now, when we humans look at this particular lambda expression it's easy for us to see that no matter what object we try to reduce it with, we'll always arrive at contradiction, but that's not part of the actual rules of the lambda expressions themselves, so no, this isn't directly equivalent to contradiction. And again, "contradiction" can be equated with "falsehood" so even if it were, that doesn't actually invalidate its use. (After all, we still use contradictions, such as in proof by contradiction.)

    Speaking of rules, (λz z≠z)(2) does not imply 2. And 2≠2 does not imply !2. (λz z≠z)(2) reduces to 2≠2, which reduces to false.

    Also, if all reductions of a lambda expression generate a contradiction, can you tell me exactly how many reductions satisfy the expression? I mean, you have to be able to answer that question somehow. The question is meaningful and well-posed, so mu is not a valid answer here. (I'm not sure well-posed is the word I'm looking for, but I think you'll still get my point.)
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    Quote Originally Posted by MagiMaster View Post
    Lambda expressions have rigid rules, one of which is that all x's get replaced together. If you're going to ignore that rule, then you're working in a different system than the proof we're discussing and none of your conclusions will really apply.
    I presume you are referring to the view where all the x's can vary independently e.g. 4 such that 3≠5. I agree this is not the normal view but nonetheless as written λz z≠z is ambiguous without that rule you mention. Frege didn't use Lambda calculus as far as I know, that was put in by the Stanford people I think. From the book reference I gave, Frege said Nx : x≠x. The number of x such that x≠x. He likely used the rule you mention too. If he didn't then his definition has solutions. If he did then his definition is a contradiction.

    So, if we ignore that possibility, (λz z≠z) is still not a contradiction in and of itself. It is a lambda expression, which is a specific thing, again, with well-defined rules. Now, when we humans look at this particular lambda expression it's easy for us to see that no matter what object we try to reduce it with, we'll always arrive at contradiction, but that's not part of the actual rules of the lambda expressions themselves, so no, this isn't directly equivalent to contradiction. And again, "contradiction" can be equated with "falsehood" so even if it were, that doesn't actually invalidate its use. (After all, we still use contradictions, such as in proof by contradiction.)
    It doesn't matter if you call the result of each instance of z false or a contradiction. A statement is also a contradiction if it entails the false, i.e. every instance of the arguments reduces to false. So (λz z≠z) is a contradiction.

    The reductions of an expression are either 1) all false, or 2) at least one is true. If all false then the expression reduces to a contradiction. To say the expression has "no true solutions" or "the solutions are none" is simply to say case 1) is a contradiction. In case 2) the number of solutions is at least one. For case 1) all you can say is 0 is "the number of (contradiction)". Looks like that reduces to "0 is contradiction". But for case 2) we can say 2 is "the number of (x such that x^2=4).

    To say that "0 is the number of (x such that x≠x) which is none" requires a definition of "none" first. Defining "none" is equivalent to defining 0. So the saying is really "0 is the number of (x such that x≠x) which is 0" which is circular. We use many different words for 0 and this is part of the problem.

    Speaking of rules, (λz z≠z)(2) does not imply 2. And 2≠2 does not imply !2. (λz z≠z)(2) reduces to 2≠2, which reduces to false.
    If you have 2≠2 there are two cases:-

    1) The 2's are the same 2 at the same time. Then the 2 must be there and not there at the same time. This means 2 & !2 at the same time which is a contradiction.
    2) The 2's are different 2's. Maybe different locations. Maybe different times. The inequality evaluates true.

    Since you agree the inequality reduces to false then case 2) is ruled out. So the inequality is false, and a contradiction.

    Also, if all reductions of a lambda expression generate a contradiction, can you tell me exactly how many reductions satisfy the expression? I mean, you have to be able to answer that question somehow. The question is meaningful and well-posed, so mu is not a valid answer here. (I'm not sure well-posed is the word I'm looking for, but I think you'll still get my point.)
    How many reductions satisfy the expression?

    Well put. That is the crux of it.

    Since all reductions are false as you agree, then the expression is a contradiction. So it must be rewritten:-

    How many reductions satisfy the contradiction?

    Since all reductions are false then they all satisfy the contradiction.

    If I were to answer "none" I would be saying 0, but 0 is "the number of contradiction" as defined by Frege, so "none" is too.

    So my answer is "the number of contradiction, or, the number of all false reductions of the contradiction".

    Aside
    I am a programmer and have studied Mathematics. Some years ago I noticed there were many things in programming and elsewhere used for "nothingness". There are 0, undefinable, undeclared, "", null, false, empty set {}, contradiction, zero, nothing, none, no, non-existence, and on it goes. Why so many ways of stating the same nothingness? Then there is the word "not" which is somewhat special as it is used in the definition of contradiction i.e. A & not(A) being the same A at the same time in the same sense.

    In addition I found many of those to be a source of bugs in programming. Often traps had to be set for them which was extra work and they didn't seem to be of any required use. You could always program some better way without them. When I looked into it I saw the whole theory of "nothing" was flawed. In particular 0 was treated the same as undefined. So was it undefinable? If definable then it is something and nothing at the same time. That looked wrong. Hence this post. When I checked the foundational definition of 0 by Frege, I saw it was not a definition of 0.

    You look at A & not(A) and there can't be anything there. The same concept as 0. Nothingness is a contradiction.
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    You keep using contradiction to mean error. It's not. An error would be something that does not follow logically like Some dogs are brown and some animals are dogs therefore all animals are brown. That is not a contradiction, that's a logical error. A contradiction is the same as a function returning false. It's something we can continue to work with.

    0, null, false, contradiction, "", {}, etc. are all basically 0 but with different types. int x = 0, int* x = 0, set x = 0, string x = 0, etc. The errors are from C++ treating the type of 0 as indeterminate (something that's changed in C++11 with nullptr).

    The definition of zero in this case doesn't even seem to be that circular to me. The way you keep stating it is, but I doubt that's the way it was originally stated. If you say "zero is defined to be the number of ways of satisfying a contradiction" there's no circularity there. It relies on there being a concept of numbers in general, but not zero in specific, which is fine.

    1) The 2's are the same 2 at the same time. Then the 2 must be there and not there at the same time. This means 2 & !2 at the same time which is a contradiction.
    This doesn't make a whole lot of sense. 2 is 2. There is no other 2. That's what 2=2 means. And !2 doesn't come in to this anywhere. !2 doesn't even really make sense in logic since it's an atom not a preposition. I'm sure it can be given a rigorous definition, but I don't think standard logic does. "Doesn't equal" is not the same as "equals not." "Doesn't equal" just answers the question is the left object difference from the right object?
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    You agreed that every reduction of the expression (x such that x≠x) is false. This entails that the expression is a contradiction. That means Frege's definition is that 0 is the number of (contradiction), which means that 0 is a contradiction.

    Do you agree that?
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    0 as a non-sequitur is prevalence in theory, and genius in execution.

    (0 1 is what conforms into Chaos Theory)
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    Please stop posting rubbish.
    "[Dywyddyr] makes a grumpy bastard like me seem like a happy go lucky scamp" - PhDemon
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    Is read writ rubbish?

    Key -

    Before 0, in logic, is xy - a mathematical recursive numerical element, used as a system setup in Game Theory.

    Chaos Theory expands into what underlies the system - a qual state, used for the magnification of the sub-atomic in a numerical equation.
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    More rubbish.
    Please stop it.
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    Quote Originally Posted by JohnMiddlemas View Post
    You agreed that every reduction of the expression (x such that x≠x) is false. This entails that the expression is a contradiction. That means Frege's definition is that 0 is the number of (contradiction), which means that 0 is a contradiction.

    Do you agree that?
    I'm not sure which meaning of contradiction you're using. Can you define it?

    Also, to make sure we're on the same page, are you trying to work this out in lambda calculus, or in second-order logic? (The original proof seems to be in second-order logic.)
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    Only as a semantic expression is 0 a contradictive element, in mathematical analysis, 0 is a polynomeric fault, akin to expressing 1.

    Maybe I enjoy writing in duck.
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    Quote Originally Posted by MagiMaster View Post
    Quote Originally Posted by JohnMiddlemas View Post
    You agreed that every reduction of the expression (x such that x≠x) is false. This entails that the expression is a contradiction. That means Frege's definition is that 0 is the number of (contradiction), which means that 0 is a contradiction.

    Do you agree that?
    I'm not sure which meaning of contradiction you're using. Can you define it?

    Also, to make sure we're on the same page, are you trying to work this out in lambda calculus, or in second-order logic? (The original proof seems to be in second-order logic.)
    Frege used predicate logic a.k.a first order logic, where a proposition is defined to be a contradiction if and only if the proposition entails false. Entailing false means that all variable assignments to the proposition produce false, which they do in this case.
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    Predicate logic isn't exactly the same as first-order logic. From further down that page...

    Quote Originally Posted by Wikipedia
    Frege's logic, now known as second-order logic
    Anyway, I don't agree that (the number of [false/contradiction]) implies (contradiction). "The number of" isn't a predicate and doesn't return a truth value.
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    Quote Originally Posted by MagiMaster View Post
    Predicate logic isn't exactly the same as first-order logic. From further down that page...

    Quote Originally Posted by Wikipedia
    Frege's logic, now known as second-order logic
    Anyway, I don't agree that (the number of [false/contradiction]) implies (contradiction). "The number of" isn't a predicate and doesn't return a truth value.
    Do you agree that Frege's definition is as the number of (contradiction)?
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    Yeah, I think that basically works, although I'd write it as "the number of ways to satisfy (false/contradiction)." I'm not sure whether or not you can really say that about the lambda version, but from what I can tell, the original definition wasn't in terms of a lambda expression. (It's a subtle difference that I don't think has any effect on the proof as a whole.)
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    Quote Originally Posted by MagiMaster View Post
    Yeah, I think that basically works, although I'd write it as "the number of ways to satisfy (false/contradiction)." I'm not sure whether or not you can really say that about the lambda version, but from what I can tell, the original definition wasn't in terms of a lambda expression. (It's a subtle difference that I don't think has any effect on the proof as a whole.)
    Hey, we're getting somewhere.

    Just the last bit now.

    So we have the following:-

    Let n be the number of ways to satisfy (false/contradiction)

    By Frege, 0 is n

    By you, n is "none"

    Eliminating the n gives "0 is none". This is what you believe.

    Only one problem. The n wasn't used to get that result since it was eliminated. That means the Lambda expression or any other version of it wasn't used either.

    You base your belief in Frege's definition on your own belief in "none" and your interpretation of the expression. There is nothing mathematical in that at all. It's all just your subjective belief.

    Frege could equally have said n is the number of ways you can twist an elephant in two, or anything else that looks impossible.

    Then there is another significant problem. In order to have your belief you are assuming a definition of "number". But no such definition has been given. It is what you assume a number to be. But such assumption is not the basis of a serious mathematical definition. Frege's definition is also ruled out for that reason too. 0 is a number he says but without defining what a number is, and even if he did define what a number is he would have had to define the number 0 elsewhere previously to be able to use the word number. Having defined 0 previously then why is he defining it again in his Lemma? All numbers are defined using 0 as basis so you can't use the word "number" in a definition of the basis of numbers! That is infinite regress or circularity. With the empty set that's how they define numbers too. 0={}, 1={{}}, 2={{}, {{}}}, etc. This means the empty set is equally an impossible idea as 0.
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    I think you need to look up what an axiom is. That there are things called numbers is an axiom. That 0 is a number is also an axiom. (And none is a synonym for 0.) And yes, the number of ways of satisfying (any contradictory expression) is 0 no matter which contradictory expression you use.
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    Quote Originally Posted by MagiMaster View Post
    I think you need to look up what an axiom is. That there are things called numbers is an axiom. That 0 is a number is also an axiom. (And none is a synonym for 0.) And yes, the number of ways of satisfying (any contradictory expression) is 0 no matter which contradictory expression you use.
    I know what an axiom is thanks. It is an assumption without proof.

    Frege is trying to define zero. Why would he try that if numbers were already assumed by axioms? This post is about the definition of zero rather than any axioms (assumptions). That is why Frege attempts definition using that expression. The zero is required for later definitions of the other numbers. In set theory for example 0={}, 1={{}}, 2={{},{{}}}, etc. The numbers are defined using the empty set which is defined using a similar expression to what Frege used.

    You use the symbol 0 but are unaware what it means. If it was a number then it would be a number of something. The number of nothing is nothing, not a number. A number refers to real things, 1 apple for example.

    A contradictory expression is an absurdity and that is its only property. To imagine any number associated with absurdity is also absurd. I can equally say that a contradictory expression is satisfied by all the false solutions because those are indeed those which give birth to the contradiction in the first place so they must satisfy their mother. In no case does a zero come into this.

    In all of this you have shown only that you fail to understand 0, provided little of any merit, and persist in your irrational belief in zero despite being shown that it is a contradiction.

    Carry on believing in contradictions but be aware that those responsible for supporting contradictions in science share the blame.
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    John - rather than insulting members, I suggest you revise some axiomatic set theory and how these can be used to construct the ordinal numbers. I can show it in gory detail if you want - it really is very straightforward.

    For example, the empty set follows trivially from the set theoretic axiom of specification, and the ordinals follow from the definition of the set function called the successor function.

    It only remains to simplify notation (a valid re-labeling exercise) and so on

    As you easily see, an ordinal number contains all its predecessors as subsets which clearly excludes the empty set (alias zero) as an ordinal umber - exactly as expected.

    So your point is obscure to say the least
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    Quote Originally Posted by JohnMiddlemas View Post
    Quote Originally Posted by MagiMaster View Post
    I think you need to look up what an axiom is. That there are things called numbers is an axiom. That 0 is a number is also an axiom. (And none is a synonym for 0.) And yes, the number of ways of satisfying (any contradictory expression) is 0 no matter which contradictory expression you use.
    I know what an axiom is thanks. It is an assumption without proof.

    Frege is trying to define zero. Why would he try that if numbers were already assumed by axioms? This post is about the definition of zero rather than any axioms (assumptions). That is why Frege attempts definition using that expression. The zero is required for later definitions of the other numbers. In set theory for example 0={}, 1={{}}, 2={{},{{}}}, etc. The numbers are defined using the empty set which is defined using a similar expression to what Frege used.

    You use the symbol 0 but are unaware what it means. If it was a number then it would be a number of something. The number of nothing is nothing, not a number. A number refers to real things, 1 apple for example.

    A contradictory expression is an absurdity and that is its only property. To imagine any number associated with absurdity is also absurd. I can equally say that a contradictory expression is satisfied by all the false solutions because those are indeed those which give birth to the contradiction in the first place so they must satisfy their mother. In no case does a zero come into this.

    In all of this you have shown only that you fail to understand 0, provided little of any merit, and persist in your irrational belief in zero despite being shown that it is a contradiction.

    Carry on believing in contradictions but be aware that those responsible for supporting contradictions in science share the blame.
    You seem to have too many ingrained misconceptions here for me to really sort through them all, and based on past experience it wouldn't really do any good anyway, but there are two points here that I think are worth mentioning.

    A contradiction is not an absurdity. The only way to get an absurdity in math is to not follow the rules, and contradictions do follow the rules. The closest thing to an absurdity would be assuming a contradiction is true, though that would still fall under not following the rules.

    Second, zero is as real as any pure number. Show me a one. Not one apple, or one dollar, just a one. And zero is just as useful as any other number in the real world. How many apples do you have in your hand right now? How much money would you have if you payed off your debt with your last cent? What's the voltage across a dead battery?

    I'll also say that "numbers are a thing" and "here are all the numbers" are two very different axioms.
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    Quote Originally Posted by Guitarist View Post
    John - rather than insulting members, I suggest you revise some axiomatic set theory and how these can be used to construct the ordinal numbers. I can show it in gory detail if you want - it really is very straightforward.

    For example, the empty set follows trivially from the set theoretic axiom of specification, and the ordinals follow from the definition of the set function called the successor function.

    It only remains to simplify notation (a valid re-labeling exercise) and so on

    As you easily see, an ordinal number contains all its predecessors as subsets which clearly excludes the empty set (alias zero) as an ordinal umber - exactly as expected.

    So your point is obscure to say the least
    Apologies for that, I was getting a little terse. It's because this topic is so frustrating. To me it is plain and concrete that the idea of zero is a mistake. This is for many reasons over a long period and I will never change my view. So it is frustrating to see so many intelligent people accept this idea and stick to it. It's like I live in my world and they live in theirs. No matter what anybody says nor with justification by set theory nor ordinal numbers, will detract from the simple observation that nothing is simply nothing and so cannot be used for anything. You can give it a symbol 0 if you like but all that is is a symbol. As soon as you say 1+0=1 then you try to give meaning to the meaningless 0. You introduce an absurdity to the system, like a virus. That's how I see it. The only thing that 1+0 equals is 1+0. How can a thing like 0 just vanish. It's absurd. Things don't do that. Nor can 0 ever be defined otherwise it would be something rather than nothing. That is how I know that Frege's definition of 0 is wrong. You cannot locate the unlocatable nothing with a symbol 0. Unlocatable means just that but the symbol locates nothing right where the symbol is written.

    In one version of set theory they use the axiom of the empty set:-



    Axioms as I understand them are supposed to be sensible but I claim that x here is a contradiction and so not an empty set nor even a set at all, just a contradiction. Any axiom or definition regarding nothing must be a contradiction. What does it say in words:-

    There exists a set x such that for all y it is not the case that y is a member of x.

    Therefore for every single y possible it is false that y is a member of x. Since all possibilities contained in x are false then by the principle of entailment this requires that x is the set of contradiction.

    x = {contradiction} = {absurdity}

    There is nothing "empty" about x at all, it is chock-a-block full of all those failed y's which gave rise to and satisfy the absurd contradiction.

    The set of absurdity is absurdity so x itself is absurdity. x=absurdity.

    This is virtually the identical same disproof as for Frege's 0 Lemma.
    Last edited by JohnMiddlemas; August 16th, 2014 at 02:28 AM.
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    Quote Originally Posted by JohnMiddlemas
    To me it is plain and concrete that the idea of zero is a mistake. This is for many reasons over a long period and I will never change my view. So it is frustrating to see so many intelligent people accept this idea and stick to it.
    Why are you even bothering to post here then? What do you think your chances are of convincing us that the last three to four thousand years of mathematics have been wrong on such a fundamental issue?

    Answer this one simple question: If you have a $1000 loan (short term, no interest if you pay it back fast enough), and you take your last $1000 dollars to the bank, how much money do you have when you pay off the loan?
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    Quote Originally Posted by MagiMaster View Post
    You seem to have too many ingrained misconceptions here for me to really sort through them all, and based on past experience it wouldn't really do any good anyway, but there are two points here that I think are worth mentioning.

    A contradiction is not an absurdity. The only way to get an absurdity in math is to not follow the rules, and contradictions do follow the rules. The closest thing to an absurdity would be assuming a contradiction is true, though that would still fall under not following the rules.

    Second, zero is as real as any pure number. Show me a one. Not one apple, or one dollar, just a one. And zero is just as useful as any other number in the real world. How many apples do you have in your hand right now? How much money would you have if you payed off your debt with your last cent? What's the voltage across a dead battery?

    I'll also say that "numbers are a thing" and "here are all the numbers" are two very different axioms.
    I apologise for being somewhat critical in the prior post. However, for you to now say that I have misconceptions is also a criticism I object to because I am very sure of what I am saying having researched it extensively. It is better for both of us not to criticise at all and just stick to issues.

    The issue of contradiction is a very important one. More important probably than the zero. It is "One cannot say of something that it is and that it is not in the same respect and at the same time."

    That could be written (A & !A) with the understanding that the A's are the same A at the same time.

    It is perfectly possible that (A & !A) is a contradiction and an absurdity too. For example if it can be shown that any one of 'A', '&', or '!' is an absurdity or meaningless then contradiction would be too.

    Now the symbol ! is 'not'. The meaning of this word is highly debatable. A whole book has been written on the topic, called "A Natural History of Negation". 'Not' could be an absurdity (like zero is).

    Axioms or definitions can be written in the theory about these meanings and of course in the custom of logic and mathematics any axiom at all can be made up without proof. However, if an axiom is self contradictory or absurd then that is poor practice. One absurd axiom surely leads to an absurd theory in total.

    What then does the '&' mean? This is the intersection or overlap of two sets. If A={1, 2, 3} and B={2, 3, 4} A & B is {2, 3}. This is an immediately dubious concept to me since the 2 and 3 in A are different from the 2 and 3 in B. They are 4 different objects because they are placed in different sets. To say one thing is in two places at the same time is absurd. Yet that is exactly what is being done here. The '3' is in two places at the same time, namely set A and set B.

    Therefore the concept of intersection (&) as defined by set theory is an absurd concept.

    Therefore the concept of contradiction is also absurd since it contains an absurdity.

    It's not just the zero that's wrong, it's logic too. There is no such thing as contradiction because there is not such thing as intersection. The & is just the beginning. The 'not' is more tricky.

    Second, zero is as real as any pure number. Show me a one. Not one apple, or one dollar, just a one.
    Great question! Where did the idea of 1,2,3 come from? Counting fingers probably. Counting real things. There never was pure number in the beginning. Number always came with object(s) and never zero because you can't count zero. They (the Indians) stuck the zero in much much later.

    Show you a pure one without a thing.

    This goes back to foundation of reality. What do we know for certain certain? Is it "I think therefore I am" via Descartes? But what is the I? I prefer something simpler:-

    Certainty: "One indivisible thing exists"

    That certainty is probably the only one that really can't be doubted.

    If it was divisible then it would be more than 1 thing.

    We start the scientific system of things with one indivisible thing, one certainty, one foundation. There's your 1 but it's still related to a thing. The smallest piece of reality possible. Imagine it has a length, then the length is 1. If it has a volume then that volume is the 1 smallest volume possible. All other volumes are multiples of the 1. All distances are positive integer multiples of the 1 smallest distance. Bigger things are made from lots of the foundational thing. Now you say but it's still a thing so it's not a pure number. Hang on though maybe there are no things at all, maybe we are in a computer simulation. In which case the 1 "thing" is just a number in the heavenly computer and we are all made of numbers. That should be pure enough.

    Anyway the main point here is the zero. You are trying to claim zero and 1 are equally pure abstract numbers but there is a very big difference:-

    I have zero apples This is nonsense. You cannot "have" zero apples since there's nothing there!
    I have one apple This is possible

    The abstract zero cannot refer to an object under any circumstances but the abstract 1 can.

    The 1 has the possibility to represent an object. A number must be able to represent objects but the zero represents only illusion. It is not a number.

    If you have no apples then where are the apples you don't have, and why have you mentioned apples at all since you have none.

    How much money would you have if you payed off your debt with your last cent?
    I am parted from money at present.

    What's the voltage across a dead battery?
    The battery is parted from voltage at present.
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    Quote Originally Posted by MagiMaster View Post
    Quote Originally Posted by JohnMiddlemas
    To me it is plain and concrete that the idea of zero is a mistake. This is for many reasons over a long period and I will never change my view. So it is frustrating to see so many intelligent people accept this idea and stick to it.
    Why are you even bothering to post here then? What do you think your chances are of convincing us that the last three to four thousand years of mathematics have been wrong on such a fundamental issue?

    Answer this one simple question: If you have a $1000 loan (short term, no interest if you pay it back fast enough), and you take your last $1000 dollars to the bank, how much money do you have when you pay off the loan?
    Better to try than to sit on it. What if I was right! Actually the zero is a latecomer, started around 500 AD I believe.

    My bank account is parted from money. It is not necessary to mention an amount in the case of zero since there is no amount. Your bank statement could just show blank, if there's nothing there why put a number to it, there's no point.
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    Quote Originally Posted by JohnMiddlemas View Post

    To me it is plain and concrete that the idea of zero is a mistake. This is for many reasons over a long period and I will never change my view.
    Then there is no discussion to be had, right?

    John, for this reason I am a finger away from locking this thread - let us know if you want a discussion or a just personal rant

    PS Your grasp of mathematics is tenuous to say the least. I suggest you learn the utility of zero in defining the negative integers, as the simplest possible example, though I could cite many others. While you're at it you may come across the perfectly sound assertion that the empty set contains as elements ALL those objects which have no possibility of existing - like square circles, say.
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    Quote Originally Posted by JohnMiddlemas View Post
    My bank account is parted from money. It is not necessary to mention an amount in the case of zero since there is no amount. Your bank statement could just show blank, if there's nothing there why put a number to it, there's no point.
    This is a silly argument. In particular...

    Quote Originally Posted by JohnMiddlemas View Post
    Your bank statement could just show blank
    You are in effect arguing over the shape of the "0" character. Whether the statement show a "" (a whitespace or blank character) or a "0" doesn't alter that the balance is numerically zero.
    There are no paradoxes in relativity, just people's misunderstandings of it.
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    Yeah well, the way bank balances are written is entirely arbitrary, as is almost any other "everyday" application of zero AS IT IS WRITTEN.

    The following is not a strict proof, though it would be if I could be bothered to prove the background...........

    Consider the integers and let me assume that, as a set, it is a total order - that is one and only only one of the following must be true







    Then the subset - the positive integers - is the set of all upper bounds of the subset - the negative integers, which are in turn the set of all lower bounds of the set

    But not only is a total order, it is also a well-order - that is every (non-empty) subset has a least element.

    Suppose the least element . This is then a lower bound for which implies

    Therefore is both a positive integer and a negative integer or, what operationally amounts to the same thing, neither positive nor negative (as you may easily check using grade arithmetic) It is a matter of notational convention ONLY to set this particular

    Now let me insist that and are truly disjoint - that is . Since is neither positive nor negative i .e. it seems perfectly natural to equate
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    Quote Originally Posted by Guitarist View Post
    Quote Originally Posted by JohnMiddlemas View Post

    To me it is plain and concrete that the idea of zero is a mistake. This is for many reasons over a long period and I will never change my view.
    Then there is no discussion to be had, right?

    John, for this reason I am a finger away from locking this thread - let us know if you want a discussion or a just personal rant

    PS Your grasp of mathematics is tenuous to say the least. I suggest you learn the utility of zero in defining the negative integers, as the simplest possible example, though I could cite many others. While you're at it you may come across the perfectly sound assertion that the empty set contains as elements ALL those objects which have no possibility of existing - like square circles, say.
    Just because I have a concrete view does not mean the topic cannot be discussed. If someone could say something persuasive then obviously I could be persuaded but that doesn't seem to be happening. In fact I would be very happy to change my view since then I would learn something. This is not a rant. It is a serious comment on possible problems in mathematics and science. Frege's definition of 0 does not make sense as I have repeatedly demonstrated and nobody has been able to show otherwise convincingly. Now if the 0 cannot be defined then neither the 1, nor 2 etc since they depend on the zero definition according to set theory. This is a highly serious problem at the root of the foundations of science.

    Just because you don't agree with my observations is no reason to lock this thread. Are alternative to mainstream views prohibited here on this forum?

    As for my grasp of Mathematics, I have a 1st class BSc Hons degree in mathematics from Hatfield, UK. Feel free to check. Why would you then think my grasp of it is tenous?

    I subscribe to strict finitism which means I only accept the existence of finite mathematical objects. The completed infinite is unacceptable also in accordance with the great philosopher Wittgenstein. For me this means everything starts with 1. No zero, and no negative numbers, and no continuum. Everything is based on positive integers. Everything is discrete not continuous. That answers your point about the zero and the negative numbers. I also do not accept negative numbers. All these views are a perfectly acceptable alternative branch of mathematics and there have been various recent studies regarding finitism.

    In accordance with my finitism also real numbers are not accepted, only positive integers. The world is made of indivisibles, incredibly small things, maybe Planck length maybe not, but extremely small. That's where the 1 comes from, 1 indivisible. The idea of something becoming smaller and smaller without limit is wrong in my opinion. That is like an infinite regress and doesn't make sense to me. Even if it was the case it would still not approach zero since zero is nothing at all, but an infinitessimal is always something no matter how small. So for me, to say the limit as dx tends to zero is incorrect. It should be said the limit as dx becomes very small. Once again the zero is unnecessary.

    The zero in numbers is also unnecessary. There is a perfectly acceptable decimal alternative called the bijective decimal system which uses 'A' for 10, '1A' for 20, etc.

    Regarding the empty set please see my analysis of it above. Not only does it contain all objects with no possibility of existing it also contains all objects period. It is not empty. If you disagree then please address my comment about it.
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    Quote Originally Posted by KJW View Post
    Quote Originally Posted by JohnMiddlemas View Post
    My bank account is parted from money. It is not necessary to mention an amount in the case of zero since there is no amount. Your bank statement could just show blank, if there's nothing there why put a number to it, there's no point.
    This is a silly argument. In particular...
    There is no need to use insulting text (silly). Wouldn't that be a case for the moderator?

    Quote Originally Posted by JohnMiddlemas View Post
    Your bank statement could just show blank
    You are in effect arguing over the shape of the "0" character. Whether the statement show a "" (a whitespace or blank character) or a "0" doesn't alter that the balance is numerically zero.
    No I'm not. So long as there is nothing to indicate the position of the balance then that is what I mean by blank. Unlocatable since there is none. In other words where it says balance there is nothing surrounding that looks different from the background. You can't tell where the balance is. Because that is exactly what a zero balance means. It's gone away.
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    Quote Originally Posted by Guitarist View Post
    Yeah well, the way bank balances are written is entirely arbitrary, as is almost any other "everyday" application of zero AS IT IS WRITTEN.

    The following is not a strict proof, though it would be if I could be bothered to prove the background...........

    Consider the integers and let me assume that, as a set, it is a total order - that is one and only only one of the following must be true







    Then the subset - the positive integers - is the set of all upper bounds of the subset - the negative integers, which are in turn the set of all lower bounds of the set

    But not only is a total order, it is also a well-order - that is every (non-empty) subset has a least element.

    Suppose the least element . This is then a lower bound for which implies

    Therefore is both a positive integer and a negative integer or, what operationally amounts to the same thing, neither positive nor negative (as you may easily check using grade arithmetic) It is a matter of notational convention ONLY to set this particular

    Now let me insist that and are truly disjoint - that is . Since is neither positive nor negative i .e. it seems perfectly natural to equate
    In the above you have assumed the integers but they require definition. Such definition uses 0. Therefore your conclusions about 0 are entirely circular.

    Negative integers are defined using 0 as you yourself said in another comment here. However, 0 is undefinable as I have demonstrated by showing Frege's defintion to be false. Simply said, if zero could be defined, then it would be something, rather than nothing, an obvious contradiction.
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    Yeah, there's no point arguing with you. There's nothing anyone can say that'd change your mind. But for everyone else reading, I have to point out that you're taking a philosophical position (finitism, etc.) and trying to cram it in to math and physics where it doesn't belong. Whether or not you believe 0 has any existence in the real world, you cannot exclude it from (for example) Peano arithmetic. In Peano arithmetic its existence is one of the axioms, and if you change any of the axioms, you are no longer talking about Peano arithmetic. The same goes for all the mathematics built on Peano arithmetic.

    As to why everyone uses zero, it's really simple. It's useful. There's nothing else to it. Whether or not it's real is a moot point.

    Now, if you want to actually take your philosophy and make it mathematical, you should come up with your own set of axioms and then prove what you can about them. If you can re-prove some interesting results without the use of zero, you might even get some attention. For a parallel, see all the work that's been done trying to show what is or isn't provable if you exclude the axiom of choice.

    I also want to add that there's a significant difference between a blank check and a check for zero dollars.
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    Quote Originally Posted by MagiMaster View Post
    Yeah, there's no point arguing with you. There's nothing anyone can say that'd change your mind.
    Why would you want to change my mind? Just state your case and we can discuss it from there. If the case is convincing then of course I must change my mind. When I said I will "never change my mind" that was obviously incorrect if viewed strictly since nobody can predict the future exactly like that. I should have said "I believe I will never change my mind" or just not said anything at all.

    But for everyone else reading, I have to point out that you're taking a philosophical position (finitism, etc.) and trying to cram it in to math and physics where it doesn't belong.
    My feelings for zero do not arise from finitism. The finitism came after I questioned the zero many years ago from the simple observation that if zero could be defined then any definition must be a contradiction because zero defined is something and zero means nothing. The Frege analysis backs this up rigorously and it is clear that his zero definition is contradictory or includes a contradiction so allowing anything to follow as per the principle of explosion. If you then introduce the zero anywhere in mathematics then by the same principle of explosion anything whatsoever can be proved. That makes mathematics invalid I am afraid.

    Whether or not you believe 0 has any existence in the real world, you cannot exclude it from (for example) Peano arithmetic. In Peano arithmetic its existence is one of the axioms, and if you change any of the axioms, you are no longer talking about Peano arithmetic. The same goes for all the mathematics built on Peano arithmetic.
    Unfortunately Peano arithmetic is invalid for the above reasons I gave. Any axiom proposing a 0 proposes a contradiction. There must be a proper definition of zero. An axiom alone without some justification has no validity and my word is as good as anybody's. I can just say the axiom is wrong. You say it is right. So there is no evidence whatsoever Peano's zero is allowable. Therefore the whole of arithmetic is based on a guess!

    As to why everyone uses zero, it's really simple. It's useful. There's nothing else to it. Whether or not it's real is a moot point.
    You tried to show that before with your bank balance questions. I answered those questions but you did not reply further. How is it useful? In programming I have found it only to be a persistent source of bugs and entirely useless in fact. Equally the "empty string" is a right pain. If you ever "have" nothing then why mention it at all!

    Now, if you want to actually take your philosophy and make it mathematical, you should come up with your own set of axioms and then prove what you can about them. If you can re-prove some interesting results without the use of zero, you might even get some attention.
    Rather than Philosophy, this is plain common sense and intuition. The stuff axioms should be made of. You can't take 1 away from itself. How will you do that? Are you moving the 1 or destroying it? If I take 1 away from two, I am moving a 1 aside and leaving a 1 remainder, that makes sense. Where is the destruction? So if I take 1 away from itself all I can possibly be doing is moving it. So what did I take it away from? It's just a twist of the brain to think there is a zero remainder. All you did was move something. When your bank balance has 5 dollars and you spend the 5 all you did was move the 5 out. The 5 still exists elsewhere. Where is the destruction and the nothing remainder? What is left in the account is the same space that was there when the 5 was. How can you "find nothing" in that space and give a symbol to it? Where in the space is the nothing? Space requires real objects for reference to provide location. If you do give it a symbol then where was that symbol when the 5 was there?

    I am working on some axioms and have made some progress. First a definition of 1 is required without using the zero. That is not easy. For the moment I am concentrating on making sure my case for eliminating the zero is sound. This thread and others are confirming that since nobody can come up with anything convincing so far.

    For a parallel, see all the work that's been done trying to show what is or isn't provable if you exclude the axiom of choice.
    I will have a look at that. Thanks.

    I also want to add that there's a significant difference between a blank check and a check for zero dollars.
    Given that zero is a contradiction then anything can be proved. The amount could be anything at all even it says zero. That could be tried in Court. Since zero does not exist then a check for 0 is the same amount as for a blank check. You could cross the zero out as invalid and write in any amount you liked. They couldn't prove otherwise since zero is undefinable as I said. You should win! They won't be able to define the zero. Except they might try and claim it was a spoiled check.
    Last edited by JohnMiddlemas; August 20th, 2014 at 02:39 AM.
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    I say there's no point arguing with you not just because you said you won't change your mind, but also from all the other things you've said. You've taken a rigid philosophical stance and are trying to apply it to everything. What exactly can I say that would change your mind? (The whole point of arguing is to convince the other person that your position is correct, and thus change their mind. Although on a public forum like this, it's also to convince the spectators that your position is correct.)

    Now, back to the actual argument, like I said, assuming a contradiction is true causes the principle of explosion. That much I agree with. But using contradictions in other ways is still valid. For example,
    - Is there a greatest integer?
    1) Assume there is one. Call it N.
    2) But if you add one to N, you get a bigger number.
    3) (1) and (2) together is a contradiction, therefore either one or both must be wrong. (2) seems pretty watertight, so (1) must be wrong.

    Are you saying that there is a greatest integer? Contradictions are still valid parts of logic. You can't assume it is true, but you can still use it.

    Second, like I said before "the number of ways of satisfying a contradiction" is not a contradiction, just like "the number of ways of satisfying preposition p" isn't a preposition and it definitely isn't p.

    Third, Peano arithmetic cannot be invalidated by a philosophical argument. You can say it doesn't apply to the real world, maybe. But it is definitely not invalid. The only way it could be invalid is if it's axioms contradicted themselves, which they provable don't. "Assume 0 is a number" is not a contradiction. You could easily make an axiom that says "assume unicorns are invisible." I doubt it'd be useful, but as far as logic is concerned, it's a valid axiom and anything proven from that axiom would also be valid.

    Fourth, common sense and intuition are nearly worthless and relying on them is pretty much the start of every crank I've ever seen on here. Human brains aren't built for logical thinking.
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    I do believe there is no point in arguing.

    Frege's zero Lemma is not a definition of 0: Philosophy Forums

    It is on the other forum, but I thought to post it anyway.
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    I think one needs to be clear upon why a definition is required and what it is purported to achieve and how one can recognise when a satisfactory definition has been achieved.

    In my opinion, in the domain of mathematics , people have been trying to demonstrate the 'existence' of mathematical objects. This would seem to be quite absurd when it is apparent that pure mathematics is an entirely abstract discipline. It has no direct reference to anything in the real world.

    So I suggest that a sufficient definition for zero is one that enables its usage and function to be used in mathematical statements clearly and unambiguously.

    So why not just define zero as : ' 1 - 1 = 0 ' ?

    Obviously the symbols '1' '-' and '=' would also have to be defined in some simple and unambiguous way, but as long as there no ambiguity or contradiction, then the only requirement is that they be self-consistent, as it is only defining an abstract system.
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    Quote Originally Posted by A_Seagull View Post
    I think one needs to be clear upon why a definition is required and what it is purported to achieve and how one can recognise when a satisfactory definition has been achieved.
    Clear definitions are always important. Otherwise you get bogged down as soon as two people try to communicate using different internal definitions.

    Quote Originally Posted by A_Seagull View Post
    In my opinion, in the domain of mathematics , people have been trying to demonstrate the 'existence' of mathematical objects. This would seem to be quite absurd when it is apparent that pure mathematics is an entirely abstract discipline. It has no direct reference to anything in the real world.
    That's a mostly philosophical question. I would say I'm more of a mathematical realist than not, but that has no effect on how math actually works. (More details: https://www.youtube.com/watch?v=TbNymweHW4E)

    Quote Originally Posted by A_Seagull View Post
    So I suggest that a sufficient definition for zero is one that enables its usage and function to be used in mathematical statements clearly and unambiguously.
    The typical way to do this is to use the Peano axioms. One of those axioms is that zero is a number. The other axioms explain how to use that to get the rest of the numbers.

    Quote Originally Posted by A_Seagull View Post
    So why not just define zero as : ' 1 - 1 = 0 ' ?
    So how do you define 1? Basically, 1 or 0, you've got to start somewhere.

    Quote Originally Posted by A_Seagull View Post
    Obviously the symbols '1' '-' and '=' would also have to be defined in some simple and unambiguous way, but as long as there no ambiguity or contradiction, then the only requirement is that they be self-consistent, as it is only defining an abstract system.
    The Peano axioms are well established, proven to be self-consistent and proven to be useful (different definitions of proven there). If you want to use different axioms, among other things you're going to have to show that your axioms are more useful than the Peano axioms.
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    Quote Originally Posted by A_Seagull View Post
    In my opinion, in the domain of mathematics , people have been trying to demonstrate the 'existence' of mathematical objects. This would seem to be quite absurd when it is apparent that pure mathematics is an entirely abstract discipline. It has no direct reference to anything in the real world.
    I agree. I personally adopt the principle that all mathematical objects exist by default unless it can be proven that their existence is impossible. Thus, it is never necessary to prove the existence of a mathematical object that can be defined to exist. However, this is not to be confused with the notion of proving the existence of a non-empty subset of a well-defined set that satisfies some property. In this case, the subsets already exist and it is a merely a matter of proving whether or not any of them satisfy the particular property.
    There are no paradoxes in relativity, just people's misunderstandings of it.
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    Quote Originally Posted by MagiMaster
    That's a mostly philosophical question. I would say I'm more of a mathematical realist than not, but that has no effect on how math actually works. (More details: https://www.youtube.com/watch?v=TbNymweHW4E)
    Cool video! I'm most definitely a mathematical realist then. This is the same thing as saying that I'm a mathematical Platonist? Right?
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    Beats me. I don't know much about philosophy.
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    Neither do I, but my understanding is thus. The Platonic idea of forms is that there is a "realm" of idealized forms. That all incidences of these forms in the "real world" are imperfect or one-off copies of the Platonic realm form. I myself intentionally assign no physical ontology to this realm of forms. But I believe the gist is that mathematical concepts come from this Platonic realm and are not mere whole-cloth inventions of any mind.
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    I think that might be a more specific version of mathematical realism. Personally I just think that if aliens had math at all, they'd have the same math as us, so math is something apart from the human mind. (Many animals can count, and not all of them do it just because we taught them.)
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    Quote Originally Posted by MagiMaster View Post
    Quote Originally Posted by A_Seagull View Post
    I think one needs to be clear upon why a definition is required and what it is purported to achieve and how one can recognise when a satisfactory definition has been achieved.
    Clear definitions are always important. Otherwise you get bogged down as soon as two people try to communicate using different internal definitions.

    Quote Originally Posted by A_Seagull View Post
    In my opinion, in the domain of mathematics , people have been trying to demonstrate the 'existence' of mathematical objects. This would seem to be quite absurd when it is apparent that pure mathematics is an entirely abstract discipline. It has no direct reference to anything in the real world.
    That's a mostly philosophical question. I would say I'm more of a mathematical realist than not, but that has no effect on how math actually works. (More details: https://www.youtube.com/watch?v=TbNymweHW4E)

    Quote Originally Posted by A_Seagull View Post
    So I suggest that a sufficient definition for zero is one that enables its usage and function to be used in mathematical statements clearly and unambiguously.
    The typical way to do this is to use the Peano axioms. One of those axioms is that zero is a number. The other axioms explain how to use that to get the rest of the numbers.

    Quote Originally Posted by A_Seagull View Post
    So why not just define zero as : ' 1 - 1 = 0 ' ?
    So how do you define 1? Basically, 1 or 0, you've got to start somewhere.

    Quote Originally Posted by A_Seagull View Post
    Obviously the symbols '1' '-' and '=' would also have to be defined in some simple and unambiguous way, but as long as there no ambiguity or contradiction, then the only requirement is that they be self-consistent, as it is only defining an abstract system.
    The Peano axioms are well established, proven to be self-consistent and proven to be useful (different definitions of proven there). If you want to use different axioms, among other things you're going to have to show that your axioms are more useful than the Peano axioms.
    There are problems with Peano's axioms:
    For reference:The five Peano axioms are:

    • Zero is a natural number.
    • Every natural number has a successor in the natural numbers.
    • Zero is not the successor of any natural number.
    • If the successor of two natural numbers is the same, then the two original numbers are the same.
    • If a set contains zero and the successor of every number is in the set, then the set contains the natural numbers.

    (Taken from Encyclopedia Brittanica)

    There are problems with Peano's axioms:
    1. They do not define words like 'number', 'set' etc but presume instead that they are already understood terms.
    2. They do not contain information regarding that one number is bigger or smaller than another number, only that they can be ordered in a sequence. It does not define the criteria for that sequence.

    Computers are well able to do mathematics without any 'knowledge' of Peano's axioms. They work instead on a system of elements and operations that are consistent and yet entirely abstract.

    I am not aware of any 'usefulness' of Peano's axioms that would not be better served by using some other axiomatic basis.

    Perhaps a basis for an alternative axiomatic system would be to start with arithmetic in base 1. ie: 111+11=11111.
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    The terms Platonism and realism are both used inside and outside of a mathematical context. Inside of the mathematical context they seem to be equivalent. I'm just glad to be able to attach a new label to something I already know of, as opposed to having to learn a whole new thing. Although it seems that a lot of mathematics is complex structures of slightly less complex sub-structures on through possibly a few more reductive iterations till we finally get to a few basic pieces. Such as Euclid writing thirteen books from a handful of axioms.
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    Quote Originally Posted by A_Seagull View Post
    There are problems with Peano's axioms:
    For reference:The five Peano axioms are:

    • Zero is a natural number.
    • Every natural number has a successor in the natural numbers.
    • Zero is not the successor of any natural number.
    • If the successor of two natural numbers is the same, then the two original numbers are the same.
    • If a set contains zero and the successor of every number is in the set, then the set contains the natural numbers.

    (Taken from Encyclopedia Brittanica)

    There are problems with Peano's axioms:
    1. They do not define words like 'number', 'set' etc but presume instead that they are already understood terms.
    2. They do not contain information regarding that one number is bigger or smaller than another number, only that they can be ordered in a sequence. It does not define the criteria for that sequence.
    The Peano axioms as a whole are a definition of natural numbers. And no, they don't define set. That's another collection of axioms. (Usually ZF set theory.)

    And they actually do contain information on the ordering. A number is defined to be greater than another if you can get from the lesser to the greater by the successor function. So 2 is greater than 1 because 2 = S(1), and 3 is greater than 1 because 3 = S(S(1)). (Peano axioms - Wikipedia, the free encyclopedia)

    Quote Originally Posted by A_Seagull
    Computers are well able to do mathematics without any 'knowledge' of Peano's axioms. They work instead on a system of elements and operations that are consistent and yet entirely abstract.

    I am not aware of any 'usefulness' of Peano's axioms that would not be better served by using some other axiomatic basis.

    Perhaps a basis for an alternative axiomatic system would be to start with arithmetic in base 1. ie: 111+11=11111.
    So because you're not aware of any usefulness, it's not useful?

    The Peano axioms are already base 1. The entire thing is built on 0 and the successor function. 1 is defined as S(0). 3 is defined as S(S(S(0))). Etc.
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    Quote Originally Posted by MagiMaster View Post
    Quote Originally Posted by A_Seagull View Post
    There are problems with Peano's axioms:
    For reference:The five Peano axioms are:

    • Zero is a natural number.
    • Every natural number has a successor in the natural numbers.
    • Zero is not the successor of any natural number.
    • If the successor of two natural numbers is the same, then the two original numbers are the same.
    • If a set contains zero and the successor of every number is in the set, then the set contains the natural numbers.

    (Taken from Encyclopedia Brittanica)

    There are problems with Peano's axioms:
    1. They do not define words like 'number', 'set' etc but presume instead that they are already understood terms.
    2. They do not contain information regarding that one number is bigger or smaller than another number, only that they can be ordered in a sequence. It does not define the criteria for that sequence.
    The Peano axioms as a whole are a definition of natural numbers. And no, they don't define set. That's another collection of axioms. (Usually ZF set theory.)

    And they actually do contain information on the ordering. A number is defined to be greater than another if you can get from the lesser to the greater by the successor function. So 2 is greater than 1 because 2 = S(1), and 3 is greater than 1 because 3 = S(S(1)). (Peano axioms - Wikipedia, the free encyclopedia)

    There is nothing in the quoted axioms that refer to lessor or greater, it only refers to 'successor' which is quite different.


    Quote Originally Posted by A_Seagull
    Computers are well able to do mathematics without any 'knowledge' of Peano's axioms. They work instead on a system of elements and operations that are consistent and yet entirely abstract.

    I am not aware of any 'usefulness' of Peano's axioms that would not be better served by using some other axiomatic basis.

    Perhaps a basis for an alternative axiomatic system would be to start with arithmetic in base 1. ie: 111+11=11111.
    So because you're not aware of any usefulness, it's not useful?

    That does not follow logically. If I had wanted to say that I would have done so. But it does mean that if you want to aver that they are intrinsically useful that you will have to prove your case.

    The Peano axioms are already base 1. The entire thing is built on 0 and the successor function. 1 is defined as S(0). 3 is defined as S(S(S(0))). Etc.
    I was referring to a basis for mathematics rather than just the natural numbers.
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    Quote Originally Posted by JohnMiddlemas
    I was referring to a basis for mathematics rather than just the natural numbers.
    Um... yea, that would be set theory; Set theory - Wikipedia, the free encyclopedia

    Oh, by the way, there's really no reason to yell.
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    Quote Originally Posted by A_Seagull View Post
    There is nothing in the quoted axioms that refer to lessor or greater, it only refers to 'successor' which is quite different.
    Perhaps the font size buttons should be removed from the forum.

    Anyway, define greater. Just accepting an intuitive notion is not good enough when you're talking about axiomization.

    And yes, you implied that the Peano axioms were not useful. I never claimed they were "intrinsically useful" whatever that means, but they are useful in the sense that they provide a very concise and generally well accepted set of axioms for the construction and use of the natural numbers (which also provides a basis for many other types of numbers).
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    I had no intention of 'yelling' , I just wanted to differentiate my reply from the previous post. Normally I would use a colour variation, but that option does not seem to be available on this forum. And the FAQ seems a little sparse on information about disseminating replies from earlier posts.

    And, yes I know that Peano's axioms are popular with many mathematicians, but this does not mean that they are the last word in describing the foundations of mathematics.
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    Quote Originally Posted by JohnMiddlemas
    And, yes I know that Peano's axioms are popular with many mathematicians, but this does not mean that they are the last word in describing the foundations of mathematics.
    It is very likely that in the future there will be a better description of the foundations of mathematics. But that result will come from people who have made an effort to understand the thing they are trying to provide a superior description of, not people who haven't made that effort.

    Oh, by the way. The font color option being turned off saved you some embarrassment. It is considered a sign of beyond the pale eccentricity.
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    Quote Originally Posted by A_Seagull View Post
    I had no intention of 'yelling' , I just wanted to differentiate my reply from the previous post. Normally I would use a colour variation, but that option does not seem to be available on this forum. And the FAQ seems a little sparse on information about disseminating replies from earlier posts.
    Just use the quote tag. That's what it's there for.

    Quote Originally Posted by A_Seagull View Post
    And, yes I know that Peano's axioms are popular with many mathematicians, but this does not mean that they are the last word in describing the foundations of mathematics.
    I don't think I ever said they were. But if you want to replace them, you're going to have to offer something equally useful in their place. They're popular for a reason.
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    OK, a few comments, some of which I dare say will make me unpopular

    First, learning mathematics from Britannica (or even Wikipedia) is no education at all

    Second, mathematics (by and large) admits of no taste, opinion, or preference - if some statement follows logically from a set of coherent axioms, then, within the axiom set, it can be taken to be true.

    Axioms can be replaced or refined, as has been done many times in the past, but it is usual to present a very strong case for doing so

    So I agree totally with MagiMaster as far as the mathematics goes. I will expand.

    I suppose a set and define a "successor" function by It follows that every contains every "predecessor" set as a subset, which, for every in turn induces a chain of subsets (I have been through this before, so I won't repeat the argument)

    Now since notation in mathematics is entirely arbitrary, it is legitimate to rename the objects so described as "numbers" and to replace the symbol for a total order by set inclusion (as shown) by the symbol .which you can take to mean "less than".

    What;s the problem?
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    Quote Originally Posted by GiantEvil View Post
    Quote Originally Posted by JohnMiddlemas
    And, yes I know that Peano's axioms are popular with many mathematicians, but this does not mean that they are the last word in describing the foundations of mathematics.
    It is very likely that in the future there will be a better description of the foundations of mathematics. But that result will come from people who have made an effort to understand the thing they are trying to provide a superior description of, not people who haven't made that effort.

    Oh, by the way. The font color option being turned off saved you some embarrassment. It is considered a sign of beyond the pale eccentricity.

    Well I must be eccentric then

    It is all too easy to defend the status quo as it does not require any imagination and only minimal brain activity.
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    Quote Originally Posted by MagiMaster View Post
    Quote Originally Posted by A_Seagull View Post
    I had no intention of 'yelling' , I just wanted to differentiate my reply from the previous post. Normally I would use a colour variation, but that option does not seem to be available on this forum. And the FAQ seems a little sparse on information about disseminating replies from earlier posts.
    Just use the quote tag. That's what it's there for.

    Quote Originally Posted by A_Seagull View Post
    And, yes I know that Peano's axioms are popular with many mathematicians, but this does not mean that they are the last word in describing the foundations of mathematics.
    I don't think I ever said they were. But if you want to replace them, you're going to have to offer something equally useful in their place. They're popular for a reason.
    I consider Peano's axioms to be redundant and as pointed out earlier computers can carry out mathematical operations without them. So I suggest that they are replaced with nothing at all. Mathematics is an abstract system and requires no foundations. It only needs to be a self-consistent system.
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    It's also all too easy to say something's wrong without actually providing any alternative or even offering any evidence why it's wrong.

    And yes, computers operate on the laws of physics, not the laws of mathematics. But you seem to have completely missed the point of the Peano axioms. They are not a recipe for how to build computers, or even how to do arithmetic. They are an attempt to formalize such things. Just accepting intuitive notions of how things ought to work used to cause massive headaches every time someone bumped in to some paradox. Eventually someone had the idea to actually write down rules instead of just throwing their hands up in the air and saying what's the point. They could then look at the rules and prove they were self consistent and then work out the consequences those rules had for all those paradoxes they kept running in to.

    Math does need a foundation. Since it is so abstract, it can't take it's foundation from reality. Instead, you get to define your own foundation and see what you can build on it. The Peano axioms are a very succinct foundation for the rules of arithmetic.

    Again, your entire argument seems to boil down to "I don't have a use for this, so why should any one else?"
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    Quote Originally Posted by A_Seagull View Post
    I consider Peano's axioms to be redundant and as pointed out earlier computers can carry out mathematical operations without them.
    How do you know the computers are carrying out the mathematical operations correctly? How do you know you are carrying out them out correctly?
    There are no paradoxes in relativity, just people's misunderstandings of it.
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