If we take a block of cheese a foot squared. And we say each infinite point is 1 gram in weight then that means the block of cheese must weigh infinite grams. AS long as each point has a value more than zero you will always get this. — albie

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.

And what does infinitely small mean? — albie

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.

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. — Echarmion

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.

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. — 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.

Two points, in a quantum universe, our reality could be not much more than a giant movie screen, and there can be plane after plane after plane of other realities. This could be really cool, but it is an awful concept to argue anything from, because it allows for every possibility of everything. There are no flying teapots, there could be on universe X, next. Secondly, the real issue in physics now is general relativity works, for everything, right up until it doesn't, an then quantum mechanics kicks in. For applied physics, not much of an issue, pick the right tool out of the bag, plug in the numbers and do the calculations. But theoretical physics has the same problem you have in point one, if everything is possible, nothing definitive is. They need a bridge between GR and quantum to link the 2 worlds. Not there yet. So, in the interim, I would suggest we do what applied physicists do, and unless working in the quantum level, we work in the world of GR.

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. — SophistiCat

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.

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. — SophistiCat

A fair point. I think we don't actually disagree on very much, we only have a slightly different perspective.

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. — 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.

Nor is it even equivalent to your vaguer observable-in-principle criterion. — 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?

We routinely form beliefs about things that cannot be verified by direct observation - for example, things that have occurred in the past. — SophistiCat

Sure, but these beliefs should still be based on indirect observation, that is archaeological evidence, textual evidence, etc.

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. — 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.

And this is why science doesn't really have a problem with an infinity of physical things. — SophistiCat

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.

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. — Echarmion

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.

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. — SophistiCat

Ok, this is convincing. We need tools in addition to just observation (or falsification through observation) in order to formulate general theories.

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. — 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?

The business of science is to come up with theories, and the best theories win more peer approval. What is the "best" theory? Ideally - one that has the most theoretical virtues. As we have discussed, providing a good fit to data is an important, but not the only virtue. Otherwise the best theory would just be an enumeration of all known observations and measurements: that would guarantee maximum fitness. But the best theories can actually sacrifice some fitness in favor of other virtues, such as simplicity, and of course they venture to extrapolate beyond available observations. That latter feature is pretty much a sine qua non for a scientific theory: if it does not offer theoretical predictions that go beyond what has already been observed, then it is not much of a theory.

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.