Why might some mathematical truths be unknowable?

April 10, 2013 5:03 PM

For all we know, some mathematical truths might be unknowable. E.g., for all we know Goldbach's conjecture (or its negation) is this way. Yet, if there are unknowable mathematical truths, why might this be? Here are some initial considerations/hacks in the dark: (a) Presumably any computing device would have a finite number of parts. Maybe this would limit its processing power in some relevant way; (b) maybe some of the proofs are too long.
posted by Eiwalker to Grab Bag (20 answers total)

Is this the sort of thing you're looking for? Or are you aiming for some sort of 'why' to the Incompleteness Theorems?
posted by hoyland at 5:09 PM on April 10, 2013

The field you're interested in is called computablitity theory. In general, when you're working within a particular mathematical system (set of rules for proving conjectures), it's hard to reason about the system itself. As hoyland mentioned, there are theorems like Godel's incompleteness theorem that formalize what we can and can't prove
To answer your specific hypotheses:
a) Yes! Our model of computing is based on an idea called a Turing machine, which is a computer that has a finite amount of working memory (state) and an infinite amount of potential storage and makes decisions at a finite speed. Turing machines are limited in some well defined ways. Specifically, they can't solve the halting problem which means that you can't build a turing machine that makes predictions about the results of other turing machines.

b) Not really that I know of. Mathematicians don't care too much about things that are arbitrarily long (but still finite). Causing contradictions is a bigger concern.

Was that what you were asking? I can give you slightly more thorough descriptions about some of these things, but I'll admit that it's been a few years since I had to discuss them in a mathematically formal way.
posted by martinX's bellbottoms at 5:38 PM on April 10, 2013

The question is: if there are unknowable mathematical truths, why might this be?

As for (b): what about if the proof had to be infinitely long?
posted by Eiwalker at 5:47 PM on April 10, 2013

What is your definition of a mathematical proof that allows infinitely long proofs?
posted by aubilenon at 5:50 PM on April 10, 2013

An infinitely long proof probably doesn't make sense. So, maybe in cases where a given mathematical claim would require an infinitely long proof (or disproof), there is simply no mathematical truth (or falsity) in such cases. E.g., if Goldbach's conjecture (or its negation) is that way, then there's simply no fact of the matter about whether Goldbach's conjecture is true.
posted by Eiwalker at 5:55 PM on April 10, 2013

The question is: if there are unknowable mathematical truths, why might this be?
In that case, Godel's incompleteness theorem is what you should really look into. In plain English, it says basically that you can't prove or disprove the statement "this statement is false," and that for any nontrivial set of axioms there are statements that are equivalent to "this statement is false."

On the topic of infinitely long proofs, the only way to prove anything about how a Turing machine runs is to actually run it. Hence, the proof that a Turing machine runs for an infinite amount of time would be infinitely long, which is why we can't prove whether a particular Turing machine will halt or not.
posted by martinX's bellbottoms at 6:14 PM on April 10, 2013

The question is: if there are unknowable mathematical truths, why might this be?

I don't think it's even meaningful to ask "why" when it comes to things like this. It just is like that, that's all.

Also, be wary of the idea that "if we can't do it now we never will". Why can't we trisect the angle? Well, we can trisect an angle; we just can't do so with the traditional straight-edge-and-compass.

There are things which are unprovable within a particular mathematical system but which may be provable in some larger mathematical system which contains the former. Godel showed that any given system had limits, but they aren't necessarily the same limits.
posted by Chocolate Pickle at 6:15 PM on April 10, 2013

Because otherwise the system (provided it contains multiplication or something analogous to it in a specific way) would be inconsistent.

This is what Gödel's first theorem says. This is a good book explaining it.
posted by wayland at 6:20 PM on April 10, 2013

E.g., if Goldbach's conjecture (or its negation) is that way, then there's simply no fact of the matter about whether Goldbach's conjecture is true.

There's a lot of discussion in the history of mathematics and philosophy about this sort of thing. You had/have intuitionists who would hold to what you said: there's simply no truth-value assigned to Goldbach's conjecture obtaining.

However, the common mathematical/philosophical opinion is that there is a truth to the matter about Goldbach's conjecture, regardless of contingent matters of if we will ever solve it or fundamental matters of if it is provably undecidable.

You might want to look at Shapiro's, Thinking about Mathematics.
posted by SollosQ at 6:26 PM on April 10, 2013

Thanks for the book recommendations. The reviews on them look great. I will check them out.

In the meantime, it is difficult to see how a mathematical claim might be true if it is not computable. By virtue of what would it be true?
posted by Eiwalker at 6:58 PM on April 10, 2013

Something might be true by metamathematical reasoning but not provably true in the system.
posted by wayland at 7:25 PM on April 10, 2013

See, this is where the really fun stuff lives, where philosophy and math bleed into each other.

"Hilbert believed that all mathematical problems were solvable, but in the 1930's Gödel, Turing, and Church showed that this is not the case."

There. That link should get you started. It will be a long journey...

More fodder:

Undecideable Problem.

Turing Degree.
posted by Slap*Happy at 7:45 PM on April 10, 2013

Something might be true by metamathematical reasoning but not provably true in the system.

Would the metamathematical reasoning have to be part of a system that is either incomplete or inconsistent?
posted by Eiwalker at 8:05 PM on April 10, 2013

Well, on some level, we're allowed to just declare some things (axioms) to be true because we feel like it. The Wikipedia article on the Axiom of Choice attributes the following to Jerry Bona: "The Axiom of Choice is obviously true, the well-ordering principle obviously false, and who can tell about Zorn's lemma?" The joke is that all three are equivalent. The combination of the axiom of choice feeling very natural and the fact that you need it to prove things we want to be true means people by and large just deal with the fact that it's equivalent to something that seems like it ought to be false. (This is very much not my corner of math. Whether we should believe the axiom of choice may be a burning question in some quarters, but not in my world.) Which I suppose raises the question of how we decide which things we want to be true. Some combination of intuition and an expectation that things that seem to describe the real world be true, I suppose. (Which is not to say math needs to describe the world around us in some clear way. I'd be at a bit of a loose end if you insist on obvious, immediate applications.)

This is all super hand-wavy, if that's not apparent.
posted by hoyland at 8:16 PM on April 10, 2013

Would the metamathematical reasoning have to be part of a system that is either incomplete or inconsistent?

It's tricky. Wikipedia actually is good for this stuff, from an interested layman's perspective: Proof Theory.

A lot of this is still up in the air, waiting for better math or better philosophy.
posted by Slap*Happy at 8:21 PM on April 10, 2013

I agree that axioms of math are conventions, not facts, and that some mathematical systems have practical applications and others don't. For instance, there are a lot of different geometrical systems, but not all of them accurately describe our world. But is this relevant? At any rate, Godel doesn't seem to care whether anything is true in some system-independent sense. He merely says that some statements won't be provable in their own systems (if those systems are consistent).
posted by Eiwalker at 8:31 PM on April 10, 2013

This is getting rather chatty. But consider this question, Does mathematics exist independently of our ability to think about it?

To put it a different way, do we create mathematics or discover it?

My own opinion is that mathematics does exist independently of our ability to think about it, and we discover it, we don't create it.

In the meantime, it is difficult to see how a mathematical claim might be true if it is not computable. By virtue of what would it be true?

If mathematics exists without us, then the truth of a claim also exists even if we can't prove it (yet).
posted by Chocolate Pickle at 10:38 PM on April 10, 2013

Godel was a Platonist, and he thought his incompleteness proofs gave support to this view.

See also the halting problem and undecidability.

Check out the Putnam/Benaceraff eds. volume on philosophy of mathematics.

See This as well.
posted by professor plum with a rope at 12:43 AM on April 11, 2013

Infinitely long proofs seem to make perfectly good sense (although appearances are sometimes deceiving). For example, supposing the Goldbach conjecture is correct, an infinite proof might consist in a list of all the even natural numbers x that are greater than two together with the two primes whose sum is x. Such a proof makes just as much sense as the idea of completed infinite sets, like the set of all natural numbers.

Actually writing out such a proof is a supertask. (See the SEP entry for more.) If reality were cooperative enough, supertasks might be physically possible. Here (pdf), for example, is a nice paper on the possibility of supertasks in some weird spacetimes.
posted by Jonathan Livengood at 1:02 AM on April 11, 2013

Would the metamathematical reasoning have to be part of a system that is either incomplete or inconsistent?

Yes.

Btw, a kind of infinite proof in this case would be saying "we can prove this system is consistent by metamathematical reasoning, and we can prove that that is consistent by metametamathematical reasoning, ad infinitum". That's not allowed (or very satisfying), though.
posted by wayland at 3:13 AM on April 11, 2013