Can you imagine a world where nothing it in can be numbered or counted? Can you imagine a world where events take place but they have no causes? Can you imagine a world outside of time?

Kind of, yes, kind of. This argument has a few major flaws: 1. the problem of other minds suggests that even if you (or I) can’t imagine something that it might still be imaginable to someone (or some thing). 2. What we can imagine has little bearing on what is actually true (I almost said “no bearing,” but the fact that we can imagine in the first place probably entails a few truths). 3. It’s an argument from “lack of imagination” which creationists use all the time but which aren’t at all normative (largely due to (2)).

At any rate, not being able to imagine something doesn’t provide grounds for ruling out the possibility. I have trouble imagining a universe where things cannot be counted (a difficulty that hunter gatherer tribes probably don’t share, incidentally) but that doesn’t mean it’s impossible.

Any world we imagine must meet certain conditions for us to know it, and by “must meet” I mean they are a priori and necessary. To follow your analogy, we cannot listen to the world unless it makes sounds.

Again, “imagine” makes the argument problematic — the universe is not bounded by our imaginations. But there are other problems here. You argue that because the universe is knowable there must be a priori and necessary truths. But how do you know the universe is knowable? Again, following the same analogy, we can only know about those parts of the universe that make sounds (and it’s not necessary that every part of the universe should make sounds) and we have to trust that those sounds aren’t lies.

And I have yet to hear an example of an “a priori truth” that wasn’t either a mathematical statement (the truth of which depends on the axioms of some motivating mathematical theory) or “bachelors are unmarried” (which is true by virtue of a contingent definition rather than the structure of the universe itself). While I wouldn’t rule out a priori and necessary truths any more than I would rule out an uncountable universe, I’ve certainly never heard any convincing examples of either. Compared to the successes of empiricism and scientific reasoning, “a priori” and “necessary” are just about useless concepts for determining anything about the universe, at least as far as I’ve seen.

There was one question where you kind of punted, and I wished you had had more to say. Gironi asked you (p. 320) to describe what it was that non-mathematically trained people were missing when they only read the popularized versions of scientific theories/discoveries. As a fellow popularizer of physics, I think this is a very interesting question, and one that is hard to answer. (One is tempted to respond, “Well, just learn the math and then you’ll know what you’re missing.”) It’s particularly important for philosophers, if they are going to try to take ideas from modern physics and do philosophy with them.

So here’s a nudge to get you to address this issue more fully in some future blog post.

Any world we imagine must meet certain conditions for us to know it, and by “must meet” I mean they are a priori and necessary. To follow your analogy, we cannot listen to the world unless it makes sounds.

Okay – if the laws of our universe vary in time or space, you can develop an equivalent set of invariant laws that describe those variations.

SO – does there exist *any* set of observations which can’t be described mathematically?

If not, then what is really so unreasonable about the effectiveness of mathematics in the natural sciences (in reference to Eugene Wigner’s article with that title)?

Mathematics describes the universe that we observe…but it also describes an infinite number of universes that we don’t observe. Mathematics is just a descriptive language par excellence…yes?

Maybe?

Thoughts?

Maybe there is the possibility of getting around this by proposing some set of laws whose variations cannot, even in principle, be described by an invariant theory. But I am strongly skeptical that that is possible.

Now, if we were talking about the effective laws of physics as opposed to fundamental ones, then that is a more interesting question. But I think that at present, it is more or less solved by the existence of spontaneous symmetry breaking in the standard model of particle physics. With the existence of spontaneous symmetry breaking, it is no longer a question of whether the effective laws of physics vary, but instead a question of how much.

Differently put, if you found a law that seemed varied across space and time (e.g. the metric tensor in GR), wouldn’t you go looking for some deeper law that was able to describe the variation (in the example, the Einstein field equations)? Maybe the metric tensor could have been just be random with no underlying law to be found, but that would be hard to imagine (which brings us back to your point that we cannot make a priori statements about the properties of the world, I suppose).

Or maybe I am just lacking the imagination to think of the kind of inhomogeneity you had in mind.