Math of infinity

Reply Fri 15 Aug, 2014 08:24 pm
First of all infinity is not a numeral like 7 or 1292029. Infinity is not a measurement as 12 apples or 3 liters or a bazillion light years, it is a concept. Two ways to think about it is: increase without measure or as big/small as you need. Infinity can be thought of as developed as a way to maintain mathematical validity while allowing us to get closer to the limits of our human construct capabilities than we could get without its use. It allows us to get closer to the (currently) insolvable paradoxes or other limits of the current formulation of mathematics, while not trashing mathematics validity along the way.

I would hope we could agree that the numeral 7 if written in Arabic numerals or in bumps on the page of a brail book or any other representation is completely understandable and we can all agree what it means, it does not depend on any other information. The numeral six comes before the numeral seven and after the numeral five (in current convention) for example.

I would also hope we could agree the measure seven liters or seven birds or seven galaxies are equivalent in as much as we can put them into one to one correspondence.

Again I would hope we can agree that no matter how we display the numeral seven, it is the same old numeral seven. Further, that we can agree that no matter how we come to the measures above they must provide the same answer, i.e. any valid method of measuring seven birds would all agree on what that means and if we got another value it must be in error.

This is not the same as the concept of infinity. Unless infinity is defined within the context used we would be hard pressed to agree on what that means. You can count or point to seven birds you cannot point to or count to infinity.

Further the concept of infinity does not have to be and usually is not always the same. For example: you get interested in astronomy and buy yourself a telescope which then is badly jarred. Now you want to know if the optical elements were jarred so hard that their spacing has changed. You look in a book and it says “a simple test is to point your telescope at an object at infinity, then attempt to get a clear focus of the object, if you can do this the spacing is still ok.”

Now the problem is not only what object is “at infinity” but how can I see it in a telescope when no signal including light can reach me from infinity. Hmmmm.

You call your friend who is more experienced and who got you into this mess and ask him what object you can point to and see that is “at infinity.” Your friend says “either point it at the stop sign down the street or at one of your next-door neighbor’s lug nuts on the car in his driveway.”

You think your friend has been smoking one of those “left-handed” cigarettes. Now you may not know exactly how far “at infinity” is but it sure the heck ain’t next door or down the street. You don’t know what your friend knows – that in this context “at infinity” is defined as when all light paths from the object hitting the primary’s optical plane are parallel to each other.

You then think, big help, how in the heck far away is that exactly. Your friend tells you that there is no one answer, it depends. It depends in some very small amount on the type of optical system but mostly on the focal length of your scope. Here is where the concept of infinity diverges from a numeral or measure. An object 100 feet or a bazillion light years away from any telescope is 100 feet or a bazillion light years away, period. However, until infinity has been defined for the specifics of the scenario it is a moot concept.

[This is where I am having a premature “senior moment” and cannot remember exactly so I think I am correct with greater than 2 times the focal length but to play it safe let’s say this is an example.]

In an optical system such as a telescope those infinite paths occur, and so an object it “at infinity”, for any object greater than about 2 times the focal length of your scope. You have a 4 inch refractor working at f/15 so your focal length is 60 inches, meaning any object greater than 120 inches away (in this example) is “at infinity.” Your friend’s scope has a focal length of 600 inches so at infinity for him is at greater than 1200 inches. Your infinities certainly do not match, and you can chuck a rock as far as “at infinity.” Yet an object one hundred feet from either scope is 100 feet away for either scope.

In addition if you add a 3x tele-extender to your scope its focal length increases to 180 inches so now “at infinity” for you is not 120 inches but rather 360 inches. You have changed what “at infinity” means using the same scope but simply changing its focal length. Clearly infinity is not the same as a numeral or measure.

While others probably have already given you better explanations than mine I would hope you can see from this explanation that until you can change your concept of what infinity is you will never be able to understand a simple optics concept, let alone try to discuss esoteric cutting edge applications of this concept or have someone explain them to you. It is an exercise in futility (as if that isn’t very clear already).
Reply Sat 16 Aug, 2014 08:05 am
Thank you Zara for that posting. I think I understand it and have no quarrel with it but I can't see how it bears upon my OP

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Reply Sat 16 Aug, 2014 11:56 am
...until you can change your concept of what infinity is...
Zara I'm not entirely sure, not being a mathematician nor physicist. I'd suppose in an infinite Universe if one could travel in a straight line he'd proceed forever without returning, observing something entirely new with each passage through an area containing matter

..the q before the house whether by the sheer exercise of chance he'd eventually come upon another environment very similar or nearly identical to the one we're seeing now, then later maybe another such, ad infinitum

Maybe your apparent misunderstanding arises from my use of the term "Math" with regard to the concept of infinity. Please see esp items (1) thru (4) at the end of my OP
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cicerone imposter
Reply Sat 16 Aug, 2014 02:40 pm
The only factual truth is that there is always change.
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