Wikipedia:Reference desk/Archives/Mathematics/2013 March 28

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March 28

Division by infinity

Why is it that something that is mathimatically sound, can be madness when applied to real life? Example: I have a cake, I have an infinite number of guests. If I devide the cake evenly, then each guest receives exactly no cake. Where did all the cake go? Plasmic Physics (talk) 03:29, 28 March 2013 (UTC)

How is an infinite number of guests either mathematically sound or applicable to real life? Do you have a request for references about some definable topic here? μηδείς (talk) 03:37, 28 March 2013 (UTC)

On the one hand you have a finite cake, but infinite guests. That's a bit like having your cake and eating it too! [I will understand any groan responses] Shadowjams (talk) 06:29, 28 March 2013 (UTC)

It can hardly be called "madness", since it describes very practically and accurately what happens in "real life" when a large number of people are forced to share a very small and limited amount of resources. It's all very simple and down to earth. — 79.113.209.3 (talk) 19:40, 28 March 2013 (UTC)

I guess a more serious response to your question would be that your error consists in thinking that 0 x Infinity = 0. The product 0 x Infinity is undefined. Another more practical approach would be to say that if each person were to consume a single atom or sub-atomic particle of a pie, that is the same thing as if each were to eat nothing. Yet the pie still is consumed, although each person in particular, taken individually, consumed nothing, and remained just as hungry as before, the total number of persons did however consume the whole pie. — 79.113.219.21 (talk) 06:34, 28 March 2013 (UTC)

I never took the product of anything. Plasmic Physics (talk) 07:29, 28 March 2013 (UTC)

You did, though perhaps you didn't quite realize it, when you wrote: "each guest receives exactly no cake. Where did all the cake go?". This is the mathematical expression of your thought: 0 [pie/guest] x Infinity [guests] = 0 [pies eaten]. 1 [pie] - 0 [pies] = 1 [whole pie left], ergo "Where did all the cake go?". Just think about it. — 79.113.219.21 (talk) 08:07, 28 March 2013 (UTC)

No, that is not a correct assumption. x/infinity (x ≠ 0) is a non-injective function. Plasmic Physics (talk) 08:24, 28 March 2013 (UTC)

What exactly "is not a correct assumption" ? — 79.113.209.3 (talk) 15:40, 28 March 2013 (UTC)

The assumption that the x/infinity, is an injective function. Assuming that the inverse function yields x and only x, which it does not. Plasmic Physics (talk) 18:31, 28 March 2013 (UTC)

Your words, when expressed mathematically, lead us to think otherwise. If not, then please try to express them in a mathematical manner, and prove that I was wrong in describing your thought-process, as opposed to merely asserting as such. — 79.113.209.3 (talk) 19:31, 28 March 2013 (UTC)

Better publish that quick. You just licensed your Fields Medal winning discovery under the CC-BY-SA 3.0 License! Shadowjams (talk) 08:38, 28 March 2013 (UTC)

I think the OP's first and fatal mistake is assuming infinity is a number - an extremely large number, but still a number, one that obeys all the laws of mathematics. That is simply not the case. No matter how extremely large you set it at, there's always a number higher than that. Think of it this way: If you invited every single person in the world to your party, and they all turned up, firstly there'd be a massive traffic jam, but secondly, when it came time to divide the cake evenly among the 6 billion guests, how much would each get? Far less than a crumb, far less than a grain of flour, but still something (assuming you could manage to dish it up evenly; and assuming even a portion as small as a few molecules is still not nothing, and hence something). What you've done is taken the extra logical step and increased the guest list to infinity, thus reducing the portion size to zero, thus causing the entire cake to vanish. Well, it might feel logical, but you can't do that, at least not in the world of mathematics. Science fiction is where this belongs. -- Jack of Oz ^[Talk] 08:46, 28 March 2013 (UTC)

What is the answer, you've said my logic is incorrect, without giving the true answer. Plasmic Physics (talk) 09:01, 28 March 2013 (UTC)

I told you. Infinity is not a number. You can only divide a number by another number, and infinity is not a number. Division by infinity has as much meaning as division by an elephant, or division by Beethoven's 5th Symphony, or division by your sexual orientation. -- Jack of Oz ^[Talk] 11:43, 28 March 2013 (UTC)

That's not true, infinity is within the domain of maths, unlike elephants etc. Moreover, I had to solve many (x → infinity) limits in a recent assignment. If I can do no arrithmatic on infinity, then these limit calculations would also be senseless. Plasmic Physics (talk) 18:38, 28 March 2013 (UTC)

It depends on what you're talking about. There are structures (see for example real projective line) where infinity is a first-class citizen of the structure, and where division by infinity is defined and gives 0. However this particular one, at least, is more an abstraction from real-number division than from integer division. So it doesn't really have an interpretation as cutting things up into infinitely many pieces. --Trovatore (talk) 18:55, 28 March 2013 (UTC)

I think that the operation ${\tfrac {x}{\infty }}$ is undefined if x is a fixed number and is only defined when dealing with the limit of a function. -- Toshio Yamaguchi 09:14, 28 March 2013 (UTC)

See L'Hôpital's rule. It's not actually the same as dealing with fixed number, though. -- Toshio Yamaguchi 09:47, 28 March 2013 (UTC)

I know that rule, it doesn't give me an answer different from zero. Let the numerator equal one, then, as the denominator tends to infinity, the function tends to zero. Note: neither zero, nor infinity share all the properties of numbers. In my view, zero is not really a number at all, much like infinity; instead, it is a placeholder which represents nothing. Zero, and infinity represents concepts, not numbers. Plasmic Physics (talk) 10:01, 28 March 2013 (UTC)

" Where did all the cake go?": Presumably into energy as you split the atoms and subatomic particles. -- Q Chris (talk) 10:33, 28 March 2013 (UTC)

Actually you do not have an infinite number of guests, so your premise is false, and any conclusion is possible. For example that each guest gets a whole cake. Then you may ask, where did all the cake come from? Both questions are nonsensical. Bo Jacoby (talk) 12:43, 28 March 2013 (UTC).

Plasmic Physics, from a measure-theoretic point of view, if we interpret a "cake" as a volume of finite positive measure in three-dimensional space, it is logically impossible to divide the cake evenly among countably infinitely many people (for example, if your "people" are the natural numbers 1, 2, 3, …), and the reason is exactly the paradox you describe. You can divide it evenly among uncountably infinitely many people (for example, if your "people" are the real numbers), and in that case every person gets a volume of zero cake, but not necessarily the empty set, just a set that has zero volume (a null set, like a single point, for example). However, if you have uncountably infinitely many people, you can no longer add the volumes of the individual portions together to get the volume of the whole cake, just as you cannot add the volumes of the individual points in a sphere to get the volume of the whole sphere. —Bkell (talk) 16:04, 28 March 2013 (UTC)

• Yes, this is the best answer so far. Measure theory is the natural setting for this kind of problem, not limits of functions/arithmetic. This topic is also related to the Banach-Tarski paradox, which depends on the axiom of choice. My recollection is that the B-T is not available if we only allow the axiom of countable choice. So perhaps the OP's dilemma comes down to the axiom(s) of choice, which is a perfectly reasonable place put our uncertainty and doubt :) SemanticMantis (talk) 16:19, 28 March 2013 (UTC)

(...and I want my measure-zero piece of cake shaped like the Cantor set, please) SemanticMantis (talk) 16:21, 28 March 2013 (UTC)

The axiom of choice is not necessary for anything that I said, nor does it cause or resolve Plasmic Physics' paradox. Nothing that I said relies on the existence or nonexistence of nonmeasurable sets or any other consequences of AC. The axiom of choice is irrelevant as far as Plasmic Physics' question and my answer are concerned. —Bkell (talk) 16:31, 28 March 2013 (UTC)

Right, AC does not come into play for your answer, but I do think that the B-T paradox is an interesting related problem. Maybe I was reading to much in to the original question, but I thought I'd err on the side of providing interesting links for further reading on related topics. SemanticMantis (talk) 16:42, 28 March 2013 (UTC)

No reasonable "uncertainty and doubt" about AC, though. AC is obviously true. --Trovatore (talk) 16:59, 28 March 2013 (UTC)

(off-topic rebuttal follows, skip if not interested in the axiom of choice)

“It is a peculiar fact that all the transfinite axioms are deducible from a single one, the axiom of choice, — the most challenged axiom in the mathematical literature.” -- David Hilbert (1926), more interesting quotes about and challenges to AC found here[1]

So, even Hilbert acknowledged that AC was widely challenged. Here's my response: AC is true if you accept it. It's an axiom. Its truth is by fiat. Our article explains "most mathematicians accept the axiom of choice as a valid principle for proving new results in mathematics." -- and I agree with that, so at least AC is "true" in that sense. But you cannot prove that AC is true, nor can you measure something in the world that will tell you it's true. That's why we call it an axiom, no? Really, mathematicians use it because it is useful and convenient, and just makes a lot things nicer. (Who wants to work in Hilbert spaces that lack orthonormal bases?) Of course, strange things like the B-T paradox also come along with AC, and many people find that problematic (e.g. all the people publishing results based on countable choice).

You can dismiss AC as "obviously true" if you want, and "uncertainty and doubt" was probably a poor choice of words on my part. But great minds have wrestled with the value, "correctness", and implications of AC (or weaker versions, or assuming the negation, etc) for about 100 years. So I think "obviously true" is selling things a bit short. (I am happy to discuss further, but please direct further comments to my talk page so as to avoid further derailing of this question.) SemanticMantis (talk) 19:15, 28 March 2013 (UTC)

No no no. Axiomatics has nothing whatever to do with it. AC is true in the real universe of sets, the von Neumann universe, which exists independently of our reasoning about it. And yes, this is quite obvious. --Trovatore (talk) 23:01, 28 March 2013 (UTC)

If you are invoking truth of AC in V, then you are claiming truth by assumption or fiat. Now that I know what you mean, I happily agree. SemanticMantis (talk) 01:21, 29 March 2013 (UTC)

No, I'm not sure I'm getting my point across. I'm not assuming anything, and it's not my fiat. I am making a claim that this is true in the real world (correspondence theory of truth), and asserting that the way I know that is by self-evidence). --Trovatore (talk) 01:23, 29 March 2013 (UTC)

It doesn't seem to have been mentioned explicitly in this thread, although implicitly SemanticMantis may have already pointed it out through his reference to the Banach-Tarski paradox, that with the axiom of choice (and assuming Bkell's interpretation of "cake" and "cut"), it is possible to cut the cake into countably many pieces (non-measurable pieces), then rearrange those pieces into cakes that each has the same volume as the original cake. (This also explicitly contradicts Bkell's assertion that it is logically impossible to evenly divide the cake among a countable infinity of guests. This would be true if you assumed that the cuts had to be measurable subsets.) Sławomir Biały (talk) 20:53, 28 March 2013 (UTC)

I don't think the word "evenly" is meaningful if you are trying to apply it to nonmeasurable subsets. I suppose you could argue that the concept of an "even" division of the cake could include a partition into (nonmeasurable) subsets in which each subset is a translation of every other subset, modulo the boundary of the cake (whatever that should mean), as in the construction of a Vitali set. But that bizarre extended definition of "even" wasn't what I meant when I wrote about dividing the cake evenly, and I doubt it's what Plasmic Physics had in mind. I interpreted the phrase "divide the cake evenly" to mean "partition the cake into subsets that have equal volume," and with that interpretation what I said is true. —Bkell (talk) 04:09, 29 March 2013 (UTC)

I think you missed a subtle point here. Each guest will receive a piece of cake that is identical to the original cake (in particular, the piece is measurable). The construction, however, requires an intermediate stage where the cake is cut into countably many non-measurable pieces. These pieces are then rearranged (by Euclidean motions) into countably many measurable cakes of equal measure. Sławomir Biały (talk) 11:07, 29 March 2013 (UTC)

Oh, yeah, I misunderstood what you were saying. Okay, that's another bizarre extended definition of "divide the cake evenly" that I didn't mean. :-) —Bkell (talk) 15:05, 29 March 2013 (UTC)

Conceptually, since I tend to avoid equating infinitesimals with zero, each person gets an infinitesimal amount of cake. Thus, these infinite number of infinitesimal pieces should add up to a cake (if it still existed and wasn't rapidly being consumed! :-)) -Modocc (talk) 19:10, 28 March 2013 (UTC)

One should stick to ultrafinitism. There are only a finite number of states available for the entire observable universe, so it doesn't make sense to describe the physical world using infinite sets, let alone uncountable ones. Count Iblis (talk) 19:33, 28 March 2013 (UTC)

Certainly we should at least try to let the other more distant guys (beyond the observable universe) eat cake too. I don't suppose anyone is up to the task though of simultaneously cutting all the slices and sending them. And I'm wondering how many candles we ought to include with each? ;) --Modocc (talk) 20:01, 28 March 2013 (UTC)

If "state" is an element of a Hilbert space, then you are already need an infinite set to describe a single "state". Sławomir Biały (talk) 21:00, 28 March 2013 (UTC)

If you don't impose an upper limit to the energy of a system (in a finite volume). Too much energy in a finite volume will lead to gravitational collapse and a black hole there. When you take gravity into account, that has consequences even for the total number of physically possible states below an energy E; this is proportional to the radius instead of the volume, see Bekenstein bound. Count Iblis (talk) 23:10, 28 March 2013 (UTC)

Even assuming a compact topology for the universe, so that the Bekenstein bound implies that the universe can be described by a finite-dimensional Hilbert space, what is the cardinality of this Hilbert space? Sławomir Biały (talk) 23:22, 28 March 2013 (UTC)

Yes, the cardinality is infite. However, there are only a finite number distinguishable states, in the sense that any measurement can only yield a finite number of different outcomes. A simple example is the 2 dimensional Hilbert space describing the spin of an electron. While you can put the electron an an infinite number of different superpositions of spin up and spin down, that doesn't allow you to use the electron as a magical device that can store an unlimited amount of information. Count Iblis (talk)

Then perhaps I am misunderstanding what you mean when you say " it doesn't make sense to describe the physical world using infinite sets", since here you are using an infinite set to describe a single state. Sławomir Biały (talk) 13:39, 29 March 2013 (UTC)

The issue for me is that while we can use infinite sets to describe the World, you only need a finite amount of information. So, you can imagine simulating the Earth including the people who live on it using a huge computer. While that computer is a finite state machine, it will simulate how the virtual mathematicians and physicists use infinite and even uncountable sets to describe the finite reality that the computer is rendering. Count Iblis (talk) 14:40, 29 March 2013 (UTC)

Note that I just reverted a recent edit of the infinitesimal article [2] primarily because it stated that an infinitesimal is equal to the difference between ".999..." and 1 which is zero, but an infinitesimal is an infinitely small nonzero number. --Modocc (talk) 22:52, 28 March 2013 (UTC)

The cake can only be divided up a finite number of times until each guest receives one atom . After that the cake can be divided no further so the premise of this question is flawed if you are applying to a "real life" situation. Ap-uk (talk) 11:53, 29 March 2013 (UTC)

Computing in infinite matrix groups

I have a finite set of integral matrices generating an infinite group. I'm trying to find relations in the group, but have no desire to do this by hand. Does anyone know of a program which will systematically search for relations? The only programs I'm aware of have difficulty when working with infinite groups... Thanks, Icthyos (talk) 12:35, 28 March 2013 (UTC)