The Nature of Infinity in Mathematics and Reality

upgrade

Hello all!

In the past few months I've stumbled upon an issue that has played games with my mind. I feel I need some help to solve this, as I've tried various other sources and remain without answers.

Firstly, I was confronted with a mathematical proof which states that 0.9~ (to infinity) = 1. The proof is simply:
x = 0.9~
10x = 9.9~
9x = 9
x = 1
This bugs me, for from a purely logical standpoint it seems ludicrous to claim that two different numerical representations are representative of the same quantitative value.

As a result of studying this issue I have also run into a problem with my whole perception of reality. All my life, I have considered mathematics to be an ultimate truth which has certain affinity with reality. Thus, what is proven in mathematics, has for me been enough proof for such a truth in reality. In mathematics, infinity exists and is essential to the functionality of the number system. In reality, however, I struggle to find an example of infinity which can be tied directly to its existence in mathematics.

To further this conundrum, I have realized that, should there be infinite distance between any two given points in reality, there can be no points to begin with, for a point which is endlessly broken down ad infinitum has no end. My entire perception of time and space has been confused, and I'd like some help to understand why.


Thank you in advance.

pwsnafu

1/2 = 2/4

Be aware that the whole point of the equals sign is this.

We can use mathematical models to study the real world, but mathematics itself is only applying logic to axioms. It is independent of reality. You can take something which is impossible in real life and study it mathematically. Banach-Tarski theorem is a good example.

Warning: snarky answer!

Are you part of reality? If you are, and if you have a mental intuitive category called "infinity" somewhere in your in mind, then that philosophical construct is also part of reality. We can study that using mathematics.

In mathematics, the concept of distance is usually codified by what we call a metric.
One property of a metric is that between any two points, the distance is always finite.

The idea that points can be infinitely far away is called "extended metric", but every extended metric can be converted into a normal metric which retains the important concepts of convergence and continuity, so its not studied that often.

The continuum is just a model of space. We don't know what RL space actually looks like, and most scientists don't care.

micromass

See our FAQ on 0.999... and infinity: http://www.physicsforums.com/forumdisplay.php?f=207

upgrade

I apologize for I misspoke. I should have said that no two decimal representations are representative of the same value. For example, there is no other decimal representation of ∏ than 3.14159... correct? If not, what other equivalent decimal representation is there?

Is this not a contradiction? To assert, on the one hand, that mathematics is independent of reality, but on the other hand that anything which can be conceived of is part of reality, seems illogical. For is not mathematics a concept? I think it is important in this situation to acknowledge a certain mystery known as mind-matter dualism. Could this not be the mystery which hinders our ability to answer this very basic question?

Apart from this, though, I find it incredibly hard to believe that mathematics is independent of reality, for to claim such would be to brush aside known affinity between the two as mere coincidence. For instance, if one adds 1 rock to pile already containing 1 rock, he now has 2 rocks, which remains a truth in and of itself, apart from how we choose to address it with language.

I'll read up more on this point because I'm having trouble understanding the "metric" concept. Thanks for the topic!

I'm not sure what you're saying here. Is it untrue to claim that there can not be a point to begin with, since all points can be broken down further to infinity?

Thank you! That does answer a few of my questions!

Number Nine

A "purely logical point of view" suggests no such thing. That's ridiculous.
Is the statement [itex]2+3=4+1[/itex] ludicrous? Is the statement [itex]y=x^{2}[/itex] ludicrous? That [itex]0.999... = 1[/itex] is a fundamental property of the real number system. It can be shows to be true with absolute metaphysical certitude given the axioms of the real numbers.

upgrade

As I clarified above, I should have said that no two decimal representations are representative of the same value. Could you give another decimal representation of 3.14159... (∏) that is its equivalent?

micromass

No, the decimal representation of pi is unique. But why is unique decimal representation so important anyway?? I don't see why...

phinds

You are missing the point. The fact that SOME numbers have only representation has no effect on the fact that some ALSO have more than one representation. As has been said already, this is fundamental to the real number system.

upgrade

Ah I see, this was just a dumb misconception I had going on in which I forgot there are rational and irrational numbers! Thanks, sorry for wasting your time on that tangent.

disregardthat

It's actually no particular reason to allow an infinite sequence of 9's as part of a decimal expansion. It doesn't appear in any division process; it doesn't appear "naturally". But we put it in to make any infinite sequence of digits define a number. But some infinite sequences (which incidentally don't appear in the division process) coincide with others (which do).

Deveno

perhaps you should ask yourself (apparently no one ever does), why:

1 = 1.00000000~ (an infinite string of zeros).

certainly, as strings of symbols, these are not the same string (they have differing lengths). it is not just "decimal representations ending in 9's eventually" that have non-unique representation. EVERY finite decimal has a non-unique representation (an infinite number of finite representations, and two infinite ones).

the reason for this peculiarity is "the decimal system", which is base 10. and 9 is just 1 (the smallest digit available) away from 10. similar problem arise if you switch to another base:

in base 3:

1 = 0.2222222222222222~

as far as "proving in reality by proving in mathematics", the situation is thus:

we have a "real problem" (perhaps an engineering problem, for example). we "abstract the situation (model it)" and re-formulate it as a mathematical problem. we then use the techniques of logical formalism math so excels at, to (hopefully) arrive at a simpler expression of our problem (the beam will carry 500 pounds per foot, for example), and then apply this to our "real world".

the question is: our are models appropriate, and accurate? and the best answer we have, is: "it depends". for some things, there appears to be a high degree of agreement between the model, and the "real situation". companies that run state lotteries, for example, use statistics and probablilty very effectively to avoid losing money (so do casinos). the utility of arithmetic in figuring such things as taxes, balancing one's checkbook, and computing how much paint to buy to paint a room is undeniable.

bolstered by such satisfying results, one wonders if it might be possible to model EVERYTHING mathematically. but here, the best minds of our species have run into some snags. the exact nature of the very large, and the very small, are not entirely known. we HOPE that such things can indeed be modelled by mathematics, and indeed many working cosmologists, and sub-atomic physicists, begin by formulating a mathematical model, and checking to see if that is a) consistent with what we know so far, and b) predictive of things we do not yet know. there have been some successes in this vein, and some believe it is the best available approach. realize, however, that you are now in the realm of philosophy, it could very well be the case, that there is something about the universe which we are not capable of modelling (perhaps not even a limit of mathematics, per se, but of our limited ability to apprehend the world).

it is not known if the universe is "bounded" (finite), or not. some theorize it is. some disagree. so while one cannot say for sure if "a completed infinity" exists in nature, one certainly does in mathematics (unless you are an ultrafinitist, which most consider an extreme and untenable position). in fact, one of the basic "axioms" of set theory (the most currently popular foundation for mathematics), is simply "an infinte set exists" (this set is often identified with the natural numbers).

mathematics is "bigger" than reality, in the sense that one can do perfectly acceptable mathematics, without any regard for some "instantiation" of it in the real world. on the other hand, we are in the world, and mathematics is in our heads, which leaves us in the peculiar position of being able to imagine we are bigger than we are. while this leads to no great crisis of faith when we play online role-playing games (imagination in fiction doesn't seem to pose the same sort of epistemological paradoxes), it does lead us to wonder about how we can think about things with such a tangled hierachy of "in".