This is, of course, an impossible challenge.
Any two things that you think about or otherwise experience are, at the very least, connected to you (and consequently to each other).
This isn’t a particularly complicated argument: Everything you experience is connected (to some degree) as evidenced by you experiencing them.
We can make this tautologically true by saying that the definition of ‘connection’ is your ability to experience. If you can experience something then it is connected to everything else that you can experience.
Axiomatic mathematics (in the broadest sense) is the part of mathematics that provides proofs. One branch of axiomatic mathematics is First Order Logic.
All of axiomatic mathematics is pure mathematics. This means that axiomatic mathematics is not an empirical science. Potentially, a given piece of axiomatic mathematics need have nothing to do with real world observations.
Axiomatic mathematics is abstract in so far as it is not (supposed to be) restricted by the physics of the universe.
Whereas empirical sciences are tested against the universe, axiomatic mathematics needs some other measure of value.
This measure is consistency/inconsistency and is based solely on whether a given argument contains contradictions, or not. (A typical example of a contradiction in mathematics is a statement that turns out to be both true and false at the same time).
Any argument that contains a contradiction can ‘prove’ anything/everything and is consequently as useless as not proving anything (The Principle of Explosion). Such an argument is considered inconsistent.
It is really hard (impossible) to prove that a given argument is consistent.
In practice, a mathematical proof is one where nobody has yet found a contradiction. A proof is typically strengthened if it is based on an existing proof that has already been thoroughly examined for any inconsistency (contradictions).
One such system is ZFC Set Theory (Zermelo–Fraenkel set theory with the axiom of choice).
ZFC Set Theory is an axiomatic system that has been extensively examined by mathematicians and is believed to be consistent (no contradictions have been found after some really smart people looked very hard).
Any statement made using ZFC Set Theory is assumed to be proven automatically by virtue of being made within this ‘known’ consistent axiomatic system.
It is, of course, trivially easy to show that ZFC Set Theory is, in fact, inconsistent, along with every other axiomatic system and all of First Order Logic.
Any contradiction within an argument (mathematical theory, axiomatic system) renders that whole argument inconsistent.
Any particular contradiction found within an argument is a symptom rather than the disease. One contradiction means that every statement within the argument can be contradicted.
There are no half measures. An argument is either consistent or entirely inconsistent. Any contradiction in any part of an argument renders the entire argument inconsistent.
As such, it is vitally important that two different arguments have no overlap with each other lest a contradiction in one argument renders both arguments inconsistent.
Naturally enough, mathematicians assert that different mathematical arguments (axiomatic systems and logic systems) are unconnected. Each system is judged consistent or inconsistent independent of any other arguments.
This seems fair. Holding a mathematical proof responsible for the faults of a family argument during the holidays seems a bit extreme. Just because one argument is shown to be inconsistent can’t mean that every argument is inconsistent, can it?
Well… yes… if any (mathematical) argument is inconsistent then every mathematical argument is inconsistent.
First, there are many (infinitely many) inconsistent mathematical arguments. It is trivially easy to create known inconsistent mathematical arguments. There is no question that some (mathematical) arguments are inconsistent.
The question is whether one contradiction in one argument applies to all arguments. If all arguments are connected, then any contradiction anywhere means that all arguments are inconsistent.
Given the tautological definition of ‘connection’ given at the top, then clearly all arguments are connected and thus all arguments are inconsistent.
“There is no way that saying all arguments are connected can demolish huge swathes of modern mathematics (including logic), is there?”
There are a few things to unpack here…
Mathematics is (in part) an attempt to create formal languages that avoid some of the messy miscommunications of natural languages (such as English). As part of that process, mathematics attempts to formalise rules of language and follow those rules far more rigorously than is typical of a normal conversation.
The issue here is not arguments themselves – it is the rules that mathematics thinks should apply to arguments that are at fault.
If you follow the rules of axiomatic mathematics to their logical conclusion, we find that those rules, themselves, are inconsistent by the standards those rules try to impose.
The reason this problem hasn’t previously been glaringly obvious has to do with a confusion between the stated rules of mathematics and the actual mechanics of language.
Language does work – it just doesn’t work in the way that (axiomatic) mathematics would like language to work. As such, much of mathematics works by following the mechanics of natural languages despite claiming to follow a different set of (unworkable) rules. A problem exacerbated by the impression that the rules of mathematics are merely a formalisation of the actual mechanics of language.
Definition of ‘connected’
In natural language we have no problem considering two separate arguments to be unconnected.
Your family squabble over who last saw the TV remote is clearly different to a discussion about how to get the best thrust to weight ratio for a rocket engine.
Even knowing that both those arguments are connected by being between humans living on planet Earth doesn’t change your perception that they are two different arguments.
The rules of mathematics, however, are rather more stringent. Specifically, if something is true and false at the same time it is inconsistent – useless.
In this case we know that all arguments are connected (by the definition of connected given above). We also know that mathematicians have declared particular arguments to be unconnected.
If this holds then within mathematics, arguments are both connected and unconnected. This is a contradiction and renders anything and everything that follows from that contradiction inconsistent.
Formal languages are, by intent, quite technical in so much as some words/meanings have a special and/or narrow meaning that doesn’t necessarily translate well to more natural uses of those words.
Perhaps mathematics has a more rigorous/technical definition of ‘connection’.
It is certainly the case that I chose a (tautological) definition of ‘connected’ that most clearly makes the point.
Dirty tricks – or how to be too clever by half
Definitions are tricky. Clearly one of the intents of mathematics is to clearly define terms, but this turns out to be (again, impossibly) hard.
There is, however a workaround.
Mathematicians don’t need to define what they mean by ‘connected’ or ‘unconnected’. Instead they can declare that there is a definition of ‘connected’ that is consistent with their requirements.
They can then use that abstract definition and, so long as that abstract definition doesn’t lead to a contradiction, then they can assume that their declaration is consistent and consequently legitimate.
There are many, many contradictions that follow from the rules of axiomatic mathematics. But since we can declare each of those contradictions to be unconnected to anything else (including the rules of mathematics), those contradictions don’t count.
So long as there might be a definition of ‘connected’ that serves to isolate a given contradiction from a given argument then no contradiction disproves axiomatic mathematics.
Bonus points, because ‘connected’ hasn’t actually been defined, any definition that shows axiomatic mathematics to be inconsistent is the wrong definition of ‘connected’ and can be disregarded.
Even better, ‘connected’ can mean whatever anyone needs at any time because there is no way to show that it doesn’t mean whatever you want it to mean. There is no way for anyone to prove that you are using the word incorrectly or inconsistently.
In short, it is completely impossible to challenge the foundations of axiomatic mathematics using the standard tools of logic (a branch of axiomatic mathematics).
To be fair
Asserting that there is a definition that satisfies the requirements is quite clever.
It really is remarkably hard (impossible) to nail a single, unambiguous definition to a word. At the same time it seems perfectly obvious that we do define words. There just appears to be an annoying gap between fairly fuzzy natural language definitions and the really rigorous and precise definitions that would be so useful for formal languages.
Assuming the existence of the desired definition and then behaving as if we had that definition seemed to be working remarkably well.
Really, seriously. Mathematicians know that the approach isn’t perfect (The Foundational Crises in Mathematics). However, quite a lot of mathematics seems to be working really well; and whatever slight drawbacks it may have, what is the alternative? Nobody is throwing away something (however flawed) in favour of nothing.
Mathematics had to do some creative accounting just to get started (the process was somewhat more ad hoc with quite a lot of this being justification after the fact but…).
On the other hand
Mathematics finds itself locked into a set of mistaken assumptions that are immune to the very tools of mathematics that are supposed to expose mistaken assumptions
The standard tool of reason (logic) is incapable of a critical level of self analysis.
It is a bit of a hint in itself that axiomatic mathematics has to be made immune to the rules of axiomatic mathematics before it even gets started.
In any case, there cannot be proof that the assumptions of axiomatic mathematics are false because those assumptions have been isolated from the mechanisms of proof.
You can, however, experience for yourself the evidence that everything you experience is connected.
Try the headline challenge for yourself. Feel free to change the definition of ‘connected’.
You will find that it is obvious/clear that by any reasonable definition, all of your experiences are connected to each other to some degree.
It is quite impossible to create a definition of connection that you can
- Communicate to other people.
- That clearly shows some statements are connected within an argument.
- Shows that any given pair of arguments are not connected.