41 - Richard Sorabji on Time and Eternity in Aristotle
Posted on 10 July 2011
Peter talks to Richard Sorabji about Aristotle's physics, focusing on the definition of time and the eternity of the universe.
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Further Reading:
• R. Sorabji, Matter, Space, and Motion (London: 1988).
• R. Sorabji, Time, Creation, and the Continuum (London: 1983).
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Golden Mountains
Warning: Not much philosophy here :-)
I feel that Richard dismissed the possibility of a golden mountain out of hand without any basis.
It seems to me that in a universe infinite in time and very large in volume, populated with biological beings like humans, magpies and ants the eventuality that one being, or group of beings, will place enough gold in a single location to be termed a mountain is exceedingly easy to imagine.
Just to prove to every body else on your planet that you are the biggest poo-bah or whatever!
However ... I may now have to alter my view, since wikipedia tells me that 'only' 165,000 tonnes of gold have been mined by humans and that this represents a cube of approx 20 metres on a side (I checked the maths and it is correct despite initial scepticism).
Therefore if we wanted to contruct a gold mountain of, say, 600m^2 base and 300m in height we would need the gold from 4240 earths! (700 million tonnes)
Therefore it seems unlikely that they would be enough gold in a single solar system to make a gold mountain.
But wait! The BBC tells us that the asteroid Eros is thought to contain "20,000 million tonnes of aluminium along with similar amounts of gold, platinum and other rarer metals". http://news.bbc.co.uk/1/hi/sci/tech/401227.stm
So all that is required is for one enterprising sentient being to capture a large asteroid and seperate it in to its various constituents in large piles and the golden mountain would be a reality! (28 times over)
Golden mountain
Hi Felix,
Sounds like a worthy project for a government grant application.
Peter
A comment about space and necessity.
I have a comment about the discussion at the end of the episode regarding empty space and the distinction between ability and opportunity. Richard seems to argue that space beyond the stars might have the ability to receive matter, but not the opportunity to do so, since the stars are confined to circular motion. But if space is an eternal thing, then, by Aristotle's conception of necessity, the fact that space beyond the stars is eternally without matter implies that it is necessarily without matter, and thus that it lacks the ability to receive matter.
Space outside the cosmos
Dear Henry,
Yes, that's a nice point. Remember though that according to Aristotle there is no "space" (or in his terms place, empty or otherwise) outside the cosmos -- there is absolutely nothing outside, not even emptiness. And Richard was trying to explain what he takes to be a reason Aristotle thinks that, namely that a place is something that can be occupied. Hence:
1. A place is where a body can be
2. There is never a body outside the cosmos [because of nature of heavenly motion]
3. What never happens is impossible
4. It is impossible for a body to be outside the cosmos [from 2 and 3]
5. There is no place outside the cosmos
If I'm right that this is what Richard was getting at, the argument he was ascribing to Aristotle actually presupposes the principle you mention, in point 3.
Thanks for the comment!
Peter
Thanks, and Golden Mountain Question
Hello -
Thank you so much for this podcast series. It has come at exactly the right moment for me. I only discovered the history of philosophy recently (as part of a mid-life crisis). I have read several overviews and was looking for something more detailed. This podcast is perfect.
I have a question about the Richard Sorabji episode on Aristotle's view of Time and Eternity.
Would it be possible to expand on what Richard' said about monkeys writing Shakespeare or a golden mountain existing given infinite time. He said "that looks completely implausible and so it is mistaken", but didn't say why. This came up in a conversation I was having recently so I'd really like to know the philosophical argument for it being false.
I understand why it isn't certain given modern physics (because we only have 10^100 years until the heat death of the universe ;-) ...but if Aristotle believed that all matter was contained in a fixed area, was constantly being rearranged and would continue to do so for an infinite amount of time, and if there is nothing in principle to prevent golden mountains from existing, why wouldn't it be certain to happen eventually? In fact why wouldn't there be an infinite number of golden mountains?
Thanks
- Jon
mathematics doesn't apply
There problem with 100 monkeys is typical of the seeking of understanding thru a model with no reality.
First of all there is the matter of the word processors, and their creation and maintenance.
There is the matter of feeding the monkeys.
Monkeys could not and would not sustain their typing for any period of time.
Etc. Simply, they would never succeed.
Infinity
I agree, thinking about infinity in physical contexts is slippery. We could change the example to make it a bit easier to implement (e.g. replace the monkeys on typewriters with computers randomly producing strings of characters) but it is easy to forget, when constructing a thought experiment, that infinite sets cannot actually be physically implemented and that will apply to any such scenario. The question is what that means -- it seems to me that one can ask, "ok, this isn't physically possible, but ignore that -- what would happen if it were to occur?" We're being asked to focus on the mathematical or probability issue and ignore the implementation issue. There's a deeper question here about the role of thought experiments in philosophy, and in particular, impossible thought experiments.
Aristotle himself insisted that infinity is not physically realizable, which is one reason he insists that the cosmos cannot be infinitely large. But he doesn't think that eternal time counts as a physical realization of infinity, since the infinite moments/motions are not all present simultaneously. As we'll be seeing in episodes on late antiquity this position did not go unchallenged.
Infinite monkeys
Hi Jon,
Glad you are enjoying the podcasts! To be honest I wondered that when I was doing the interview with Richard, too. I think one might need to ask a mathematician and not a philosopher (or at least, not this philosopher). I know a mathematician and I'll ask him. But one thought might be this: if there are N possibilities, that doesn't mean that given an N number of trials, every possibility will be realized. For example if you roll a six-sided dice, you won't necessarily get each of the six numbers. Indeed, in theory there is nothing to prevent rolling a six-sided dice, say, 100 million times, without ever getting the number 4 to come up. That would be astronomically unlikely, but not impossible.
Now in the infinity case our intuitions tend to fail because infinity is so tricky. But it seems like it is just the same kind of case: there are an infinity of possible situations and an infinity of times when situations can obtain. Thus we have N possible outcomes and N trials or opportunities. As with the dice case it does not follow that every one of the N outcomes will arise.
Does that help?
Peter
Re: infinite monkeys
Thanks
It helps a bit, but I'm still not sure. The more outcomes there are, the closer the probability comes to 100%. I understand that if there are a vast but finite number of outcomes the probability never quite reaches 100%, but it's infinity which throws me. It sounds like over an infinite period of time the probability that the monkeys will write the entire works of Shakespeare (many times over) is infinitely close to 100%. At that point the difference between that number and 100% doesn't seem meaningful.
Thanks again
- Jon
Re: Infinite monkeys
I think that there are many examples where you might have never-ceasing motion, and yet this doesn't mean that every possible arrangement will eventually occur.
For example, consider a pendulum, that moves forever (say in zero gravity or whatever). As long as the string is taut at the beginning, it will always remain taut. So the weight will always remain at a fixed distance from the pivot. Although it is possible is for the string to bend and the weight touch the pivot (a possible arrangement), this will never occur.
A simpler example: the motion of a planet, which always remains in its orbit, forever (not really, but whatever). And I think that it's more general: usually when we describe motion by a mathematical equation, the object's location is very restricted, although it might in theory move forever. (Ignoring examples of chaos theory.)
Even more infinite monkeys
Thanks, that's helpful: so the point is that if there are constraints, it doesn't matter how long you keep trying, you will never get certain outcomes. The planet example is of course Aristotle's picture: he thinks that the heavens rotate eternally and for a variety of reasons nothing can stop them from doing this. So for him the "probability" of the heavens suddenly stopping is zero and you can wait as long as you want, it will never occur.
I think there's still a lingering question along the lines put by Jon: if you have an outcome which _does_ have a positive probability, so that it is not constrained not to happen, then intuitively, the more trials you run the more likely it is that this outcome will occur at some point. (So compare flipping a coin once to flipping it 100 times: the probability of getting heads after one trial is .5, the probability after 100 times is close to 1.) Jon's reasoning, I take it, is that anything with a positive probability should therefore occur given an infinite amount of trials (times where things can happen). So even something very improbable, like monkeys banging out Shakespeare, will have a probability of 1 or approaching 1 over an infinite number of trials/amount of time.
My response to that was that if there are an infinite number of possible outcomes, which there are in the case of something like the monkeys on the typewriters (unless we set a limit to the length of what they're writing, or something), then the probabilistic reasoning won't work. Let's say that each outcome has an equal probability: then it will be positive, but infinitely small. So again, it's just like doing something like rolling a 6-sided die 6 times: you have N equally possible outcomes and N trials, and this certainly doesn't guarantee that each possible outcome will occur.
"(unless we set a limit to
"(unless we set a limit to the length of what they're writing, or something)"
Since the number of words in the works of Shakespeare (or the number or atoms in a golden mountain) is not infinite each of these events should come to pass eventually if you agree that they have a positive probability; which seems to mean - if you have not used the fact that X can never happen as one of your starting assumptions.
I still agree with Jon but I was surprised by the excellent arguments against the intuitive position.
Those darn monkeys
Peter's suggestion that with infinite possible events, individual events can have infinitesimal probabilities, which don't necessarily sum to 1, is surely right. In fact, standard probability theory has to deal with this thing all the time. Usually we avoid the language of infinitesimals in mathematics, but we could rephrase it without infinitesimals in this case and put it on solid ground. But instead, I want to discuss a different approach, that doesn't require Peter's suggestion.
The thing is, the probability of having the works of Shakespeare is NOT a constant, like 0.5. Rather, it depends on the current situation. It's higher when a lucky monkey has managed to type a couple words correctly, and lower after the typing the next word, "gbbbfsbs". With a variable probability, the chance of getting Shakespeare does not necessarily have to approach 100%. Specifically:
Let p(n) be the probability of obtaining the goal during time n. Let q(n)=1-p(n). Then the probability of never obtaining Shakespeare is the infinite product, q(1)q(2)q(3)...q(n)..., so the probability of getting Shakespeare is 1 minus that. If p(n) is any constant, say p(n)=0.01, then q(n) is a constant 0.99, so the infinite product q(1)q(2)q(3)...q(n)... will approach 0, so the probability of getting Shakespeare will approach 1, which is 100%, as we expect.
But if p(n) is not constant, q(1)q(2)q(3)...q(n)... might approach something other than 1. For example, if it approaches 1/2, then the probability of getting Shakespeare would approach (1 - 1/2) = 1/2, which is only 50%. To give you a concrete example demonstrating this, let p(n) = 1/(n+1)^2. Then
q(n) = 1 - 1/(n+1)^2
= n(n+2) / (n+1)^2
= [(n+2)/(n+1)] / [(n+1)/n ]
= [1 + 1/(n+1)] / [1 + 1/n],
so the product q(1)q(2)q(3)...q(n) = [1 + 1/(n+1)] / [1 + 1/1] = (1/2) [1 + 1/(n+1)], which approaches 1/2 as n goes to infinity.
(By the way, there's nothing special about "1/2"; I could have set it up to approach any constant between 0 and 1. It's more general than that; the only real requirements are that the product q(1)q(2)q(3)...q(n) must be between 0 and 1 and always be decreasing.)
Finally: Jon seemed to have another question: what does it _mean_ to say that a probability of something approaches 100% as time goes to infinity: does it mean it is guaranteed to happen "eventually, after an infinite amount of time", or does the probability go to 100% with some mysteriously improbable but real chance that it might never happen? Here, math doesn't help. When you look at the formal definition we use for a "limit", the whole point is to sidestep that sort of issue, while still being able to do whatever calculations we want to do and have it be correct (no matter how the issue is decided).
Monkey puzzles
Thanks Mike, that's brilliant. Just to take this back to Aristotle for a second, I find it interesting that he seems to have not only the (apparently false) intuition that everything possible will happen given an infinite time, but also the countervaling intuition that randomness _doesn't_ give you apparently ordered results. Hence his refutation of Empedocles which, to oversimplify, is just to say that a chance process will never give you a result that looks overall well-ordered.
infintely confused
Hi Mike-
Thanks for the detailed answer
I'm not a mathematician, but I think I understood your infinite products and the reason something which wasn't a constant wouldn't necessarily tend towards 1, but I'm not sure I understand why the probability of typing the works of Shakespeare in a given time is not a constant.
To take a shorter example, if there are 4 keys on the keyboard (k=4 eg. A, B, C and D) and the text we are looking for is 3 letters long (t=3 eg. "BAD") then I think the probability of getting it right in exactly t keystrokes is (1/k)^t (in this example, 1/4 * 1/4 * 1/4 = 1/64)
So the probability of failure in any 3 keystrokes is 63/64 and the probability of failure in repeated sets of 3 keystrokes over an extended period of time q(1)q(2)q(3)...q(n) is (63/64)^n ...which will tend towards zero (as you said constants would)
When you said the probability of success was higher after typing a correct word than a gibberish word, I assume you meant that the full text could appear anywhere in a longer series of keystrokes. After a correct word we would be part of the way there and after gibberish we would have to start again. In that case the maths is beyond me, but it sounds like the result would be more likely to appear sooner than in my first example (there are only 128 ways of getting BAD in two sets of three keystrokes (64 x "BAD???" and 64 x "???BAD"), but 256 ways in 6 keystrokes (64 x "BAD???", 64 x "?BAD??", 64 x "??BAD?" and 64 x ???BAD)
Thanks again
And Peter I'm sorry for hijacking your Aristotle comments
- Jon
oops
Hi, Jon. Yes, the gibberish word example probably wasn't my smartest move. If you have equal (constant!) probability of hitting each key each second, then the chance of getting Shakespeare will go to 1, regardless of the fact that sometimes there are setbacks. So let me back off that exact example.
There are other reasons that probability might not be constant. Maybe the monkeys lose interest in the "a" key gradually, so the probability of hitting "a" gradually decreases. If it decreases fast enough, then you will get the kind of effect I described in the earlier post.
Is it fair to modify the infinite monkey scenario in that way? Actually, I think it's not fair to assume otherwise, that is, to assume that the monkeys will hit each key with a fixed probability, independent of the keys they hit previously. We are really only interested in monkeys insofar as it is a metaphor for the universe, and why should the universe be like that? That is, why should the universe be describable as process over a finite set of equally likely states?
While that is how probability works with coins, dice and cards, it's not true in general. In particular, when there are infinitely many possibilities (as mentioned already by Peter), then the likelihood of each possibility may well be infinitely small, in which case it isn't even obvious whether "equally likely" can be a sensible notion anymore. (In fact there is a reasonable analogue, but it's not immediate, and it's not always appropriate.)
Let's start over. A more general approach is to think of the universe (say, which includes a room of monkeys) as a "random process". At any time, the universe is in some state, and in the next moment it changes to another state according to a certain probability. For example, if the monkeys have written "To b" so far, then in the next second the paper could be "To ba", "To bb", "To bc", ..., "To bz", or "To b ", etc., or the monkey might rip out the paper and starts over. Each new state has a certainly probability of occurring. The probability of the next letter being "a" changes over time, and depends also on the current state: for example, I'd guess that repeating a letter is more likely than not, and ripping out the paper becomes more probable as time goes on. I think that this is a better model for the universe than assuming that the one where the letter "a" always appears with the same probability.
But there's more. When we have "To b", the probability of "To ba" won't equal to the probability of "To bb", but what may be more interesting is that the probability of getting "Ham sandwich" is zero. The universe is like this, too: I could move 10cm left or 10cm right, but I can't suddenly reappear in Wichita. So, in general, from a given state, you can only go to a small set of other states, and there are many states that you cannot get to immediately. But then it's quite possible that that there could be two states X,Y, such that if you start from state X, you will never (after any number of steps) get to state Y. For example (maybe too simple): while it might be possible for a moon to be in any location, once it settles into a particular orbit, it will never suddenly jump away from it (assuming ideal circumstances), so it will never end up in a different orbit.
Apologies, I write too much.
Michael, interesting
Michael,
interesting discussion!
Couple of things - you mentioned again "infinitely many possibilities" but I don't think that applies in a finite universe? But that is just nit-picking.
In terms of the monkeys, even if the probability of hitting the 'a' key decreases over time, or if the monkeys might rip the paper out half way through King Lear, my intuition tells me that as long as the probability of each of these is respectively greater than zero, less than one, then in infinite time the Complete Works would still be completed.
Apologies if you think you nailed this with you earlier maths-heavy comment - I confess that I did not fully grasp that. But I would suggest the following:
You said
"The thing is, the probability of having the works of Shakespeare is NOT a constant, like 0.5. Rather, it depends on the current situation. It's higher when a lucky monkey has managed to type a couple words correctly, and lower after the typing the next word, "gbbbfsbs"."
But if we take the base probability as being that at the start when zero correct characters have been typed, call it p0(n), any subsequent values of p(n) will be equal to or higher than p0(n).
Then applying that thought to your following paragraph:
"If p(n) is any constant, say p(n)=0.01, then q(n) is a constant 0.99, so the infinite product q(1)q(2)q(3)...q(n)... will approach 0, so the probability of getting Shakespeare will approach 1"
p0(n) is not constant, but it is better than constant since it can only vary in a manner which increases the probability of attaining the goal.
Changing tack slightly, I wonder how the idea of certain states that are unobtainable once an intermediate state has been reached (rather thanunobtainable a priori) relates to the Second Law of thermodynamics - Entropy always increases.
E.g even in an infinte period of time all the matter in the universe will not coalesce into the smallest possible region of space having once left that configuration at the big bang.
Presumably the 2nd law is just a statement of probability which does not hold when infinite time is available?
less confusion?
Hi. I think that the confusion may have started when I wrote "obtaining the goal during time n". And maybe some other stuff I wrote. Anyway, our confusion about the math is due to different ways of setting up the problem.
The way I set it up, it makes sense if we are saying that during each time period, there is a certain fixed chance of writing Shakespeare. For example, maybe the time period is 1 day, and at the end of each day we clean the monkey cage and start over. (Or, instead of a time period, we start over whenever the end of a page is reached - same result, but it seems a bit more natural.)
John's description focuses on probabilities for getting a single letter at a time, and then goes from there.
Felix, your "p(n)" is the probability of obtaining Shakespeare (for the first time?) at time n, when we are observing what has happened since the experiment has begun.
Although we are all using the letters "p" and/or "q", we have defined them differently for each of us. That's why they behave differently.
Got It
Hi -
Thanks Michael - I have it now :-)
If a series of events are truly random, unconstrained and unconnected (eg. we are looking at a random selection of letters) then over an extended period of time the probability that a particular sequence has occured will tend towards 1.
But if a series of events is non-random, interconnected in some way or regulated by natural forces (as almost all macroscopic events will be) then the probability of a particular sequence will trend to something less than 1. I'm happy with that.
Thanks again
- Jon
Re: Got it
Good! Although "will trend to" should be "may trend to"; it depends on how the numbers work out.
-Michael