Misunderstanding basic math concepts, help please?

[Treatid]

I'm getting basic mathematical concepts wrong which is causing me to be confused (which I then have a tendency to try to inflict on other people).

The first thing I'm getting wrong is Axiomatic Systems. Perhaps “models” or “theories” are better names?

Logical systems are more self-contained than you're making them out to be, Treatid. A system comes equipped with:

1) Its alphabet, that is, the symbols that can be used to make its well-formed formulas (the logical sentences that could or could not be provable)
2) Its grammar, which is the collection of well-formed formulas, that is, the collection of sentences that the system could apply to
3) Its axioms, that is, the collection of well-formed formulas that are taken as proved
4) Its inference rules, that is, the mechanisms by which we determine which other statements are provable besides the axioms.

I think I'm understanding up to this point.

We need something to represent what we are talking about (symbols). A set of rules applying to the symbols (inference rules). A starting point (axioms). And the rules need to apply to the specific symbols we are interested in (grammar).

If we are missing any of these components – or the components don't match up (e.g. the rules don't apply to the symbols) then we aren't (cannot be) describing a single, definite, unambiguous system.

If we do have all these components then we are able to describe an axiomatic system.

It is the next bit where I'm getting lost.

It looks to me as though this divides everything up into one of two sets:

1. Well formed axiomatic descriptions.
2. Not unambiguous descriptions (sorry for the double negative).

And here I run into trouble. My thinking is as follows:

Given that axiomatic mathematics as a whole exists; it must belong in one of these two sets:

A. If axiomatic mathematics is not a well formed axiomatic system then we can't be sure what is being described.

B. If axiomatic mathematics is a well formed axiomatic system then it is subject to the Principle of Explosion.

However, since axiomatic mathematics is definitely a thing, and the principle of explosion doesn't apply to axiomatic mathematics as a whole there must be another option that I'm missing. But I can't see what it could be.

An obvious possibility is that there are other types of systems other than axiomatic description. Except that the basic assumptions of axiomatic mathematics (axioms?) as described by Cauchy above appear to me to preclude an alternative method of description.

a. If we have symbols, rules and a starting point then it is axiomatic mathematics.

b. If we lack symbols or rules or a starting point (or the rules don't match the symbols) then we have nothing.

What am I missing/misunderstanding?

[Xanthir]

I'll quote from your previous thread, to show you the part where you keep going wrong:

* There is no possible subset of an inconsistent system that is consistent.
* Every possible subset of an inconsistent system is also inconsistent.
* If any superset-of-a-system is inconsistent then the system (set) is inconsistent

This is the part that's wrong, not your understanding of the mechanics of logic or the PoE.

Once again, "subsets" or "supersets" of a system are independent systems on their own; things that are true or false there might be the opposite in the "base" system. Consistency is not a conserved property across the subset/superset relationship; it's a derived property of the precise axioms and reasoning rules allowed. Changing either of those can change whether or not a system is inconsistent.

I'll repeat a concrete example, just to make it clear.

Assume standard propositional logic: you've got statements, the operators "not", "and", "or", "conditional" (if), and "biconditional" (if and only if), and the standard inference rules (here's a good example that matches what I was taught). Inconsistencies are shown by constructing a statement like "(A and not A)".

We just need to fill in the axioms. Here's a very simple consistent system, with a single axiom: "A". You cannot form a contradiction in this system using the inference rules described above. If you disagree, please show a proof; the system is extremely trivial, so any proof you might care to construct should only be a few lines at most.

Here's a very simple superset of that system, which is inconsistent: "A" and "not A". You can immediately form a contradiction by applying the "conjunction introduction" inference rule to form "A and not A". Thus the system is proven to be inconsistent.

The first system is clearly a subset of the second; it's identical in every way to the second, just with a subset of the axioms. But it's consistent, and the "superset" system is inconsistent.

[rmsgrey]

Also, I think you're misunderstanding "grammar".

If you have a simple arithmetic modulo ten, the alphabet is ten numbers, four operators, two parentheses, and an equals sign: 0123456789+-*/()=
For axioms, we can use four hundred strings of the form "<number><operator><number>=<number>" and ten strings of the form "<number>=<number>"
For a inference rules:
1) for (possibly empty) strings A, B, C and D, if "D=B" and "ABC" are both proven, then "A(D)C" is provable.
2) for strings A and B, if "A=B" is proven, then "B=A" is provable.

That's enough to pick out all strings that are both meaningful and true in this system, but it lacks any means of distinguishing strings that are meaningful but false ("2+2=5") from strings that are meaningless ("+2+=5") - that's where the grammar comes in, letting us determine which strings are meaningful sentences, and which are not - for this example, using N for numbers, O for operators, T for terms, and E for expressions, the sentences are things of form "E=E" where expressions are things of form "T" or "TOT" and terms are things of form "N" or "(E)". These substitution rules allow you to construct all sentences (meaningful strings) of this example system. Some strings that are generally accepted are not counted as sentences of this system - for example, "1+1+1=3" is not a sentence, largely because I don't want to put in the effort to figure out how to handle associativity properly.

[arbiteroftruth]

I think it would be good, early on in this thread, to avoid using the word "system", as that term is ambiguous and has likely been part of Treatid's confusion in the past.

An alphabet (symbols we are allowed to use) together with a grammar (rules for which combinations of symbols make meaningful sentences, regardless of whether those sentences are true or false) and rules of inference (rules for how to derive new sentences from old ones) constitute a language. Most of mathematics is carried out in the language called "first-order logic".

Notice that at this point we haven't declared any specific sentences to be true. That's what axioms are for, and they'll come later. But even without axioms, there are some sentences we can derive. Sentences that can derived in a given language without axioms are called tautologies of that language. A simple example is "If A then A", which can be derived without dependence on any axioms.

The principle of explosion is an informal observation about certain languages. It is essentially the observation that for any well-formed formulas A and B, "If A and not A, then B" is a tautology of the language. Meaning that if you ever derived a contradiction ("A and not A"), you could then derive whatever else you wanted ("B"). This is a property of the language, and not all languages have this property. Languages that don't have this property are called "para-consistent". As I said, most of math uses first-order logic as its language, and first-order logic is subject to the principle of explosion.

A language L together with some axioms that are well-formed formulas in L constitute a theory. Here is where it starts to make sense to talk about subsets and supersets, referring to the sets of axioms that are used for different theories. In any theory T that uses a language L that is subject to the principle of explosion, the following observations are true:
If the set of axioms of T contains a contradiction, then T is inconsistent.
If any subset of the axioms of T contains a contradiction, then T is inconsistent.
If the set of sentences that are derivable from the axioms of T given the inference rules of L contains a contradiction, then T is inconsistent.

That last one is especially important. Treatid, in the past you've suggested that "If any superset-of-a-(set of axioms) is inconsistent then the (theory) is inconsistent", and that's not correct. Consistency of a theory does not depend on arbitrary supersets of the set of axioms. It only depends on the particular superset consisting of sentences derivable from the axioms. This is why you can have two theories T1 and T2, with T2 using a superset of the axioms of T1 and being inconsistent, while T1 is consistent. Because the extra axioms that T2 has may not be derivable in T1, but are only there in T2 because they were explicitly added in as new axioms. If the inference rules of the language do not allow those new axioms to be derived using only the axioms of T1, then any inconsistency they cause in T2 does not affect T1.

Regarding your post here, it comes down to what you mean by the principle of explosion "applying to" axiomatic mathematics. The PoE is a true observation about the language of first-order logic, which is the language used for most mathematical theories. But that observation only becomes relevant to a given theory if you can derive a contradiction within that theory. Because the PoE renders such a theory trivial and useless, we try to stick to theories in which it's impossible to derive a contradiction, and thus the PoE never gets an opportunity to cripple that theory. So the PoE "applies to" math in the sense that it's a true observation about the language, but hopefully it does not "apply to" our preferred theories in the sense that a contradiction never arises and thus the PoE never gets the chance to be invoked on that theory.

[LaserGuy]

A. If axiomatic mathematics is not a well formed axiomatic system then we can't be sure what is being described.

B. If axiomatic mathematics is a well formed axiomatic system then it is subject to the Principle of Explosion.

However, since axiomatic mathematics is definitely a thing, and the principle of explosion doesn't apply to axiomatic mathematics as a whole there must be another option that I'm missing. But I can't see what it could be.

I think what you're missing is that the Principle of Explosion is a conditional. It says basically, if you have a certain type of logical system (of which axiomatic is definitely one) and if it is possible to derive a contradiction, then you can derive any other premise within the system. Systems that satisfy all of the conditions are essentially worthless. Axiomatic mathematics satisfies one part of the conditional, but not the other. At least, as far as we know.

[Treatid]

You are right - I was getting hung-up on supersets/subsets and changing axiom context without seeing the other part of that equation.

Despite the whole of Axiomatic Mathematics being inconsistent, it is only inconsistent from the perspective of the whole of Axiomatic Mathematics.

There exist other perspectives (context, language) that are consistent and allow us to define axioms and rules. Each use of axioms is independent of all other uses, because each use of axioms occurs in a distinct context.

I find it odd that this workaround is valid for Axiomatic Mathematics as a whole but isn't applied to any other inconsistent systems. Even given this workaround, from the perspective of Axiomatic Mathematics anything can be proven, just like any other inconsistent system. So anything we do prove in Axiomatic Mathematics is arbitrary, except insofar as it is actually useful in real life. But I do accept that there are perspectives that are consistent and can be used to describe other consistent systems.

What I am having trouble getting my head around is how changing context can be done.

1. The only way to change the context of a set of axioms is to change the language that describes those axioms, as far as I know. As such, each time axioms change context, it is, in fact, the language describing the axioms that is changing. The trouble is we have no way of specifying a new axiomatic system without having an existing language capable of describing the axioms and rules for the new system.

2. In order to start describing an axiomatic system we need a definite, known set of axioms and a known set of rules. The only thing capable of being not ambiguous is an axiomatic system (cf. same reasoning as for Axiomatic Mathematics as a whole). {I understand that a distinction is usually made between Languages and axiomatic systems - I don't understand the justification for that distinction}.

[Xanthir]

You are right - I was getting hung-up on supersets/subsets and changing axiom context without seeing the other part of that equation.

Despite the whole of Axiomatic Mathematics being inconsistent, it is only inconsistent from the perspective of the whole of Axiomatic Mathematics.

Only if by "the whole of Axiomatic Mathematics" you mean "what happens if I just take every single possible axiom and mash them together".

"Axiomatic Mathematics" is a description of how to use logical inference to do math. It is not, itself, a logical system.

I find it odd that this workaround is valid for Axiomatic Mathematics as a whole but isn't applied to any other inconsistent systems.

What do you think is the "workaround"?

Even given this workaround, from the perspective of Axiomatic Mathematics anything can be proven, just like any other inconsistent system.

Incorrect. Again, "axiomatic mathematics" is not itself a logical system. (Or if it is, it's brecause the term is being used informally to refer to a *specific* logical system that is assumed to be clear from context, and which is presumably consistent.)

1. The only way to change the context of a set of axioms is to change the language that describes those axioms, as far as I know. As such, each time axioms change context, it is, in fact, the language describing the axioms that is changing. The trouble is we have no way of specifying a new axiomatic system without having an existing language capable of describing the axioms and rules for the new system.

2. In order to start describing an axiomatic system we need a definite, known set of axioms and a known set of rules. The only thing capable of being not ambiguous is an axiomatic system (cf. same reasoning as for Axiomatic Mathematics as a whole). {I understand that a distinction is usually made between Languages and axiomatic systems - I don't understand the justification for that distinction}.

You're still wrong at the very base of your understanding. These issues are non sequiturs.

[arbiteroftruth]

You are right - I was getting hung-up on supersets/subsets and changing axiom context without seeing the other part of that equation.

Despite the whole of Axiomatic Mathematics being inconsistent, it is only inconsistent from the perspective of the whole of Axiomatic Mathematics.

We need to be clear what you're referring to when you use the term "axiomatic mathematics". I see three possibilities.

1.) Xanthir's suggestion that you're referring to a particular mathematical theory that results from combining the axioms of every mathematical theory ever conceived. If that's what you're referring to, then the resulting theory is indeed inconsistent and worthless. But it's also not something that any mathematician actually uses.

2.) You're referring to the language of First Order Logic that is shared by most mathematical theories. If that's the case, remember that consistency/inconsistency are properties of theories, not languages. A language is neither consistent nor inconsistent. The language is just the messenger of what the theories say; the language itself makes no claims about what is or isn't true and thus cannot be inconsistent.

3.) You're referring to some specific foundational theory like ZFC or Category Theory. In that case, your assertion that it's inconsistent is simply false (we think).

Edit:

I understand that a distinction is usually made between Languages and axiomatic systems - I don't understand the justification for that distinction

Again, a language describes which sentences are meaningful and what they mean. A theory asserts particular sentences to be true. A language distinguishes between "2+2=5" being a meaningful sentence and "+2+=5" being a meaningless sentence. A theory distinguishes between "2+2=5" being false and "2+2=4" being true. Languages deal with meaning, while theories deal with truth. Consistency/inconsistency is a property of truth, and is thus relevant to theories, but not to languages.

[Meteoric]

. Again, "axiomatic mathematics" is not itself a logical system.

Would it be reasonable to phrase this as "There are many distinct axiomatic mathematics"?

[Treatid]

Ah. Okay.

It looks like the core of my problem is that I think things are axiomatic systems that are not axiomatic systems. This is pretty fundamental, and until I understand this I'm not going to make any head way.

Obviously we usually come at axiomatic systems from the front end - we start with some axioms, rules and symbols.

I think we can also recognise axiomatic systems by looking at the properties of a system.

If a thing is ambiguous it is not an axiomatic system.

If a thing is not ambiguous then it is an axiomatic system because an axiomatic system is the only thing capable of being not ambiguous.

I agree that you have shown that languages/first order logic are exempt from the rules of axiomatic systems.

But I also think that only an axiomatic system can define axioms and rules. Anything that isn't an axiomatic system is ambiguous. So any definite definition of axioms, rules and symbols can only be done within the scope of an axiomatic system.

There is something wrong with the method of identifying axiomatic systems I've described. It contradicts well established and evidenced mathematics.

What you are saying is correct, but it doesn't help me understand why this isn't a legitimate method of identifying axiomatic systems.

[Twistar]

Ah. Okay.

It looks like the core of my problem is that I think things are axiomatic systems that are not axiomatic systems. This is pretty fundamental, and until I understand this I'm not going to make any head way.

I'm going to bring up the rule of not using the word "system" here because it is again leading to confusion. You are correct that you are characterizing axiomatic "systems". You should be thinking about axiomatic "theories", and we can help identify those.

Obviously we usually come at axiomatic systems from the front end - we start with some axioms, rules and symbols.

Not always. It sounds like YOU always come at thinking about languages, theories, etc. from the front end. That is not always the case, and in fact this is one of the big things leading to your confusion. arbiteroftruth defines things from the backend. STARTING with language and only towards the end arriving at theory with axioms and semantics.

I think we can also recognise axiomatic systems by looking at the properties of a system.

Probably, but you don't get it quite right below.

If a thing is ambiguous it is not an axiomatic system.

That is not the criteria for whether a system is an axiomatic system. Again the word system is confounding but just a few notes. I'm going to again go back to using the word theory. There are things which are not ambiguous which have nothing to do with formal logic which negates one of the directions of your conditional statement. There are also mathematical theories which lead prove both "X" and "~X". Those theories are 'inconsistent', so they could in a very loose sense be interpreted as 'ambiguous', so this negats the other direction of your conditional statement.

Bear in mind there are other theories which do have true or false as the truth value for any statement with no ambiguity.

In other words this is a very bad test for whether something is a mathematical theory.

If a thing is not ambiguous then it is an axiomatic system because an axiomatic system is the only thing capable of being not ambiguous.

Again, no. If I'm driving to work and my tire blows out in the middle of traffic there's no ambiguity about the fact that I'm going to be late to work. This has nothing to do with axiomatic "systems" or rather this has nothing to do with formal logic. In other words, there are other things which can be not ambiguous. I think you're trying to give these things called "axiomatic systems" some privileged status in the architecture of human thinking but they're probably not quite as special as you would like them to be.

I agree that you have shown that languages/first order logic are exempt from the rules of axiomatic systems.

Right. The rules defining a language are different than the rules defining a formal THEORY. The use of the word exempt is a little weird. The way I think about it is that we first define a set of rules for the language. Then we define some rules for what a theory is. Then using the language we have created and the rules for what a theory is we pick out axioms and make a theory. In other words they just both have different rules which apply. It's not that one is exempt from the rules of the other. It would be sort of like saying when I'm walking in public I'm exempt from the no-travelling rule of basketball.

But I also think that only an axiomatic system can define axioms and rules. Anything that isn't an axiomatic system is ambiguous. So any definite definition of axioms, rules and symbols can only be done within the scope of an axiomatic system.

Again, ambiguity has nothing to do with whether a system is an axiomatic system. Something is a mathematical theory if it is defined in such a way that it satisfies all the rules needed to define a mathematical theory. Those rules are as follows:

-The theory is a collection of syntacticly valid sentences in a given formal language.
-Some of those sentences are designated as axioms
-The rest of those sentences can be logically deduced (using the laws of the deduction for the formal language) from the axiomatic sentences.

If what you have follows these rules then it is a theory. If what you have doesn't follow these rules it is not a theory.

There is something wrong with the method of identifying axiomatic systems I've described. It contradicts well established and evidenced mathematics.

What you are saying is correct, but it doesn't help me understand why this isn't a legitimate method of identifying axiomatic systems.

The point is that the definition of a mathematical theory has nothing to do with ambiguity. Rather a mathematical theory is just something which is defined in such a way as to give us a rigorous, consistent, and repeatable way to manipulate logical objects according to some set of rules.

edit: To respond to meteoric:

. Again, "axiomatic mathematics" is not itself a logical system.

Would it be reasonable to phrase this as "There are many distinct axiomatic mathematics"?

Let's avoid the use of the word system. When we say axiomatic mathematics we are typically referring to a particular theory written in a particular language. It is true that much of mathematics the we use has been formulated in different theories in a few different languages. That is to say in some sense it is correct that there are multiple distinct axiomatic mathematics. For purposes of this conversation we could restrict ourselves to one of those. Say ZFC in language of first order logic.

[arbiteroftruth]

If a thing is ambiguous it is not an axiomatic system. If a thing is not ambiguous then it is an axiomatic system because an axiomatic system is the only thing capable of being not ambiguous.

A more accurate phrase for what I think you mean would be "If a thing is not rigorous it is not a mathematical theory, and a mathematical theory is the only thing capable of being rigorous". People might still debate whether that's strictly true, but it seems reasonable enough to me.

I agree that you have shown that languages/first order logic are exempt from the rules of axiomatic systems.

Sort of. It goes back to something we've discussed in your earlier threads. You can't be rigorous right from the start, because at some point you have to communicate ideas to human brains through messy human language and the messy physical world. So at some level you're forced to start with something informal and take it for granted that it works. One of the goals (sort of) of mathematical rigor is to minimize the amount that we take for granted by using only some of the most ingrained intuitions in the way the human mind works. Things like basic logical inference (If I know 'A' is true and I also know "if A then B", then I can conclude 'B' is true). So we create formal logic as a language that expresses only these most basic intuitions, take it largely for granted that formal logic works, and go from there.

But there is a little loophole (sort of). Once you have a language in place and can start stating axioms of theories, it's possible to create a theory whose axioms mimic the rules of grammar and rules of inference of your language. In that sense there is a similarity between languages and theories, because you can make a theory that models the rules of a given language, and in that way it can make sense to talk about "consistency" of a language by talking about consistency of a theory that models that language. This allows us to gain some additional confidence that the language we defined really does work, by studying the corresponding theory and failing to find any inconsistencies in it.

But there are three important things to note here. 1) In order to get to the point of constructing that model of the language, you still have to take the language itself for granted in order to start stating axioms. 2) The model of the language is still just one specific theory. The consistency of that specific theory may reflect on the integrity of the language, but other theories do not. 3) The fact that your theory is indeed an accurate model of your language is just an informal observation. If you want to formalize that observation, then you need to discuss the structure of the language directly in order to compare it to the model and prove equivalence. And once you add in the ability to discuss the structure of the language directly, you've defined a new language and are no longer actually using the old one. If you try to get around this by defining a language that can discuss its own structure, you're likely to find that the corresponding theory is indeed inconsistent and thus not a good language for formal reasoning.

My point here is that although there are similarities, there will always be a significant distinction between the language you use and the theories that can be stated in that language, and in general the consistency of the theories has no bearing on the integrity of the language.

[Xanthir]

I think we can also recognise axiomatic systems by looking at the properties of a system.

No, you can't. "Axiomatic system" has a definition; it's a collection of particular types of objects (described earlier). That system has some properties, but you can't *recognize* those properties until you know that it *is* a logical system. You can't just go willy-nilly looking for "ambiguity" - that's not an intrinsic quality of reality that you can recognize in arbitrary things. (It's not similar to, say, the color of physical objects.)

There is something wrong with the method of identifying axiomatic systems I've described. It contradicts well established and evidenced mathematics.

Yup, what I just described above is what's wrong.

[dalcde]

One of the goals (sort of) of mathematical rigor is to minimize the amount that we take for granted by using only some of the most ingrained intuitions in the way the human mind works. Things like basic logical inference (If I know 'A' is true and I also know "if A then B", then I can conclude 'B' is true). So we create formal logic as a language that expresses only these most basic intuitions, take it largely for granted that formal logic works, and go from there.

I disagree with this. Formal logic take these as axioms (or rules of inference). We do not take for granted these are true, just as we don't believe that "rings are commutative" is a fundamental fact of nature when we study commutative rings. It's just that we take these as the axioms of formal logic, and see what we can prove from these axioms. It is certainly reasonable to study logical systems where modus ponens is not valid (not sure how much you can prove from this, but rejecting the law of excludded middle, which is "obviously true", is rather common).

What we do take for granted, which is the "non-rigorous" part, is that people will agree that the following is not an application of modus ponens:
- q and (p => q)
- p
(assuming p and q are distinct).

[Twistar]

One of the goals (sort of) of mathematical rigor is to minimize the amount that we take for granted by using only some of the most ingrained intuitions in the way the human mind works. Things like basic logical inference (If I know 'A' is true and I also know "if A then B", then I can conclude 'B' is true). So we create formal logic as a language that expresses only these most basic intuitions, take it largely for granted that formal logic works, and go from there.

I disagree with this. Formal logic take these as axioms (or rules of inference). We do not take for granted these are true, just as we don't believe that "rings are commutative" is a fundamental fact of nature when we study commutative rings. It's just that we take these as the axioms of formal logic, and see what we can prove from these axioms. It is certainly reasonable to study logical systems where modus ponens is not valid (not sure how much you can prove from this, but rejecting the law of excludded middle, which is "obviously true", is rather common).

What we do take for granted, which is the "non-rigorous" part, is that people will agree that the following is not an application of modus ponens:
- q and (p => q)
- p
(assuming p and q are distinct).

I think I agree, but on this point we need to be very careful. The word "Axiom" is getting thrown around a bit loosely here. I think we should reserve the word "axiom" to mean the following:

Assume we have a formal langauge with symbols and syntax.
Construct a number of well formed sentences in that language. Call this set of well formed sentences the set SEN.

Now if in addition to the rules of syntax, we can add semantics. That is, we can deem certain sentences to be true. Furthermore, we can develop rules of deduction by which we can derive truth values for previously undetermined sentences from the truth values of sentences which have known truth values.

Now, deem all of the sentences in set SEN to be true. This set of sentences together with the semantics indicating those sentences are true constitutes the set of AXIOMS for a THEORY.

What is being discussed in the previous few posts does NOT regard axioms in this sense. But rather, what is being discussed above regards the RULES OF DEDUCTION.

So to restate what others have said: The non-rigorous part of formal language is the understanding that other people will agree to and abide by the agreed upon rules of deduction and rules of syntax.

Once that idea is understood or accepted the process of choosing axioms and generating a theory is entirely rigorous and deterministic.

[dalcde]

I think I agree, but on this point we need to be very careful. The word "Axiom" is getting thrown around a bit loosely here. I think we should reserve the word "axiom" to mean the following:

Assume we have a formal langauge with symbols and syntax.
Construct a number of well formed sentences in that language. Call this set of well formed sentences the set SEN.

Now if in addition to the rules of syntax, we can add semantics. That is, we can deem certain sentences to be true. Furthermore, we can develop rules of deduction by which we can derive truth values for previously undetermined sentences from the truth values of sentences which have known truth values.

Now, deem all of the sentences in set SEN to be true. This set of sentences together with the semantics indicating those sentences are true constitutes the set of AXIOMS for a THEORY.

What is being discussed in the previous few posts does NOT regard axioms in this sense. But rather, what is being discussed above regards the RULES OF DEDUCTION.

So to restate what others have said: The non-rigorous part of formal language is the understanding that other people will agree to and abide by the agreed upon rules of deduction and rules of syntax.

Once that idea is understood or accepted the process of choosing axioms and generating a theory is entirely rigorous and deterministic.

This depends on what logical system you are using. Under Hilbert-style systems, there are axioms of first order logic (like p => (q => p)) and deduction rules (usually just MP and maybe generalization). Under, say, natural deduction, I guess these are all considered as rules of deduction. I did put rules of inference in brackets at the beginning, but wasn't bothered to afterwards.

[Treatid]

Yes, words are important, and I'll happily use any terminology agreed to. However, I think it is the concept rather than the labels that I'm getting wrong.

I think that pretty much everything is an Axiomatic System or part thereof. Whether this idea is labelled as a logical system or a theory - I still think that everything is a logical system or a theory and that the rules of Axiomatic Systems apply to all instances.

I muddled my explanation with 'ambiguity'. I'll try to clarify.

Consider the set of all things that cannot possibly be Axiomatic Systems. This set consists of objects that cannot be described by any set of symbols, rules and axioms.

Now the inverse set is things-that-can-be-Axiomatic-Systems.

As Arbiteroftruth and Twistar point out, 'ambiguous' could be ambiguous. There are two distinct varieties here:

1. A system that cannot be expressed by any possible set of symbols, rules and axioms. Such a system cannot be known or communicated.
2. An axiomatic system whose precise rules, axioms and symbols may be in doubt at this moment - but can, in principle, be expressed by some combination of rules, axioms and symbols.

The first sense is the one I have been intending.

If everything that can be communicated, can be expressed as an axiomatic system, then everything is an axiomatic system (or part thereof).

This is probably why I appear blasé about calling everything an Axiomatic System and dismissive of statements that languages are distinct from axiomatic systems. Because I really do think that every system we encounter must be an axiomatic system, languages and first order logic included.

This is also why I feel confident in recognising an axiomatic system from its properties. If we can talk about it, it is an axiomatic system.

To me, the definition of Axiomatic Systems (see original post) provides us with a tautology. The only way to define a definite (rigorous, repeatable, consistent) system is as an Axiomatic System. Therefore anything that is defined in a rigorous, repeatable and consistent way is an Axiomatic System.

It is little wonder that I come off as a freak when my basic understanding of what an axiomatic system is is so far from the actual meaning.

Could anyone clarify for me how it is possible to have a non-axiomatic system? Presumably a non-axiomatic system is one that cannot be described by any possible combination of rules, symbols and axioms - but doesn't that make all non-axiomatic systems inherently indescribable?

[Twistar]

Could anyone clarify for me how it is possible to have a non-axiomatic system? Presumably a non-axiomatic system is one that cannot be described by any possible combination of rules, symbols and axioms - but doesn't that make all non-axiomatic systems inherently indescribable?

English is a non-axiomatic system. There are some rules but they can be broken and the point will still get across. Sometimes the point won't get across even if the rules are followed. That sucks when that happens but we can always try harder.

3 questions.
1) Do you think English exists? i.e. do we "have" English as a system?
2) Do you think English is an axiomatic system?
3) Do you think English is inherently indescribable?

[arbiteroftruth]

It is certainly reasonable to study logical systems where modus ponens is not valid (not sure how much you can prove from this, but rejecting the law of excludded middle, which is "obviously true", is rather common).

Even if your formal system system doesn't include modus ponens, your human understanding of that system certainly will. Any set of rules of logic you can possibly come up with ultimately comes down to 1) saying that "if these circumstances apply, then I can make this deduction", then 2) saying "the circumstances apply", then 3) making the appropriate deduction. That's why I used modus ponens as my example rather than the law of the excluded middle. The law of the excluded middle can be rejected by saying there are more possibilities than just 'true' or 'false'. But to reject the concept of modus ponens is to reject anything that could reasonably be called logic.

But aside from the specifics, even though we can and do construct logical languages formally for the sake of studying them, at the foundational level it seems clear to me that it's a matter of accepting certain intuitions about truth and deduction and formalizing them for the sake of explicity. In the context of studying first order logic itself, it's just another theory with appropriate axioms. But in the context of using first order logic as the language in which axioms are stated in the first place, it's taken as an accurate representation of human intuitions about truth and deduction.

Could anyone clarify for me how it is possible to have a non-axiomatic system? Presumably a non-axiomatic system is one that cannot be described by any possible combination of rules, symbols and axioms - but doesn't that make all non-axiomatic systems inherently indescribable?

I wouldn't say a non-axiomatic system is one that absolutely can't be described precisely; just one that isn't typically used that way. English is a good example. If you really wanted to, you might be able to come up with some huge list of axioms for every possible rule of communication in English, but no one actually treats English that way. It's fuzzy, and messy, and much more a product of human psychology than of rigorous obedience to rules. But it mostly works anyway, because it's a product of human psychology meant for consumption by human psychology. To explain English axiomatically would amount to explaining every nuance of human psychology axiomatically. It might be possible in principle, but in practice it's an insurmountable task.

[elasto]

Indeed. When Watson wins Jeopardy, it doesn't do it by reducing English to a set of describable axioms; It embraces the fuzzy and imprecise nature of natural language.

That means it can never 'mathematically prove' that its answer is guaranteed to be correct (whatever that might mean). But it can none-the-less be as sure of the correctness of any of its answers as any human would be.

[Treatid]

It looks like I'm being excessively black and white when it comes to description.

A set of rules, axioms and symbols that doesn't rigorously describe a system might still non-rigorously describe a system.

While a complete and rigorous description of a system may not be possible, a partial description is a good second best option.

It isn't immediately clear to me what exactly a partial (non-rigorous) description is.

The following is why I don't think a non-rigorous description exists as a middle-ground between a definite description and no description at all:

Given a system that cannot be described in rigorous way by any possible combination of rules, symbols and axioms; any description we do have for that system must be partial/incomplete.

There must be some degree to which the system can be described and some degree to which the system is indescribable.

The nature of the element that can't be described is.... well.... can't be described. That element is wholly indescribable (the bits we can describe aren't the indescribable bits).

Thus we have a system containing two elements. One element is rigorously describable (a thing that is rigorously describable is an Axiomatic System), the other element is not describable but interacts with the describable element in an indescribable way.

So I'm left with just two types of thing - axiomatic objects and things that are utterly indescribable.

Is splitting something that can't be described rigorously into what can be described vs what can't be described a valid choice?

It might be possible in principle, but in practice it's an insurmountable task.

Aren't we talking 'in principle' at the moment? It looks like you are agreeing with me that, in principle, English is an instance of an axiomatic system.

@Twistar:
1) Yes, I think English exists as a system.
2) Yes, I think given the definition of axiomatic systems, anything we can communicate must be an axiomatic system or part thereof.
3) No, I do not think English is inherently indescribable.

[Sizik]

It might be possible in principle, but in practice it's an insurmountable task.

Aren't we talking 'in principle' at the moment? It looks like you are agreeing with me that, in principle, English is an instance of an axiomatic system.

@Twistar:
1) Yes, I think English exists as a system.
2) Yes, I think given the definition of axiomatic systems, anything we can communicate must be an axiomatic system or part thereof.
3) No, I do not think English is inherently indescribable.

I suppose the followup question would be: What are (some of) the axioms of English?

[arbiteroftruth]

Treatid, you're treating the term "axiomatic system" as meaning "anything that can be described", which we might reasonably condense to "anything". That is, it's a useless term, because it applies to literally anything.

Generally, you have whatever thing you want to describe, and then there may be a mathematical theory that serves as the description. That is, generally, things being described aren't mathematical theories. Mathematical theories are the descriptions, not the things themselves. So the physical world for example is not a mathematical theory. But the currently known laws of physics, which describe the physical world, can be formulated as mathematical theories.

This distinction seems crucial to most of your confusion. So I want to particularly draw your attention to it. Not everything that can be described is a mathematical theory. The description may be a mathematical theory, but the thing itself typically isn't. A mathematical theory is an abstract description that may or may not have any relevance to things that actually exist.

[Twistar]

It looks like I'm being excessively black and white when it comes to description.

A set of rules, axioms and symbols that doesn't rigorously describe a system might still non-rigorously describe a system.

While a complete and rigorous description of a system may not be possible, a partial description is a good second best option.

It isn't immediately clear to me what exactly a partial (non-rigorous) description is.

The following is why I don't think a non-rigorous description exists as a middle-ground between a definite description and no description at all:

Given a system that cannot be described in rigorous way by any possible combination of rules, symbols and axioms; any description we do have for that system must be partial/incomplete.

There must be some degree to which the system can be described and some degree to which the system is indescribable.

The nature of the element that can't be described is.... well.... can't be described. That element is wholly indescribable (the bits we can describe aren't the indescribable bits).

Thus we have a system containing two elements. One element is rigorously describable (a thing that is rigorously describable is an Axiomatic System), the other element is not describable but interacts with the describable element in an indescribable way.

So I'm left with just two types of thing - axiomatic objects and things that are utterly indescribable.

Is splitting something that can't be described rigorously into what can be described vs what can't be described a valid choice?

It might be possible in principle, but in practice it's an insurmountable task.

Aren't we talking 'in principle' at the moment? It looks like you are agreeing with me that, in principle, English is an instance of an axiomatic system.

@Twistar:
1) Yes, I think English exists as a system.
2) Yes, I think given the definition of axiomatic systems, anything we can communicate must be an axiomatic system or part thereof.
3) No, I do not think English is inherently indescribable.

Ok, so yeah. Again. Let's back off from the words "axiomatic system" again. I think this distinction between the describable and indescribable aspects of a thing are more central to the discussion now.

My gut reaction is that you have and idea for something being describable which is actually too strong. In other words, my suspicion is that nothing can ever be described to the level that would qualify as "describable" as you were using it in your last post.

-You have set a VERY high bar for the word 'describable' and you believe some things are above this bar. You are then getting confused about how we can decide whether something is above or below that bar. In particular, you're not happy with the idea that something like axiomatic mathematics -which should be above the bar- could be built on top of a metalanguage like English or something else -which is below the bar-.


I think others' and my point to you is that actually EVERYTHING is probably below the bar INCLUDING axiomatic mathematics. The point is that you have set the bar too high and even axiomatic mathematics falls below that bar.


Maybe let's put it in your language. Maybe this will get through to you. Take the following as an assumption. every THING has a describable aspect and an indescribable aspect. Well, if everything has an indescribable aspect then it is not fully describable. That means that you can't really describe the thing at all if you always miss out on some of its essential aspects so we should actually just go ahead and put the WHOLE THING into the indescribable category. Forget the describable part (whatever that means) because you're always just going to miss out on something anyways so you won't get the full idea.


Anyways, I haven't made the above point very clear but you probably get it. If an object is slightly indescribable we should place it entirely in the indescribable category. Since every object is slightly indescribable that means all objects are actually just in the indescribable category. This is my proof that EVERYTHING falls below YOUR bar for "describable".



To explore this a little more lets think about examples.

In particular, can you give us an example or two of things that are describable and an example or two of things that indescribable? Please try to avoid the things under discussion so far. That is, avoid math, avoid axioms, avoid axiomatic systems etc. Maybe talk some about why you think English is or isn't describable (or what aspects of English are describable and which aren't.)

[dalcde]

I think you should get a book and actually learn how things are actually done, instead of just speculating without actually knowing the subject you are talking about. Once you have actually learnt how logic is done in modern days, we can discuss about the technical aspects of it.

[Treatid]

Twistar, I think you have a good grasp of what I'm getting at.

You are right that the bar for 'description' is too high for axiomatic mathematics to reach (I think arbiteroftruth also agrees with this). However, that bar is set by Axiomatic Mathematics itself. Axiomatic Mathematics specifies exactly what is required to rigorously describe a system. I think the problem you have with my high bar for description is the problem I have with axiomatic mathematics's use of description.

You are also right in your characterisation of what would happen next; that partially-describable is no different than indescribable. We are left in a position where we either know everything or nothing.

You are right that this is absurd.

I think someone gently reminded me that "The Excluded Middle" is a thing. Thank you.

If Axiomatic Mathematics is a case of the excluded middle such that the only possible options are a rigorous description or no possible description; then we get absurd results. I think this is common ground.

I don't just think that axiomatic mathematics is an excluded middle system... it looks to me like axiomatic mathematics is intended to be an excluded middle system.

It seems pretty obvious to me that the definition of 'rigorous description' within axiomatic mathematics is of an absolute that can only be achieved under the specified conditions. There is an explicit distinction between things that can be rigorously described and things that cannot be rigorously described. Is and is-not.

But we agree that axiomatic mathematics being an excluded middle system leads to absurd results.

So there is no way that Axiomatic Mathematics is an excluded middle system with respect to description.

Which is where I am.

I know that Axiomatic mathematics must not be an excluded middle system with respect to description. But all the evidence I see says it is.

There must be something that shows that axiomatic mathematics includes the middle with regard to 'rigorously-describable' and 'not- rigorously-describable'. I'm just not seeing it - everything I try comes back to an excluded middle.

What is the argument that shows axiomatic mathematics is an included middle?

[Twistar]

Ok.. lets forget all of that law of the excluded middle stuff. It's just a red herring for where your real confusion lies.

Twistar, I think you have a good grasp of what I'm getting at.

You are right that the bar for 'description' is too high for axiomatic mathematics to reach (I think arbiteroftruth also agrees with this). However, that bar is set by Axiomatic Mathematics itself. Axiomatic Mathematics specifies exactly what is required to rigorously describe a system. I think the problem you have with my high bar for description is the problem I have with axiomatic mathematics's use of description.

"However that bar is set by Axiomatic Mathematics itself. Axiomatic Mathematics specifies exactly what is required to rigorously describe a system."

Where do you get that idea? You need to explain this better. Can you describe this bar to me? Can you also point me to a resource which shows text where axiomatic mathematics is supposedly specifying this bar?

Here is my understanding of "the bar" we are talking about. The bar specifies a level of description whereby I or anyone could describe a system to that level and anyone I am talking to would unambiguously understand what I am talking about with no chance of miscommunication. I think this is what you mean by "rigorously describe".

However, and here is my main point to you:
Logicians and mathematicians DO NOT CLAIM that axiomatic mathematics attains that level of description. And I'm not just making a point to you, I'm trying to point out something about which you are wrong. You have this misconception (misconception #1) that logicians purport axiomatic mathematics to describe things to this level, so you see a contradiction. HOWEVER, logicians DO NOT purport that axiomatic mathematics describes things to this level.

Now, this isn't an unreasonable misconception. You could probably easily find example of people glibly talking about how math is absolutely right and unambiguous. People making those statements probably haven't thought hard about formal logic or formal languages. I would say that they are wrong. But if you talk to people who really have thought hard about these things my bet is they would agree with what I'm saying here. But the point is, now that you're thinking about it a lot, you need to realize that this is, in fact, a misconception.

So then your next worry is that:

You are also right in your characterisation of what would happen next; that partially-describable is no different than indescribable. We are left in a position where we either know everything or nothing.

Your next misconception (misconception #2) is that if we can't describe something to the level of this "bar" that we are speaking about then we can't describe ANYTHING and we then know nothing. This is a misconception. Let me first tell you what is correct and what I agree with:
-It is correct that for any given thing we can either describe it above the level of the bar or we cannot.
-I agree with the statement that NOTHING (including axiomatic mathematics) can describe ANYTHING to the level of this bar.

But here is where I differ with you:
-However, just because we can't describe anything to the level of this bar, that DOESN'T mean we can't describe anything at all. We can describe things to lesser degrees than the bar. Yes, there will be some ambiguity, but that doesn't mean there is no description. You're thinking of it as binary. That is the wrong way to think about it. it is actually a spectrum. It's like a dial. Say we can rate a level of description from 1-100. Something that is a 1 is not describe at all. Something that is 100 would be AT the bar we are talking about. The scale doesn't go above 100 because the way we have defined the bar means there is no higher level of description. Let's talk about describing the weather tomorrow.
-Say you ask me what the weather is going to be tomorrow and I say "I dunno". That would be a 1 or 2 on the level of description. I haven't really told you anything about the weather except maybe the fact that it is unpredictable at the moment by me.
-But say instead I say "It's supposed to be nice". Now maybe I've given you a description which is at a level of 10 or 20 or something. Maybe I've given you all of the information you need and you can go on with your day. But there's still really a lot of ambiguity there.
-Now maybe I can get more descriptive. I can say "it's supposed 50 F at 06:00 rising up to 70-80 F from 11:00 - 18:00 and settling back down to 60F from 19:00 on." This is now starting to rank at maybe a 50 on the descriptive level. However there is still a lot I'm not describing.
-Now, maybe you could talk to a weather scientist or meteorologist who could tell you the absolute state of the art prediction for the weather tomorrow. This would include by the minute temperature predictions, pressure reading, wind patterns, local and global weather patterns etc. This is now getting up to maybe an 80 on the descriptive scale. There is still a lot of uncertainty but the level of description is getting higher.
-However, we can imagine a description which is even better than that. Maybe I could tell you what all of the individual atoms making up weather patterns are going to be doing tomorrow. And from that we can extract temperature information on a microscale as well as predict the answer to any question about the macroscopic weather that you might be interested in knowing about. Maybe this is a 97 or a 98 on the scale.

The key is that for the final description it is STILL not 100. Now the level of descriptiveness is not limited by our current meteorological models, it is limited by this fundamental lack of an ability to describe anything 100%. But the point with all of this is that even if we can't describe something 100% we can STILL describe it TO SOME DEGREE. It is not black or white, it is a spectrum.

So it is with axiomatic mathematics. I think on this scale most logicians would agree that the goal of axiomatic mathematics is to describe something to 99 or 99.9% or whatever. But they're not claiming it's 100%. That is the distinction.

Let's get a little epistemological for a second. Your argument is that if we don't know something 100% then we don't know it at all. If we can't know anything 100% then the composition of these two statements implies we don't ANYTHING. I disagree with the first statement. The only claim that be made is that "we don't know anything 100%". But what can still be said is: "we know a lot of things 99%". or "we know a TON of things 60%".

So to repeat your two misconceptions:
1) Logicians purport that axiomatic mathematics describes things 100%
2) If we don't know something 100% then we don't know it at all.

These are the two things which I and everyone else in this thread disagrees with. And I really think it ALL boils down to these two misconceptions that need to be worked on.
The first one is a simple matter of fact. You can look up what logicians actually say or just talk to them. It's not a statement about knowledge or mathematics or anything. It's literally a statement about what people say. The subject of the sentence is logicians, not axiomatic mathematics.
The second one is a bit philosophical and one that I have grappled with in the past. You know, maybe the notion that we can't know anything 100% throws us into a nihilistic nightmare but I don't actually think it's that bad. You just have to accept that our world just isn't like that. In the end, this idea that we can't know anything 100% doesn't change much about what actually happens in our lives. Furthermore, and this is your main concern, it doesn't change much about how math actually works either, vis-a-vis misconception #1.

[Treatid]

You tell me to forget about the excluded middle and then discuss the distinction between binary and continuous systems?

Invoking the excluded middle is explicitly specifying a binary system versus a continuous system.

But you are absolutely right that the distinction between a binary system and a continuous systems seems to be at the heart of where I'm going wrong.

You are right that continuous systems don't have the problems I'm describing.

My trouble is that Axiomatic Mathematics looks profoundly binary to me: a thing can be rigorously-defined, or it cannot be rigorously-defined; it is true or it is false; it is consistent or it is inconsistent.

Axiomatic Mathematics looks to me to be the poster child of the excluded middle - of binary choices.

It seems to me that Axiomatic Mathematics was intentionally setup as a system to give definite, yes, no; true, false; results. That is why it is considered useful, isn't it?

I agree that we do, in fact, seem to be able to have partial knowledge without it evaporating in a puff of logic in front of our faces. I agree that Axiomatic Mathematics must not be an excluded middle with respect to the precise qualities it specifies (whatever anyone thinks they may be at a given moment).

So, avoiding worrying about what exactly it is that axiomatic mathematics does:

Logicians should be able to give a definite answer to the question of whether Axiomatic Mathematics is binary with respect to the properties it specifies? Is Axiomatic Mathematics designed to be an excluded middle system?

Hopefully show why Axiomatic Mathematics is a continuous system and not a binary, true/false, system?

[arbiteroftruth]

You tell me to forget about the excluded middle and then discuss the distinction between binary and continuous systems?

Invoking the excluded middle is explicitly specifying a binary system versus a continuous system.

The law of the excluded middle asserts a binary formal language regarding the possible truth values of a statement in that formal language. Twistar wasn't talking about truth values or about statements in a formal language. He was talking about descriptions of things as a general human endeavor. Read his post again.

[rmsgrey]

Is "Axiomatic Mathematics" a specific system? Google doesn't offer a specific system with that name - trying fairly quickly leads to searches that return this thread as their first hit. My feeling is that it's a class of systems - that "axiomatic mathematics" is the subclass of "mathematics" that is constructed from axioms.

Boolean logic is a class of axiomatic systems which have binary truth values.

Fuzzy logic is a class of axiomatic systems which have continuous truth values.

Both are mathematics, and both are axiomatic. Are any of the systems involved "Axiomatic Mathematics"?

[Gwydion]

Treatid, this conversation isn't going very far because you're not conceptualizing what everyone else seems to be saying in the same way. You're attributing qualities to "Axiomatic Mathematics" that no one else seems to be asserting, because you believe it should have those qualities, but then shooting your own assertions down when it fails to live up to your expectations. It's like claiming that Usain Bolt isn't very fast because he can't teleport, and everyone knows that if something moves from point A to point B, then it either teleports or doesn't move at all, because if you aren't teleporting then you can break down the movement into partially teleporting and partially staying still, but it's impossible to "partially stay still" so if you're partly staying still then you're just completely staying still so if you can't teleport you can't move.

The fact is, human beings are unable to communicate "perfectly" in the way you seem to want us to. The fact that we communicate at all is sometimes miraculous. I'm reminded of a Penny Arcade posting in which Tycho discussed the phrase "the die is cast" - in his mind, that phrase meant that a die is created by its manufacturer, and therefore can not be unmade; a friend believed the phrase was "the dye is cast" and that it referred to indelible paint being thrown on a canvas. Both were "wrong" in the sense that this is not the way the phrase is traditionally understood, but they both got the same symbolic point, which is awesome.

Given that human beings rely on human languages to communicate, it is not ever possible to achieve the bar of "rigor" that you ask for. There are a limited number of words in our languages, and it is not possible to define them all in the same language without some circularity. The true rigor that axiomatic mathematics provides, is in the idea that from very simple statements and a few rules governing structure and inference, one can derive exceptionally complicated results.

[Twistar]

Is "Axiomatic Mathematics" a specific system? Google doesn't offer a specific system with that name - trying fairly quickly leads to searches that return this thread as their first hit. My feeling is that it's a class of systems - that "axiomatic mathematics" is the subclass of "mathematics" that is constructed from axioms.

Boolean logic is a class of axiomatic systems which have binary truth values.

Fuzzy logic is a class of axiomatic systems which have continuous truth values.

Both are mathematics, and both are axiomatic. Are any of the systems involved "Axiomatic Mathematics"?

We decided we're talking about ZF or ZFC set theory in this thread.


@Treatid: I'm not responding to any of the excluded middle stuff. See arbiteroftruth's post. The law of excluded middle is something that applies to statements within a formal language. I am speaking to you in plain English. I'm trying to have a regular English conversation but you keep over complicating with misapplications of properties of formal languages.

My trouble is that Axiomatic Mathematics looks profoundly binary to me: a thing can be rigorously-defined, or it cannot be rigorously-defined; it is true or it is false; it is consistent or it is inconsistent.

I've bolded the important parts here. It is YOUR trouble and it looks binary to YOU. But in fact, that is because you are thinking about it in a slightly lopsided way. If you were thinking about this the way I, or others in this thread, or other logicians think about it this wouldn't be a problem. Here's the issue. And it has to do with misconception #1 from above.

It depends what you mean by rigorous. Using my language from my post before, right now you are taking rigorous to mean 100% described. If that is what we're talking about then here is my position: Mathematics is NOT NOT NOT NOT NOT NOT rigorous. If you disagree then find a reference other than your feelings defending the statement that mathematics delivers absolute truth.

However, When mathematicians talk about rigorous they are actually talking about a level of description that is at like 99%. For example, I can make a mathematical statement such as:

"The integral of a function is the anti-derivative of that function."

This is a statement of the fundamental theorem of calculus. However, it's not very rigorous. Maybe as far as the statement of the theorem I've only been 50% descriptive in what I'm even trying to say. As far as the proof of the statement I've been 0% descriptive. So even though I'm making a mathematical statement I'm not being up to the 99% level of rigorous. I could bring in a lot more symbols and define terms blah blah blah and then maybe I could eventually get up to 99.5% descriptive. At that point any mathematician would look at my work and say "oh yeah, that's a pretty rigorous proof of the fundamental theorem of calculus." Or maybe it would be at the 97% level of descriptiveness. If that was the case maybe mathematicians would look at it and say "well, I think I see what you're getting at but it's not quite rigorous yet." Maybe I left a step or two for the reader to figure out him or herself. But the point is that no matter what it STILL wouldn't be 100% descriptive because that is impossible.

Again, this is nothing but misconception #1. YOU think that mathematics is supposed to be 100% descriptive. Further more, you think that OTHER people* think this as well. But you are wrong. That is why it is a misconception that I'm trying to address. If you want to convince me that I am wrong then you HAVE to provide a reference source OTHER than yourself. Furthermore, if you continue to purport this position without providing this reference then I will leave this thread immediately.

It seems to me that Axiomatic Mathematics was intentionally setup as a system to give definite, yes, no; true, false; results. That is why it is considered useful, isn't it?

At the level of discussion we're having here the answer is NO. Mathematics does not give absolute yes, no, true, false results. The reason axiomatic mathematics is useful is because it sets a bar of descriptiveness. That bar is at the 99% level of descriptiveness. If I make a mathematical statement to you, it is often times not that difficult to tell whether the statement I have made is above or below this 99% bar. If it is at the 99.5% level then it is easy to test whether the statement I have made is correct within the rules of axiomatic mathematics. Note that that doesn't mean I have given a 100% level description. That is impossible. Also, if my statement is below the 99% bar it will be easy to tell at which point it means I haven't been descriptive enough of what I'm trying to say and the onus is on me to try to increase my level of descriptiveness so you can more clearly see what I'm trying to say to see if it is correct with respect to the rules of axiomatic mathematics or not.

TLDR: Axiomatic mathematics is useful because it sets a well-defined bar for rigor. But that bar is at a DIFFERENT level than the bar we were talking about before. Like I said, the bar we were talking about before is too high.



*Footnoted because I'm talking about other qualified people. It is true that there are probably a lot of people who haven't thought about axioms or formal logic or anything as deeply as we're talking about it here. They probably share the misconception that mathematics is 100% rigorous. That is why I said it was a reasonable misconception to have, but now we are delving deeper and we are trying to overcome that misconception.

[Cauchy]

What do "rigorously defined" and "unambiguous" mean when you use them, Treatid? What is "Axiomatic Mathematics"? What are "binary" and "continuous systems"? When you say "Axiomatic Mathematics must not be an excluded middle with respect to the precise qualities it specifies", what is "an excluded middle", and what are "the precise qualities"?

I feel like you use a lot of these wishy-washy terms that no one else is using (except to talk with you) to hide your lack of understanding. If your goal is to actually understand, then I think we need to unpack these terms to address the misunderstandings that they cover up.

[Treatid]

While I was working on your responses I was struck by the following:

In logic, the law of excluded middle (or the principle of excluded middle) is the third of the three classic laws of thought. It states that for any proposition, either that proposition is true, or its negation is true.-

This seems to be as black and white as it gets.

If I'm reading this right, it is a fundamental assumption of mathematics that all propositions are excluded middle.

There is no question that axiomatic mathematics (in all possible forms) is an excluded middle. The foundational laws of logical thought explicitly state that all mathematical propositions are excluded middle.

So I have a definitive answer to the question of whether axiomatic mathematics (in all possible forms) is an excluded middle. For any proposition that axiomatic mathematics makes, the only inverse is not-proposition. For any x in mathematics, the inverse of x is always and only !x. There is never a partial-x (real world experience notwithstanding).

This is a pretty large step I'm taking. I want to examine the consequences we didn't resolve upstream as we focused in on the question of whether axiomatic mathematics is an excluded middle system. But me claiming that Axiomatic Mathematics (in all possible forms) is an excluded middle; is not the same as us agreeing that Axiomatic Mathematics is an excluded middle, albeit it looks like it is specified by one of the three classic laws of thought.

[arbiteroftruth]

While I was working on your responses I was struck by the following:

In logic, the law of excluded middle (or the principle of excluded middle) is the third of the three classic laws of thought. It states that for any proposition, either that proposition is true, or its negation is true.-

This seems to be as black and white as it gets.

If I'm reading this right, it is a fundamental assumption of mathematics that all propositions are excluded middle.

You're not reading it right. It's not talking about a fundamental assumption. It's talking about a rule of classical logic. Classical logic is only one kind. There's also intuitionistic logic where the law of the excluded middle is rejected.

But that's also irrelevant to the point at hand. All this business about the law of the excluded middle has absolutely nothing whatsoever to do with fundamental questions about describability.

Seriously, drop the stuff about the law of the excluded middle, and read what people are actually saying to you about where your misconceptions are. Hint, most of your misconceptions have nothing to do with the excluded middle, except for your evident misconception that the excluded middle is somehow relevant here.

[rmsgrey]

While I was working on your responses I was struck by the following:

In logic, the law of excluded middle (or the principle of excluded middle) is the third of the three classic laws of thought. It states that for any proposition, either that proposition is true, or its negation is true.-

This seems to be as black and white as it gets.

If I'm reading this right, it is a fundamental assumption of mathematics that all propositions are excluded middle.

There is no question that axiomatic mathematics (in all possible forms) is an excluded middle. The foundational laws of logical thought explicitly state that all mathematical propositions are excluded middle.

So I have a definitive answer to the question of whether axiomatic mathematics (in all possible forms) is an excluded middle. For any proposition that axiomatic mathematics makes, the only inverse is not-proposition. For any x in mathematics, the inverse of x is always and only !x. There is never a partial-x (real world experience notwithstanding).

The Law of the Excluded Middle founders on the string "This proposition is not true", but there are axiomatic systems that can handle that string just fine.

[Twistar]

While I was working on your responses I was struck by the following:

In logic, the law of excluded middle (or the principle of excluded middle) is the third of the three classic laws of thought. It states that for any proposition, either that proposition is true, or its negation is true.-

This seems to be as black and white as it gets.

If I'm reading this right, it is a fundamental assumption of mathematics that all propositions are excluded middle.

There is no question that axiomatic mathematics (in all possible forms) is an excluded middle. The foundational laws of logical thought explicitly state that all mathematical propositions are excluded middle.

So I have a definitive answer to the question of whether axiomatic mathematics (in all possible forms) is an excluded middle. For any proposition that axiomatic mathematics makes, the only inverse is not-proposition. For any x in mathematics, the inverse of x is always and only !x. There is never a partial-x (real world experience notwithstanding).

This is a pretty large step I'm taking. I want to examine the consequences we didn't resolve upstream as we focused in on the question of whether axiomatic mathematics is an excluded middle system. But me claiming that Axiomatic Mathematics (in all possible forms) is an excluded middle; is not the same as us agreeing that Axiomatic Mathematics is an excluded middle, albeit it looks like it is specified by one of the three classic laws of thought.

Misconception #3: Thinking rule/laws/principles of formal logic apply outside the scope of formal logic.
Misconception #4: Thinking something can be an excluded middle. "The law of the excluded middle" is a rule in a formal theory that allows one to derive a new proposition from an existing proposition under certain conditions. "The law of the excluded middle" is a single thing. There is no definition for "excluded middle" or "excluded middle system".

Sorry to load it on but I'm not going to respond unless you correctly answer 2 of the problems on this logic worksheet. http://cw.routledge.com/textbooks/9780415997454/printable-exercise-worksheets/chapter4/web-ch4-4-exer-D.pdf

edit: Actually don't do that worksheet unless you want to. We don't need quantifier logic for this discussion. Try 2 problems from each page of this one. http://www.luc.edu/faculty/avande1/logi ... oblems.pdf

Sorry if I'm being a major asshole for making you do worksheet assignments but in probably over a year of you posting these sorts of things on the forums occasionally you have never once shown evidence that you can actually solve these sorts of simple logic proofs. If you can't then it's not surprising you have no idea what we (or you) are talking about. If you can then that's great and we can go from there! If you can't then that tells us all something about the next steps to think about.

edit2: For reference and to make everyone's lives easier here's a list of the inference rules. http://www.luc.edu/faculty/avande1/logi ... -rules.pdf

[dalcde]

Sorry to load it on but I'm not going to respond unless you correctly answer 2 of the problems on this logic worksheet. http://cw.routledge.com/textbooks/9780415997454/printable-exercise-worksheets/chapter4/web-ch4-4-exer-D.pdf

edit: Actually don't do that worksheet unless you want to. We don't need quantifier logic for this discussion. Try 2 problems from each page of this one. http://www.luc.edu/faculty/avande1/logi ... oblems.pdf

It is very unclear what the worksheet wants you to do.

[Twistar]

Sorry to load it on but I'm not going to respond unless you correctly answer 2 of the problems on this logic worksheet. http://cw.routledge.com/textbooks/9780415997454/printable-exercise-worksheets/chapter4/web-ch4-4-exer-D.pdf

edit: Actually don't do that worksheet unless you want to. We don't need quantifier logic for this discussion. Try 2 problems from each page of this one. http://www.luc.edu/faculty/avande1/logi ... oblems.pdf

It is very unclear what the worksheet wants you to do.

For each problem you are supposed to use the assumed statements (which say prem next to them) to derive the conclusion statement next to the 3 dots using the rules of deduction linked later in my edits.

edit: Also Treatid, it's ok if you don't correctly answer some of the problems but you should at least attempt to answer them. Basically anyone who can't solve these sorts of logic problems shouldn't really be taken seriously in this conversation, at least at the depth of discussion we're trying to have. That's why I posted it. I think we're all sort of taking these sorts of problems as a base level of understanding/common communication point so if we differ on this we won't be able to get anywhere.

I'll write up and post my own example in a couple mins.

edit 2: Here's my answer to #4 on the worksheet. I prefer the rightarrow notation for implication to the sideways U notation used on the sheet.

[Treatid]

In logic, the law of excluded middle (or the principle of excluded middle) is the third of the three classic laws of thought. It states that for any proposition, either that proposition is true, or its negation is true.-

Before we get on to more complicated logic... it appears I can't read this simple passage properly.

I was under the impression that mathematics is built upon thought. That the laws of thought are the laws of mathematics. Am I mistaken in this?

Arbiterofthruth reminds us of the exception that proves the rules. There is indeed a branch of mathematics that explicitly subverts the law of the excluded middle.

If mathematics is founded in something other than thought as defined by the three classic laws; that could go a long way to explaining where I'm going wrong.

What is the basis for mathematical thought? How does this differ from that specified by the three classic laws?

Edit: I'm away for a bit and we be delayed in responding...