Linked by Thom Holwerda on Fri 22nd Mar 2013 10:02 UTC
"But a powerful new type of computer that is about to be commercially deployed by a major American military contractor is taking computing into the strange, subatomic realm of quantum mechanics. In that infinitesimal neighborhood, common sense logic no longer seems to apply. A one can be a one, or it can be a one and a zero and everything in between - all at the same time. [...] Now, Lockheed Martin - which bought an early version of such a computer from the Canadian company D-Wave Systems two years ago - is confident enough in the technology to upgrade it to commercial scale, becoming the first company to use quantum computing as part of its business." I always get a bit skeptical whenever I hear the words 'quantum computing', but according to NewScientist, this is pretty legit.
RE[2]: later
by Alfman on Sun 24th Mar 2013 13:39 UTC in reply to "RE: later"

Member since:
2011-01-28

I think that's tripping some people up is treating "infinity" as though it were a discrete number that can be compared. In discrete calculus, we were always careful to say a sequence could "approach" infinity faster than another sequence, which is both valid and fairly easy to understand.

The moment you treat "infinity" like a discrete number and manipulate it with discrete operators like comparison, you break the concept of infinity. Nothing is bigger than infinity. Infinity plus one isn't a discrete number, neither is infinity minus one. Sequences do not "equal" infinity because the transient property of equality would imply that all sequences approaching infinity are equal, which they're not.

One might be tempted to say infinity minus infinity is zero, but that's not semantically valid because infinity isn't a discrete number.

S1=1+2+3+4...
S2=2*S1

Both sequences are infinite, but neither are equal, nor do they "equal infinity". S2-S1 doesn't equal zero, it equals S1.

RE[3]: later
by BeamishBoy on Sun 24th Mar 2013 18:13 in reply to "RE[2]: later"
Member since:
2010-10-27

The moment you treat "infinity" like a discrete number and manipulate it with discrete operators like comparison, you break the concept of infinity.

This is completely incorrect.

Infinity is a well-defined concept. It's so well defined that one can talk about (i) different kinds of infinity, and (ii) how one version of infinity is "larger" than another.

I addressed this in another post by pointing out that the number of elements contained in the set of real numbers is one kind of infinity, and how this is smaller than the number of elements contained in the set of all subsets of real numbers.

I know this stuff is confusing but it's been understood by mathematicians for over a century.

RE[4]: later
by Alfman on Sun 24th Mar 2013 20:53 in reply to "RE[3]: later"
Member since:
2011-01-28

BeamishBoy,

"Infinity is a well-defined concept. It's so well defined that one can talk about (i) different kinds of infinity, and (ii) how one version of infinity is 'larger' than another."

I didn't say it wasn't a well defined concept. It's just that the concept is being abused when we treat infinity as a discrete number as though it could be compared.

"I know this stuff is confusing but it's been understood by mathematicians for over a century."

It's really not that confusing, if you attempt to solve a discrete number for "infinity", anyone else can conceive of a number a number which is factually higher, therefor the concept of infinity rules out the possibility of any discrete number equaling infinity.

What we can do is compare sums of series which approach infinity at different rates, like you did earlier. Discrete calculus (which as you noted is well understood) allows us to solve the rate for any given iteration of the sequence and confirm via inductive proofs that it continues infinitely.

There are an infinite number of unique sequences who's sums add up towards infinity given an infinite number of iterations. Because of the transitive properties of mathematical equality, one cannot claim any of these infinite sequence sums "equal infinity", for the simple reason that they are not equal to each other.

We might be careless and say a given sum "equals infinity" and still understand one another, but we don't actually mean it in the true mathematical sense.

This is what my earlier example was trying to illustrate. If infinity could be treated as a discrete number, then mathematically this would be sound:
A=infinity
B=infinity
B-A = 0

But in fact the earlier example showed that B-A=A.

To be sure, I am arguing semantics in the conversation as a whole and not just you. I don't want us to talk over each other, and I don't think we have all that much to disagree on, despite your bolded statement that "This is completely incorrect."

Edited 2013-03-24 20:57 UTC

RE[3]: later
by Krul on Sun 24th Mar 2013 18:58 in reply to "RE[2]: later"
Member since:
2009-01-29

Well, first of all, there are two different concepts of infinity that had been discussed in some of this threads. One is the idea of infinity as a number that is larger than all the real numbers. The other is the idea of a set having an infinite number of elements. They are not the same concept. Let me give a brief description of both to see the difference:

1.- For certain applications, it is convenient to extend the real numbers to allow the values infinity and -infinity. (For example, this is a very common thing to do in measure theory and integration.) In this case infinity is (in a sense) a quantity that is bigger than all others quantities. The drawback of this approach is that infinity doesn't behave well under the usual arithmetic operations. For example infinity+1=infinity, while infinity-infinity is always undefined. In this setting you can't have anything bigger than infinity. Trying to manipulate divergent sequences as if they were the quantity infinity and doing algebraic manipulations with them is incorrect, and can quickly get you to absurd statements. For example, Riemann showed that if you have a conditionally convergent sequence, then you can rearrange its terms to make it have any limit you want.

2.- In set theory there is the notion of the "cardinality" of a set, which is a measure of how big it is. Two sets are said to be of the same cardinality if you can give a bijection between them. Of course, given any set you can always add one element to it to make it bigger. (This is based on the fact that there is no such a thing as "the set of all sets".) But having a set with more elements doesn't automatically means that it has a higher cardinality. For example the set of positive integers has the same cardinality as the set of even positive integers. (You just enumerate your positive even integers to get the bijection.) However, Cantor showed that the set of real numbers does have a bigger cardinality than the set of integers. This is very shocking. It means that it doesn't matter how you try to enumerate the real numbers, at the end there is always an infinite number of real numbers that you left out. Given any set, the set of all its subsets always has a higher cardinality, so there is no biggest cardinal number. (Again this is related to the fact that there is no "set of all sets".)