Monday, August 29, 2011

Actual infinities in physics

This is a continuation of my series, "Here are a few things that are wrong about the cosmological argument." Previously, I tried to define, precisely the distinction between "actual" and "potential" infinities.  Now, I will try to provide examples of actual infinities in physics.  But before we get on with the examples, I will outline the possible conclusions.
  1. My definition of actual and potential infinities is no good.  Bring your objections to my previous post.
  2. I've somehow misapplied the definition.
  3. The objects in question are not "real" or "existing" objects.  Many people contend, for instance, that the set of all positions is not a set of real objects.
  4. The theory of physics I'm describing must be incorrect (not just possibly incorrect).
  5. Contrary to William Lane Craig's argument, sets of real objects in the real world may be actually infinite.
In this post, I will not argue for any particular conclusion (though I claim that 2 is incorrect).  But I would definitely like to hear what you think for each example.

A. Real Numbers

Real numbers are abstract entities, so you might wonder why I am including them in a set of physics examples.  The reason is that real numbers are so ubiquitous in physics.  Nearly every quantity in physics lives in the set of real numbers, including time, distance, energy, electric field, temperature, brightness, pressure, to name a few.

Note that there are two ways in which the real numbers have an infinite cardinality.  First, it extends to arbitrarily large numbers, just like the set of natural numbers.  Second, between any two real numbers (say, between zero and one), an infinite number of real numbers are packed in between.

Now, as to whether this is a potential or actual infinity is a bit tricky.  It depends what we are talking about.  If I'm talking about the set of all possible distances, this is a set with infinite cardinality, and thus actually infinite.

However, if I talk about one particular distance, say the distance between two stars, this is not an infinite set.  The distance between the two stars has only one value.  And yet, perhaps there is some possible world where they are further apart.  Perhaps the two stars can, in the set of all possible worlds, be arbitrarily far apart.  If this is the case, then the distance between the two stars is potentially infinite.

But what if the two stars are moving towards each other?  Then the distance between the two stars really does have an infinite cardinality of values (in this world, not in other possible worlds), though in different moments of time.  The set of distances between the two stars would be actually infinite.

And if we're talking about points in space themselves (or moments in time, or events in space-time), these are of course actually infinite.  In standard physics, space is infinite in extent, and events are densely packed.  Yes, I know that there are theories in which space is not infinite, and theories in which space is made up of indivisible atoms.  The question is, are you willing to accept these theories on a purely philosophical basis without any empirical evidence?

One last note is that the very idea of taking a derivative relies on actual infinities.  For example, if I consider the velocity (which is the derivative of position with respect to time) of an object, the idea of a "velocity" is only coherent because the object exists in an infinite number of locations at different instants.

B. Fields

A field is something that has a value at every point in space.  There are scalar fields, whose value at every point is a real number, and vector fields, whose value at every point is a magnitude and direction.  For example, temperature is a scalar field and wind velocity is a vector field.  Of course, these two examples don't really exist at every point, since on the smallest of scales, there is no temperature or wind velocity between the air molecules.

But physics has many examples that are more fundamental.  One example is the electric potential.  The electric potential has a value at every point of space.  And thus, just like the points in space are actually infinite, the values of the electric potential are actually infinite.  Furthermore, we also speak of the electric field, which comes from the derivative of the electric potential.  If the electric potential did not exist at an infinite number of points, this concept would not even be coherent.

Fields are also a fundamental part of other fields of physics.  There is the gravitational potential and gravitational field.  In general relativity, we have the metric tensor.  In quantum physics, we have the wavefunction.  In particle physics, every one of the different particles is associated with a different field.  It is not a stretch to say that fields are one of the most fundamental components of physics.

C. The singularity

The singularity is a hypothetical point in time at the "beginning" of the big bang.  Note that Big Bang theory has little to do with the singularity, and is merely the theory in which distances between all galaxies is increasing over time.  But if you naively follow the trajectories of these galaxies backwards through time, then they all cross at one point.  My expert opinion is that the existence of the singularity is disputed, but for now I will analyze it as if it really existed.

I'm going to point to this video debunking the Kalam cosmological argument, starting at 3:38.  Monica points out that William Lane Craig's position is inconsistent, because he denies the existence of infinities in realities, and yet asserts the existence of the singularity, which he himself describes as having infinite density, pressure, temperature and curvature.

I'm going to disagree with Monica.  WLC's position, in this case, is consistent.  The density, pressure, temperature, and curvature of the singularity are only potentially infinite, at least by the definitions we've discussed.  My argument?  None of these properties correspond to an infinite set.

By infinite density, we really mean that there is some set of massive objects in zero volume.  Unless this set of massive objects is itself an infinite set (see example D), then there is no actual infinity to speak of.  Instead, the density is potentially infinite, because for any finite value M, we can find an instant in time where the density is greater than M.  At the singularity itself, the density is technically undefined, divide by zero.

(Of course, the set of all densities is actually infinite, but then this is going back to example A.)

Similarly, pressure and temperature do not correspond to infinite sets.  I'm going to admit that I am ignorant as to whether curvature represents an infinite set, but I suspect not.

D. Uniform cosmology

You may have heard that the universe is finite in size.  But actually, you misheard.  The correct statement is that the observable universe is finite in size.  The speed of light is finite, and has had a finite amount of time to travel through space.  When we look far into space, we are looking far into the past.  If you look too far back in time, you reach the "time of last scattering", which is when the universe became transparent.*  But theoretically, if we had an omniscient view of everything, we would find that the universe goes on forever.

*Note that this is not the same time as the Big Bang.  It occurs 380,000 years afterwards.  This is where the Cosmic Microwave Background Radiation comes from.

In standard cosmology, not only does the universe go on forever, but it is uniform on the largest scales.  On a small scale, the universe is not uniform, because the sun is not at all like the earth, which is not at all like the space between them.  But on a very large scale, larger than galaxies, larger than galaxy clusters, larger than galaxy filaments, the universe is uniform.  That is, the density of matter is the same everywhere throughout the infinite universe.

A corrolary of uniform cosmology is that there are an actual infinite number of particles, stars, galaxies, and galaxy clusters.  This is not a potential infinite, because the infinite set exists entirely in this universe, not in some possible universe.

Mind you, there are theories of cosmology in which the universe is not uniform or not infinite.  For instance, I recall that blogger Sean Carroll once authored a paper proposing that the universe is just a bit lopsided.  However, the universe would still have an infinite amount of matter in this scenario.

So, what are your conclusions?  1, 2, 3, 4, or 5?  Myself I would agree with different conclusions for different examples.

"A few things wrong about the cosmological argument"
1. Actual and potential infinities
2. Actual infinities in physics
3. What is real?
4. The "absurdity" of Hilbert's Hotel  
5. Interlude: God is infinite
6. Forming Infinity, one by one
7. Uncertain beginnings
8. Entropy: The unsolved problem 
9. Kalam as an inductive argument
10. Getting from First Cause to God  


    drransom said...

    I don't quite think I agree with your idea that taking a derivative requires actual infinities. Actual infinities are required for the derivative-based theory to be completely accurate, but it can be a useful approximation on an abservable scale provided that the number of entities involved is sufficiently large. In classical mechanics, F = m*dv/dt + v*dm/dt, and yet nobody suggests that Newtonian physics is categorically invalidated by the fact that mass is quantized.

    thetruthaboutGENETICS said...

    Hard to get my head around it. Zero volume? How is that even possible?!?!

    miller said...

    Yes, drransom, I agree. If space and time were discrete, we would simply replace derivatives with differences.

    In fact it's not clear that containing all that mass in zero volume is possible. When you confine matter to a smaller space, its momentum becomes more uncertain, by the uncertainty principle. Higher momentum uncertainty requires higher energy. So basically, it requires an infinite amount of energy to squeeze matter into a point.

    The problem is Big Bang theory is based on General Relativity, which is a non-quantum theory of gravity. We need a quantum theory of gravity to really say whether infinite density is possible.

    Charles said...

    Number 3 is correct.

    For the record, Craig does not assert the existence of the singularity. He says, and I paraphrase, "The ontological status of the Big Bang singularity is a metaphysical question concerning which one will be hard-pressed to find a discussion in scientific literature... the ontological status of this boundary point is virtually never discussed... a good case can be made for the assertion that this singularity point is ontologically equivalent to nothing... instants and points seem to me to be mathematical fictions."

    miller said...


    For future posts, I'm happy to use (short) articles provided by opponents as references. However, the link you provided doesn't work for non-members.

    Charles said...

    Registration is free if you wish to read it, but it's also in his book (of which the "short article" is an excerpt) Theism, Atheism, and Big Bang Cosmology (page 259).

    Helenski said...

    It seems as though most of your examples (excluding D.) are dependient on a point continuum. Some may say that space and time may ultimately be discrete, thus following 3.

    As for myself, I am not convinced by the whole idea that space and time must be discrete at something like the planck scale (which is apparently where the argument is usually derived from), but I am also not fond of the idea of a point continuum either (the concept may be practically useful though). I would like to think of continuity as something that can never be truly divided into parts (like atomless gunk). As space has no points and time has no instants, there would be no true complete division, which fits into the intuitive idea of "infinite divisibility". But as a result though, I would follow Aristotle on this one, and say that it is only potentially infinitely divisible, since by definition such gunk has no ultimate parts and thus also adopt 3 (albeit for different reasons from the above).

    valjok said...


    I suspect that the direction is produced by multiple dimensions. Take one dimension. Can you say what is the direction for a 1-component vector? It should make difference between real number and identical vector. Can you say that a scalar is a 1-dimensional vector? Thanks.

    miller said...

    That's a point tangential to this post, but actually no. One of the defining features of a vector field is the way it transforms under rotations. If a vector has fewer dimensions than three, it's basically a completely different mathematical object. Maybe you could call a scalar a one-component vector, but it's not much like a vector.

    valjok said...

    Gilbert Strang says that rotation applies universally, no matter how many dimentions you have. Rotation is just one of linear operations and there is no limit on which operations you can apply to which dimensions. The only vector that has no direction, is null-vector (from 0-dimentional space). But scalar, as 1-dimensional object, has a direction.

    miller said...

    Rotating a vector field involves using the same linear transformation on both the vectors and on space. So it needs to be 3D rotation.

    valjok said...

    But, this says that vectors in the vector fields (must) have their dimensionality matched with the dimensionality of the space. IMO, this looks much better than the tradition to say that "scalars do not have any direction".

    valjok said...

    Your answer gets a lot of downvotes in

    Is it because it was delivered by as stupid person as me?

    miller said...

    I'm not sure my answer is appropriate for that context. Here, I am mainly focused on WLC's argument, which states that actual infinities are impossible in principle. I point out there are lots of actual infinities in our models, so WLC is forced to say our models are impossible on a priori grounds. Of course, it could be that the models are incorrect, and that the underlying reality is discrete, but I think it's frankly anti-scientific to declare that this must be true.