1=0.999... infinities and box of chocolates..Phliosophy of Math...

That isn't nonsense. I can easily abstract the notion of a set containing a finite number of camels to a set containing all the finite sets of camels. If I then ask how many camels are in this set of "all camels", I also have to think about what kind of number is the sum of all finite numbers, of camels or any other kinds of discrete objects.

What I think is the real problem here is an inability to abstract. That would be: from your corner.

 

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Hi arfa. Had more time to edit typos, and saw this from you...

See, right there you have an abstraction/philosophizing 'modeling exercise' which loses effective information/reality meaning in the process!

The real camels have 3-D extension parameters and effectiveness, while the points you abstract them to in your 'sets' have NO SUCH information about their 3-D extension properties. Pure 'numerology' and 'philosophical' set construct which ends up as an ipso facto GIGO system of 'maths', and not real rigorous maths based on real axioms and real objects from which you just 'abstract and lose' information that the further manipulations can NEVER RECOVER. Hence the unreal and incomplete status that inevitably results from the maths-axioms-as-is situation.

If you can't still understand that the real camels will OVERLAP/SUPERPOSE with each other if only their NOTIONAL DIMENSIONLESS POINT CENTERS are used for the 'abstraction to sets of dimensionless points', then you miss what I and others have been pointing out about the 'infinitesimal' and the 'zero' problem as exists in the current maths abstractions that are pure philosophy when examined closely.

If you have time, go back and find Quantum Quack's excellent discussion points regarding the physically 'non-nothing' (non-dimensionless) zero to which his physical sphere reduces without ever actually reducing to 'nothing' and disappearing in the reality physical context. Then go back and read all the stuff I and others have been pointing out, which you apparently have missed, that highlights the subtle nature of the INFORMATION/REALITY aspects which the the current maths axioms/practices/techniques of abstraction and philosophy are EFFECTIVELY ELIMINATING in the process;hence the dooming to incompleteness thereby.

Your above example says it all. The real camel's 3-D extension parameters/information is LOST when you do that current-maths-practice of abstracting the camels into a purely limited INCOMPLETE construct 'set of dimensionless points'. See the problem? One of the many problems already covered even if you have been missing them?

Ok. The guest is nigh, so think about that one point above regarding the loss of information when abstraction/phiolosophy insidiously enters the supposedly 'rigorous' maths-as-is-developed/practiced-to-date.

See ya round, arfa, everyone!

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Going backwards:

Information is just something we define, again this is completely arbitrary. If I want the information to be the total number of camels I own, then I can safely ignore that each camel is 3-dimensional, occupies space etc. That is, it's perfectly safe to count up a total number of elements in some set, be those elements camels, past presidents of the US, whatever they are. They can also be numbers and therefore not physical, but abstract.

If you really think Quantum Quack's "model" hasn't been demolished conclusively (it describes nothing useful), then you have a long way to go holding up your "math is flawed" theory.

It doesn't lose the information I want to recover from a set of objects named camels, which is: how many camels are there? I don't want to know how much each camel weighs, how many humps it has, what sex it is, or any other information than a single number: the size of my set of camels. You present a non-argument.

 

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Hi arfa.

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Sorry for my tardiness in getting back to you.

See how seductive and insidious the human mind's 'preference' for abstraction/philosophy is? It just led you to lose touch with the 'camels in reality' units/dynamics/properties etc, and dropped you in a fantasy 'dimensionless points infinities' abstracted-number-sets 'world' of your own making. And do so without demure, or even a murmur or a fleeting questioning of that process inbuilt into the human psyche and our currently all too human maths based on abstraction and philosophy from the outset and throughout (as I have pointed out already).

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And no; information is related to the actual existence in energy-space dynamics of features having discernible properties, parameters and interaction profiles and extents in 3-D physical universal phenomena.

Once you abstract the camels into NOTIONAL DIMENSIONLESS CENTERS via sterile 'labels/numbers', then you are no longer able to MODEL REALITY of the camels' individual and collective extents, interactions and superposition properties and effects in the camel dynamics which occur in reality between real camels and not 'dimensionless points' abstractions represented by numbers/symbols which have lost all reality context of 'camels' from which you abstracted the dimensionless abstractions.

The problems are all of a piece from the 'dimensionless point' notional starting axioms, and all the obvious non-sequitur results which follow (undefined, undetermined, infinity upon infinity etc), as I already mentioned more than once before now.

Get it? The very abstraction into numbers makes the 'set' applicable to ANY number of dimensionless points, irrespective of the original 'units' from which the numbers were abstracted.

Sure, its useful to have 'techniques' which apply to some GENERAL case, irrespective of the origins/units involved in reality physically, BUT the technique is obviously limited and flawed when it comes to all those things I have been pointing to that need to be catered for in the starting axioms if we are going to COMPLETE both the maths and physics ToE models.

That's where I am coming from and going to; looking/reviewing to identify and remove all purely philosophical notions from the get-go of the maths construct/practice/outputs/modelings. While it's a lot of 'fun' exploring fantasy abstract/philosophy based 'worlds/models', it is not what the MAIN GAME of both maths and physics is about: The REALITY context and how best to model/explain it without unnecessary confusing/contradictory and just plain 'un-real' overlays. The maths-as-a-gaming tool of fantasy is OK in its place, but that place does not extend to the reality.

Let's have two maths systems, one for all the fantasy/abstraction 'gamers', and one for th reality freaks wanting to complete the REAL maths and physics consistent and without undetermined etc 'axiomatic nonsense outputs' and so-called 'unreal physics explanations' which are now holding us back from completing the reality math/physics ToE.

Sorry I can't stay and chat. See ya again whenever, arfa, everyone!

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Yes, anyone who counts things like camels, is mapping an abstract set of numbers to a set of real things. This is well understood. And the numbers can be mapped to any kind of things including themselves.

You're saying that counting things is a problem of some kind.

As to being able to model reality, if I have n camels and count them and still have n camels, what have I failed to model? As long as I don't lose any camels then I have all the "camel information" I need. How is any context or information lost if I count n camels? I know what a camel is so I don't need to take pictures to remind myself, or have a taxonomist check each animal and confirm each is indeed a camel. I don't need to do anything ridiculous.

What you seem to be saying is clearly ridiculous, so are you saying counting things is a problem--information is somehow lost--or what?
Have you started to notice yet that your ideas seem, hmm, more absurd than not being able to divide by zero?

 

Hi arfa.

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Another flying visit for other reasons. Can't stay long, so briefly...

Sure, like I said, let's have two maths, one for minimalist/abstract purposes which some may like in certain contexts/games etc., and another for the full and complete reality modeling requirement if the physics reality ToE is to be modeled properly and completely consistent. No sweat. There are already different maths/techniques for different domains/disciplines of applicability/expediency. Why not go the whole hog and come up with another maths that starts and ends in reality context and suitable for the ToE modeling required?

And sure, if all you want is the number of camels info, no problem. But if you want to know what the camels are getting up to and GETTING PREGNANT via social interaction, you may still have the same 'number' of camels for a while, but you want to know if you are going to have more than what you had before soon, and to plan/control which camel mates with which to control for inbreeding/type of camel wanted by the buyers in a few months time and if you can supply extra camels or not from the breeding rate/efficacy etc etc.

See? The real life uses of all the real life modeling is crucial if what you want/need is much more than just sterile/static numbering/labeling of your camels just for the abstract exercise of it?

That is what reality modeling for the complete and consistent real life universal phenomena is all about. We need a more contextually complete and non-nonsense-outputting axioms/maths system for the ToE to be possible to complete in real explanatory terms and not just more abstractions upon abstractions without any underlying information/understandings of what is actually happening and what the actual 'units of physical effectiveness quanta' entities (ie, as in the 'camel units' of your exercise

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) exist and interact at the quantum level and the macro level too. G'night!

 

mmm... one hump or two...

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@Arfa_brane,

To demolish something conclusively as you suggest requires at least a refutation yes?
I have seen no refutation of the claim that a ball reduced to a diameter of infinitesimal still retains 3 dimensions yet contains non-existent space as volume.
If there is such a refutation please provide a link to clarify the situation.

Or would you like to have a go at refuting it now?

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...and if you can not fathom the usefulness of the above then that's your problem not mine.

 

Certain contexts such as counting things?

Look, the guy counting camels only wants to know how many there are "now", it's not a complicated exercise.
It's fairly likely they know that camels can produce more camels, but they're probably able to count them all before this happens.

Counting or tallying animals in a herd isn't an abstract exercise; farmers and stockmen do it as part of their job.

And yet, farmers and stockmen have all they need to count animals, something humans have done for thousands of years. How is the number system they use an "incomplete" system? What does it fail to do, given that it's been fit for the purpose for as long as it has?

You can't really say, right?

 

No, you've failed to see any refutation, probably because you don't want to.

A sphere cannot have both a radius and zero volume unless the radius is zero. Your claim is a contradiction and refutes itself.

 

no .. it is your math that exemplifies the unavoidable paradox.
Maybe if you could tell us all what the volume of the ball/sphere with a diameter of infinitesimal is and show that it is >0.? I am all ears seriously...

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and if the diameter is infintesimal ? what then?

 

An infinitesimal is hard to define because there is no real number greater than zero which is smaller than any other real number greater than zero.

But suppose you can define a sphere with an infinitesimal radius. Then it must have a nonzero radius, so it must have a nonzero volume.
A sphere is what it is, and if it has a zero radius it has a zero volume, i.e. is the 0-sphere and equivalent to a Euclidean point.
There is no unavoidable paradox.

 

yeah... keep going... we still have a sphere with an infinitesimal diameter that has a volume of what?
It logically can not be an infinitesimal volume as that would mean that infinitesimal is less than infinitesimal.
It can only be zero or non-existence... thus proving the paradox exist.

 

Sorry.
\(0 \leq r_1 \lt r_2 \quad \Rightarrow \quad 0 \leq \frac{4 \pi r_1^3}{3} \lt \frac{4 \pi r_2^3}{3} \)
and thus it follows that
\(0 \leq r \lt \sqrt{\frac{3}{4 \pi}} \quad \Rightarrow \quad 0 \leq \frac{4 \pi r^3}{3} \lt \sqrt{\frac{3}{4 \pi}} \quad \Rightarrow \quad 0 \leq \frac{4 \pi r^3}{3} \lt r\) where the rightmost inequality cannot possibly be a statement of geometric truth since \(\sqrt{\frac{3}{4 \pi}}\) is number, not a length or volume.
So when x and r are restricted to be lengths, it follows that
\(\forall x \left( 0 \lt r \lt x \quad \not\Rightarrow \quad 0 \lt \frac{4 \pi r^3}{3} \lt r \right)\)
because length and volume are expressed in different units and so the expression \(\frac{4 \pi r^3}{3} \lt r \) is necessarily geometrically absurd.

You have misapplied algebra in a question of algebraic geometry and thus have failed to demonstrate that geometry requires a least magnitude positive number.

 

So in summary, what do you calculate to be the volume of a sphere that has a diameter of infinitesimal?
Possibly to be even more specific: if the surface area of a sphere is infinitesimal what is it's volume?
it must be less than infinitesimal unless I am missing something...

 

If you are in a system of numbers that includes non-zero infinitesimals like the hyperreals or like the surreals, then it follows that if \(\epsilon\) is a positive number less than any positive rational number, then \(0 \lt \epsilon^4 \lt \epsilon^3 \lt \frac{4 \pi}{3} \epsilon^3 \lt \epsilon^2 \lt \epsilon \lt \frac{21}{43} \lt \sqrt{\frac{3}{4 \pi}}\).
Thus \(\epsilon\), if positive, is not the smallest positive number.

If you are in a system of numbers that does not include non-zero infinitesimals like the real numbers, then it follows that if \(\epsilon\) is a positive number less than \(\sqrt{\frac{3}{4 \pi}}\), then \(0 \lt \epsilon^4 \lt \epsilon^3 \lt \frac{4 \pi}{3} \epsilon^3 \lt \epsilon^2 \lt \epsilon \lt \sqrt{\frac{3}{4 \pi}}\).
Thus \(\epsilon\), if positive, is not the smallest positive number.

If you want to propose a closed system of number \(\mathbb{X}\) where \(\mathbb{N}_0 \subset \mathbb{X}\) and \(\forall n \in \mathbb{N}_0 \quad \left( \frac{1}{n + 1} \in \mathbb{X} \right)\) and \(\exists \epsilon \in \mathbb{X} \; \forall n \in \mathbb{N}_0 \quad \left( 0 \lt \epsilon \lt \frac{1}{n+2} \lt 1 \right)\) and \(\forall x \in \mathbb{X} \forall y \in \mathbb{X} \quad \left( 0 \lt x \lt 1 \wedge 0 \lt y \lt 1 \; \rightarrow \; 0 \lt x \times y \lt x \wedge 0 \lt x \times y \lt y \right)\) then it immediate follows that \(\epsilon\), if positive, is not the smallest positive number.

So your mistake is assuming that you can have one and only one positive infinitesimal and still have a self-consistent theory of ordered numbers.

 

ok... Thanks for that... so what can I say is the volume of a sphere that has a surface area of infinitesimal?
or are you saying that the sphere is impossible according to those number systems available?
Please excuse my inability to read your notations.

 

I'm saying to you "infinitesimal" is not a number even in number systems that allow infinitesimals. You have to be specific about which infinitesimal you mean at every step. That's why "x" exists in algebra -- it exists to name a specific number which you know by it's properties. (Say \(x^2 = x + 1\) and \(x > 1\). ) You name the number from it's properties and use algebraic manipulation to determine the identity of the number.

But just calling something "infinitesimal" is not a property that pins down a number to be specific. You have always acted like there must be one such smallest positive number, but that is not a general property of numbers less than 1. Therefore it follows if there is one such positive number smaller than any positive rational number there must be an unending progression of numbers with the same properties.

This follows because the square of a positive number is a positive number and the product of any two non-zero numbers of magnitude less than 1 is a third number with magnitude smaller than either of the first two.

Even if you want to claim there is a number \(0 \lt \epsilon \lt 1\) such that \(\epsilon^2 = \epsilon \) and \(\frac{1}{2} \epsilon = \epsilon \) then you haven't made \(\epsilon\) a number because \(\epsilon = \frac{\epsilon + \epsilon}{2} = \frac{\epsilon}{2} + \frac{\epsilon}{2} = \epsilon + \epsilon \). And thus \(\epsilon = 0\) contrary to hypothesis.

Likewise assuming \(\epsilon^2 = 0\) and \(\frac{1}{2} \epsilon = 0 \) leads to \(\epsilon = 0\) contrary to hypothesis.

If \(x\) is name of the surface area of a particular sphere, then it follows that \(\frac{x^{\tiny \frac{3}{2}}}{6 \sqrt{ \pi }}\) is the name of the volume. And if \( 0 \lt x \lt 36 \pi \lt 113 + \frac{4}{41}\) then it follows that \(0 \lt \frac{x^{\tiny \frac{3}{2}}}{6 \sqrt{ \pi }} \lt x\).

 

Ok.. thanks. It appears the problem is directly related to the use of the term infinitesimal.

What say we swap this term with the fundamental Planck length instead?

What would the volume of a sphere be if the diameter was 1 Plank length?
Would it change the math at all?

re: wiki http://en.wikipedia.org/wiki/Planck_length

which is probably closer to the mark with regards to the Uncertainty Principle and the reality of the the zero point.

edit: @ alfa_brane, rpenner

I consider the notion I put forward to be utterly and conclusively refuted ... thanks for your patience.

 

Hi arfa, everyone.

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While mathematicians may be satisfied to just count the things, physicists like to know ALL ABOUT what the real things are, in all their physically EFFECTIVE states and interactions and effects, so that ALL the physical universal phenomena can be explained and not just 'abstracted' only to the abstract 'counting points' stage.

I'm with the physicists.

That's it for now from me on this, arfa, everyone. I have said all I wanted/can say for the time being (pun intended). Have fun in your discussions, guys! Cheers!