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

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Neither, but yes there can be many names for the same object (in math or in the real word).

What I said, clearly, was that two (or more) definitions can point to the same object, set, thing, or point. I don't see how anyone fluent in English could mis-understand, especially as I gave the example of the evening star (defined obviously as the bright light point in the evening sky near where the sun was) was a different concept, differently defined, with different things that could truthfully be said about it than the morning star, even though in fact both definitions point to or are other names for Venus.

 

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I read both 146 and 152. from 146 you correctly IMHO say:
"As to what the "0.9999..." example represents, from my observations of present and past discussions about this very thing, it appears that that example is only consistent/applicable to results/treatments within the axiomatic system where it arises as part of the conventions and assumptions therein. It is not intended, as far as I can gather, that it should be taken 'out of context' from the relevant maths-only construct/concept abstractions into the reality context as such. "

But then ignore this in most posts, like in your now red text quoted above, seeming to insist the math applies to reality, even when doing that produces a conflict.

There is no Philosophy in math only logical statements that follow from the axioms* (assumptions). One of the usually axioms of the geometry part of math is that parallel lines never intersect / cross each other, but if that axiom (assumption) is not postulated, then a differ geometry follows. There is NO reason to think either MUST apply to the "real world."

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Consider a mathematical “pulsating sphere.” I. e. one whose radius is given as a function of time by r(t) = 1+ sin(wt) where the w has been chosen such that the pulsation period is one second. Those who know that the time derivative of r is the surface speed, S, will confirm that for this pulsating sphere: S = (w)cos(wt) which has it maximum speed, w (or -w) when t = 0 and every half second after wards. The surface is instantaneously stationary, S = 0, half way between these speed max points in time. The volume, V = (3/4)(pi)r^3, is max when r = 2 and half second later is the least. The max volume is 6pi and the least volume is zero.

These are true statements one can make about this sphere that one can not make about any point, which has only location, not volume or surface area. They are not invalidated because the sphere's volume is instantaneously zero at one second intervals. It is non-sense to claim that based on your intuition when the math PROVES otherwise.

* They are given this special name, not just called "assumptions," because we don't assume they are true (what ever that might mean) in the "real world."
THEY ARE POSTULATED TO BE A "VALID BASIS" FOR WHATEVER CAN LOGICALLY FOLLOW.

 

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You made the claim that a sphere of zero radius has many points all occupying the same location at the center. Is that what you believe? Did you want me to go back and find your own quote and show it to you?

 

Yes. Please do that as I do not think (recall that) I said anything about how many points were on the zero area surface of a zero radius sphere.

I think their number could still be infinite, but agree there is no way to distinguish them from the point which is the center of the sphere as points have only location.

The problem is quite like the point where two lines intersect. There too I would tend to say that each line has a point at that same location. Do you have any reason to think that wrong?
I. e. to think that points defined differently can not be at the same location? If you do, does that intersection point cease to exist for line A or for line B?

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One divided by three. You can't divide one by three, nor can you divide 10 tenths equally into 3 piles, or 100 hundredths equally into 3 piles etc. The operation of division is not completed, which is represented as .... after the progress of the operation, ie .9999999...

You can have a pile of 100 hundredths and try to divide that number of hundredths into three equal parts, but there is always a remainder of 1, just as in my penny example where you can make 3 piles of 33 but there is always one penny left over to split into 3, and the operation continues, and it will never be completed due to the remainder (1) not being equally divisible by 3.

This is not just a problem of the math not matching the reality, it is a problem that clearly shows the fact that math can not divide 1 by 3 without a remainder that needs to be split into three once again...

 

Who falsely said Math matches real world "reality?" Beliveing that it should is one more of your basic problems. Math is a closed tautology. Closed means only things logically derived from the axioms can be stated - nothing added from outside by MD is allowed.

 

So you are trying to have your cake and eat it too?

Don't bring up any math in discussions about the real world because it's not applicable to the real world and will only be severely misleading to those that think math is applicable to the real world. It's convenient for you to brush off the issues math has as "math doesn't have to match reality" but then later when someone asks a question about physical systems in reality you respond with...you guessed it...math! Either keep your math separate from discussions about real physical phenomena or admit that your math needs to be fixed before it can attempt to correctly address reality!

 

Yet, eighth graders know how to divide a circle into three equal parts exactly. Why can't you?
Yet, eighth graders know how to divide an equilateral triangle into three equal parts exactly. Why can't you?

 

The circle has an area of 1 m^2. You divide it into 3 parts. What is the area of each part and what is the remainder left over that has to be divided by 3? You know that 1.0 equals 10 tenths, right? You know that 9 tenths is less than 10 tenths, right? You know that regardless of how many 9's are after the 9 in the tenths column that it will never be equal to a 1 in the ones column, right? 1.0 is greater than .9, it doesn't matter how many 9's follow the .9, it will always be less than a 1 in the ones column. 9 tenths and some change doesn't make 10 tenths, it is LESS THAN 10 tenths! It is only 9 tenths and some change. The only way to complete the operation is to make a pile of 100 hundredths into three piles, one pile having 33 hundredths, one pile having 33 hundredths, and one pile having 34 hundredths, for a combined total of 100 hundredths. The three piles are not equal! Do you know how to count?

 

This explains it, you never got past 5-th grade, where they teach integer division. One day, you might get into sixth grade and , if you manage to abandon your kooky ideas, you may even get into 8-th grade (in the very distant future). There, you would learn how to divide ANY circle into three equal parts.

 

What is the area of each of your "equal" parts? If they don't total 1 m^2, why not? Did you quit dividing the remainder at some point and give up and just put a ... at the end of your answer for your less than equal pieces that don't total 100%?

 

Hi Billy T.

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Mate, I'm NOT trying to use my "intuition" at all in this. I am philosophically examining the axiomatic gaps (as highlighted by "undefined" and " Limits/Infinity" PHILOSOPHICAL aspects ALREADY brought BY THE PROFESSIONAL MATHEMATICIANS over recent centuries into the current maths lexicon/practice, seemingly UNNOTICED by anyone that these things ARE THEMSELVES "philosophical" in nature).

Ok? So please calm down and stop ascribing things/stances to me which are NOT AS YOU MAKE OUT they are. OK? I trust that is now clear and NO MORE MISUNDERSTANDING will persist about where I AM COMING FROM in these discussions? Yes?

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Anyhow, since the discussion IS the question of the CURRENT AXIOMATIC SUFFICIENCY/DEFICIENCY when it comes TO trying to apply maths in reality contexts (such as the examples in post #146 and #152), then it is ENTIRELY PROPER that we examine those current axioms THEMSELVES, and NOT just keep repeating (like you are doing again) exercises and 'proofs' which FOLLOW those current axioms.

Get it, mate? We already KNOW what comes of following the current axioms. That is the point. We are discussing what can be done to ENHANCE those axioms so that in future we may follow BETTER more contextually grounded axioms which will NOT give the usual "undefined" and require "limits/Infinity" to paper over the gaps resulting from following the current axioms as they stand.

The point of these discussions is to GO BACK TO SCRATCH to see what may have been missed during the INITIAL philosophical PROCESS which GAVE RISE TO the INITIAL AXIOMS which has led on to the current axiomatic SET/FORMULATION.

Ok, mate? We are NOT JUST DOING MATHS following current axioms. We are examining those same axioms ANEW and see where it leads. So PLEASE stop ascribing "intuition" etc to me/others; and stop dragging us back into the current maths axiomatic practice/results which we all know about by now; and INSTEAD just concentrate on understanding/assisting the THRUST of these examination/discussions of current axioms with a view to RE-exploring their foundations and hopefully thereby CONTEXTUALLY enhancing them to the point they no longer give the usual current "undefined" et outputs which arise because they are not contextually complete. That is the point of the discussion here.


Anyhow, thanks for your contributions to date, Billy T; they have been useful to both clarify and distinguish the various subtleties involved in discussions/explorations such as these! Cheers, mate.

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You will learn that in 6-th grade, where they teach you division of real numbers. So far, you are still stuck with integer division. (5-th grade).

 

Tach, can you take us all through the reality process of 'sharing' a DIMENSIONLESS 'endpoint' in reality context? Thanks.

PS: Just making abstract axiomatic assumptions is not what is asked for. What is asked for is the reality process of effecting such a 'sharing' of such an 'endpoint' in this reality context. Thanks.

 

Billy T you have suggested that a zero rad sphere would have zero volume.
If it is a 3 dim. sphere then how can it not have volume?
or in a reality context:
If the diameter of the sphere is 1/infinity [infinitesimal] therefore having a 3 dimensional infinitesimal material existence,
What is it's volume?

The reason why this issue is so critical is that in philosophical terms and in physics terms we have the issue of ex-nihilo, "something from nothing".
A pulsating sphere that reduces to zero is in effect saying that it just emerges from nothing to become something and then back again.

The clarification regarding the "real" existence of an immaterial void as compared to an abstracted zero [ maths ] is the reason for my inquiry.
[and why the axiomatic use of zero is in question]

 

As soon as you manage to solve the 8-th grade exercise of dividing a circle in three equal parts using a ruler and a compass. You still owe the solution to the exercise.

 

Hi Tach.

This was a question put to you pursuant to YOUR 'suggestion' to rpenner that you could 'share' the endpoint. I therefore asked YOU as follows::

Now your answer below is an EVASION pure and simple...

Please remember that you have been banned before for such dishonest evasion etc tactics.

So please answer the question asked because YOU made, to rpenner as above quoted, the suggestion of 'sharing' the 'endpoint'. Please show how this 'sharing' of a DIMENSIONLESS (as rpenner confirmed) abstract 'point' is done, precisely, in reality.

Any more dishonest evasions and you will be reported. Thankyou.

 

If anyone is evasive, that one is you. I have given you the simple exercise of dividing a circle in 3quite a long time ago. It's been 4 days and you still don't know how to solve it. You have a lot of difficulty with an exercise that is solved routinely by 8-th graders, yet you pretend (much like chinglu) to be discussing higher level concepts.

 

The ONLY improvements possible to existing axioms would be to remove any ambiguities, of which AFAIK there are none. You could remove an axiom and have different consequences follow, or add new ones. There is a whole lot of literature on the geometry that result when the axiom that parallel line do not intersect is removed to allow that they can.

With in the very useful normal set of axioms new concepts, like imaginary numbers, have been well defined and are quite useful in the real world, especially when describing the behavior of voltage and current relationship in circuits with reactance; however there is no claim within the mathematics that imaginary numbers will have any real world application. Some electrical engineers have discovered that they do (and that the response of linear circuits to complex wave forms is most easily achieved by expressing those forms in terms of their Fourier* components). Likewise quantum physics makes great use of mathematics, but mathematics it self is a CLOSED TAUTOLOGY and like all such closed tautologies can make NO CLAIMS of application outside its closed domain. There are whole fields of mathematics that AFAIK are of interest only to mathematicians. (Perhaps parts of number theory are totally without any real world application, but large prime numbers are very valuable for making nearly unbreakable codes.)

If you apply some math to a real world problem and it is helpful, fine, but if in some situations doing that leads to a paradox or confusion, the fault is with you, not mathematics. Mathematicians warned you up front that it was a CLOSED TAUTOLOGY, based on an arbitrary set of postulated axioms. For example 41/333 is completely well defined both in the real world and in mathematics when also expressed as a decimal too. I. e. as the infinitely long sequence 0.123123123123... but the real world has problem with 41/333 expressed that way as all calculations are finite.

SUMMARY:
It is non-sense to speak of "CURRENT AXIOMATIC SUFFICIENCY/DEFICIENCY." (Sufficiency or deficiency for what?) Yes, the logical deductions from the current, most commonly postulated axioms can be misapplied.
All I have seen posted here faulting mathematics is ignorance and wasted time. If they don't like the results the current most common set of axiom produces, then suggest changes in the current set, either by adding new axioms or deleting one or more.

* Fourier died in 1830. AFAIK, his transforms had no real world application for ~100 years. Mathematician tend to be "Pencil & Paper" explorers. - That reminds me of old joke: University Administrator called head of Physics Department in complaining about their large budget request asking: "Why can't you be more like the Math department?" Physics head said: "We need lots of experimental equipment. All they need is paper and pencils and their greatest expense is for waste baskets."