The Last Conformist
Irresistibly Attractive
Unable to form a mental image of anything? I certainly cannot imagine what living like that would be ...
Infinity...
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Adso de Fimnu
Prince
Conception is exactly what I'm trying to get at here. How did we get our concept of infinity? Was there some greek who thought it up, or what? (I realized there's a connection with math in Cartesian philosophy, but I'm just not that interested in it.) So does anyone have a theory as to how we have the concept of an infinite 'something'.Plotinus said:
Mise
isle of lucy
I don't think you can talk about infinity without talking about mathematics, since "infinity" is a mathematical term, conceived by mathematicians, who also happened to be philosophers. Many philosophers have concerned themselves with the concept of infinity, but only in a mathematical context does any of it make sense. Zeno, for example, made lots of very interesting ... umm... things (I don't know the word for it, hypotheses? theorems?) about infinite divisibility. They're interesting reads. Why would you need to use it infinity anything other than maths? It's just not something that comes up normally.
Marla_Singer
United in diversity
When I try to imagine the mathematical infinite, I'm explaining it to me this way : "Try to imagine the largest figure imaginable and, relatively to the infinity, it will be smaller than the smallest figure imaginable"
Erik Mesoy
Core Tester / Intern
I don't know where the first notion of infinity came from, but I can tell you that the ancient Greeks regarded infinity as imperfect, because it is undefined (to be perfect, you would have to be a perfect *something*). That's why they thought that God was limited. The early Christians believed the same thing - Origen, in the third century, expressly said that God was limited, because otherwise he would not be perfect. To be perfect, he thought, you must be intelligible, and the infinite cannot be comprehended.
Some pagan theologians were developing the notion that God cannot be understood, that he transcends all earthly categories, and that we cannot therefore say anything positive about him - all we can say are negatives (God is not like this, not like that, etc.). As far as I recall, none of them explicitly deduced from this that he was infinite.
St Gregory of Nyssa, in the fourth century, who was deeply influenced by some of those pagan theologians, was the first Christian to say that God is infinite, and he devoted some time to arguing for the belief, because it was decidedly controversial.
Not remotely relevant to the question, but I'll never miss the opportunity for a patristic excursus!
Some pagan theologians were developing the notion that God cannot be understood, that he transcends all earthly categories, and that we cannot therefore say anything positive about him - all we can say are negatives (God is not like this, not like that, etc.). As far as I recall, none of them explicitly deduced from this that he was infinite.
St Gregory of Nyssa, in the fourth century, who was deeply influenced by some of those pagan theologians, was the first Christian to say that God is infinite, and he devoted some time to arguing for the belief, because it was decidedly controversial.
Not remotely relevant to the question, but I'll never miss the opportunity for a patristic excursus!
Yeah, I knew you weren't making it up, but I'm still confused. "The limit"? How do you know what the limit is? For example, in your problem you say "the limit" is one-tenth, but I'm not quite sure why you say that.Erik Mesoy said:
Also, I assume in your ASCII art the denominator should be 1, not 5?
So are you saying that infinitesimals aren't *really* zero, but for all practical purposes they should be assumed as such? That makes sense, and really the only reason I was ever confused is because I thought that mathematicians think they really ARE zero.Erik Mesoy said:
newfangle
hates you.
Here's what a limit is WillJ:
lim f(x)=L
x->a
In words, "The limit of a function f as x approaches a is equal to L.
And the mathematical meaning:
For every positive epsilon, there exists a positive delta, such that if 0 < |x-a|< delta, then |f(x)-L| < epsilon.
What this is saying is that if I select a y value on a graph, it corresponds to a specific x value. As I get arbitrarily close to this x value, I get arbitrarily close to the corresponding y-value.
lim f(x)=L
x->a
In words, "The limit of a function f as x approaches a is equal to L.
And the mathematical meaning:
For every positive epsilon, there exists a positive delta, such that if 0 < |x-a|< delta, then |f(x)-L| < epsilon.
What this is saying is that if I select a y value on a graph, it corresponds to a specific x value. As I get arbitrarily close to this x value, I get arbitrarily close to the corresponding y-value.
toh6wy
Emperor
All right, WillJ. Here's a simple way to think about a number being infinitely close:
What happens when you divide 1 by 3? Well, there are 2 answers - the fractional answer, 1/3, and the decimal answer, 0.33333.... 1/3 times 3 = 3/3, which simplifies to 1. 0.33333... times 3 = 0.999999... Since that's techically the same thing, (just the decimal form versus the fractional form) 1 must equal 0.999999... and so a number that is infinitely close to another number is that number.
What happens when you divide 1 by 3? Well, there are 2 answers - the fractional answer, 1/3, and the decimal answer, 0.33333.... 1/3 times 3 = 3/3, which simplifies to 1. 0.33333... times 3 = 0.999999... Since that's techically the same thing, (just the decimal form versus the fractional form) 1 must equal 0.999999... and so a number that is infinitely close to another number is that number.
pboily
fingerlickinmathematickin
The epsilon-delta definition, while definitely valid, is sometimes considered a little bit circular, as one usually needs to know the limit before computing it (we can solve this problem using a variety of theorems, but it's an inelegant approach).
There is a rigourous (axiomatic) mathematical theory of calculus (and so also a theory of "limits" or "approaching") built, in the 1960s, on the concept of the infinitesimal, by Canadian mathematician A. Robinson.
In it, one defines an infinitesimal as a non-zero quantity whose higher positive powers are all 0 (in effect, these quantities are "nilpotent" real numbers). All the standard results of univariate and multivariate calculus hold, and so do slightly more complex theories, like differential geometry.
The point of all this? It is possible to define infinity (or something very much like it) as the quotient of a real number over an infinitesimal, and that definition is as valid as the definition using the limit...
If anyone is interested, check out
"A Primer of Infinitesimal Analysis", by J.L. Bell, for instance.
There is a rigourous (axiomatic) mathematical theory of calculus (and so also a theory of "limits" or "approaching") built, in the 1960s, on the concept of the infinitesimal, by Canadian mathematician A. Robinson.
In it, one defines an infinitesimal as a non-zero quantity whose higher positive powers are all 0 (in effect, these quantities are "nilpotent" real numbers). All the standard results of univariate and multivariate calculus hold, and so do slightly more complex theories, like differential geometry.
The point of all this? It is possible to define infinity (or something very much like it) as the quotient of a real number over an infinitesimal, and that definition is as valid as the definition using the limit...
If anyone is interested, check out
"A Primer of Infinitesimal Analysis", by J.L. Bell, for instance.
Aphex_Twin
Evergreen
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On infinity, I did recently find a very interesting article on Wikipedia.
Hilbert's paradox of the Grand Hotel
In a hotel with a finite number of rooms, once it is full, no more guests can be accommodated. Now imagine a hotel with an infinite number of rooms. You might assume that the same problem will arise when all the rooms are taken. However, there is a way to solve this: if you move the guest occupying room 1 to room 2, the guest occupying room 2 to room 3, etc., you can fit the newcomer into room 1. Note that such a movement of guests would constitute a supertask.
It is even possible to make place for an infinite (countable) number of new clients: just move the person occupying room 1 to room 2, occupying room 2 to room 4, occupying room 3 to room 6, etc., and all the odd-numbered new rooms will be free for the new guests.
If an infinite (countable) number of coaches arrive, each with an infinite (countable) number of passengers, you can even deal with that: first empty the odd numbered rooms as above, then put the first coach's load in rooms 3n for n = 1, 2, 3, ..., the second coach's load in rooms 5n for n = 1, 2, ... and so on; for coach number i we use the rooms pn where p is the i+1-st prime number. You can also solve the problem by looking at the license plate numbers on the coaches and the seat numbers for the passengers (if the seats are not numbered, number them). Regard the hotel as coach #0. Interleave the digits of the coach numbers and the seat numbers to get the room numbers for the guests. The guest in room number 1729 moves to room 1070209. The passenger on seat 8234 of coach 56719 goes to room 5068721394 of the hotel.
This state of affairs is not really paradoxical but just profoundly counterintuitive. It is difficult to come to grips with infinite 'collections of things', as their properties are quite different from the properties of ordinary 'collections of things'. In an ordinary hotel, the number of odd-numbered rooms is obviously smaller than the total number of rooms. However, in Hilbert's aptly named Grand Hotel the 'number' of odd-numbered rooms is as 'large' as the total 'number' of rooms. In mathematical terms, this would be expressed as follows: the cardinality of the subset containing the odd-numbered rooms is the same as the cardinality of the set of all rooms. In fact, infinite sets are characterized as sets that have proper subsets of the same cardinality. For countable sets this cardinality is called .
An even stranger story regarding this hotel shows that mathematical induction only works in one direction. No cigars may be brought into the hotel. Yet each of the guests (all rooms had guests at the time) got a cigar while in the hotel. How is this? The guest in Room 1 got a cigar from the guest in Room 2. The guest in Room 2 had previously received two cigars from the guest in Room 3. The guest in Room 3 had previously received three cigars from the guest in Room 4, etc. Each guest kept one cigar and passed the remainder to the guest in the next-lower-numbered room.
[edit]
Application to the Cosmological Argument for the Existence of God
A number of defenders of the cosmological argument, among others William Lane Craig, for the existence of God have attempted to use Hilbert's hotel as an argument for the physical impossibility of the existence of an actual infinity. Their argument is that, although there is nothing mathematically impossible about the existence of the hotel (or any other infinite object), intuitively (they claim) we know that no such hotel could ever actually exist in reality, and that this intuition is a specific case of the broader intuition that no actual infinite could exist. They argue that a temporal sequence receding infinitely into the past would constitute such an actual infinite.
However, the paradox of Hilbert's hotel involves not just an actual infinite, but also supertasks; it is unclear whether this claimed intuition is really the physical impossibility of an actual infinite, or merely the physical impossibility of a supertask. A causal chain receding infinitely into the past need not involve any supertasks.
It should also be noted that the addition of guests to a full Hilbert's would require infinitely fast communication, in order for every guest to tell the next guest to move one up in a finite amount of time. Thus, a universe could contain an actual infinite hotel, but with a finite speed of light, and hence it would not be able to contain any more guests even if it was full.
This is debatableif the infinite hotel was a single row of rooms, connected by a long hall, each new guest could be placed in the room nearest the front desk, and told to instruct the inhabitant of room 0 to move to room 1, and pass along a similar message, in a way similar to the working of an infinite systolic array. Then, more guests could always be addedbut at a rate limited by the walking speed of the guests, their ability to gather their possessions, etc. However, in that case, the hotel would always be overfull (in an overfull, the number of rooms has exhausted, but guests have been forced to share rooms, or some are deprived of rooms). And of course, you can add more guests to a full finite hotel as well, making it overfull.
Hilbert's paradox of the Grand Hotel
In a hotel with a finite number of rooms, once it is full, no more guests can be accommodated. Now imagine a hotel with an infinite number of rooms. You might assume that the same problem will arise when all the rooms are taken. However, there is a way to solve this: if you move the guest occupying room 1 to room 2, the guest occupying room 2 to room 3, etc., you can fit the newcomer into room 1. Note that such a movement of guests would constitute a supertask.
It is even possible to make place for an infinite (countable) number of new clients: just move the person occupying room 1 to room 2, occupying room 2 to room 4, occupying room 3 to room 6, etc., and all the odd-numbered new rooms will be free for the new guests.
If an infinite (countable) number of coaches arrive, each with an infinite (countable) number of passengers, you can even deal with that: first empty the odd numbered rooms as above, then put the first coach's load in rooms 3n for n = 1, 2, 3, ..., the second coach's load in rooms 5n for n = 1, 2, ... and so on; for coach number i we use the rooms pn where p is the i+1-st prime number. You can also solve the problem by looking at the license plate numbers on the coaches and the seat numbers for the passengers (if the seats are not numbered, number them). Regard the hotel as coach #0. Interleave the digits of the coach numbers and the seat numbers to get the room numbers for the guests. The guest in room number 1729 moves to room 1070209. The passenger on seat 8234 of coach 56719 goes to room 5068721394 of the hotel.
This state of affairs is not really paradoxical but just profoundly counterintuitive. It is difficult to come to grips with infinite 'collections of things', as their properties are quite different from the properties of ordinary 'collections of things'. In an ordinary hotel, the number of odd-numbered rooms is obviously smaller than the total number of rooms. However, in Hilbert's aptly named Grand Hotel the 'number' of odd-numbered rooms is as 'large' as the total 'number' of rooms. In mathematical terms, this would be expressed as follows: the cardinality of the subset containing the odd-numbered rooms is the same as the cardinality of the set of all rooms. In fact, infinite sets are characterized as sets that have proper subsets of the same cardinality. For countable sets this cardinality is called .
An even stranger story regarding this hotel shows that mathematical induction only works in one direction. No cigars may be brought into the hotel. Yet each of the guests (all rooms had guests at the time) got a cigar while in the hotel. How is this? The guest in Room 1 got a cigar from the guest in Room 2. The guest in Room 2 had previously received two cigars from the guest in Room 3. The guest in Room 3 had previously received three cigars from the guest in Room 4, etc. Each guest kept one cigar and passed the remainder to the guest in the next-lower-numbered room.
[edit]
Application to the Cosmological Argument for the Existence of God
A number of defenders of the cosmological argument, among others William Lane Craig, for the existence of God have attempted to use Hilbert's hotel as an argument for the physical impossibility of the existence of an actual infinity. Their argument is that, although there is nothing mathematically impossible about the existence of the hotel (or any other infinite object), intuitively (they claim) we know that no such hotel could ever actually exist in reality, and that this intuition is a specific case of the broader intuition that no actual infinite could exist. They argue that a temporal sequence receding infinitely into the past would constitute such an actual infinite.
However, the paradox of Hilbert's hotel involves not just an actual infinite, but also supertasks; it is unclear whether this claimed intuition is really the physical impossibility of an actual infinite, or merely the physical impossibility of a supertask. A causal chain receding infinitely into the past need not involve any supertasks.
It should also be noted that the addition of guests to a full Hilbert's would require infinitely fast communication, in order for every guest to tell the next guest to move one up in a finite amount of time. Thus, a universe could contain an actual infinite hotel, but with a finite speed of light, and hence it would not be able to contain any more guests even if it was full.
This is debatableif the infinite hotel was a single row of rooms, connected by a long hall, each new guest could be placed in the room nearest the front desk, and told to instruct the inhabitant of room 0 to move to room 1, and pass along a similar message, in a way similar to the working of an infinite systolic array. Then, more guests could always be addedbut at a rate limited by the walking speed of the guests, their ability to gather their possessions, etc. However, in that case, the hotel would always be overfull (in an overfull, the number of rooms has exhausted, but guests have been forced to share rooms, or some are deprived of rooms). And of course, you can add more guests to a full finite hotel as well, making it overfull.
Sorry, but I'm still confused. I think the only way I'll understand this is through a calculus class.newfangle said:
Yep, and in fact I was the one that mentioned that in the .999... =? 1 thread.toh6wy said:
Still, though, that seems like an example of mathematical reasoning getting in the way of common sense. I mean, .9999... is different from 1 to an infinite amount of digits; saying that they are ultimately equal, no matter what mathematical proof you have, is just odd....Similar to how, considering 1/infinity can be assumed to be zero, as can (3 billion)/infinity, the probability of randomly picking a number from the set of all real numbers and arriving at 5 is 0%, despite being possible. And if you randomly pick a number fifteen billion times, it's still 0%! Arrgh!
Vancouver 2010
Getting Paid to Code
Aphex_Twin said:
Ow! I have a headache now...
Duddha
His Dudeness
It would be impossible to imagine infinity in its entirety when infinity has no entirety. Why do we care anyways, the universe is finite.
The Last Conformist
Irresistibly Attractive
Evidence? Present cosmological models suggests it's infinite in both time and space.Duddha said:
Erik Mesoy
Core Tester / Intern
The point is that it isn't different at any point!
1-0.9 = 0.1
1-0.99 = 0.01
1-0.999 = 0.001
...
...
...
1-0.999999999999999999999999...999999999... = 0.00000000000000000000000...
No difference from zero at any point.
Just as 0+(1/Inf) has no difference from 0 at any point, nor is 1-(1/Inf) different from 1 at any point.
There are two models of the universe. In the one, which is "closed", trying to reach the egde of the univers would be like trying to reach the edge of a 4D ball- we'd find ourselves back where we started.The Last Conformist said:
In the other, which is "open", the universe is unlimited but finite. We can travel an infinite distance in any direction, but the point being that due to the speed limit, we won't ever gt anywhere infinity. Since the universe came into being at one point (don't bring up the how), it's possible expansion since then has been ((C*Age)+Starting Area), and is limited in that area.
There might be infinite amounts of vaccum out there which simply don't exist until we get there.
newfangle
hates you.
Infintiy = 1/(1-0.999999999999999999..........)
The Last Conformist
Irresistibly Attractive
Yes. Current observational evidence, however, suggests that this model is wrong.Erik Mesoy said:
Of course, we can't infinitely far away without infinite speed (which can fairly safely be considered impossible). That doesn't, however, mean, that the universe isn't infinite. The obervable universe will always remain infinite, but that doesn't mean the actual universe can't be infinite.In the other, which is "open", the universe is unlimited but finite. We can travel an infinite distance in any direction, but the point being that due to the speed limit, we won't ever gt anywhere infinity. Since the universe came into being at one point (don't bring up the how), it's possible expansion since then has been ((C*Age)+Starting Area), and is limited in that area.
There might be infinite amounts of vaccum out there which simply don't exist until we get there.
Jorge
Yo mismo
I think infinity comes as an answer to the question: what's next?
Can you imagine a finite universe? And if it's finite, it must have an end. But what is there beyond?
Can you imagine a finite universe? And if it's finite, it must have an end. But what is there beyond?
Iggy
Deity
As a six year old I understood the concept of infinity / eternity when being taken to the cinema by my mum to see the Sound of Music. The film seemed to go on and on and on.....
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