Why is mathematics limited/incomplete?

[Estanho]

I can understand that we can have limitations on physics, as we are trying to describe nature and so constraints can naturally occur - even meta-constraints that limit our ability to model nature to an arbitrary degree.

However, why does that applies to a synthetic system such as mathematics as well? From my understandings of computability, there are several things that we cannot do with mathematics - and that's completely proven. But that seems to be generic enough that any system that we could come up with will always have those limitations.

So I guess my question is twofold:

• If we scraped our mathematics development completely and started from scratch completely out of the box, coming up with a totally different system for building abstract models and computation, would we still probably hit equivalent milestones and in the end have the same limitations (e.g. would there be calculus, or something equivalent but totally different?)
• What is the reason why such limitations exist? Is it some sort of physical constraint that our universe impose? For example, the fact that we can use mathematics to model nature, maybe some natural limitations are transferred to it. It's kind of clear that there are some limitations in information theory maybe due to its more applied nature that are due to physical constraints, such as uncertainty or relativity.

 

[PeroK]

Check out Goedel's Incompleteness Theorems.

It's nothing to do with relativity, physics or the universe.

 

[fresh_42]

I can understand that we can have limitations on physics, as we are trying to describe nature and so constraints can naturally occur - even meta-constraints that limit our ability to model nature to an arbitrary degree.

However, why does that applies to a synthetic system such as mathematics as well? From my understandings of computability, there are several things that we cannot do with mathematics - and that's completely proven. But that seems to be generic enough that any system that we could come up with will always have those limitations.

So I guess my question is twofold:

• If we scraped our mathematics development completely and started from scratch completely out of the box, coming up with a totally different system for building abstract models and computation, would we still probably hit equivalent milestones and in the end have the same limitations (e.g. would there be calculus, or something equivalent but totally different?)

This cannot be answered. How can we make any statements about a subject that does not only not exists, but isn't described at all? All we can say is, that there will be the same limitations, i.e. unprovable truths and undecidable theorems. Goedel's proofs will almost certainly apply in such a system, too.

• What is the reason why such limitations exist? Is it some sort of physical constraint that our universe impose? For example, the fact that we can use mathematics to model nature, maybe some natural limitations are transferred to it. It's kind of clear that there are some limitations in information theory maybe due to its more applied nature that are due to physical constraints, such as uncertainty or relativity.

The nature of mathematical limitations is not of physical nature. The limitations are of logical nature. There is no such thing as the set of all sets. It cannot be proven that arithmetic is free of contradictions by merely arithmetic methods.

Such restrictions will remain. You need a meta-level to discuss a subject. And to discuss the meta-level on a subject, you will need a meta-meta-level. And so on. There is a system imminent distinction between an argument and the question of whether it can be applied or not.

 

[Estanho]

Hmm I am aware of Gödel's theorems and I should probably have mentioned it on the post, although I don't possesses the technical knowledge to follow the proofs on a fundamental level.

What I'm looking for is more on the level as to why such limitations exist. Sorry for repeating this example again but physical limitations make sense due to the external factor that nature imposes constraints into any models we might come up with, and that's outside of our control. But why then should there be limitations in any logic system we come up with, on a fundamental level?

 

[PeroK]

Hmm I am aware of Gödel's theorems ...

What I'm looking for is more on the level as to why such limitations exist.

Goedel's theorems are the reason.

 

[fresh_42]

What I'm looking for is more on the level as to why such limitations exist. Sorry for repeating this example again but physical limitations make sense due to the external factor that nature imposes constraints into any models we might come up with, and that's outside of our control. But why then should there be limitations in any logic system we come up with, on a fundamental level?

We have logical limitations in mathematics: meta-levels, independent, possibly undecidable statements like AC. Those are constraints due to the facts, that it makes a difference to reason within a system or about a system, and that proof of existence is a purely logical construction, none that necessarily can be worked out in an algorithm.

We have also created nonbinary logics, but that didn't really solve the problems.

 

[phyzguy]

I think what Goedel's theorems show is that any system where you start with a set of axioms and apply a consistent set of algorithmic procedures has limitations. Because of the rigidity of algorithmic procedures, to me it does seem too surprising that there is more to logic than algorithmic procedures. For example, neural networks like our brains are not algorithmic in nature, and can potentially "see" solutions to problems that are not accessible with algorithmic procedures. Penrose makes the point (perhaps in "The Emperor's New Mind", I've forgotten the reference) That we should view Goedel's theorems positively, not negatively. It means we have the ability to deduce things beyond what rigid algorithms can tell us.

 

[jack action]

But why then should there be limitations in any logic system we come up with, on a fundamental level?

The limitations are what define the system. No limitations, no system.

Here's an example with a classic connect-the-dots game. To win, you have to connect all nine dots using only four straight lines without lifting your pencil off the paper:

Where the answer is:

The constraints are:

1. connect all nine dots;
2. using only four straight lines;
3. without lifting your pencil off the paper.

The point of the game is to notice that you DO NOT have a constraint that restricts you to change direction only on a dot.

But if you remove all constraints, then there is no game. All you are left with is: To win, you have to do something ... or nothing.

If you keep only the first constraint, it is still a valid game. It is just a much much easier problem to solve because there is an infinite amount of solutions.

 

[jbriggs444]

What I'm looking for is more on the level as to why such limitations exist.

At a hand wavy level, the limitations exist because of self reference. If you design a system capable of "understanding" or "encompassing" everything then you can ask it questions about itself.

The self-reference is not always obvious. Mathematicians are clever folks who know better than to use a language that allows direct paradoxes like "this sentence is a lie". But they do use formal systems that can contemplate things like "all strings of any finite length". With a little bit of cleverness, mathematicians can embed a functional copy of a formal system within itself so that the theorems of the system can be expressed as strings of finite length within the system.

Then you can have a theorem whose truth or provability correlates problematically with the "truth" or "provability" of a string corresponding to itself.

 

[Hornbein]

I can understand that we can have limitations on physics, as we are trying to describe nature and so constraints can naturally occur - even meta-constraints that limit our ability to model nature to an arbitrary degree.

However, why does that applies to a synthetic system such as mathematics as well? From my understandings of computability, there are several things that we cannot do with mathematics - and that's completely proven. But that seems to be generic enough that any system that we could come up with will always have those limitations.

So I guess my question is twofold:

• If we scraped our mathematics development completely and started from scratch completely out of the box, coming up with a totally different system for building abstract models and computation, would we still probably hit equivalent milestones and in the end have the same limitations (e.g. would there be calculus, or something equivalent but totally different?)
• What is the reason why such limitations exist? Is it some sort of physical constraint that our universe impose? For example, the fact that we can use mathematics to model nature, maybe some natural limitations are transferred to it. It's kind of clear that there are some limitations in information theory maybe due to its more applied nature that are due to physical constraints, such as uncertainty or relativity.

It's pretty much the same as why you can't have a set of all sets. It couldn't contain itself. In my opinion it isn't all that profound.

 

[MidgetDwarf]

At a hand wavy level, the limitations exist because of self reference. If you design a system capable of "understanding" or "encompassing" everything then you can ask it questions about itself.

The self-reference is not always obvious. Mathematicians are clever folks who know better than to use a language that allows direct paradoxes like "this sentence is a lie". But they do use formal systems that can contemplate things like "all strings of any finite length". With a little bit of cleverness, mathematicians can embed a functional copy of a formal system within itself so that the theorems of the system can be expressed as strings of finite length within the system.

Then you can have a theorem whose truth or provability correlates problematically with the "truth" or "provability" of a string corresponding to itself.

This is one of the most illuminating post I have read in a while on this website. Not to mention I always learn something new here.

 

[PeroK]

It's pretty much the same as why you can't have a set of all sets. It couldn't contain itself. In my opinion it isn't all that profound.

I guess you wouldn't have been fooled like Hilbert and all the rest of them before Goedel surprised the mathematical world.

 

[jbriggs444]

It's pretty much the same as why you can't have a set of all sets. It couldn't contain itself. In my opinion it isn't all that profound.

In mathematics, we try to reason formally. Intuition is nice, but it is not infallible.

We have this collection of objects called "sets". And we have this relation "is a member of". There is nothing immediate that precludes a "set" from having this relationship with itself.

It takes more than just self-membership in order to instantiate Russell's paradox. Some form of the axiom of comprehension is also required (e.g. "for any predicate and any set there is a subset which contains exactly those elements that satisfy the predicate"). With that form of the axiom, I believe that you still need a "set of all sets" to get a paradox. A mere set containing itself won't do.