Actual Infinite?

[Truth_Seeker]

I am trying my hardest to come up with an example of an actual infinite, versus just a potential infinite, without having a platonist rationale. One book I was reading suggested the number of positions that, say, an electron could occupy on a yardstick.

 

I guess it could truly be anything: a ping pong ball and football field, or even a golf ball and a line segment 1 mm longer than the golf ball's diameter. But, my hangup in this, is if there are truly an infinite number of points it could occupy, then how could you roll that ball the entire length of your measure and cover every position if it truly had an infinite number of positions?

 

Like I said, my thinking is more nominalist than platonist, so I'm holding the position that points or numbers would be classified as potential versus actual.

 

Where is my inconsistency? Thanks in advance for any ideas.

[the tree]

The short awnser is that we don't deal with an actual infinity.

 

Imagine the golf ball travelling at exactly 1ms^-1, then pause it at any point, such as after a billonth of a second where it will have travelled a billionth of a metre.

When treating time as just another axis, you will find just the same amount of points that you could mention, so there shouldn't be a problem.

 

Incedently, in the real-world (why people bother so much with this is beyond me) you are actually constrained to Planck units, so there isn't so much an infinite amount of points anyway.

[bascule]

Stanford mathematician/computer scientist Donald Knuth gave a lecture in which he described what he considered the impossibility of distinguishing the infinite/infinitessimal from the finite via physical measurement, due to the unbounded nature of finite numbers. One can only prove finiteness by reaching finite limits, but how can infinite quantities ever be distinguished from finite ones? Knuth's point was that they can't, you will simply continue to measure endlessly.

[Rocket Man]

the problem i see with defining infinites, is that we have a digital number system, you cant make an infinitely acurate measurement with a digital system unless the measurement is like an integer.

essentially im talking about the difficulty in defining .9 recurring as different to one.

the difference IS equal to the inverse of infinite, which is by definition, 0.

but an analogue number system (impossible by all practical counts) would define an infinitely small difference by using infinite precision.

(the .9 recurring argument is a quirk of the base 10 system, a base 3 system would define this equality beyond ANY doubt what so ever)

 

mathematics says there can be infinitely small differences between numbers. thats what a surd is all about, it requires infinite precision to define in a base 10 system, so they take one step back and define it with the root of a number.

[the tree]

Our number system is analouge, we just a digital way of representing said numbers. Besides, our digital system is bloody good: I'm sure there are no real numbers that our system of notation cannot represent acurately, and I doubt you'd be able to think of any other numbers that our system can't manage either.

 

(Oh, and just to clarify, there isn't any doubt about the 1=1 thing, for those who know what they are talking about.)

[Sisyphus]

But' date=' my hangup in this, is if there are truly an infinite number of points it could occupy, then how could you roll that ball the entire length of your measure and cover every position if it truly had an infinite number of positions?

[/quote']

 

This and several, similar variations are what's known as Zeno's Paradoxes. They all have to deal with the trickiness of infinite divisibility. One way to deal with it is to say that potential infinity (i.e. infinite divisibility) is not the same as actual infinity. There's no limit to the number of positions you can find, but there isn't an actual infinity of things there inherently, just a single, continuous motion. That's a simplified Aristotle's solution. Newton, basically, says that yes, there's an infinite number of positions, but also an infinite amount of time to do it in (since both time and space are continuous and infinitely divisible), so there's no problem.

[Truth_Seeker]

Thank you very much. You all have been very helpful and given me some great things to think about.

[Rocket Man]

Our number system is analouge' date=' we just a digital way of representing said numbers. Besides, our digital system is bloody good: I'm sure there are no real numbers that our system of notation cannot represent acurately, and I doubt you'd be able to think of any other numbers that our system can't manage either.

 

(Oh, and just to clarify, there isn't any doubt about the 1=1 thing, for those who know what they are talking about.)[/quote']

 

i was thinking of numbers like e and pi. theres no way to define them exactly with a base ten system unles you use an infinite number of decimal places or the limit of a function (useful in most instances).

 

our digital system has had over 30 centuries to become this good, lets all have a minutes silence...

 

you can apply the same mathematical rules to binary, so with enough power a computer can give an analogue signal?

there are key diferences that musnt be forgotten, our system is not analogue but the vast number of notation types available allow us to define such numbers with precision.

[CanadaAotS]

Hmm... this is a digression, but talking about number systems

 

Has anyone heard of the "infiger" system.

 

It was a math problem I came across about some race of aliens starting up their own number system. They were doing well enough with integers up to 10, but they soon needed fractions and such...

 

The race didnt believe in things having no end (aka .33333...) so they made them START with an inifinite amount of numbers, and all the numbers would be integers (no decimals) hence the name infigers.

 

like 1/3 was ...6667 I think

 

5 would be ....0005

 

-7 would be ....9992.

 

I'm getting to into to this, but if you want to know more, google it you'll probably find something. I may just make a thread on it too.

[the tree]

i was thinking of numbers like e and pi. theres no way to define them exactly with a base ten system unles you use an infinite number of decimal places or the limit of a function

There are plenty of ways, here is one for pi: [math]2\int_{-1}^1 \sqrt{1-x^2}\cdot dx[/math]

you can apply the same mathematical rules to binary, so with enough power a computer can give an analogue signal?

No. Computers don't just use binary as a number system, that would be silly. Computers have wound up using binary because of the sheer usefulness of transitors and valves which in turn make a genuinely analouge signal impossible.

our system is not analogue but the vast number of notation types available allow us to define such numbers with precision

If our number system is not analouge then tell me, exactly how many numbers are there between 2 and 3?