Hi rpenner. Sincerely and with respect, thanks for your time and trouble in providing your learned opinions/comments re the humble observations/suggestions and questions/points I raised in the spirit of discussing (not dictating) these things in this context. Much appreciated.
Still missing the target, Khan.
Who is this "Khan"? Another instance of mistaken identity on your part? I trust you won't let your personal 'baggage' with this "Khan" color your attitude towards me and my respectful and sincere/humble comments/suggestions?
Anyhow, briefly responding to your comments...
This is baseless since the number line is a bijection that preserves affine addition and scalar multiplication between the Euclidean line and the set of real numbers. Therefore fixing the map for any two distinct number-point pairs establishes the map for all points and numbers.
So if $$f(A) = a$$ and $$f(B) = b$$ and $$A\neq B$$ and $$a \neq b$$ then it follows that $$\forall x \in \mathbb{R} \quad f^{\tiny -1}(x) = A + \frac{x-a}{b-a} \left(B - A)$$ and that specifically $$f(A + \frac{a}{a-b} (B - A)) = f(A) +\frac{a}{a-b} \left(f(B) - f(A)\right) = a +\frac{a}{a-b} (b - a) = a - a = 0$$.
My aim was to point out the initial situation regarding the reality 'zero' status before all the higher abstractions status given to it after further axioms and manipulations come into play which thereby 'define' what zero will be rather than 'discover' what it started out as in reality.
Zero is an essential part of the definition of any number system which follows the logical axioms that define a
ring because zero plays the role of the identity element of the
Abelian group that defines what addition means in the ring.
The ring involves negative numbers. For pairing negative numbers to give 'zero', the actual 'operation' must be 'subtraction'. Naturally the subtraction of like from like will give a zero as a balance state/condition OR it is a trivial a=priori situation which has been 'decomposed' into a negative and positive number of equal value (which is the TRIVIAL 'construction process' for the number line extending in both directions from a 'common origin' point 'labeled' zero but not actually a number, just an 'origin' for the abstraction of negative numbers which would otherwise represent in reality some 'opposite force' or 'opposite state' physically, and hence cannot be said to be
'purely mathematical' in axiom or application/logic. You can't have it both ways. Either you are discussing zero as an axiomatic symbol; or as a real physical state/condition (as I already observed was the case IF zero had some real significance apart from mere abstraction used and abused to suit various axiomatic/proof 'definitions' which become thereby circuitous since they are defined by the very same axioms/usages which I have been questioning).
That is an baseless assertion, not an argument. You are trying to define the term of the conversation while evading any burden of proof.
If maths can 'define' zero according to its circuitous axiomatic/proofing needs, why cannot one define it according to its reality status as I have observed, and see where that leads in discussion? My aim is to arrive at a more reality-contextual mathematical system via discussing the reality aspects of all things involved, including the starting axioms and any 'definitions' and 'undefined' parts (grey areas) still afflicting both the maths 'abstract constructs' and the physics 'reality models'.
This is shabby false dilemma. Positive means greater than zero. Negative means smaller than zero. Zero is both non-positive and non-negative, but that does not mean it is both positive and negative at the same time. It is the midpoint between +1 and -1.
But because you didn't discuss what positive and negative mean, you didn't actually argue your false dilemma. You simply asserted falsely that zero is considered to be both positive and negative and did not explain why that had to be a contradiction. I could be black and jewish. I could be buddist and shintoist. I could be conservative financially and liberal socially. I could be both rich and poor if you use different standards of comparison. But the reason that negative and positive are considered opposites comes from the axiom of trichotomy.
A real number can be less than, equal to, or greater than a real number. A real number can be less than, equal to, or greater than zero. A real number can be negative or zero or positive.
You just said it yourself!
That zero is neither positive nor negative. Hence it cannot be on the number line that starts either 'arm' of that line. If it isn't the first negative number, then it's not at the beginning point of that negative arm. If it's not a positive number, then it's not at the beginning point of that positive arm. If it's actually IN BETWEEN, then its an abstract origin point label/symbol for EACH arm as well as a TRANSITION state for the transition of the actual numbers from positive to negative and back.
But I contend that 'negative numbers' are abstractions and not real things, so there is no problem IF we can consider the zero to be a label for the zeroeth positive number ONLY. In this contention the negative numbers merely represent an axiomatic 'shorthand' for what SUBTRACTION operations can be done using the positive numbers and applying a "-" sign for EACH operation so as to distinguish which positive number is being deducted from which other positive number.
This implies that 'absence' of value' is just that, an absence of POSITIVE value....ie, that some CHANGE has occurred to make the situation of OPPOSING number values change from an effect in one direction to another opposite direction. That implication is also extended to the case where two EQUAL positive value numbers/forces etc act so as to create a BALANCED STATE (such as the origin on the xyz co-ordinate system implies as a starting state for all analyses of values departing therefrom).
Seeing as you are asserting your definitions in defiance of history, convention, logic and civility, there is no surprise that you equate you misunderstanding with a impossibility.
Civility? Since when did 'in defiance of civility' come into it? I just wished for your opinions/comments on my humble observations/suggestions in the spirit of discussion. No more, no less, let alone any intention of 'defiance of civility'. As for the rest, naturally these are the very things under question/discussion. One can't make new omelets without possibly breaking some old eggs, hey?
That is not argued for, but nakedly asserted. And as you introduce a lot of terms which have no applicable definitions anywhere in the literature, it follows that this isn't a coherently expressed idea. Thus your premise isn't even a communicable thought. Numbers are numbers, they aren't "states". They have no "transitions", no movement. You can capitalize "Origin Symbol" but it means nothing as the line works equally well as a vector space with an origin and an affine space without notion of origin.
I merely made the humble observation and posted the argument for it from my reality perspective as written. This is the discussion stage where I merely asked for your own counter argument/views on that. So why the 'naked assertion' accusations?
And the maths/physics literature did not always contain all the terms/concepts/understandings etc which they do at present status quo. They were INTRODUCED at various stages, they did not just appear full blown and complete. So just as they appeared piecemeal, they can also be challenged and 'varied' piecemeal as better approaches and understandings come along requiring review/replacement as appropriate. The only way we will know when the literature/maths/physics is COMPLETE is when no possibility of new avenues/challenges remains to be discussed. Until then we are just discussing politely and objectively without rancour or personal attachments/resentments of any kind, Yes?
This combines the fallacy of argument from personal ignorance and false dilemma. Creationists don't win their arguments by default and neither do you.
Why bring 'creationists' into it? I am objective atheist and scientist since age 9. Please don't introduce such emotive irrelevancies into a polite maths/science discourse. Thanks.
I am following the principles of substitution of equals and therefore I could have done many things. I choose to make a coherent argument that the hypothesis $$\frac{0}{0} = 1$$ was incompatible with established axioms of arithmetic.
When it is those very axioms brought into question by my humble observations/suggestions as made, is it appropriate to depend on those same axioms when trying to justify using them in your counter-arguments like that? In any case, I only referred to the 0/0 case
for the sake of argument assuming your use of zero as a number....and then made the point that it is TRIVIAL and unnecessary to 'decompose' such 'like/like' construct/entity (of ANY 'like/like' numbers used, including zeros). The only reason to decompose it trivially is to equally trivially 'prove' something 'contrived' rather than fundamentally important which would be evident if we could treat such like/like entities as complete and whole (by using parentheses to make them be evaluated before any other mathematical operation is done (meaning that your separating them to use in your proof exercise was trivial and not representative of the fundamental situation which is only 'hidden' by such trivial 'decompositions' as you used in your 'proof').
The principle of parsimony does not play a role in logical argument. It is a scientific principle not a logical necessity or mathematical axiom.
When the axioms are brought into question by new observations/suggestions, then everything is again on the table until the dust settles anew. Trying to limit the scope of exploring the status quo is akin to religious dogmatic adherence to exclusion of hearing anything against that status quo being discussed/explored with an open mind free of personal attachments and reputations etc. Anyhow, I made no claims other than that it seems worthy of polite and searching discussion to see what results therefrom. Any 'helpful principal', be it scientific or other kind, is also not to be sneezed at when everything is being reviewed with an open mind 'from scratch' to see where we may possibly have thrown out the reality baby with the abstraction bathwaters. We are obviously coming at this discussion from quite different angles. But that's ok, I will try not to let your own 'angles' create any feelings of personal offense at my end, mate! It takes all kinds to make for interesting searching discussion, hey!
Adopting such terms would make the "/" in "0/0" meaningless and thus "0/0=1" would be as sterile and vacuous as asserting "1=1" for if "0/0" is a monolithic indecomposible single symbol then it is merely 1 written in a different language and not a mathematical claim.
Exactly my observation! Thanks for understanding my point about the trivial/meaninglessness of using 'like/like' constructions/entities in the first place; and then just as trivially 'decomposing' them when it suits 'spurious proofs' depending on said trivial 'decomposition' (instead of properly leaving such entities whole in parentheses as I humbly suggested before?).
$$\forall x \in \mathbb{R} \backslash { 0 } \quad \frac{x}{x}= 1$$ is trivial, but why is it "invalidly trivial?" It's not invalid at all. I think you have let delusions of authority distract you from actually arguing for your thesis.
http://us.metamath.org/mpegif/divid.html is a theorem. And far from being invalidly trivial, it forms the basis of proving other theorems.
http://us.metamath.org/mpegif/divdivdiv.html
What 'authority' is claimed? I only asked for your learned opinions/comments and counterarguments on my humble observations/suggestion as put. No more, no less than my opinion has been contended. Just because these contentions challenge certain axioms/usages currently in vogue does not make it somehow delusional. I already argued the point regarding the triviality and invalidity given that viewpoint I put. The rest is discussion stances, not 'authoritative' claims.
Not a principle of mathematics from any source. Also not compatible with the theory of limits or calculus. Also not argued from any basis.
No; just observations in context as given. That context being that those very same axioms etc are what are being questioned in this context as given. That is the whole point of the discussion; ie: Are those axioms/principles, developments/theorems etc etc more fundamental than the observations made about zero in the contexts given? The arguments about these things are continuing in that given context, and not in the self-selecting context of those same axioms being used/questioned in the discussion.
It is a number if numbers satisfy the axioms of rings.
Again, "rings" are abstract 'constructs' based on equally abstract 'axioms' FOR those very same "rings" abstract constructs. What is required is for something EXTERNAL from such self-selecting definitional/axiomatic constructs if the true nature of zero and its valid/invalid uses are to be properly identified independent of such self-selecting approaches to the subject of zero in reality context. Otherwise we will only be forever chasing our own abstract tails until we disappear up our own equally abstract backsides. That way lay madness of a particularly subtle but insidious kind, yes?
That axiom is not anywhere accepted in mathematical sources and is also the point of contention on this thread, so its unlikely that you will settle this argument by adopting the hypothesis you seek to prove as an axiom.
But IF "0" IS a number, then my humbly suggested treatment of "0/0" (ie, keeping (0/0) in parentheses throughout) would put the "undefined" status of "(0/0)" to bed, wouldn't it? And would not conflict with the non-trivial entity of "1" for any and all such "(like/like) constructs involving any and all the other numbers in such obvious "unitary" term to begin with?
Mathematical arguments only seem trivial to those without a stake in the game.
Some things are trivial per se, irrespective of the attempted 'cover' given them by spuriously self-selecting definitional abstractions which are tailored to suit/hide that triviality under a snowstorm of trivial manipulations that are not necessary fundamentally.
It is my opinion that you are beyond the reach of charitable posters and moderators to correct. I believe the best thing for all readers is for the moderators to ban you, delete all your posts and until such time all right-minded people to put you on ignore. I considerate it a major failing of the Obama presidency that you are allowed internet access. It arguably is a failure of your parents not to have christened you with the middle name Bloviating. Those are my opinions and you are welcome to them.
Your personal opinion regarding the person presenting the humble observations/suggestion is noted. Thanks. But I prefer to stick to the objective discussion of the objective essentials involved, not the persons/source.
What say you about sockpuppets?
In case you and others are still uninformed, the admin know who I am and let me remain because it was not me who was the problem, but the trolls who baited, personalized and framed their victims and then colluded with certain mod to ban the victim instead of the trolls in question (although the situation has since been remedied and the site is now excellent example of true science and humanity discourse site with fair rules and mods applying same). So 'sockpuppets' are what the trolls use. The victims merely come back when the trolls have been taken properly to task and the past injustices have been remedied (evidence my continuing presence here as "Undefined" with the full knowledge and approval of admin because I was not the problem. Thanks again, fair admin!).
But let's not bring personal baggage etc into it, hey? Admin frowns on that sort of thread-derailing tactics now, and you risk a ban if you and others persist in bringing such things up again and again in order to 'justify' your own personal characterization/baiting tactics etc for whatever personal reasons you may have, but which are not relevant to the maths/science points per se under discussion.
This:
It's not enough to quote. You have to read, understand and respond in detail to what you quote or you look like you are ignorant and unhelpful.
I said what I wanted to say in context. What you made of it is your affair. The context was clear as to what I addressed.
Indeed, some argue from Category Theory. But the concept of sets are found in predicate calculus and category theory both. So it's not a matter of quibbling with
my choice of axioms but rather for you to state clearly what your axioms are and defend your system as one that corresponds well with the historical definitions of which you seem ignorant. If you are trying to define division or zero or number in a new way without relying on the authority of textbooks you are responsible for carrying the burden of proving that your new definitions are useful. Instead you have retreated to saying "0/0" is just a indecomposable symbol that doesn't say anything about zero or division.
I have been keeping to the most fundamental stages, away from the higher abstract constructions/systems which bury the fundamentals way down and lose sight of them (like the "undefined" 0/0 and the nature of zero itself etc as I have observed/argued already).
I have no other observations to make. The 0/0 does not make sense unless zero is a number. If it IS a number, then any like/like construction (including 0/0) must be a number IF we can find a way of treating it such that it is NOT "undefined" but a whole entity in parentheses) which equates to unitary value just like any other like/like construction using any other like numbers in that way.
That was all I suggested.
It was a way to get around the "undefined" ambiguity which 0/0 currently is treated as by the axioms which do not cover such a construct like 0/0...UNLESS it is put in parentheses as I suggested....such that it joins all the other like constructs as "1" (as long as it is kept in parentheses throughout).
Whether the professional mathematical fraternity/sorority adopts this new HUMBLY SUGGESTED convention to avoid the "undefined" status of 0/0 is up to them. I humbly make that suggestion since it satisfies the 'reality check' test for my purposes of overall contextual real consistency (irrespective of whatever arbitrary axiomatic limitations are being currently encountered in any one particular abstract system/model or other which requires "undefined" as an ongoing status for any entity/action etc). That is my 'angle' in this discussion.
Not a coherently expressed communication.
Whenever one introduces new ideas/terminology etc to convey/explain new things which may require changes to conventional status quo/literature, it is an unavoidable occupational hazard that he/she be accused of that. I'm sure Einstein and all the other pioneers/explorers of new paradigms took some flak of that sort. Naturally, if the conventional discourse was already complete, there would be no need for new and possibly 'halting' attempts at conveying new things which may at first blush come across as 'incoherent' perforce of being exploratory and not settled conversation process. Give and take is called for, without personal feelings getting in the way of tolerance and understanding when new ideas/discussion points are still at that "up in the air" stage.
Perhaps to YOUR satisfaction maybe, but obviously not to mine. That is the discussion and question here, that maybe it is not after all as fundamentally based as you seem to believe?
Nope. Being demonstrably more correct than those that came before is what matters in the sciences. And you can't be demonstrably more correct if you don't communicate to your fellow human beings.
I made observations and suggestions with accompanying arguments in support. The outcome of the discussion will tell. No more than that is attempted in this discussion. My work overall is more comprehensive in scope and consistency, but I am not presenting that complete works here, just presenting the points/suggestion made in context for your comment etc. Whatever else you feel I am trying to do more than that is your construction, not my actual intention as made clear already, just limited to the two aspects of zero and like/like etc.
Not a coherently expressed communication.
That was posted to chinglu in the context of our exchange, so as long as he understood in the context, then it will do.
That was not chinglu's point. He was arguing that the map that takes x to the product of x times zero was in fact a function. You didn't not argue against that point.
I said I wasn't interested in that approach at all. I made clear what I was after with my approach to the zero and 0/0 aspects treatments etc. So no problem/arguments from me on what he and you are discussing in that particular vein.
Arguing zero is a symbol is mistaking the the map for the landscape.
"0" is a symbol. So is "zero" or "1". But "zero is a number" is a mathematical claim that depends crucially on the definition of "number" which I have staked a position on and you have ignored.
I have not ignored it. I even argued my view on that, as well as the case IF it IS a number for the sake of argument. Did you miss it all?
There is no physicality to Euclidean geometry or the concept of polynomials, so I think you aren't arguing mathematics at all.
I am arguing/exploring the treatment/identification etc of axioms and fundamental entities of all kinds, including in this particular case the nature of zero and the fundamental 0/0 "unitary" construct which may not be "undefined" after all IF we keep it in parentheses at all times (as my suggested NEW CONVENTION). I've made/argued my observation/suggestion/viewpoint about these things already. Can't do more than that.
If it is trivial how can it be not proper? If it is not proper how can it be trivial?
The point is not semantics, but the thrust of what I said. The funamentality of anything can be hidden by trivial/improper treatments which are so precisely because they only cover rather than consistently treat the fundamentality of what is being obscured by abstraction upon higher abstraction away from the essentials in reality. That is the thrust.
You quote this term like it has a commonly understood meaning. Source?
Surely you jest.
Evaluating terms in parentheses before any other operation is carried out. That was the whole point. It's using "brackets" of all sorts for such hierarchical treatment of terms according to whatever "Rules of Evaluation" are implemented by use of such bracket notation enclosing the expressions to be treated in that order of preference). But you were joking, right! lol.
A ridiculous claim supported by nothing.
The claim that
0 is a real number is a theorem which follows from the basic axioms of real and complex numbers.
First of all, 1 is a real number. Secondly, whenever there is a real number there exists a real number such that their sum is zero. Thirdly, the set of real numbers is closed under the operation of addition. Therefore since 0 is expressible as the sum of real numbers it too must be a real number. Q.E.D.
Again, any two numbers that add up to zero must be 'like value' numbers which are involved in SUBTRACTION as such, and not some higher abstract 'sum' action which hides that fundamental fact/action.
And since the numbers are like negative and positive, they subtract trivially (just as '0/0' or any other 'like/like' construct) divide trivially.
In which case the resultant zero either represents a 'balance' situation between two real opposing forces/values OR the subtraction of like but 'opposiing 'positive' numbers is also a trivial expressions which I also suggest as a convention should be treated 'whole' inside parentheses (like I suggested for the 0/0 expression) so that the fundamental nature of the term/actions is not hidden under the abstractions of higher treatment/axioms that leave the real essentials behind).
Anyhow, that's my perspective on these things, and the end of my contribution to the thread/discussion because I've observed/suggested all that I wanted to do on that. Thanks again sincerely for your time and trouble in responding, rpenner.
Good luck with your own perspective/discussions, rpenner, chinglu, everyone!