Good Morning, Trippy.
I really don't have time to read through your exceedingly long posts that largely just seem to state the same things eight different ways and somehow manage to come close to (or even crossing the threshold of) word salad.
Urmmm, Trippy, may I remind you that this is the "Alternative Theories" section, not the "Established Maths" section. Here participants are exploring and discussing alternative views with a view to further refining/replacing established understandings, if possible. Yes?
Moreover, whenever the discussion is moved back to the very Axiomatic/Conventionality 'beginnings', and the reasons/validity/consistency OF those in the 'larger picture' context (and not being just constrained to 'blindly following' of the consequences OF those very same axioms/conventions once 'introduced'), then it is unavoidable to end up with WORDS to arrive at some sort of MUTUAL 'contextual understandings' rather than just being constrained to self-justifying circularity of just repeating the 'blindly following' numerical 'proofs' manipulations which depend on certain 'exceptions' in order to make the axioms/conventionality consistent in a self-serving way rather than exploring the basis for the whole system 'from scratch'.
And that is what I am doing (politely and 'alternatively' exploring/discussing) here certain claims made by you and others IN THE CONTEXT OF EXAMINING whence such claims arose and whether they are self-consistent in both the limited axioms AND in the 'bigger picture' which I have already made clear I am 'coming from' in this discussion.
Given this section, and the nature of the discussion as I point out, it is a bit ungenerous to characterize the INESCAPABLE USE of EXPLORATORY and necessary WORDS and discussion modes which must necessarily precede any 'blindly following' and 'self-serving' REPETITION of 'numerical proofs' and all those very things which this discussion (from where I am coming) are all in question precisely because of the 'from scratch' exploration/discussion points/words presented to treat what went before the axioms/conventions rather than just what follows them (which we all agree DOES blindly follow 'given' those axioms/conventions, but which I now question the 'completeness' of those axioms/conventions THEMSELVES).
So your ungenerous and 'personal' attitude characterizing others use (perforce of the nature of this discussion) of their polite and on-topic descriptive words in fair context, as 'word salad', is not really playing cricket, is it. It brings a certain prejudicial air to your reading in the context, which only makes the common understanding part of a conversation difficult from the very start, and failure to fully understand the other guys' stance AS EXPLAINED in their (so-called by you) 'word salad' indicates laziness and/or reading bias and automatic resort to kneejerking rather than fair and exhaustive conversation on point as presented/explained in context. Yes?
If you do not read this reply like many others) properly and fairly, then you automatically disqualify yourself from, and maybe even distort, the conversation based on your own personal attitudes to 'the source' rather than fairly and considering the points put in context of THIS Alternative Theories section/discussion. Yes?
First off, the zeroes are not trivial - that's not what significant means in this context.
With respect, but you seem not to have read or understood my quoted statement (to Pete) recognizing the differences IN CONTEXT.
If you had read/understood properly, then you would have gathered that I said zeros can be trivial in some cases and not trivial in other cases. Context indicates there significance or their non-significance etc. That is what I said. So your attempt at characterizing that I did not recognize/understand the difference is sailing pretty close to a 'strawman'. But perhaps it isn't meant to be a strawman; perhaps you just didn't read/understand that relevant post/comment properly and IN FULL. I will assume so rather than take it as an intentional strawman (since I know you would never do such a thing intentionally). :
Secondly the 'formatting device' is the agreement not to write the non-significant zeroes because it's cumbersome, not the other way around. If you want to deal in fundamentals then you have to realize that it 1.0(0) not 1. The zeroes exist whether you write them or not. Do I need to explain to you what numbers actually mean?
The decimal point is fundamental even though the convention is not to write it unless the zeroes after it. If we follow your logic, then rather than writing "A circle of 565m has an area of 1.00 km[sup]2[/sup]" I should just write "A circle of 565m has an area of 1km[sup]2[/sup]". There's a problem with this - a circle with a radius of 560m also has an area of 1km[sup]2[/sup].
The fundamental fact is that the zeroes before and after the 1 exist whether we choose to write them or not.
The convention is to not write zeroes unless they're significant.
I can't say it any more directly than that.
Ok. Conventions. Let's assume (for the sake of argument) that I agree with your insistence on a "pre-existing zeros" convention; ie:
"0.00000 ->" in an unending series of 'pre-existing' zeros to the right of the decimal point. Where do we get to with that convention?
The example I initially questioned, IN THAT CONTEXT only,was Pete's use of a 'proof' involving the step/operation :
10 x 0.99999... = 9.9999...
I pointed out that fundamentally we added the zero from the 10 to the end of the original 0.9999... unending string, which "forced" the unending string to move the leading "9" one position to the left, past the decimal point, resulting in:
9.9999...0
Pete's 'convention' was to interpret that 'result' as being the (in my view TRIVIAL convention) "action" of "shifting the decimal point to the right".
Whereas I argued the "alternative" view (from observation of the real fundamentals I perceive); ie:
That in fact the REAL action was to add a zero (from the real number 10 factor/operation/entity) to the end of the unending string, and so by that actual action forced the ACTUAL NUMBER leading 9 to move to the left, rather than merely trivially and conventionally making some NON-real "action/operation" trivially described as "moving the decimal point to the right".
If zero is a number; and if that zero IN CONTEXT IS 'significant' and not just trivial, THEN I merely observe that the 'real action involving that adding a zero at the end is MORE fundamental and significant than any mere convention of "moving the decimal point to the right" as per his argument/proof in that case/context.
Oh, and before you come back with the obvious trivial retort that 'the pre-existing zeros' and 'the repeating 9 string never 'ends', you would be trying to have it both ways: ie, such strings are conventionally presumed (via the Limits argument) to END at some limiting 'state' in the context of 'infinity' arguments/conventions; and yet you adjured QQ to "try long division" etc which never ends. So I leave it to the more 'real action' context to tell which 'take' is the real way to understand this aspect being re-examined in this 'alternative theory' section/discussion.
Anyhow, that's it, that's where I am coming from in this discussion of alternative exploration 'from scratch' context, mate. No more, no less than that.
If it's still all 'word salad' to you, or if you don't bother reading all the arguments in full, then you leave yourself open to creating further UNINTENTIONAL 'strawmen'.
In any case, Trippy, thanks sincerely for your responses regardless; much appreciated all the same.
