Descriptive vs deductive logic

I've written this previously elsewhere on the internet, but it's something I've had to point out several times in several contexts, so I thought I should write it down here.

There's this really confusing thing, that trips a lot of people up -- mathematicians included, indeed I might say mathematicians especially, as others don't have reason to deal with this -- which is that the word "logic" in math seems to actually refer to two different things. Well, sort of; you’ll see what I mean. But for now, let's call them "descriptive logic" and "deductive logic".

Descriptive logic is about what structures (models) satisfy what statements. So, you define a language, you define how to interpret the connectives and such of that language, and then if you have a would-be model with meanings assigned to the primitives, you can say which statements it makes true and which statements it makes false. Whereas by deductive logic I mean, what are the rules of inference, and what sets of statements prove what other sets of statements under these rules.

It seems to me that this distinction is often not well-appreciated? I mean yes, you have model theory vs proof theory, but putting it that way frames it as two ways of studying at the same thing, when really there appear to be two different things. Now by Gödel’s completeness theorem, for first-order logic these are actually the same, but... well, it’s not quite that simple.

So here's the confusing thing. I said above that the word "logic" refers to two things. But as best I can tell, in technical mathematical usage, the word "logic" refers purely to what I called "descriptive logic". Which is confusing, because going the ordinary usage of the word, you would expect it to mean "deductive logic". But actually in technical usage that seems to be called a "proof theory" or a "proof calculus"?

Now as mentioned above, in the case of first-order logic, Gödel's completeness theorem lets us elide the distinction. The term "first-order logic" technically refers to the descriptive logic, not the deductive logic! And indeed there are various proof calculi for first-order logic, but the distinction ends up not being important to anyone other than proof theorists, because they’re all equivalent. So, whatever; you can refer to any of these proof theories as simply "first-order logic" and people will get the point. They won't know which one in particular you're referring to, but, again, it won't matter.

Where this becomes a problem, then, is when people want to use second-order logic. Because for second-order logic we don't have any of this. There's no accepted proof theory for second-order logic, and given that Gödel's completeness thoerem doesn't hold for second-order logic, there arguably can't be a satisfying one. So "second-order logic" really refers to descriptive logic, and only to descriptive logic. But I think a lot of people don't appreciate this, and so suggest using second-order logic in places where it makes no sense (because you need deductive logic); and other people don't call them on it, because they don't appreciate it either.

This is why so many suggestions to use second-order logic actually present more problems than they solve, and why we found mathematics on first-order logic and set theory (or equivalent), emulating second-order stuff that way, rather than on second-order logic.

-Harry

The physics rant

Well, joshuazelinsky has asked for this to be a blog entry, so here we go.

Now, I am not the best person to write this rant. I don't know physics well enough. But Nic isn't writing it, and Josh wants to see it, so I guess I will.

So, people who aren't mathematicians seem to think of physics as this, like, really mathematically rigorous thing. And as best I can tell as a mathematician -- no. It isn't.

Now most of physics prior to, like, quantum field theory can be quite well mathematized. (Note: I guess I'm really primarily talking about fundamental physics here. Less the purelyBut QFT is another matter, and putting it on a mathematically rigorous footing remains a famous unsolved problem.

This is all well-known. But I think the situation is worse than is generally appreciated. Like, given that people keep on doing QFT despite its mathematical foundation not being fully settled, you might think that this is some minor matter. Like, yeah, the foundation will be worked out later, but if you handwave the foundations things are basically fine, right? I mean, we get numbers out of it that have been confirmed by experiment to ridiculous precision.

But no! Things are not fine! Yes, getting out numbers you can compare to experiment is important -- otherwise you're not doing physics. But it is not everything, and it does not mean that there is no problem. It's a little funny, because you see a lot of complaints about physicists ignoring the whole empirical reality part and just doing math that is disconnected from the physical world; and these complaints are also correct, but they're a separate matter, unrelated to this as far as I know.

Because you see, while QFT may be the big thing that lacks mathematical foundation, the problem is much broader than that. It's about how physicists think about their models generally. QFT is just where this gap has most clearly manifested into a visible problem.

So just what is the problem? Well, let me put it this way. When you ask a physicist to give you a model of something, you'll typically notice something big missing: There's no ontology. They just launch straight into equations.

Like -- normally if you ask for a mathematical model of something, the way it should begin is with something like, "OK, let's say M is a Riemannian manifold and p is a point in it..." or "OK, let's suppose G is a finite connected graph and T is a spanning tree for it..." or what have you. You start by introducing all the mathematical objects that the model includes, and what sort of mathematical objects they are. This is what I mean by an "ontology" in this context. Once you've done that, only then can you meaningfully discuss the relations between them, possibly in the form of equations. Right? How can you talk about a mathematical object, if you don't even know what sort of mathematical object it is?

And yet when you ask a physicist for a model, typically they skip this step. Now you might just think that, oh, that's because they consider all this stuff implicit and obvious and don't want to waste your time with it. Because that's the obvious, charitable, assumption to make, right? You're talking to somebody who knows the subject, you're going to assume that they, y'know, know it.

But if, getting confused, you actually ask about these parts that have been left implicit, what you will find is that the physicist you're talking to likely can't answer these questions. They haven't been left implicit -- they're missing. They didn't leave out telling you the ontology; they honestly don't have one.

Now obviously they have some idea what types of objects they're using, in that they can typically tell you what set any given map is mapping to (this is a scalar-valued function vs this is a vector-valued function, etc); they just have no idea what it's mapping from!

So, mathematics is about the manipulation of mathematical objects, but that's not what they're doing. They're just manipulating equations relating numbers. What do the numbers mean? What spaces do these functions live in? Who knows! Certainly not the people working with them. The thing that physicists refer to as a "model" is not, in fact, a model. When physicists claim to have a model of something, typically, they don't, not a real one.

And this is important! Most especially for fundamental physics. Because the goal of fundamental physics should be to build a mathematical model that can describe the universe. Not just to get numbers out, but to say, hey, here are what types of mathematical objects physical reality consists of -- to provide an ontology. Like, y'know, any model has. Fundamental physics is not solved until you can do that, until you can describe the universe by saying, "We'll take the universe to be a tuple consisting of...".

(Yes, of course there can be multiple isomorphic descriptions. I'm not claiming that there will be a unique such model. Just that to solve fundamental physics means, in part, to find at least one.)

What I find is that physicists don't seem to be aware that there's anything missing here. And it goes beyond that, to how they think about other things as well. Like, they just don't really care about any distinction between what's fundamental and what's not; what applies truly universally, and what requires assumptions. Like, entropy. Is entropy an objective thing that can be defined for any state of the universe, without assumptions, and does the second law of thermodynamics truly always hold; or is it a leaky abstraction that only works on certain contexts, or is dependent on one's epistemic state, and all of our statements about it only hold when the necessary assumptions are satisfied? Ask a physicist, they typically won't have an answer to this!

So what looks at first just like this problem of, oh, QFT still needs to have its foundation filled in, turns out to be just the biggest red flag of a much broader problem, that physicists have basically entirely the wrong attitude to the mathematics that they do, to the point that they can't even answer the most basic questions about it. I'm not even sure I'd say they're truly doing mathematics at all, so much as mathematics-inspired manipulation of equations.

Now I've observed all this on my own, but Nic -- who's learned more physics than I have -- tells me it's even worse than I know. As an example, he tells me, He says, nobody actually knows how to write down a one-proton state. Like, a proton is made of three quarks, right? So there should be some way of writing down a one-proton state that is somehow a combination of three one-quark states. Except, nobody knows how to do this! Turns out once you manage to make the problem clear, it's an unsolved problem.

But, he says, it's even worse than that, because it's hard even making it clear to physicists what you're asking. Because -- surprise, surprise -- they don't think that way. Like, if you ask a physicist, how do you write down a one-proton state, they'll just, like, write down "|p⟩" or something. I mean, it's a proton, right? It takes quite a bit of work to explain what the problem is.

So, yeah. That's the physics rant, more or less. joshuazelinsky, I hope that answers your questions here.

But let me also take this time to point out that Nic has been writing a series of articles on the web called "Physics for Mathematicians", that aims to be better about this sort of thing. It's far from done, but I suggest reading it. Now parts of it get pretty advanced and frankly I don't understand all of this, and I think there there are definitely parts there I skimmed quite a bit. And also there are parts that I think could use a fair bit more explanation. Still, on the whole, it's pretty good, and quite enlightening, and I definitely recommend it. In particular, reading the thermodynamics one should probably do a fair bit to clear up the question of what the hell is up with entropy.

Anyway, yeah...

Addendum: Oddly, I get the impression (and again this is just an impression because I haven't studied this much) that economics of all fields is better about actually using math correctly. I don't know what to make of that...

Addendum later: See also this entry where I discuss another example.

-Harry