at least in one sense, I am a Platonist afterall

[gusl]

Whether your own philosophy of mathematics is Platonism or not, can be easily determined by using the following test. Let us consider the twin prime number sequence:

(3, 5), (5, 7), (11, 13), (17, 19), (29, 31), (41, 43), (59, 61), (71,73), (101, 103), (107,109), (137, 139), (149, 151), (179, 181), (191, 193), ..., (1787, 1789), ..., (1871, 1873), ..., (1931, 1933), (1949, 1951), (1997, 1999), (2027, 2029), ...

It is believed that there are infinitely many twin pairs (the famous twin prime conjecture), yet the problem remains unsolved up to day. Suppose, someone has proved that the twin prime conjecture is unprovable in set theory. Do you believe that, still, the twin prime conjecture possesses an "objective truth value"? Imagine, you are moving along the natural number system:

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, ...

And you meet twin pairs in it from time to time: (3, 5), (5, 7), (11, 13), (17, 19), (29, 31), (41, 43), (59, 61), (71,73), ... It seems there are only two possibilities:

a) We meet the last pair and after that moving forward we do not meet any twin pairs (i.e. the twin prime conjecture is false),

b) Twin pairs appear over and again (i.e. the twin prime conjecture is true).

It seems impossible to imagine a third possibility...

If you think so, you are, in fact, a Platonist. You are used to treat natural number system as a specific "world", very like the world of your everyday practice. You are used to think that any sufficiently definite assertion about things in this world must be either true or false. And, if you regard natural number system as a specific "world", you cannot imagine a third possibility: maybe, the twin prime conjecture is neither true nor false. Still, such a possibility would not surprise you if you would realize that the natural number system contains not only some information about real things of human practice, it also contains many elements of fantasy. Why do you think that such a fantastic "world" (a kind of Disneyland) should be completely perfect? To continue click here.

My response is that this problem CAN'T be unprovable in set theory. This may be very naïve of me, given Gödel's incompleteness theorem, but this strikes me as the kind of number theoretic proposition which MUST be decided (although I guess intuitively-concrete Diophantine equations are likewise undecidable). And if it's not decidable in some particular set theory, that set theory is WRONG (in the sense of being a bad model), because twin primes *EXIST*, at least in my intuition.

Perhaps the lesson to learn from Gödel's theorem is that even though no axiomatization of number theory can be complete, we can always progress. Perhaps the union of the provable propositions in those "conservative" axiomatizations correspond to the "intuitively true" or "really true" propositions.

But that begs the question: so why don't we use the union of those axioms.... which would get us in trouble again.

I still have to struggle with Gödel's theorem, before I feel comfortable with it. I wonder how many experts feel philosophically comfortable with it and how many just gave up and got used to it.

Comments

[parakleta.livejournal.com]

I thought Gödel's theorem merely stated that there exist problems for which we cannot know the answer, even though their answer may be as simple as true or false. Take for example the halting problem, it has been proven that it isn't possible to know the answer, even though it is as simple as true or false.

Thinking about it, I suspect that it make be possible to generalise the prime pair problem to a version of the halting problem, and I'm pretty sure that's exactly what it is. I'm pretty sure the twin prime conjecture must be either true or false, simply that it isn't possible for us to ever be certain one way or the other.

[gustavolacerda.livejournal.com]

Take for example the halting problem, it has been proven that it isn't possible to know the answer, even though it is as simple as true or false.
What answer? You can make a machine which is guaranteed to conservatively determine whether another machine will halt in some cases (in fact, we are such machines when are debugging programs for infinite loops). But no machine can decide it for every machine.

[parakleta.livejournal.com]

But no machine can decide it for every machine.
That's exactly the problem, and why it's undecidable. It's about the only grasp on Gödel I have so far.

[jcreed.livejournal.com]

In my own wrestlings with G%ouml;del's incompleteness theorems, I've come to believe that among other things, what they say is that we do not, and can not ever definitively understand what it means to talk about an arbitrary finite quantity.

The the twin prime conjecture can be phrased as the question "does there exist a (finite) n such that (n,n+2) is the last prime pair?" The platonist believes the question is determined because she has in mind the model of the theory of natural numbers, the "one true" model that contains all (and only the) finite steps on the path to infinity.

But your (a) and (b) to a more skeptical mathematician would be phrased as

a) We meet the last pair in all models of PA

or

b) Twin primes appear over and over again in all models of PA

and the third possibility opens up that

c) Twin primes go on forever in some models, but not in others.

[rdore.livejournal.com]

This is acutally interesting, because I was reading "The Future of Set Theory" by Shelah yesterday because I was just wasting time online. In there he at some point tries to defend how one can both be a platonist AND Not seem to believe that everything has an answer. Maybe I misread or something though, I didn't read that section very carefully.

I guess I'm just generally finding your platonist test dicdes things into two simple of a dichotomy.

Of course, in the typically pramatist sort of why, what's the difference if an "ultimate truth" we don't actually know exists, if all we can really expect to come up with is good approximations anyways.

I've never really been convinced that mathematics is some innate part of the universe, since so much of the way we percieve and use it is based on how we think anyway.

[easwaran.livejournal.com]

The problem with the intuition is that we don't know if (a) has happened, and don't know if (b) is going to continue happening. Neither of them has definitively happened after any finite number of steps. Thus, we are always in some sort of third possibility.

Anyway, I see no reason to think that set theory would decide all concrete statements of number theory. This is basically the point of Godel's Incompleteness Theorem, that even in something as simple as number theory there are always undecidable statements. Godel took the same intuition that you have and suggested that we can eventually come up with more axioms for set theory (though of course, never enough to complete it). I suppose these would be things like large cardinal axioms, and maybe things like V=L, which are certainly undecidable (not even provably consistent). This idea ends up seeing math becoming more of an empirical science, as we decide new axioms based on our intuitive perception or understanding of these abstract systems.

On a Platonist account, it seems one must end up with this sort of empiricism, since the mathematical entities are seen as having actual existence and truths that we just have to access somehow, through our intuitions or something.

[aiprofessional.livejournal.com]

My understanding of Godel's theorem is this: If you use too few axioms to characterize number theory, then there will be propositions that are undecidable, if you add enough axioms so you can characterize every proposition, then you will get contradictions. So you cannot have a non-contradictory set of axioms that characterize completely all propositions of number theory, you are stuck with either incompleteness or contradictions. Furthermore, I think there is a theorem somewhere that any sufficiently powerful axiomatic system is homomorphic to number theory, so Godel's theorem is supposed to apply to all sufficiently powerful axiomatic systems.

Because of this reason I do have a problem with Godel's theorem, I think it has to be wrong, to accept it would entail accepting that knowledge is unattainable in a lot of areas. I haven't had time to look into it yet, but I suspect that the derivation of the theorem itself depends heavily on set theory, and I think that is where the problem comes from. I started suspecting that set theory might be wrong from learning about mereology, the study of the parts, which is used in AI for building ontologies. Mereology rejects the notion that the set containing a set is different from the contained set itself, and that avoids things like uncountable infinities. From a practical purpose it was more adequate for the ontology building domain than set theory. I guess at some point these two were competitive, but set theory was so successful in several areas of mathematics that mereology got forgotten.

Of course taking on set theory would be a huge undertaking, which at some point I want to take the time to go over, but I haven't had time yet, this is just an idea I've had for a while.

[spoonless.livejournal.com]

Very interesting. I probably need to take a look at mereology.

Godel's Theorem is bothersome, but only from a platonist perspective of math. You could interpret it as "proof" of mathematical formalism; but it would be nice to be able to think about numbers in the same way we think about physical objects. Which is why I think your questioning of set theory is a good idea.

I noticed you have a fair amount of similar friends and interests too, so I went ahead and added you.

[aiprofessional.livejournal.com]

It is bothersome also from an objectivist perspective of math. From this perspective numbers are conceptual abstractions of reality, they are not platonic objects that exist by themselves on an ideal dimension, but they are not an arbitrary model in a subjective game of logic that mathematicians play either. If numbers are abstractions of reality, then they represent reality and the prospect of having either incompleteness or contradictions is extremely bothersome because that would mean that there are contradictions or incompleteness in reality. Contradictions in reality is definitely out of the question, but incompleteness would mean that there are things that we can never know in principle and I am not ready to accept something like that without questioning every step of the reasoning.

I've added you to my list of friends as well.

[spoonless.livejournal.com]

It is bothersome also from an objectivist perspective of math. From this perspective numbers are conceptual abstractions of reality,

My view is that math is (loosely) based on conceptual abstractions of reality; however it's always possible to combine abstractions in a new way that hasn't ever been seen in reality (and may or may not ever be seen). A lot of math invovles this type of process. Nobody has ever seen 10^(10^100) of something, so the only sense in which it's an abstraction of reality is that we've taken the process of counting (which works well for small numbers) and defined it to continue forever, well past anything anyone's observed. For nubmers it seems especially intuitive that you should be able to continue counting forever if you lived long enough. But with other more complex mathematical objects, it gets a lot sketchier. You start having to make decisions about what to define as true. And it's not always a meaningful process that represents concrete objects even though you can't find a direct contradiction in the axioms themselves.

[r6.livejournal.com]

I believe Gödel’s theorem is provable in Peano Arithmetic, and probably even weaker systems.

Gödel’s theorem is the same as the Halting Problem in some fundamental way, although I don’t really understand the relationship well enough to explain it.