It would seem that people are being blinded by the baggage certain symbols carry.
Let us substitute in some more neutral symbols:
for any A, A = B iff all C = B
all C are A
All C are in D
All D != B
You can see that the problem is structural. The specific meaning of the words is irrelevant. A never equals B.
For those who want to see an actual contradiction: "A statement has been defined": An A = B.
It doesn't matter what you think a 'definition' or 'meaning' is/are. It doesn't matter what you think a statement is.
It also doesn't matter how many levels of indirection you insert.
This structure goes nowhere. It does nothing.
The symbol 'meaning' has two* distinct meanings. There is the axiomatic 'meaning' and the informal 'meaning'. These two meanings are defined to be different. As much as some similarity or overlap might be desired (or even actually exist) - they are specified to not be equal to each other. Informal meaning is not the same object as axiomatic meaning.
If you want informal meaning to interact with axiomatic meaning - you must define that relationship. Axiomatic mathematics simultaneously says that meaning is specifically not informal - and then uses informal meaning as if it was axiomatic meaning.
*'meaning' is a symbol. We can map anything to that symbol. 'meaning' can have an unconstrained number of different meanings. The label is just a label, it is not the territory.
Treatid wrote:2. A statement only has meaning with respect to a (defined) set of axioms.
False. Informal language has meaning. Not rigorous, "axiomatic" meaning, but meaning nonetheless.
a. I specified that 2. was an axiom. The question of truth doesn't apply to axioms themselves. Axioms need to be consistent, they don't need to be a-priori 'true' however that might be specified. We are using standard rules of mathematics to show that the standard rules of mathematics don't work. Standard rules say that axioms don't have to be justified/true. They just have to be consistent in context.
b. A symbol may have any number of different meanings. The informal meaning of 'meaning' and the axiomatic meaning of 'meaning' are defined to be distinct. Changing from one to the other half way through your chain of logic is only going to lead to confusion.
c. Your point is an illustration of the problem with axiomatic mathematics. Any given symbol can have an unlimited number of meanings. We have no way of specifying which meaning is intended. The labels we use are just labels. We can only manipulate the labels. We can't (that we know of) directly manipulate the concepts behind those labels.
Having said that - I'm attempting to show that an axiomatic approach to knowledge doesn't work; cannot work. But I think there are approaches to knowledge that do work. We can't have an absolute idea of what 'meaning' means. We can
have a sufficiently agreed perception of what 'meaning' is to communicate - and to refine any given conception.
But your argument fails because your second premise is false.
False doesn't apply to the second premise (axiom). Moreover, the apparent untruth only arises because you conflate two distinct uses of the same symbol.
Where does this leave us?
Forest Goose wrote:^
5 isn't true either. Every axiom of ZFC is in the language of set theory, which is a formal language.
Who are you trying to fool? Me or yourself?
You know that (in theory) the axioms of any axiomatic system can be expressed in terms of any other (sufficiently flexible) axiomatic system, in addition to informal language(s)
. Specifically, The axioms of ZFC have been expressed in multiple languages including
You can, rightly, point out that, at some point, we need to use informal terms to get the process started
You directly concede that this is just indirection in any case - that sooner or later we end up back with the starting point based on informal language. So what is the point?
That we have to use informal terms to say things like what a string of symbols are, or that conjunction is commutative, or etc. doesn't really seem a problem.
Except for the fact that it creates a contradiction within axioms. Which means that every statement of axioms can be contradicted. Which means that axioms are a null system.
But other than that - no biggie...
2 is false too - this isn't the time of Euclid. Statements in formal languages don't have "meanings", that's not really how they work - the best sense of meaning I can see existing in axiomatic mathematics is in terms of models, and, in that sense, one major part of foundations is that ZFC has tons and tons of models. Heck, there's a model of ZFC + ~Con(ZFC), even; have fun with that:-)
It doesn't matter what the specific concepts involved are. It is the structure that doesn't work. The axiomatic approach to anything, of any kind whatsoever, cannot work. It doesn't do anything. The practical use of axioms has A=B alongside A!=B.
Either informal meaning is equal to axiomatic meaning; or it isn't. Similar, related, even same-ballpark doesn't cut it. The definitions say that they are explicitly not the same thing.
Maybe some progress could be made if the distinction between the two meanings of 'meaning' could be quantified. However, right now, the construction of axioms requires equality - not similarity.
4: neither are formal ones. However, theories in informal languages do use the axiomatic method, so there's that - heck, you're using the axiomatic method as we speak
Yes - I am using the existing concepts of mathematics to show that (some of) the existing concepts of mathematics are mistaken.
The argument has come up before that if I use axiomatic logic to show that axiomatic logic doesn't work - then my proof also doesn't work.
So... what next? The proof doesn't work - axiomatic logic pops back into existence. Ah - but then the disproof also pops back into existence. Obviously axiomatic logic works half the time - it oscillates between true and false - I guess you just have to catch it in alternate states?
By stating 3, you're giving yourself some problems, friend: one of the biggest bits of informal language is defining what a statement is - since you are, apparently, okay with statements (and, clearly, with using logic), I'm confused as to what you're disagreeing with...or, what, are you reductio'ing logic and language, supposedly? If that's how I'm supposed to read what you're doing, then my reply to you is "Flugsnatz,,,;goob-li=li-gock". Why? Because I declare my string defeats yours, and with logic and language gone, I can do so. In other words, you're either hanging yourself or admitting you reject everything, in which case anything goes - have fun on the horns of that dilemma
See above again.
Language hasn't disappeared. Just axiomatic mathematics. Axiomatic mathematics does take logic with it. Fortunately, reason (based on consistency) is still hanging around.
We could count before we had the formal number line. We can still count now that the formal number line never existed.
Axioms don't take away anything by never existing - except for some illusions. We very much want an alternate approach to knowledge. But losing an approach to knowledge that never actually did anything doesn't take away computers, bridges and buildings. Programming languages don't stop working and massive income inequality is still a thing.
The idea of infinity and the hierarchy of infinities are well shafted. But without any real world infinities, this won't make much difference to most people's daily lives.
Yakk: This has nothing to do with the intentions of axioms. I'm not asking that axioms be justified. I'm not making the mistake of thinking axioms need to be 'true'.
The problem is that axioms are a fiction. There is nothing there. There is nothing to work with. Every axiomatic statement ever made equals 'null'.
Informal language statements are not null. But axiomatic statements are defined to be null (rather - they are defined to not have whatever property axioms are supposed to imbue).
Go up top. Look at that structure. What are axioms adding that wasn't there before?