My Opinion on Infinity
Specifically the notion that you can divide a quantity up into infinite parts.
Problem: How big are those individual parts?
They have to be zero in size, hence you are no longer dealing with the quantity in question. Ergo you cannot divide a quantity into infinite parts.
Problem: How big are those individual parts?
They have to be zero in size, hence you are no longer dealing with the quantity in question. Ergo you cannot divide a quantity into infinite parts.
Comments (39)
The argument I know is that you can divide a quantity up infinitely. That is, an arbitrarily high amount of times, but never infinite times.
y will grow towards infinity as x approaches zero. But when x is zero, y is undefined. I'm not a mathematician but a graph of the function y will never touch the x axis (will not yield an answer to 1÷0).
So, it's not zero that makes infinity. It's an arbitrarily infinitesimally small value of x.
The parts are infinitely small.
So your opinion is, because a quantity divided infinitely would have parts with zero size, no quantity can be infinitely divided?
Not being a math guy, I have to ask.....is there a rule for obtaining a zero size part from any division at all?
I’m trying to picture a guy, standing there chopping off sections of a number line of x units, each part having zero size. I understand doing so is the only possible way to divide infinitely, but you gotta admit....he isn’t really doing anything. So there does appear to be some kind of contradiction.
Do you agree with the opinion contained in the OP?
The idea of chopping something into units requires a countable number of chops. You can half, quarter etc. The real line instead is an uncountable union of real numbers, so the analogy doesn't apply.
The OP stipulates a infinitely divisible quantity. Number lines do not exist in Nature, but one can be imagined a priori, consisting of an arbitrary, progressively conceivable set of real numbers (the numerical totality of the set cannot be imagined). Because it’s an abstraction, the guy chopping off numbers one at a time is itself an abstraction, but sustains the conclusion he is not chopping off parts of zero size, because the number line must be conceived as getting shorter.
I’m gonna stop now; I don’t want to be responsible for the math guys hurting themselves laughing at me. (Grin)
Being unable to shave off parts of zero size is precisely the limitation I spoke about. You can shave off sets of zero size easy, say {x in [0,1] except for 0.5}. I'd say that since it can be done mathematically, and in a consistent manner, it's certainly conceivable, and we shouldn't therefore privilege intuitions of discreteness in nature over intuitions of continuity - what holds where and to what degree is a matter for investigation; conceptual work and experiment.
Ahhh....that’s what you meant before by involving sets or elements of sets. OK, fine. I can dig chopping off sets of zero size; that’s just an empty set. And by association, the totality of the divisible quantity is undiminished, which seems to sustain the OP.
Now that you mention it, I am favoring intuitions of discreteness, aren’t I. It never crossed my mind there was any other way to look at the a priori conceptions of “quantity”. Or the infinite for that matter. Apparently, though, I shouldn’t be, with respect to the problem at hand. So....thanks for that.
The empty set has size 0, but so does any finite or countable set as a member of the real line. Even the rationals.
Wha....wait. A finite set is has size zero? So an unpopulated empty set is the same size as a set of countable numbers? In other words, the set is what makes the size, not the members of it. But what is it about a set that determines it’s size?
{1} has cardinality 1, but measure 0 in the real line. The size depends on the measure. See this vs this.
Holy crap on a cracker.....I never even knew there was any of that stuff. Now I see where you’re coming from. I looked up some of the things you brought up, but...obviously....I didn’t get that far.
Any countable set of real numbers has Lebesgue measure 0.
.....put a measure on any set: the "size" of a subset is taken to be.....
Back to the OP. Is the opinion correct?
Well, because there are sensible ways to think of subsets of sets as having 0 size, that does go against parts (subsets) of wholes (sets) necessarily not having 0 size. Really though the formulation is wrong, because there's not just one size concept which can be neatly applied to everything.
OK. Agreed. I’m in no position to hold with the things I learned here today, even while appreciating the exposure to them. I think I’m going to stick with what I’ve convinced myself I know, and if somebody comes along and upsets my intellectual applecart as respectfully as you did......so much the better for me.
I have a friend who is into physics and he claims because you can divide a quantity up for ever that means that any quantity is made up of infinite points.
And what does infinitely small mean? It is not there.
Say what?
Quoting albie
He is correct, between any 2 real numbers, there are an infinite number of real numbers
This is possible mathematically. Physically, there can never be an infinity of anything, because observing an infinity is impossible (as it takes an infinite amount of time).
Not that I'm arguing for extant infinities, but why would whether there's an infinity of anything hinge on observation?
Well physics describes observable reality. I use it in a narrow sense here, a metaphysical infinity is theoretically possible.
No, but there is a difference between things that have not (yet) been observed and things that are unobservable in principle.
Any way you look at it, it seems that your epistemology puts a priori constraints on the world, in that it can only be such as to be "in principle" observable. It seems strange to make such egocentric demands of the world, which doesn't seem to care about you one wit.
I am not sure what is unclear about my position, but anyways "in principle" means based on the attributes of the theoretical object. A ship beyond the horizon is still a ship, which means it should for example reflect light. It is observable, even if you cannot practically observe it currently.
Quoting SophistiCat
The world in a practical sense certainly ends there, as far as current knowledge can tell us. You can still make the technical distinction between things that cannot be observed because we cannot get close enough and things that cannot be observed because of their attributes irrespective of their spatial relation to us.
Quoting SophistiCat
I do not put these constraints "on the world". Observable reality can only consist of that which is observable. I am not talking about the nature of objective reality here.
Consider a 500 gm block of cheese. Let's say we choose an arbitrary mass x gm and calculate how many x gm are in the block.
The operation would be 500 gm ÷ x gm = n
As x decreases in size (tends to 0), n tends to infinity. That's all.
It's a principle. It says that no matter how small of a thing you get, you can always get something smaller. Whether or not it's true is debatable, but I think it would be difficult to prove it, one way or the other.
OK, let's go with ships then. According to some speculative calculations in quantum cosmology (cf. Many Worlds in One by Garriga and Vilenkin) not only is the universe infinite, but it is infinitely repetitious: you might say that quantum reality is not diverse enough to come up with an infinite variety of objects, and so when it gets big enough, sooner or later it begins to repeat itself. The consequence of this is that an infinite universe contains within itself an infinite number of Earths just like ours. Of course, such twin Earths are so rare that statistically, we would expect them to be too far apart to ever make contact. There almost certainly isn't another Earth in our Hubble sphere. But we are talking in principle, right? As you say, these Earths (and any ships sailing their seas) reflect light and so are in principle observable.
So there you go, an infinity of physical objects can (in principle) exist, even by your own criteria of existence.
Quoting Echarmion
That "observable reality can only consist of that which is observable" is a truism, but remember, the question is not what is observable, the question is what beliefs about the world are warranted. I agree that our knowledge of the physical world comes primarily from observation. This necessarily constrains what warranted beliefs we can have about the world. But those constraints alone don't uniquely define an epistemology. Specifically, this broad empirical principle is not equivalent to the dictum that one can only have warranted beliefs about that which one has seen with one's own eyes. Nor is it even equivalent to your vaguer observable-in-principle criterion.
We routinely form beliefs about things that cannot be verified by direct observation - for example, things that have occurred in the past. Neither does the scientific method require that every single implication of a scientific theory be verifiable through observation. And this is why science doesn't really have a problem with an infinity of physical things.
But I didn't talk about an infinity of objects "existing in principle", did I? I think you're mixing physics and metaphysics (and arguably so do the physicists speculating about multiple realities). Even if an infinity of objects (e.g. ships) existed in objective reality, we could never observe the entirety of them. We could only ever observe a finite (but arbitrarily high) amount. As a result our experienced reality would never actually contain an infinity. Since physics (and the scientific method in general) is concerned with figuring out the rules with govern experienced (i.e. empirical) reality, it can not include an infinity of anything.
Quoting SophistiCat
A fair point. I think we don't actually disagree on very much, we only have a slightly different perspective.
Quoting SophistiCat
This is true in a sense. Of course I base all my knowledge on things I have somehow experienced, but I don't need to personally see a Blue Whale to believe they exist as part of empirical reality.
Quoting SophistiCat
This I am not so sure about. It's certainly possible I am missing something, but I think that ultimately knowledge about "the world" must reference experience, where else would we get it from?
Quoting SophistiCat
Sure, but these beliefs should still be based on indirect observation, that is archaeological evidence, textual evidence, etc.
Quoting SophistiCat
Are you sure that a scientific theory can have "implications" - which I presume means predictions - that are not verifiable through observation? If we have such a theory, how would we verify it? Specifically, how would we determine which of two theories is a more accurate descrition if they only differed in their implications for the non-observable. The Copenhagen interpretation vs. multiple worlds might be such a case, but my knowledge about quantum physics is to limited to say for sure, and I have a suspicion (though the previous disclaimer applies) that those are actually concerned with metaphysics.
I don't think this is true, I think science has a major issue with an infinity of any thing physical. Working from memory, could be wrong.
Good question (and excuse me for not quoting the rest - I believe the following will suffice to address the substance of your post). So to recap, what's at stake are our epistemic criteria for selecting among alternative beliefs - in this case, scientific theories. What are the virtues of a theory? Well, being testable is paramount. But what does that mean exactly? If a theory has any generality to speak of (we are not talking about the theory of how much change I have in my pocket right now), then chances are that as a practical matter, we can't test all of its predictions because there are too many of them and many (indeed, most) are impractical or even physically impossible to test. So, although we say that theories should be testable, we get by with testing only a manageable sample of their predictions and generalizing from that.
And how do we distinguish between theories that fit the evidence equally well? We consider other theoretical virtues: simplicity, cohesion with other theories, fecundity.
Now to take an example, forget speculative cosmology (I brought that up just for fun) and consider something much more intuitive and uncontroversial. It was long thought that space was infinite; indeed, only since advances in mathematics and Einstein's General Relativity did it become even theoretically conceivable that space might not be infinite in extent. In earlier times people worried about possible problems, such as gravitational collapse (Newton) or Olber's paradox, but in the 20th century these issues have received satisfactory resolutions. So far an infinite space remains the simplest model consistent with astronomical observations. So we are on pretty safe ground here.
If space is infinite, then how much stuff does it contain? Well, we can only observe a finite volume, but from what we can see, even this finite neighborhood looks to be pretty uniform beyond a certain scale. We could still posit that beyond the limits of observation stars and dust and all other matter end and the rest is just empty space, with out cosmic bubble being like an island in an infinite ocean. But a simpler theory says that the rest of the universe looks pretty much the same as what we see around us. Another way to put this can be expressed as the so-called Copernican principle: we have no reason to assume that the spot from which we look out at the universe is special, and so we should not so assume.
So to conclude: we can only practically observe a finite amount of things, but other theoretical considerations lead us to believe that there's a lot more stuff out there - indeed, perhaps an infinite amount. Direct observation is not the only criterion by which we determine what exists.
Ok, this is convincing. We need tools in addition to just observation (or falsification through observation) in order to formulate general theories.
Quoting SophistiCat
This is well written and I mostly agree with you. The Copernican principle seems to me an extension of the "virtue of simplicity", as you called it. We assume the universe is, on a large scale, uniform and consistent in both time and space. If it weren't, we could not make any predictions at all, so this is a necessary assumption.
The only thing I wonder if the proper conclusion is that the universe is "infinite" or that it is "indefinite". That is does it include a positive infinity or is it merely not finite, in that there is always more in space and time, but the total amount is never infinite. The question is, I think, one of the proper application of the virtue of simplicity. Is infinity "simpler" than an indefinite universe? One could argue that "infinity" includes an additional positive, and unprovable, claim, so it is more complex.
We essentially brush up against metaphysical realism vs constructivism here. A realist would, presumably, find it hard to entertain the idea of an indefinite reality, so infinity seems the only reasonable option. But is this just a metaphysical position, or does the scientific method actually provide good reasons to conclude positive infinity rather than merely the absence of a definite border?
Where can "indefiniteness" fit into all this? I can think of a few aspects. One is where a theory is altogether silent about some question, leaving it (as far as that particular theory is concerned) completely open. Another is an explicitly stochastic element of a theory, such as can be seen in classical statistical mechanics, population dynamics or quantum mechanics. Finally, there is an uncertainty associated with theory choice, which owes itself to insufficient or uncertain data or to theoretical controversies. As far as cosmology is concerned, this latter "indefiniteness" is the most relevant, I think.
The amount and the quality of data that is necessary to determine the topology of the universe is necessarily limited, nonuniform and biased. Scientific methodology, such as statistical model selection, is also somewhat controversial - no more so as when data is scarce. Astrophysicists and cosmologists understand this, but there isn't much they can do about it. I said that infinite space models are currently favored as both the simplest and the fittest, but there actually are publications in scientific journals that argue that finite topologies provide a somewhat better fit to observations. I don't have any expertise to evaluate this research, but my general impression is that if you ask most experts who are well-versed in this topic, whatever their own opinion is on the question of the size of the universe, they will freely admit that there is a lot of uncertainty here, and that this is probably how it will always be.