Monday 23 August 2021

The Case of Hilbert’s Hotel and Infinity

In Hilberts infinite hotel, a room can always be found for newly arriving guests.

Posted by Keith Tidman


‘No vacancies. Rooms available’. That might as well be the contradictory sign outside the Hilbert Hotel of legend. Yet, is the sign really as nonsensical as it first seems?


The Hilbert Hotel paradox was made famous by the German mathematician David Hilbert in the 1920s. The paradox tells of an imaginary hotel with infinite rooms. All the rooms were occupied by an infinite number of guests.


However, a traveller wondered if a room might still be available, and approached the receptionist. The receptionist answered that the hotel could indeed accommodate him. To make the solution work, the receptionist asked all the current guests simply to move to the next room, making it possible to assign the new guest to Room 1.


This was a scalable maneuver, accommodating any number of new lodgers, whether a hundred, a hundred million, or far more. Because of the infinite rooms, importantly there was no last room; the receptionist could therefore keep moving the current guests to higher room numbers.


But the challenge was to get a bit harder. What showed up next was an infinitely large coach occupied by an infinite number of vacationers. To accommodate these guests, the receptionist shifted people so that only the infinite even-numbered rooms were occupied. 


Increasingly complex scenarios arose. Such as when an infinite number of coaches, each carrying infinite travellers, pulled into the hotel’s infinite parking lot. But we don’t need, for our purposes here, to delve into all the mathematical solutions. Suffice it to say that any number of new travellers could be lodged.


The larger significance of Hilbert’s thought experiment was that an ‘actual infinite’ is indeed logically consistent, even if on the surface it’s counterintuitive. As with Hilbert’s hotel, the infinite exists. Infinity’s logical consistency has further consequence, tying the thought experiment to the cosmological notion of an infinite past. 


That is, a beginningless reality. A reality in which our own universe, like infinite other universes, is one bounded part. An unlimited reality that extends even to the ‘far side’ of the Big Bang that gave rise to our universe almost fourteen billion years ago. A universe located within the continuum of the infinite. And a universe in which change gives us the illusion of time’s passage.


A common argument in cosmology (origins) is that the string of causes must start with the Big Bang. Or, rooted in a theological origin story, that it must start with a noncontingent divine creator, or so-called ‘first cause’. The claim of such arguments is that reality doesn’t reach back indefinitely into the past, but has a starting point.


‘Our minds are finite, and yet even in these circumstances of finitude,

we are surrounded by possibilities that are infinite’.

Alfred North Whitehead, philosopher and mathematician

 There are no grounds, however, to believe that our universe, with its time-stamped beginning (the Big Bang) and its one-way life-cycle toward net disorder, is the entirety of existence. Rather, an infinite history before the Big Bang, or beginningless reality, does make sense. As does the other bookend to that reality, an endless future, where infinity describes both before and after the fleetingly present moment, or what we might think of as the ‘now’. Nothing rules out or contradicts that unlimited scope of reality.


With an unchangeable beginningless reality, there is no need to evoke the concept of ‘something coming into being from nothing’; there is no need to interrupt the different laws of physics, or of time, governing each universe’s own separate reality; there is no time zero or insupportable moment of all creation. It’s infinity ‘all the way down’, to paraphrase British philosopher Bertrand Russell’s whimsical reference to infinite regress.  


We ought to avoid conflating the emergence of things within our bounded universe (like the making of new galaxies, of which there is a finite number) with the emergence of things within the infinite (like the formation of new universes, of which there is an infinite variety, each with its unique properties, life cycle, and natural laws).


‘No other question [than the infinite] has ever moved so profoundly the spirit of man; no other idea has so fruitfully stimulated his intellect’, declared David Hilbert. Our bounded universe is simply one part of that infinite, that is, part of beginningless reality. Our universe existing among infinite others, like the infinite rooms in Hilbert’s hotel.


Roger said...

Hi. I think I mostly agree with you, but I think for a physical reason instead of a mathematical, Hilbert’s Hotel-type reason. If you don't mind, my rationale is from an interesting and very nice discussion I had with a mathematician in a comments section of a popular science magazine.

In that discussion, I suggested that there are twice the number of positive integers as odd, positive integers. My rationale was that for every odd, there are two total positive integers (the even and the odd), when the integers are kept in their sequential order (0,1,2,3…). Also, that the mathematical method of removing the subset of odds and pairing them off one-to-one with all the positive integers was like pulling some cells out of the body, studying the cells in a cell culture system, and claiming that the results reflect what happens in the intact body. That is, this method can create experimental artifacts that cause the results to not correctly describe what happens in the setting of the intact body. I hope I’m paraphrasing this correctly, but the mathematician's response was that the result that the set of odds is the same size as the set of all positive integers was true within the set of axioms that make up the abstract mathematical realm. I pointed out that not all things that are mathematically possible happen in the real, physical world. For example, you can change the value of the time variable in an equation to a negative number, and this can be mathematically allowed, but apparently time doesn't seem to go backwards in the real world. So, I said that maybe the set of axioms by which the real, physical world runs may be similar to, but not identical with, the set of axioms that are in the abstract world of mathematics. He totally agreed with that.

I think that the set of axioms that comprise mathematics may be equivalent to the realm of the cell culture system, but the set of axioms that comprise the real, physical world may be more like what happens in the intact body. So, the results about infinite sets may not be be identical to what infinities are like in the real physical world.

Sorry for the long beginning part, but it's why I’m hesitant to take the results about infinite sets in the abstract realm of mathematics like in Hilbert’s Hotel and say they apply to the real, physical world like the universe. Because of this, I think that if one could label each discrete substituent of our universe (I know space may or man not be discrete, but it’s just an example) with an integer, my bet is that in this system, the total number of substituents is twice the number of the number of odd substituents. The mathematician pointed out that this kind of reasoning was more like taking the measure of two sets (although, he pointed out that measure isn’t used much with infinite sets). So, the Hilbert’s Hotel example may be applicable to the real physical universe, but I think we need more evidence and study of this.

Martin Cohen said...

I think we defer too readily to the rules of mathematics on questions like this. For example, we allow there might be an infinite number of atoms in the universe, or infinite years in the future, but then there are clearly multiple electrons in the atoms and days in the years... As the hotel experiment says, it would seem that you can always divide up the rooms to make more space!

Keith said...

Thank you, Roger, for your interesting and informative take on infinity. Indirectly, too, you’ve underscored the fact that different people, within very different fields, have offered various takes on the subject, especially in the modern era: philosophers, theologians, mathematicians, theoretical physicists, and astrophysicists/cosmologists. It would require a much longer essay to even begin to give all that a fair shake, even at a lay level. To that extent, I propose that the subject’s aperture widens to include theories of knowledge: like what’s knowable or unknowable, as well as issues of confirmability, refutability, and probability/uncertainty, along with the healthy mutability of hypotheses and theories.

Roger said...

I agree with both of you guys in that I do think physicists defer too much to the sometimes rigid rules of mathematics at the expense of physical reality and that there are lots of different views on infinity. I hope some of these different theories will recombine and mutate to a form that everyone agrees on and that we can use to invent and do neat things!
I appreciate your site not only because of the good essays, but because you guys are polite. I check it out often. Thanks!

Thomas O. Scarborough said...

Aristotle proposed four causes, and then a fifth. The fifth cause, he noted, is uncaused. Is this cause inside or outside the other four causes? It doesn’t seem to make much sense if it is inside. He must have been thinking that there is a kind of cause that we do not understand, which includes the other four.

It has been suggested often enough that the other four on their own are just not good enough to explain things. Take Cambridge philosopher Simon Blackburn, ‘The idea that causation is a matter of discrete events joined by links is highly problematic.’ Or Ernst Mach more radically, ‘There is no cause nor effect in nature.’

Roger said...

I kind of like the idea of cause and effect. Even in the quantum world when things appear without an apparent cause, there's still the presence of the quantum fields and quantum physics that set the conditions for these things to appear seemingly with no cause. But I do think that whatever is most fundamental "causes" itself because the reason for its existence is inherent to it and not from something outside. It's self-existing, or self-defining. I wonder if that might be considered "uncaused" because nothing outside that most fundamental thing causes its existence. This kind of seems to get away from the idea of infinity, though? Although, it is very interesting!

Keith said...

In my view, Thomas, Ernst Mach’s hypothesis that ‘there is no cause nor effect in nature’, as well as Simon Blackburn’s refutation of ‘causation [as] a matter of discrete events joined by links’, are not borne out by scientific or commonsensical experience and observation within our universe.

Thomas O. Scarborough said...

In reply to Roger, ‘the reason for [a thing’s] existence is inherent to it and not from something outside’ is, I think, problematic. Suppose we remove everything extraneous to a water molecule, everything in the universe. Will it still be a water molecule? Your view of things sounds rather like Russell’s logical atomism, with which Russell himself seemed to run into internal conflicts. He wrote, ‘If the inference from cause to effect is to be indubitable, it seems that the cause can hardly stop short of the whole universe.’ In which case one surely makes a mistake to think of a thing’s existence being inherent to it.

In reply to Keith, I propose that no, yours is the position which lacks common sense. It relies on the existence of things and relations. Take relations, which would generally be thought of as the bigger problem (I think they are both equally a problem). What are relations? Fraser MacBride, a professor of logic and metaphysics, noted that relations are ‘difficult to locate’. They are not substances; they are not attributes. What are they then? In my view, they are short-circuits in thinking. I think you are trying to make a case for the non-existence of an origin, or cause behind causes.

Roger said...

Hi. When I said

“whatever is most fundamental "causes" itself because the reason for its existence is inherent to it and not from something outside. It's self-existing, or self-defining.”,

I was actually talking about a most fundamental building block of reality. That is, at some point, I think that in order to avoid an infinite regress of smaller and smaller constituents of the universe, there has to be a smallest entity (simple) whose existence can’t be explained by anything outside it. It must exist for reasons inherent to it. I think the reason might be that it’s a grouping that defines what is contained within (“nothing” in the case of the must fundamental of entities) and therefore creates a new unit whole, or existent entity. Of course, I can’t prove this by direct observation or prove there's not an infinite regress, but it seems right to me. This gets back to the discussion we had under “Why is there something rather than nothing?” in Sept. 2018.

Also, in regard to a water molecule where everything in the universe that’s extraneous to it is removed, I think it would still exist because it would still be a grouping that ties together two hydrogen atoms, an oxygen atom and the forces between them. If the rest of the universe is gone, there would be no one to label this grouping a “water molecule”, but the molecule itself would still exist, I think. I think it's real important to distinguish the mind's conception of something from the thing itself. If the universe as a unit whole also exists, I think it’s because it’s a grouping that ties stuff together and defines what’s contained within (all the stuff of the universe).

But, like I said no one can prove this stuff by direct observation. Because of this, my approach is to try and figure out what properties a smallest existent "simple" might have, and use those properties to try and build a very simple model of the universe. If it matches observation and can make testable predictions, it's science. This is, of course, very slow going! This turning of metaphysics into physics is the only way, IMHO, for anyone to ever take these kinds of ideas seriously. I learned in my day job that no matter how good an idea is, the higher ups are only convinced by working models/prototypes they can see and use.

Keith said...

‘I think you are trying to make a case for the non-existence of an origin, or cause behind causes’. Correct, Thomas; I would say yours is a fair recasting of the phrase ‘beginningless reality’ appearing in my essay.

Thomas O. Scarborough said...

Thank you Keith.

Thank you Roger. I read your comment carefully. The one question which seems to press itself forward more than any other is, what if it isn't 'a smallest existent' but something else? Say, some form of energy. You'd have a lot of friends in history, though, the oldest of them being Leucippus.

Roger said...

Hi, Thomas. Thank you and everyone for the great discussion, too! It's good to have friends, so I wouldn't object even if they have funny names like Leucippus! :-).

On the idea of a smallest existent, wouldn’t energy exist, too? And, I'm not sure why causes, being self-caused, or uncaused would apply any differently to energy as to any other existent entity? I think energy is the ability to exert a force, which is, in my view, the ability to cause some existent entity to move or change shape (and thereby do work). So, energy would be the ability of one existent entity to cause another existent entity to move or change shape. That is, energy is secondary to existent entities somehow causing one another to move or change shape. But no matter what energy is, it would still exist and I think the reason why it would exist would be the same. Because it's a grouping of something. But, I might be assuming incorrect things about what energy is.

 The grouping idea also applies to abstract concepts like the concept of a car. The concept would be a grouping of other concepts like tires, engines, a car body, movement, etc. The grouping then is the mental construct that we label “car”. I think abstract causes exist in the mind, so their cause would the human brain causing them to exist.

Many people think that abstract concepts, logic, etc. are in a Platonic realm, and exist without cause, but I have trouble with this because where is this realm? Unless someone can point it out or provide evidence for its existence, it’s just an opinion. It might be true but so might many other opinions that can’t be verified. And, I admit that my opinions are in the same category. That’s why I’m trying to use them to build a simple model to see if it matches observations about the universe and can make testable predictions. I admit, this might take a little while! :-)

I guess overall I’m sort of a materialist because I think everything has a material cause and the most fundamental material/existent entity causes itself to exist due to something inherent to that entity.

Post a Comment