## Misunderstanding basic math concepts, help please?

For the discussion of math. Duh.

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Treatid
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### Re: Misunderstanding basic math concepts, help please?

I agree with what you are saying, rmsgrey. You seem to be understanding my argument and then... well... I'm not sure what.

My whole point was that if !x does not equal everything except x then there is a problem.

You are definitely seeing that.

So you are understanding that portion of my argument - but seem to be missing that it was the argument I was making at that point.

So... Yet more "furious agreement"?

gmalivuk
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### Re: Misunderstanding basic math concepts, help please?

The thing is, ther *isn't* any conventional way to interpret "!x" when x is anything other than a statement or a set within a larger set.

When it's an axiomatic system, *every* interpretation is bizarre because the notation itself is bizarre when referring to axiomatic systems.

If you treat x (an axiomatic system) as a single thing, is that supposed to include just the axioms and rules of inference? Or those and everything provable from them? Or those and everything that is true under their framework? ANd if x includes the rules of inference, what rules of inference does !x have?

You don't get to invent new bizarre notations and then be frustrated when we don't automatically know exactly what you're trying to say.
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Cauchy
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### Re: Misunderstanding basic math concepts, help please?

Treatid wrote:Cauchy: I hope you are also a teacher outside these forums. Even when you think people are wrong - you don't tell them they are wrong, you give them a carefully measured piece of rope exquisitely judged to let them hang themselves (realise where they have made a mistake). Your consistent constructive involvement has my immense respect.

Teaching is my dream job, though market forces don't always let me realize my dreams.

As much as the first refrain of mathematics is "define your terms", no-one has ever worked out how to do that in an absolute sense - and lots of people have tried really hard - because it would make things (including mathematics) very much easier.

On the other hand, we are capable of understanding one another to some degree at least some of the time.

When I ask you what something is, you don't necessarily need to give a strict definition; if you can give examples, or the gist of your definition. But just because absolute definitions don't exist doesn't absolve you of having to describe what you mean when it's unclear.

And when a definition seems ambiguous, and I think a false conclusion is being drawn because of that ambiguity, then I'm going to press you on it until I think the problem is teased out. That's exactly how I feel about !x, which I'll address below.

The laws of thought specify a set of assumptions about identity and distinction. Basically: A thing is either itself or not itself.

Well, the classical laws of thought would say that a thing is itself, and it is not not-itself.

Except that I'm pretty much arguing the last bit. A thing - anything and everything - is entirely and solely specified through its relationships with other things.

I don't see how this is at odds with a thing necessarily being itself.

In practice, this shouldn't be too much of a shock. We are already familiar with one word being specified by reference to other words. And A ⊃ B is a standard part of logic. The idea of the context defining aspects of the content is familiar.

A ⊃ B, being a logical statement, doesn't fully specify A and B. It's certainly a relationship between A and B, but the could be fully specified outside of this statement, and then it turns out that the statement is a consequence of that specification.

So, before I continue trying to answer your questions... I have to ask whether you think it is possible to lay aside the Laws of Thought (even if only provisionally)?

I mean, the wiki page you linked to for the laws of thought suggests "Intuitionistic logic, Dialetheism and Fuzzy Logic" as logics that don't mesh perfectly with these laws of thought. Personally, I'm willing to vacillate on the Law of the Exclude Middle (that for all statements x, "x or !x" is true). I could see certain statements as being neither true nor false but undecidable. I'm pretty firm on the law of identity. A thing is itself. I don't know what it would mean for a thing to not be itself. The law of non-contradiction seems pretty solid to me, too, in terms of what I could mean by "truth", but if you want to drop it then I could walk with you down that road. Then, statements could be both true and false.

Edit: Edit: rmsgrey snuck in a post - conventionally, !x is everything except x. While I agree that those alternative interpretations can be made - if you default to assuming the most bizarre interpretations over the well established default interpretation then the other person is always going to appear a fool.

gmalivuk talked about this, too, but if !x is "everything except x", then by that very definition x and !x have no overlap. If x and !x have some overlap, then you just did a poor job of crafting !x, go back and try again. So if you're going to say that !x is everything except x and then posit that !x has some things from x in it, then I'm just going to stop you right there, because that's not a road I'm going to walk down.

Of course - I am now explicitly arguing against 'identity' as specified by the laws of thought...

You're going to have to explain to me how to not take identity as granted. Because I don't see an argument against identity, I just see you stating that you don't want to take identity.

When I am talking about axiomatic mathematics I try to use terms in the same way as a 'normal' mathematician would use those terms with respect to axiomatic mathematics - I'm not just throwing in my meanings willy nilly and expecting you to read my mind - honest...

People (or at least, I) legitimately weren't sure what you meant by !x. Since you didn't say what kind of thing x was, I went with what I considered the default for the ! symbol: x is a statement, and !x is its negation. My second guess was that you were talking about a system and its "negation", like you were before the physics conversation happened. I was never going to guess that x was an arbitrary collection of things, and that ! was the complement operator.

Treatid wrote:My whole point was that if !x does not equal everything except x then there is a problem.

Okay. !x does equal everything except x. Problem solved.

You're going to have to do more than say "I don't like the law of identity, so math has a problem". Why shouldn't the law of identity be true? Why shouldn't !x equal everything except x (especially since you said that that's what it was)? You say you're arguing against things, but I don't see those arguments.
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### Re: Misunderstanding basic math concepts, help please?

Of course "everything except x" is still unclear, as mentioned.

Everything *in what domain* except x?

If x is Peano Arithmetic, does !x include only false statements (2+2=5), or does it also include unprovable statements (the Goedel statement is true), or does it also include non-well-formed statements (225=+),or does it also include other things completely (my air-conditioner, for example, is not part of Peano Arithmetic)?
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Treatid
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### Re: Misunderstanding basic math concepts, help please?

Hmm - it looks to me as though showing reason to question the validity of the 3 laws of thought; and how we can work constructively after explicitly rejecting the assumption of the laws of thought will accomplish much of what I want to show.

I think this will take a few posts to get through.

Cauchy asks for a couple of conditions:

1. No leaping tall gulfs of understanding/description by using impossible presumptions.
2. The result should say something that can be applied: There should be mathematics involved, not 'just philosophy'.

That is, despite my protestations about there being hard limits on what can be described, there still needs to be an actual description.

I readily agree to both these conditions with the caveat that a description may look quite different to the concept of description illustrated by axiomatic mathematics.

I want to reject the laws of thought as basic assumptions. I think those assumptions are extremely mis-guided and lead to deep flaws in everything that abides by those assumptions. This needs a lot more justification. The laws of thought underlie a great deal of modern thought, not just that of mathematics. While there are some branches of mathematics that explicitly choose to subvert the laws of thought, these branches are not mainstream and generally fail to satisfy condition 2) above. They do, however, derive from the same sorts of concerns as I've expressed with regard axiomatic mathematics.

Absolutes

The second entry on my roadmap is the concept of an 'absolute'. Whether absolute knowledge, absolute truth or any other absolute.

It is here that I have my first and most fundamental objection to the three laws of thought. I perceive each of the three laws of thought as being attempts to establish absolutist positions upon which absolutist reasoning can be based.

There are compelling reasons why the laws of thought exist and why they were formalised as basic assumptions of (most) mathematics. Not least of which is that the lack of any basis for reasoning leaves us up the murky creek without oars.

My intention is to convey that we can, in fact, find solid reasoning without the laws of thought as a basis.

Twistar's mini survey on whether mathematics conveys "absolute epistemic truth" revealed a disturbing variation in responses - primarily disturbing for me in that there are several different opinions at various removes from what I'd like to describe. But it should also be disturbing for mathematicians that they appear not to agree on whether 'absolute knowledge' exists at all, and if it exists, whether it can be (or even has been) accessed, and if so, to what degree.

Cogito Ergo Sum

It was argued (I think by PsiCubed, but can't immediately find the post) that "I think, therefore I am" is evidence of an absolute - our existence is demonstrated by our awareness of our existence.

gmalivuk backed up this argument by showing instances of precise and reproducible phenomena.

Against this we have the other element of Cogito Ergo Sum in which all of our knowledge of the external world (and, I would argue, even the internal functioning of the mind) is mediated through some third party - neurons, nerve impulses, photons, electromagnetic forces and the like.

This third party element makes it impossible to be absolutely certain that the sense data we receive correlates precisely with an external reality (or an internal one).

Extreme solipsism is a hardcore way of living life. Most people are satisfied with accepting some degree of an objective reality even while we can never prove that with complete certainty.

To be clear - I don't think you are figments of my imagination - I would have imagined something different, I'm sure. I think there is an external something (typically called 'reality') that we inhabit. At the same time, we are fundamentally limited to building up a preponderance of evidence with respect to that reality - we can never prove anything as absolute epistemic certainty.

The three laws of thought

It seems clear to me that the laws of thought are specifically chosen to give us a framework in which to construct absolute epistemic certainty.

And our success at describing the world around us in ever greater detail could certainly be seen as pointing towards an end state of absolute epistemic knowledge.

However, thus far, the closest to some degree of certainty has been the afore mentioned Cogito Ergo Sum which simultaneously tells us we cannot have absolute certainty beyond the self-knowledge of our own, personal, existence.

Trying to keep it short

Next I want to consider each of the laws of thought in turn. I think each of the laws was chosen/created with careful thought. I don't think they can be dismissed out of hand. But nor do I think they have provided the foundation that they were intended to.

I haven't gotten to all the points Cauchy raised - I intend to do so.

@gmalivuk

Again, your overall argument is one I agree with and which I feel rather supports my case that axiomatic mathematics, in particular, is not as well defined as it needs to be. That there is any doubt or confusion over the domain of axiomatic mathematics strikes me as evidence that axiomatic mathematics has not bee properly defined.

That is, we would consider any function whose domain is not clearly defined to be, overall, not properly defined.

gmalivuk wrote:The thing is, ther *isn't* any conventional way to interpret "!x" when x is anything other than a statement or a set within a larger set.

When it's an axiomatic system, *every* interpretation is bizarre because the notation itself is bizarre when referring to axiomatic systems.

I don't understand this.

You are quite clearly saying that an axiomatic system is not a statement. This strikes me as being a bizarre thing to say.

I genuinely do not see why the symbols that form an axiomatic system are not a statement in the mathematical sense.

I grant that it is uncommon to refer to an axiomatic system as a single statement - but large and compound statements aren't in any way exceptional.

Can you clarify why you think that a description of an axiomatic system is not a mathematical statement?

Moreover, it is easy enough to group all axiomatic systems as "the set of all axiomatic systems". Although I grant that doing so introduces its own problems (such as having a set that contains multiple contradictions). However, from a normal mathematical perspective, I can't see the justification in rejecting this grouping - other than not liking the implications, which is hardly a mathematical justification.

elasto
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### Re: Misunderstanding basic math concepts, help please?

Treatid wrote:You are quite clearly saying that an axiomatic system is not a statement. This strikes me as being a bizarre thing to say.

I genuinely do not see why the symbols that form an axiomatic system are not a statement in the mathematical sense.

Is the set of all dogs a dog?

Why would a set of statements (or however you define an axiomatic system) ordinarily be a statement?

You need to be much more precise here - and in particular you need to use the same definitions as everyone else to have any hope of troubling established mathematics.

Sure, if you set up ambiguous or nonsensical definitions you can then demolish them, but that's a straw man attack.

---

I also agree with Cauchy that I don't know what it even means to say 'x=x' is not universally true. You'd have to do a lot to convince me that any useful mathematics that isn't a subset of current mathematics could result...

gmalivuk
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### Re: Misunderstanding basic math concepts, help please?

An axiomatic system is a set of axioms *and* the rules of inference. The axioms themselves are statements *in* the language we're using but the rules of inference are statements *about* that language, and thus cannot be lumped together as one compound statement.

If you combine just the axioms, what you get for your x is "a1 & a2 & a3 &...& aN".
Then for !x, we have !(a1 &...& aN) = (!a1)v...v(!aN).

In other words, if we treat the set of axioms as a single compoud statement, then the negation of that statement just implies the negation of *one* of the axioms, which is very definitely not what you've been talking about this whole time.
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Cauchy
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### Re: Misunderstanding basic math concepts, help please?

Treatid wrote:Cauchy asks for a couple of conditions:

1. No leaping tall gulfs of understanding/description by using impossible presumptions.
2. The result should say something that can be applied: There should be mathematics involved, not 'just philosophy'.

That's not how I'd phrase what I said at all. I want you to eventually have some sort of description (which you said after this), and I want you to actually attack the laws of thought if you claim they're invalid. I'll let you have 1. and 2. as goals, but please don't attribute them to me.

I don't understand this.

You are quite clearly saying that an axiomatic system is not a statement. This strikes me as being a bizarre thing to say.

??? An axiomatic system isn't a statement. It's a collection of symbols, rules on which strings of symbols can be statements ("well-formed formulas", as we've called them before, but when people say statement in a mathematical context that's what they mean), rules on which statements follow logically from other statements, and a collection of initially true statements. Nothing about that makes the system a statement. A system can't be true or false. It can be consistent or inconsistent, and complete or incomplete, but certainly not true or false.

I genuinely do not see why the symbols that form an axiomatic system are not a statement in the mathematical sense.

Do you mean the symbols of the system? You can make statements out of them, but they as an aggregate are not a statement. Do you mean the symbols used to describe the system, as in the English letters we're using right now? I can write a sentence defining that a system is such and such (in fact, I kinda did just a paragraph ago), but while that sentence could be true or false (setting aside whether or not definitional sentences can be false), that doesn't mean the system is true of false, or that that sentence is the system.

Can you clarify why you think that a description of an axiomatic system is not a mathematical statement?

You changed the goalposts here. A description of a thing is not the same as the thing itself.

Separately, if what you mean by !x is the negation of the description of an axiomatic system, then that doesn't itself describe a single system. So the thing described by x and the thing described by !x are now different types of objects: one's an axiomatic system, and the other is a collection of axiomatic systems.

Moreover, it is easy enough to group all axiomatic systems as "the set of all axiomatic systems". Although I grant that doing so introduces its own problems (such as having a set that contains multiple contradictions). However, from a normal mathematical perspective, I can't see the justification in rejecting this grouping - other than not liking the implications, which is hardly a mathematical justification.

Why is that the default universal set? Why not "the set of all axiomatic systems sharing the symbol set"? Why not "the set of all axiomatic systems sharing the language"? Why not "the set of all of mathematics"? Sure, you *can* say that that's your universal set. No one's stopping you. But if you *don't* say it explicitly, then there's no reason for people to assume that that's what you meant. Don't try to paint this as people "not liking the implications". It's that people legitimately didn't know what you were talking about.

This seems to be a(nother) running theme. You often claim that people are turning a blind eye to things (specifically, things you're saying) because they don't like the implications. More often, it's because people don't understand what you're talking about, which is fueled by your making assumptions regarding the meanings of expressions that no one else seems to make. If I had fished around, I might have come up with "x is a description of an axiomatic system, !x is the collection of all axiomatic systems not described by x" as one of my less likely possible interpretations of what you wrote, but I'm not sure. (And even still, I'm not confident that that's what you really mean.) You'll note that of the first three responses, I'm the only one who noticed that you snuck in a mention of the description of the system as the statement. That's how confused people get by what you write. The other two are trying to dispute what you actually said, which is that the system itself is a statement.

And I'm pretty sure I'm going to hear back about not being able to distinguish between a thing and its description, because nothing is concrete and everything is relative and how can anything even exist if it's not in relation to the descriptions coming from other things, but you know that there's a functional difference between a mathematical construct, which interacts with other mathematical constructs, and an informal language description of what that construct is, which informs us of how the construct interacts.
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Gwydion
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### Re: Misunderstanding basic math concepts, help please?

Treatid wrote:Hmm - it looks to me as though showing reason to question the validity of the 3 laws of thought; and how we can work constructively after explicitly rejecting the assumption of the laws of thought will accomplish much of what I want to show...

I want to reject the laws of thought as basic assumptions. I think those assumptions are extremely mis-guided and lead to deep flaws in everything that abides by those assumptions. This needs a lot more justification. The laws of thought underlie a great deal of modern thought, not just that of mathematics. While there are some branches of mathematics that explicitly choose to subvert the laws of thought, these branches are not mainstream and generally fail to satisfy condition 2) above. They do, however, derive from the same sorts of concerns as I've expressed with regard axiomatic mathematics.

You've made a solid claim - you believe the 3 laws of thought may not be/represent universal truths - and this may have merit. I'm looking forward to seeing how you explain yourself in the upcoming posts, because it's going to take a lot of justification - what you've provided so far is a lot of declarative statements that the laws are misguided/flawed, but not a lot of supporting evidence or argument.

It might be worth recalling that a rejection of the laws of thought in favor of other, less restrictive underlying assumptions doesn't actually equate to a rejection of "axiomatic mathematics" as a whole, simply a particular collection of axiomatic theories that hold those assumptions true. If you replace one set of axioms with a different set, you're still doing "axiomatic mathematics" just in a new theoretical framework.
It seems clear to me that the laws of thought are specifically chosen to give us a framework in which to construct absolute epistemic certainty.

Well, I'm not so sure. I think the laws of thought were specifically chosen because they represent simple concepts that are hard to conceptually disagree with, at least in specific circumstances such as discussing mathematical statements or logical reasoning. You've argued before that we're trying to build mathematics on a shaky foundation and making too many unfounded assumptions - what could be more simple than x=x?
Again, your overall argument is one I agree with and which I feel rather supports my case that axiomatic mathematics, in particular, is not as well defined as it needs to be. That there is any doubt or confusion over the domain of axiomatic mathematics strikes me as evidence that axiomatic mathematics has not bee properly defined.

That is, we would consider any function whose domain is not clearly defined to be, overall, not properly defined.

This is Strawman 101 - you've laid out an idea that is not well defined, and when we attack your concept as not well defined, you claim that it invalidates our arguments in turn. Your argument seems to be that "axiomatic mathematics" is a function with a domain and range, but as others have argued before, none of us are saying that and you need to justify why you feel it should be considered so.
Moreover, it is easy enough to group all axiomatic systems as "the set of all axiomatic systems". Although I grant that doing so introduces its own problems (such as having a set that contains multiple contradictions). However, from a normal mathematical perspective, I can't see the justification in rejecting this grouping - other than not liking the implications, which is hardly a mathematical justification.

So you're suggesting that we take the "set of all axiomatic systems" as an axiomatic system? Great, that set contains every possible statement and their contradictions, so of course such a system would be inconsistent, not to mention the fact that you haven't explained which elements of an axiomatic system are included - the language? the rules of inference? the axioms only? (BTW, none of these things would make your problem any better, the "set" would still be inconsistent and pretty uninteresting).
_____

Going back to your previous question of whether it is possible to "lay down" the Laws of Thought, I think it is perfectly reasonable to do so, and to try to build an axiomatic system that doesn't start from those basic pieces. However, you will need to explain what basic pieces you're using instead. If you don't want to use *any* basic pieces, then you need to explain much more clearly how your system works, namely what this system does and why we should care. You've built pages and pages of claims that all of the basic pieces are unfounded, but if you don't leave anything left over it's going to be hard to get anyone to care.
_____

Question for !Treatid: The three laws of thought always struck me as redundant. Namely, if the second holds, doesn't the third follow from it? Taking the statement for all x: ~(x and !x), by De Morgan's Law this is equivalent to for all x: (~x or x). Why do we need both?

rmsgrey
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### Re: Misunderstanding basic math concepts, help please?

Gwydion wrote:Question for !Treatid: The three laws of thought always struck me as redundant. Namely, if the second holds, doesn't the third follow from it? Taking the statement for all x: ~(x and !x), by De Morgan's Law this is equivalent to for all x: (~x or x). Why do we need both?

If you accept De Morgan's Laws as axioms, then, sure, you can derive excluded middle from non-contradiction (and vice versa).

But if you don't accept them as axioms, how do you establish them?

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### Re: Misunderstanding basic math concepts, help please?

The first law of thought is the law of identity. x = x.

A thing is itself.

I want to try and illustrate two extremes - I don't want to convince anyone of anything about these extremes (yet). Just show...

i. A closed system: everything about the system is within the system. We are ignoring the issue of where the language to describe the system comes from or how we are observing the system. Simply a standard closed system.

So... we have a closed system C composed of a number of sub-elements. For each possible sub-element of this system we can divide the system up into that sub-element and 'all the other elements'. So... as I've been using the term, for every possible x within C we can divide C into x and !x.

As such x + !x = C for all possible x in C.

(I'm trying to be redundantly clear - not trying to specify something strange - !x is (everything - x) and, in this case, everything is specified to be C).

Absolutist assumption

A given !x is changed. x is not directly changed. Obviously if !x is changed then C is also changed. However, x is an absolute value and does not depend on C or !x. x(before) = x(after)

Relativistic assumption

A given !x is changed. x is entirely specified by the context. if !x changes then x changes. x(after) + !x(after) = C(after) where C(after) != C(before)

Within C it is not possible to determine whether x(before) = x(after), !x(before) = !x(after) or C(before) = C(after) because the only measuring sticks to evaluate x, !x and C are x, !x and C which are themselves changed when any other aspect is changed.

The delta of a given change could be small. Changing !x doesn't necessarily mean that x(after) is entirely dis-similar to x(before). However, equality in mathematics is an absolute. The difference between (x(before) = x(after)) and (x(before) is very, very, very, very, very similar to x(after)) is... critically important.

Absolutist versus Relativistic

Right now I just want to convey the idea of what a !(x = x) thing might look like. In the absolutist system I haven't suggested how or why x might have a fixed value independent of the context. In the Relativistic system I haven't suggested how or why x depends only on the context. (yet).

Arguably, "x is entirely defined by the context" is an axiom. We'll see how that turns out.

As presented, x does equal x within a static C under the relativistic assumption. I hope to show that a static system is meaningless (see my previous argument regarding all indistinguishable things being the same thing (The Empty Set argument)). Basicly: it is impossible to measure anything in a static system. Measurement requires change, and change... see above...

Are "x is entirely defined by context" and the consequent description of relativistic C clear enough to develop further?

....

When is a statement not a statement?

We aren't connecting very well on this.

You tell me that this thing and that thing are different so it is wrong to call them both statements.

I agree that there are differences. But...

A human and an elephant are different. Obviously they are different. It would be silly to think of them as the same thing.

Well... they are both mammals. They are both vertebrates. Their skeletons are nearly bone-for-bone the same basic layout. They have the same types of organs - brain, kidney, liver, heart, digestive tract, ears, eyes, nose, blood vessels, reproductive organs. If we dig down we find cells and DNA. That DNA is identical in basic structure and very, very similar in actual protein encodings.

So - yes - an elephant is definitely different to a human. But, at the same time, there are significant similarities between an elephant and a human.

I can see the differences you are pointing out. But I'm reasonably sure, the mathematical use of 'statement' refers to a more general category (mammal, or even vertebrate) rather than a particularly narrow category (caucasian or oriental).

@gmalivuk: logical OR != logical EOR.

Yes - you only need to negate one axiom to find something that is part of the set of !(the axioms). Or you could negate all the axioms, or any combination of the axioms. What you have done is to take one instance of the set of !(the axioms) and claim that single instance as the entire set of !(the axioms). In fact, the entire set of !(the axioms) is ALL the permitted combinations of (!a1)v...v(!aN).

@Cauchy: Of course a system can be true. I just have to declare that system to be an axiom - and axioms are asserted to be true. gmalivuk did a fine job of showing how all the axioms of a system can be combined into a single statement that can, in turn, be used as the axiom of some other system.

None of the points made persuade me that I'm misusing the word 'statement' by applying it to individual axiomatic systems.

Re: The map is not the territory?...

When the map is a 1:1 model intended to describe every feature of the terrain down to the smallest detail - I find it very difficult to distinguish between the description and.... well... actually... the only thing we have is the description. Mathematics is an abstraction in which the description is the thing.

Cauchy
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### Re: Misunderstanding basic math concepts, help please?

Treatid wrote:Absolutist assumption

A given !x is changed. x is not directly changed. Obviously if !x is changed then C is also changed. However, x is an absolute value and does not depend on C or !x. x(before) = x(after)

What do you mean, !x is changed? How? Why? The complement of something (because that's what you're using !x to represent) doesn't typically just go around changing, so you're going to have to do more than just say "!x is changed" if you want people to agree that your example holds water. Until you do that, the rest of your writing is unapproachable.

You tell me that this thing and that thing are different so it is wrong to call them both statements.

I'm telling you that that thing is not a statement, so it's wrong to treat it as you would a statement.

I agree that there are differences. But...

A human and an elephant are different. Obviously they are different. It would be silly to think of them as the same thing.

Well... they are both mammals. They are both vertebrates. Their skeletons are nearly bone-for-bone the same basic layout. They have the same types of organs - brain, kidney, liver, heart, digestive tract, ears, eyes, nose, blood vessels, reproductive organs. If we dig down we find cells and DNA. That DNA is identical in basic structure and very, very similar in actual protein encodings.

So - yes - an elephant is definitely different to a human. But, at the same time, there are significant similarities between an elephant and a human.

What you're doing is saying that 2 and 3 are both integers, and 2/2 is an integer. Then you start talking about the integer 3/2, and when people correct you and say that 3/2 is not an integer, you say "of course it is, I can add it to other integers, and multiply it by integers. Yes, 2 is definitely different from 3. But at the same time, there are significant similarities between 2 and 3." Just because things are similar, it doesn't mean that you can handle them the same way and expect valid results from both. Maybe you get valid results, maybe you don't. It depends on whether the differences affect what you're trying to do. Our claim is that the differences do affect what you're trying to do.

Or to put it another way, your similarities argument is bull**** and doesn't back up your claim at all.

I can see the differences you are pointing out. But I'm reasonably sure, the mathematical use of 'statement' refers to a more general category (mammal, or even vertebrate) rather than a particularly narrow category (caucasian or oriental).

It certainly doesn't refer to axiomatic systems, that's for sure.

@Cauchy: Of course a system can be true. I just have to declare that system to be an axiom - and axioms are asserted to be true. gmalivuk did a fine job of showing how all the axioms of a system can be combined into a single statement that can, in turn, be used as the axiom of some other system.

Please write Peano Arithmetic as an axiom in some system. Saying "Axiom 1: Peano Arithmetic" is cheating. Your saying that an axiomatic system can be true that way is akin to saying that "The Simpsons" can be true because it can be an axiom of a system. And if you claim that "The Simpsons" can be true, people will rightfully think that you're being absurd.

Sure, you can write the conjunction of the axioms as an axiom in a system, but an axiomatic system is not only its axioms. You have a symbol set, a collection of wffs, and deduction rules. Please tell me how to write these as axioms.

None of the points made persuade me that I'm misusing the word 'statement' by applying it to individual axiomatic systems.

The fact that all the math people are telling you you're misusing a term doesn't give you any cause for concern?

Re: The map is not the territory?...

When the map is a 1:1 model intended to describe every feature of the terrain down to the smallest detail - I find it very difficult to distinguish between the description and.... well... actually... the only thing we have is the description. Mathematics is an abstraction in which the description is the thing.

The description is not the thing. That doesn't make any sense. The words in a novel are not the characters. The images in a movie are not the scene. The description of a mathematical object is not the object itself.
(∫|p|2)(∫|q|2) ≥ (∫|pq|)2
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### Re: Misunderstanding basic math concepts, help please?

Treatid wrote:A given !x is changed.

If '!x' is 'given', that means it has been assigned a specific, singular value. Then you're asking what happens if the value of that value changes?

It's like you're saying: '2+5=7. But what if the number 2 changed into the number 3? '

'2' has a specific meaning. That meaning is different to '3' under all non-trivial systems.

Of course if you change '2' into '3' then the axiom 'x=x' does not hold - you've specifically disregarded it. If you think you can end up showing that the law of identity is false/flawed from that then you're begging the question

Garbage in, garbage out.

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### Re: Misunderstanding basic math concepts, help please?

Treatid wrote:Re: The map is not the territory?...

When the map is a 1:1 model intended to describe every feature of the terrain down to the smallest detail - I find it very difficult to distinguish between the description and.... well... actually... the only thing we have is the description. Mathematics is an abstraction in which the description is the thing.

I've never been to Mars. I have a map of Mars. It doesn't matter how detailed my map is, I still haven't been to Mars.

Just because the actual territory is inaccessible, it doesn't mean the map suddenly becomes the territory.

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### Re: Misunderstanding basic math concepts, help please?

Cauchy wrote:The fact that all the math people are telling you you're misusing a term doesn't give you any cause for concern?

A 'statement' is an incredibly generic object right up there with 'symbol'.

gmalivuk made a mistake and you are doubling down on that mistake.

Don't try to bully me into believing you because you think you are an authority. Sure - you have a good general understanding of mathematics - but you are also human and make mistakes. I'll believe you if you give me a logical reason to do so. I won't buckle under to argument by authority just because you've gotten it into your head that you can see the emperors invisible clothes.

You are trying to cover up your mistake by casting me as the one who doesn't understand simple mathematical terms.

Of course an axiomatic system is a statement. What on earth do you think the alternative is? That an axiomatic system doesn't say (state) anything?

I honestly have no idea what you are thinking... Even if you add qualifiers like "well formed statement in ZFC" - ZFC is able to describe other axiomatic systems. (And no, the difference between a description of an axiomatic system and whatever you think an axiomatic system is when we haven't described it, isn't an excuse).

Cauchy wrote:Please write Peano Arithmetic as an axiom in some system. Saying "Axiom 1: Peano Arithmetic" is cheating. Your saying that an axiomatic system can be true that way is akin to saying that "The Simpsons" can be true because it can be an axiom of a system. And if you claim that "The Simpsons" can be true, people will rightfully think that you're being absurd.

Anything can be an axiom. And axioms are asserted to be true. These magical constraints that you are inventing don't exist in mathematics. They are a fiction that you've invented to justify an indefensible position.

Certainly mathematicians tend to choose axioms that they think are 'meaningful' or 'useful' or 'constructive' in some sense - but that is a choice - not something that is required by some law of axioms.

If someone wants to assert that "The Simpsons" is an axiom then they can do that. It may well seem absurd - but it isn't wrong.

I can only assume that you are so embarrassed about making basic errors that you are hoping to cover it up by casting me as the villain (shouldn't be too difficult - and who is going to challenge your definitions - if you say it is so then it must be so. Right?)

I've praised you previously for being a good teacher - but in this post the only thing you are trying to do is defend your ego and score points. You are making zero effort to persuade me that I'm wrong. You are just trying to tell me that I'm wrong.

This change in tone strongly suggests to me that at least a part of you knows that you are on shaky ground and can't quite find the argument that actually justifies what you are trying to say,

Get over it. I know that you have a good general understanding of mathematics. I'm not going to think less of you because you made a mistake.

I will think less of you if you continue to spout blatant nonsense about how things can only be axioms if you have personally approved of them and statements are special magical things, when, in fact, they are just a string of symbols (well formed statements have a higher standard - but just a statement is an incredibly low bar - really - go and look at any definition you like and see if you can find an exclusion for axiomatic systems - no... really... go on. Actually find something that agrees with your position. Done? Good. NOW you can make me eat my words.)

{I mean - really - Argument from authority against me? Have you not been paying attention? Reason and logic are fine - I'll listen to those. But I should believe you just because it is you saying it? I am insulted.}

rmsgrey wrote:Just because the actual territory is inaccessible, it doesn't mean the map suddenly becomes the territory.

Then what is ZFC? Where exactly does ZFC exist if not in the description? Are you saying that we have never done anything in ZFC because we have only manipulated the description - never the actual thing (whatever that might be)? Does our manipulation of the map symbols of ZFC correlate exactly with the territory of ZFC? Or is the description just an approximation? If the territory is inaccessible how do we know that the map corresponds to the territory?

If we have a 1:1 precise scale model of Mars do we not have an actual Mars?

{I'm not trying to shut you down as I am with Cauchy above. I think this question of whether a mathematical description corresponds with a knowable reality is important.}

@elasto: Even within standard mathematics things change. Functions are a fairly important part of mathematics (or so I've heard). And we can wrap a number line around a sphere and then transform that sphere into other shapes and discover that the mathematics of that number line has changed.

Even you felt the need to add the caveat "under non-trivial systems". However, there are plenty of non-trivial wrapping in which we can make 2 and 3 coincident.

Your choice of the number line as a fixed reference is interesting. Euclidean Geometry does rather assume fixed orthogonal axis. Generally we apply any transforms to the content of Euclidean space rather than to the framework - because what would our comparison be if our standard ruler (the framework) changes? (oh yes... non-Euclidean Geometry (although even a given non-Euclidean system is generally assumed to have fixed axis even where they are interestingly curved)).

elasto wrote:Of course if you change '2' into '3' then the axiom 'x=x' does not hold - you've specifically disregarded it.

Umm... yes? That is the whole point?

What did you think was going to happen if we discard the axiom x = x? Yes - if x != x then 2 != 2 and 3 != 3. At least not as a-priori assumptions.

You are correctly understanding what I'm describing and then becoming offended that I dare describe something that doesn't make the assumption that I've explicitly said I'm rejecting.

I went to some trouble to point out that I was merely showing what a lack of the axiom x = x might look like. I absolutely have not made any pretence that I have dis-proved x = x. Granted, I intend to show that x = x isn't a sustainable assumption - but at the moment I'm just trying to sketch an outline so that you have some vague idea of what I'm talking about. And you do seem to have grasped some of it even if you are then trying to stuff assertions into my mouth that I never made.

{Again, apologies if this comes across as confrontational. Your observation that I'm using 'change' to make my point rather than describing a static system is relevant. I'm aiming to expand on the points that change is needed in order to observe anything (including Cogito Ergo Sum) and that all static descriptions (descriptions of things that don't change) are equivalent.}

....

Edit:

google search for mathematical statement wrote:Definition (p. 3). A statement (or proposition) is a sentence that is either true or false (both not both). So '3 is an odd integer' is a statement. But 'π is a cool number' is not a (mathematical) statement.

I imagine that Cauchy is taking something like this as the basis for his position.

Except that it misses out some critical elements. In mathematics, a statement is only true with respect to a set of axioms. There are no statements that are just absolutely true independent of any context. Except for axioms themselves that are always true until such time as we can prove them inconsistent.

This latter part completely undercuts the distinction that is assumed in the quote. Positively cuts it off at the knees.

A piece of text is never true or false or neither except in context. It is simply wrong to declare a given statement as true, or false, or neither without also specifying the specific axioms against which that status is judged. Or... specifying that statement is an axiom and is assumed true for the purposes of axiomatic mathematics.

Given the right axioms, it is possible for any statement to be shown to be true of false with respect to those axioms.

Or we can just make a given statement an axiom; know that axioms are true by default and just stop there.

As such, I may have been overly harsh towards Cauchy. He is still wrong - but with loose statements like the above it is less surprising that someone would confuse the relative-truth-with-respect-to-a-specific-set-of-axioms with some notion of absolute truth.

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### Re: Misunderstanding basic math concepts, help please?

Oh Galileo...

Treatid
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### Re: Misunderstanding basic math concepts, help please?

Yeah... I've calmed down a bit.

I'm sorry Cauchy. I now understand the argument you were making. I disagree with it - but I now see you weren't being as arbitrarily stubborn as I perceived.

I am still convinced that it is a fundamental principle of axiomatic mathematics that anything can be proposed as an axiom and thus assumed true.

Claiming that something is absurd is not, I think, a proper mathematical argument.

And given that any string of symbols can be claimed as an axiom - then any string of symbols can be true - and any string of symbols will count as a mathematical statement under that circumstance.

Hence my belief that statement = any string of symbols (unless the specific context for the statement is specified - which has not been the case thus far).

It seems to me essential that axioms must be declared true - otherwise we cannot determine the state of any set of symbols that rely on those axioms. And this declaration of truth for axioms appears to create a loophole in the intention of referring to statements as only true of false.

As such, I do see the intention behind what you are saying - but I sincerely think that the distinction between true/false versus undefined cannot be maintained while it is possible to arbitrarily declare any set of symbols true. And it is necessary to declare axioms as true in order for axiomatic mathematics to work.

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### Re: Misunderstanding basic math concepts, help please?

That is not and has never been what anyone else means by the word "statement". You'll either have to start using words the way others use them, or admit that you're not talking about any of the same things as the rest of us.

If you do the first, then you'll be forced to admit that you're wrong about how statements work. If you do the second, then you'll be forced to admit there's no point in trying to communicate with the rest of us and you'll leave.
Unless stated otherwise, I do not care whether a statement, by itself, constitutes a persuasive political argument. I care whether it's true.
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### Re: Misunderstanding basic math concepts, help please?

A statement is only a true or false sentence within an axiomatic system. It is incorrect to refer to anything outside the scope of an axiomatic system as a mathematical statement.

Axiomatic systems themselves are neither true nor false and cannot be considered as mathematical statements. To do so was a category error on my part.

rmsgrey
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### Re: Misunderstanding basic math concepts, help please?

Treatid wrote:
rmsgrey wrote:Just because the actual territory is inaccessible, it doesn't mean the map suddenly becomes the territory.

Then what is ZFC? Where exactly does ZFC exist if not in the description? Are you saying that we have never done anything in ZFC because we have only manipulated the description - never the actual thing (whatever that might be)? Does our manipulation of the map symbols of ZFC correlate exactly with the territory of ZFC? Or is the description just an approximation? If the territory is inaccessible how do we know that the map corresponds to the territory?

If we have a 1:1 precise scale model of Mars do we not have an actual Mars?

{I'm not trying to shut you down as I am with Cauchy above. I think this question of whether a mathematical description corresponds with a knowable reality is important.}

We don't have a 1:1 model of ZFC - we can describe features of ZFC but we do that by starting with known features and figuring out what the neighbouring features would look like, but we can't just jump into the middle of ZFC and see what's there - we need to work from descriptions of known parts and figure out what they imply for the descriptions of related parts.

Even if you have a perfect duplicate of Mars, the copy is still a distinct entity to Mars itself - if you put a large lightning rod at the top of copy-Olympus Mons, then that doesn't magically spawn a lightning rod on the top of the real Olympus Mons. At best, you have a highly detailed representation of the real Mars.

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### Re: Misunderstanding basic math concepts, help please?

I'm pretty sure there is an intended difference between mathematics and physics.

A physical model/description/map has value according to how well it describes an actual physical phenomena.

Mathematical descriptions (axiomatic mathematics) are valued according to whether they are internally consistent and have no requirement to reference anything outside of that description.

I think you are suggesting that the description of ZFC is distinct from the actual ZFC; but if there is an actual ZFC it is entirely beyond our access.

I'm not sure what the point is of proposing the existence of something that we cannot ever observe, interact with or otherwise know about?

If we only ever have a description - then why propose something beyond that description? Occam's razor.

...

I get that your alteration of the Mars model is supposed to show that the model and the subject are distinct. But in order to show that distinction you have changed the model so that it is no longer an accurate model of Mars.

Suppose we want a perfect model.

If we model the planet Mars but do not place it in the same orbit as Mars it is just another planetary body - not Mars. Similarly, Mars without its moons isn't actually Mars.

So our perfect model of Mars includes Mars, its moons, the Sun, the other planets... in fact the whole solar system. And then we need to model the galaxy - we need to model the gravitational waves, individual photons impacting on Mars from distant suns...

A really pedantically perfect model of Mars requires us to also model the rest of the visible universe in perfect detail (anything less than perfection and n-body mathematics will bite us in the ass).

From which we can conclude that the only perfect model of a thing is the thing itself. Any description that isn't the thing itself is an approximation to some degree.

...

To bring this back to my argument that the only things that we can be aware of are relationships:

Even within our Mars model - the properties of rocks, soil, atmosphere all derive from the relationships of the molecules and atoms that they are composed of.

Right down at the bottom we have a handful of fundamental particles. While we assume that these particles exist and have various properties that we have measured - that measurement is done by looking at the relationships those fundamental particles have with our measuring equipment.

As Cogito Ergo Sum reminds us - we only ever see the end result of a series of relationships. Everything else is deduced from the pattern of those relationships. I suggest that an application of Occam's razor might lead us to consider the possibility that the only things that exist are relationships (that change).

In effect, I'm proposing a single fundamental element of the universe from which everything else can be constructed. As I've said, we cannot actually define this single element any more than mathematics can define a single absolute epistemic certainty.

However, because there is just one basic element we don't need to differentiate it from other fundamental elements. We don't need to define a single element in some fundamental way - we simply observe (via "I think, therefore I am") that it must exist (at least to the same extent that our self awareness exists).

We know from graph theory that it is possible to build complex structures using only vertices and edges (and Category Theory is heavily influenced towards vertices and edges as being fundamental units).

As such, there is precedent for approaching knowledge based solely on relationships with nodes being nothing more than a place holder to hang edges from.

Next up from the roadmap: Difference.

I wrote: Rough roadmap of where I think I'm going

Observation (in order to observe, the observer must be part of the system of the observed. Related: description and emulation.)
Absolute (what can 'absolute' mean given the above.)
Difference (An element of definitions is distinguishing A from B. I'll argue that the distinction between A and B is more important than A or B.)
Relationships (in this instance - the importance of a relationship is to specify not-identity - A is not identical to B but has some connection.)
Impossible (once we have ruled out the impossible - whatever is left... Without absolute definitions - there aren't many ways to specify first principles.)
Change (change implies a discernible difference between before and after. See difference above.)
Existence (I think, therefore I am. In order to exist, we need change & difference & relationships)
First principle (until we have created 'difference' - we cannot distinguish between things. If we cannot distinguish between change, difference and relationships but require these to exist, then change, difference and relationships are indistinguishable... and are our first principle even while we cannot define exactly what that first principle is.)
Orbits (Orbits of complex systems - one way of visualising change.)
Relationships that change (an internal representation of change.)
Enumerating all possible systems under this schema (Everything that is describable (and why we cannot describe anything else))
The simplest conceivable describable systems.

Edit: Next post:

There is nothing like explaining to someone else for clarifying ones own ideas.

I've previously suggested that all systems that are indistinguishable are equivalent (a la The Empty Set) and that static descriptions are indistinguishable from each other. I now retract this suggestion.

...

I previously sketched an outline of a system in which each element of the system depends entirely on the other elements of the system – no element has any properties that are independent of the rest of the system. (i.e. rejecting the law of identity)

I now want to explore in more detail the difference between the law of identity and not-the-law-of-identity.

To start, I want to look at the reasoning behind some of the axioms of set theory. (Not an attack on set theory. Trying to present a base-line against which an alternative can be compared).

Difference

The axiom of extensionality is a fairly direct statement of the law of identity as applied to set theory. This is the axiom that tells us indistinguishable sets are a single set. E.g. The Empty Set.

A prime reason for including The Empty Set as an axiom of set theory is to be able to distinguish between sets.

{Actually, a combination of The Axiom of Regularity and The Axiom Schema of Specification or equivalent axioms if we are talking about ZFC.}

If sets are allowed to contain themselves, and we do not have a known, fixed starting point (e.g. The Empty Set), and we have not yet constructed arithmetic (we can't count how many elements a set contains) it becomes impossible to distinguish one set from another set.

If all sets are indistinguishable then we only have a single set. And there wouldn't be much we could do with that.

Which is to say, the important bit is knowing that a thing is not another thing.

A ring of sets

We have a ring of sets that each contains the next set in the ring (where (x→y) means set x contains set y):

A1 → A2 → A3 → … → An → A1

This ring of sets violates the Axiom of Regularity (specifically – a set cannot contain itself) and is thus not well formed within ZFC set theory.

Each set in the ring appears identical to every other set in the ring. Where sets are indistinguishable we consider them to be the same set. In this case, that means that the ring is actually A1 → A1 which violates the Axiom of Regularity.

We could add another set off to the side of this ring. Say, A3{A4, B1} (set A3 contains sets A4 and B1).

We can now distinguish set A3 from all the other sets in the ring since it contains 2 sets rather than just 1.

There are now two possibilities for the rest of the sets in the ring:

1. We can count how many sets forward or backwards from the distinguishable set A3. Each set will have a different count for its distance from A3 and therefore be distinguishable from each of the other sets.
2. We haven't invented counting yet. All the sets with just one element still appear identical to all the other sets with just one element. We only have two sets – one with 1 element and one with two elements.

Hang on… if we haven't invented counting yet, then we can't distinguish between one set and two sets. AND if (B1 → B1) then we can't distinguish between B1 being a ring off to the side and B1 just being a repeated reference to the local ring. They are both sets that contain one element as part of a ring. We already know that all rings of sets are indistinguishable.

There is no combination of rings of sets, however we overlap them, that doesn't resolve down to a single set that contains itself (if we have no start/end points and cannot count the number of elements in a set).

In order to open up rings of sets we need to propose a start point and an end point (the start point being a set that does not contain itself directly or indirectly, the end point being The Empty Set). If we cannot positively identify a start point and an end point then we cannot distinguish our sets from a ring of sets.

This isn't anything new. This is the reason why the axiom of regularity and the axiom schema of specification (or their equivalents) are required. Without them it is impossible to distinguish between sets.

Summation

The axiom of extensionality states that all otherwise indistinguishable sets are the same set. The axiom of regularity and the axiom schema of specification are then required to create known start and end points that allow us to specify something other than indistinguishable rings of sets.

Alternatively

Having elected to see where we can get to without the law of identity – the axiom of extensionality as a representative of the law of identity is no longer an assumption. (And this is the mistake I noted at the beginning of this post – I was assuming the axiom of extensionality where I should not have been).

Without the axiom of extensionality, the axiom of regularity and the axiom schema of specification are not necessary to distinguish between sets. Or rather - having removed the concept of identity - we no longer need to worry about whether two references are identical.

This is not to say that two things are or aren't an identity. It means that the concept of identity has been set aside. It isn't part of the vocabulary any more.

Where we have a relationship, we are completely uninterested in the subjects of that relationship. The relationship is the only relevant property.

Again, this isn't that great a departure from standard mathematics. Sets in set theory are already abstract objects. Category theory makes every effort to abstract nodes even further.

We are simply going to the extreme - there is no property of a node (set) that we can know about - not even identity. This is a straightforward extension of the observation that we cannot construct a sentence that conveys absolute epistemic certainty about anything. If it is impossible to establish a single point of absolute certainty - then it would be foolish to pretend that we can.

Fortunately there really aren't many alternatives to a fixed point. The only apparent alternative is for properties to be relative.

Even if we are vague on exactly what a relationship is at a fundamental level - we come closer to actually directly observing the existence of relationships than we do to observing the constructs that we assume are connected by those relationships.

Where are we

Whereas axiomatic mathematics is deductive - my approach is to sketch a vague outline and then refine that outline by stages.

The following are reasonable outlines at this stage:

1. Category Theory (Category theory is still axiomatically founded and assumes the law of identity - but is correctly (in my estimate) shifting further towards a pure relational model - just not with sufficient conviction).
2. Set Theory without the axioms of extensionality, regularity and the axiom schema of specification (and without arithmetic).
3. a directed graph in which the nodes and edges are not labelled.

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### Re: Misunderstanding basic math concepts, help please?

Treatid, I want to start by saying that I'm not agreeing to any of the things you said just because I didn't directly challenge or question them. You wrote a lot and I don't necessarily have a cogent argument against/for all of it, but please don't take that as me implicitly agreeing with you.
Treatid wrote:The axiom of extensionality is a fairly direct statement of the law of identity as applied to set theory. This is the axiom that tells us indistinguishable sets are a single set. E.g. The Empty Set.

A prime reason for including The Empty Set as an axiom of set theory is to be able to distinguish between sets.

{Actually, a combination of The Axiom of Regularity and The Axiom Schema of Specification or equivalent axioms if we are talking about ZFC.}

If sets are allowed to contain themselves, and we do not have a known, fixed starting point (e.g. The Empty Set), and we have not yet constructed arithmetic (we can't count how many elements a set contains) it becomes impossible to distinguish one set from another set.

If all sets are indistinguishable then we only have a single set. And there wouldn't be much we could do with that.

Well, the axiom of extensionality states that sets with the same contents are the same set. And yes, two empty sets are the same by this as well. This is because there is no way to distinguish between them since all we know about sets so far is their contents and the way they relate to one another. I wouldn't necessarily say that we include the empty set so that we can distinguish between sets. Rather, an empty set *must exist* if it is possible to specify sets and do stuff with them, and because the only way to distinguish sets is by their contents, all empty sets are the same. If you decide that there is no empty set, and also decide that sets can contain themselves, and that arithmetic doesn't exist, then yes you really can't do much.
We have a ring of sets that each contains the next set in the ring (where (x→y) means set x contains set y):

A1 → A2 → A3 → … → An → A1

This ring of sets violates the Axiom of Regularity (specifically – a set cannot contain itself) and is thus not well formed within ZFC set theory.

Got it, you've created a thing that can't exist in standard ZFC. Let's think more about how this nonexistent thing behaves.
Each set in the ring appears identical to every other set in the ring. Where sets are indistinguishable we consider them to be the same set. In this case, that means that the ring is actually A1 → A1 which violates the Axiom of Regularity.

We could add another set off to the side of this ring. Say, A3{A4, B1} (set A3 contains sets A4 and B1).

We can now distinguish set A3 from all the other sets in the ring since it contains 2 sets rather than just 1.

Can we? So far, you've said that the nonexistent ring is a bunch of sets that contain themselves. If A3 contains 2 things, and A2 contains A3, then doesn't A2 contain B1 as well? And since A1 contains A2...
There is no combination of rings of sets, however we overlap them, that doesn't resolve down to a single set that contains itself (if we have no start/end points and cannot count the number of elements in a set).

In order to open up rings of sets we need to propose a start point and an end point (the start point being a set that does not contain itself directly or indirectly, the end point being The Empty Set). If we cannot positively identify a start point and an end point then we cannot distinguish our sets from a ring of sets.

This isn't anything new. This is the reason why the axiom of regularity and the axiom schema of specification (or their equivalents) are required. Without them it is impossible to distinguish between sets.

I agree wholeheartedly with how you started out here. If sets are allowed to contain themselves, and if you create a ring of sets that contain each other sequentially, then all the sets in that ring are equal and indistinguishable. However, I don't understand what you mean by "open up rings of sets" - are you saying that you want to create a mathematics in which these rings are nondegenerate? Or are you saying that all of mathematics is a giant ring until we acknowledge the existence of an empty set? Or are you saying the other axioms don't refute such a construction?
Without the axiom of extensionality, the axiom of regularity and the axiom schema of specification are not necessary to distinguish between sets. Or rather - having removed the concept of identity - we no longer need to worry about whether two references are identical.

This is not to say that two things are or aren't an identity. It means that the concept of identity has been set aside. It isn't part of the vocabulary any more.

Where we have a relationship, we are completely uninterested in the subjects of that relationship. The relationship is the only relevant property.

Again, this isn't that great a departure from standard mathematics. Sets in set theory are already abstract objects. Category theory makes every effort to abstract nodes even further.

We are simply going to the extreme - there is no property of a node (set) that we can know about - not even identity. This is a straightforward extension of the observation that we cannot construct a sentence that conveys absolute epistemic certainty about anything. If it is impossible to establish a single point of absolute certainty - then it would be foolish to pretend that we can.

Fortunately there really aren't many alternatives to a fixed point. The only apparent alternative is for properties to be relative.

Even if we are vague on exactly what a relationship is at a fundamental level - we come closer to actually directly observing the existence of relationships than we do to observing the constructs that we assume are connected by those relationships.

OK, so to sum up, you're saying that if we don't acknowledge that sets with identical contents are the same set, we can distinguish between sets by something else. Some "relationship" that exists between sets, but that we're not going to specify. Now, that means it's unfounded to talk about the concept of two sets being the same set, because same doesn't exist. We can't observe properties of sets, because apparently we can't - you're not really justifying why we can't describe properties of an object rather than relationships between objects.

We can talk about a different relationship - set A can be gribbledy to set B. Of course, nothing is fixed about sets A or B, but we can say the relationship gribbledy means "containing the same elements". Heck, we can talk about another relationship - A can be furdunken to B, where furdunken means "entirely indistinguishable". In that case, are A and B the same thing? Or is one of them Mars and the other just a model?

You seem to be frequently arguing that indistinguishable things are the same thing - you did it with the Mars model, and you did it again when explaining your rings above. Why, then, do you think that the law of identity should not hold? Are some indistinguishable things not the same as each other, but some are?

Cauchy
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### Re: Misunderstanding basic math concepts, help please?

My eyes are rolling back in my head. Your last post is so rambling that I can barely comprehend it. I'll try to, though.

Treatid wrote:I'm pretty sure there is an intended difference between mathematics and physics.

A physical model/description/map has value according to how well it describes an actual physical phenomena.

Mathematical descriptions (axiomatic mathematics) are valued according to whether they are internally consistent and have no requirement to reference anything outside of that description.

I think you are suggesting that the description of ZFC is distinct from the actual ZFC; but if there is an actual ZFC it is entirely beyond our access.

I'm not sure what the point is of proposing the existence of something that we cannot ever observe, interact with or otherwise know about?

If we only ever have a description - then why propose something beyond that description? Occam's razor.

Because descriptions describe things. Specifically, other things. They don't describe themselves, and they don't hang in a void. I've heard mathematics described as a convenient fiction, and I think that's a good analogy. Is a character in a book just the words in that book? Is the setting in a TV show just the pictures that make up the show? I'd argue 'no' in both these cases, just as I'd argue 'no' to "isn't ZFC just its description?". There is a fictional character or setting in my head that is informed by the words or images but is separate from them, just as there is a fictional structure in my head called ZFC that is delineated by its description but is separate from it.

We can know about ZFC via its description, but the description isn't the be-all end-all. Nothing in the description of ZFC says that the Continuum Hypothesis is independent of ZFC, and yet it is. There are deeper properties of ZFC than what the description lays out explicitly, and I'd say this points to a ZFC that's being described by the description, rather than ZFC being the description.

I get that your alteration of the Mars model is supposed to show that the model and the subject are distinct. But in order to show that distinction you have changed the model so that it is no longer an accurate model of Mars.

Suppose we want a perfect model.

If we model the planet Mars but do not place it in the same orbit as Mars it is just another planetary body - not Mars. Similarly, Mars without its moons isn't actually Mars.

So our perfect model of Mars includes Mars, its moons, the Sun, the other planets... in fact the whole solar system. And then we need to model the galaxy - we need to model the gravitational waves, individual photons impacting on Mars from distant suns...

A really pedantically perfect model of Mars requires us to also model the rest of the visible universe in perfect detail (anything less than perfection and n-body mathematics will bite us in the ass).

From which we can conclude that the only perfect model of a thing is the thing itself. Any description that isn't the thing itself is an approximation to some degree.

Cool. So when the description of ZFC doesn't say that it's independent of ZFC, we know that the description is an approximation to some degree and therefore not the thing itself.

To bring this back to my argument that the only things that we can be aware of are relationships:

Even within our Mars model - the properties of rocks, soil, atmosphere all derive from the relationships of the molecules and atoms that they are composed of.

Right down at the bottom we have a handful of fundamental particles. While we assume that these particles exist and have various properties that we have measured - that measurement is done by looking at the relationships those fundamental particles have with our measuring equipment.

As Cogito Ergo Sum reminds us - we only ever see the end result of a series of relationships. Everything else is deduced from the pattern of those relationships. I suggest that an application of Occam's razor might lead us to consider the possibility that the only things that exist are relationships (that change).

In effect, I'm proposing a single fundamental element of the universe from which everything else can be constructed. As I've said, we cannot actually define this single element any more than mathematics can define a single absolute epistemic certainty.

However, because there is just one basic element we don't need to differentiate it from other fundamental elements. We don't need to define a single element in some fundamental way - we simply observe (via "I think, therefore I am") that it must exist (at least to the same extent that our self awareness exists).

We know from graph theory that it is possible to build complex structures using only vertices and edges (and Category Theory is heavily influenced towards vertices and edges as being fundamental units).

As such, there is precedent for approaching knowledge based solely on relationships with nodes being nothing more than a place holder to hang edges from.

You do realize that vertices have properties, right? Have you heard of the degree of a vertex? It's the number of times an edge is incident to the vertex. It's a derived property of the interactions between vertices, but it's still a property of the vertex.

To take a different tack, the graph with 2 vertices and no edges is different from the graph with three vertices and no edges, even though both feature the same collection of interactions.

Both of these examples are meant to demonstrate that the vertices matter; they're not just placeholders to hang edges from.

Similarly, the objects in category theory can have properties, and categories with the same arrow structure but different numbers of objects are different.

Regarding your "interactions are fundamental" line of thinking: sure, you could do that, but I bet your explanations are going to be much more complicated than the current "particles and their properties are fundamental" line of thinking, since you have to explain why each interaction is what it is. If you hide behind "interactions are fundamental, so we can't describe them" then your model has less explanatory power than the currently accepted one.

The axiom of extensionality is a fairly direct statement of the law of identity as applied to set theory. This is the axiom that tells us indistinguishable sets are a single set. E.g. The Empty Set.

No, it tells us that sets the contain the same elements are the same, where "same" here is defined inductively (which we can do because of the axiom of regularity). If this is what you mean by indistinguishable, fine, but I have a sneaking suspicion that you're going to use it in a more broad way later than you're using it here.

A prime reason for including The Empty Set as an axiom of set theory is to be able to distinguish between sets.

I'd argue that the inclusion of the empty set is so that we have some set that exists, so we know that there are sets that exist. We only want models of ZFC that have sets in them. A certainly intended consequence of its inclusion is that we get the ordinals, sure.

If sets are allowed to contain themselves, and we do not have a known, fixed starting point (e.g. The Empty Set), and we have not yet constructed arithmetic (we can't count how many elements a set contains) it becomes impossible to distinguish one set from another set.

What? Different sets have different elements. If a set has some element that another set doesn't, then they're different sets, and we can distinguish them by that element. Things that are isomorphic are not necessarily the same, they just have the same structure. This is what I meant by your using indistinguishable in a broader sense than how you used it earlier.

If all sets are indistinguishable then we only have a single set. And there wouldn't be much we could do with that.

Which is to say, the important bit is knowing that a thing is not another thing.

It's *an* important bit, but I object to your calling it *the* important bit. For instance, without the axioms of empty set and infinity, ZFC could have no sets at all. So maybe the most important bit is knowing that there is a set, and another important bit is knowing that a thing is not another thing.

A ring of sets

Please don't call it a ring. A ring is a specific kind of structure, which this thing is not.

We have a ring of sets that each contains the next set in the ring (where (x→y) means set x contains set y):

A1 → A2 → A3 → … → An → A1

This ring of sets violates the Axiom of Regularity (specifically – a set cannot contain itself) and is thus not well formed within ZFC set theory.

Okay, cool, it doesn't exist. Next?

Each set in the ring appears identical to every other set in the ring. Where sets are indistinguishable we consider them to be the same set. In this case, that means that the ring is actually A1 → A1 which violates the Axiom of Regularity.

You're misusing indistinguishable, either here or back when you defined the axiom of extensionality. If A_1 and A_2 are equal, then your structure has length 1, but if they're not equal, then it doesn't. For instance, if A_2 and A_3 are different, then A_1 and A_2 would be different as well. It's internally consistent for each set to be different, just as it's internally consistent for them to all be the same. This is part of the reason why we have the axiom of regularity.

As an aside, the structure can't exist even if it doesn't collapse to one set, because the set containing each set of this structure exists by pairing and union, and that set fails regularity.

We could add another set off to the side of this ring. Say, A3{A4, B1} (set A3 contains sets A4 and B1).

We can now distinguish set A3 from all the other sets in the ring since it contains 2 sets rather than just 1.

Oh, A_3 was supposed to contain *only* A_4? You didn't say that, though it doesn't matter to what I said above.

There are now two possibilities for the rest of the sets in the ring:

1. We can count how many sets forward or backwards from the distinguishable set A3. Each set will have a different count for its distance from A3 and therefore be distinguishable from each of the other sets.
2. We haven't invented counting yet. All the sets with just one element still appear identical to all the other sets with just one element. We only have two sets – one with 1 element and one with two elements.

Counting comes out of the axiom of empty set, pairing, and union. We can construct the finite ordinals with these, which allows us to count, by putting sets in bijection with various ordinals. This is also a good time to remind you that regularity forbids rings like this from existing.

Hang on… if we haven't invented counting yet, then we can't distinguish between one set and two sets. AND if (B1 → B1) then we can't distinguish between B1 being a ring off to the side and B1 just being a repeated reference to the local ring. They are both sets that contain one element as part of a ring. We already know that all rings of sets are indistinguishable.

You're misusing indistinguishable as equal again. Just because we can't count doesn't mean that 1 = 2. They could be equal or unequal; if we can't distinguish between them, then we can't tell. Remember, equality, as set up by extensionality, has a specific meaning that is not the same as "they look the same to me".

There is no combination of rings of sets, however we overlap them, that doesn't resolve down to a single set that contains itself (if we have no start/end points and cannot count the number of elements in a set).

If we assume indistinguishable means equal, which it doesn't and that we don't have counting, which we do.

In order to open up rings of sets we need to propose a start point and an end point (the start point being a set that does not contain itself directly or indirectly, the end point being The Empty Set). If we cannot positively identify a start point and an end point then we cannot distinguish our sets from a ring of sets.

This isn't anything new. This is the reason why the axiom of regularity and the axiom schema of specification (or their equivalents) are required. Without them it is impossible to distinguish between sets.

(1) The main reason regularity exists is to ground everything in the empty set. You can't have infinite downward chains of containment due to regularity, infinity, and replacement, so every downward chain must end, necessarily at a set which contains nothing, which is the empty set. In this way, everything is built up out of the empty set. The fact that your non-existent rings of sets can't exist follows from that. Without regularity, you'd have all the normal sets, plus these extra sets that contain only themselves (called Quine atoms, apparently), and cyclical chains of containment (your loops).

(2) Your arguments here only show that sets that are parts of loops are indistinguishable (which, again, does not mean equal). How do you know that every set is part of one of these loops? You don't, because they aren't. The empty set, for instance, could never be a part of these loops, because it contains no elements.

The axiom of extensionality states that all otherwise indistinguishable sets are the same set.

No, it does not.

The axiom of regularity and the axiom schema of specification are then required to create known start and end points that allow us to specify something other than indistinguishable rings of sets.

Assuming the axiom of extensionality says what you claim it does (which it does not), you still have to show that every set is part of one of these rings to claim that they're all the same.

Having elected to see where we can get to without the law of identity – the axiom of extensionality as a representative of the law of identity is no longer an assumption. (And this is the mistake I noted at the beginning of this post – I was assuming the axiom of extensionality where I should not have been).

Without the axiom of extensionality, the axiom of regularity and the axiom schema of specification are not necessary to distinguish between sets. Or rather - having removed the concept of identity - we no longer need to worry about whether two references are identical.

This is not to say that two things are or aren't an identity. It means that the concept of identity has been set aside. It isn't part of the vocabulary any more.

Where we have a relationship, we are completely uninterested in the subjects of that relationship. The relationship is the only relevant property.

You're losing me. I don't see why we should be casting aside the axiom of extensionality. Are you saying you want to do that? That's fine, but it's not really for a good reason.

Again, this isn't that great a departure from standard mathematics. Sets in set theory are already abstract objects. Category theory makes every effort to abstract nodes even further.

We are simply going to the extreme - there is no property of a node (set) that we can know about - not even identity. This is a straightforward extension of the observation that we cannot construct a sentence that conveys absolute epistemic certainty about anything. If it is impossible to establish a single point of absolute certainty - then it would be foolish to pretend that we can.

Fortunately there really aren't many alternatives to a fixed point. The only apparent alternative is for properties to be relative.

Even if we are vague on exactly what a relationship is at a fundamental level - we come closer to actually directly observing the existence of relationships than we do to observing the constructs that we assume are connected by those relationships.

Again, I don't see why we should be doing this. Or rather, you haven't convinced me in the slightest that I should want to throw away the properties of sets. Your rings example has done nothing to convince me that identity is bad and should be removed.
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### Re: Misunderstanding basic math concepts, help please?

Relative vs Absolute

A short while ago in this thread we agreed that no one has been able to define anything in an absolute, epistemically certain, way.

While there was some discussion, the overall impression was that this was a pedantic technicality. A bothersome formal result, but without significant impact on the practice of mathematics.

IF mathematics has never defined or described anything in an absolute fashion THEN nothing that has been achieved in mathematics relies on defining or describing things in an absolute fashion.

Stop. That is everything you need to know. Everything that follows is a superfluous and redundant reiteration of this single point.

Relative vs Absolute

The fundamental limitations of mathematics have already been thoroughly explored and are even somewhat celebrated.

Indeed, I have been criticised for harping on about subjects that are already well known and established within mathematics.

We all know and agree that Gödel et al described, in formal language, strict limitations on what axiomatic mathematics is capable of doing.

On a more philosophical front, nobody is arguing against Descartes pointing out that we cannot 'know' anything directly. We know we cannot have complete epistemic certainty about anything.

With regard to these limitations I don't have to demonstrate anything. The heavy lifting has already been done and, for the most part, everyone recognises that these limits do exist.

My mistake has been in trying to persuade you that these limits exist. You already know they exist. You don't need further arguments to persuade you of the existence of something that you aren't denying.

Relative vs absolute

Whether or not you think absolutes exist... we agree that mathematics has not been able to define or describe anything in absolute terms.

So... we have the concept of 'absolute' which hasn't, itself, been defined in an absolute way; we don't have a single instance of a known absolute that we can reference. In other words, we have a concept that has not been defined in its own terms and for which we do not have a single piece of evidence - and then build axiomatic mathematics on top of this foundation.

Absolute faith without - even despite - evidence. That is religion - not science.

Somehow, the idea of 'absolutes' has become so embedded that generations of incredibly smart people have twisted or ignored the results of their own tools in order to maintain the pretence that it is possible to define things in an absolute (axiomatic) way. And felt completely justified in doing so because they just know that absolute definitions are a thing.

The idea that an incompleteness theory might be telling us something fundamental about the mechanism of knowledge has been hidden behind world class synchronised mental gymnastics.

If some two thousand years of mathematics has failed to produce a single unambiguously absolute statement then maybe, just maybe, that is a hint that your intuition about 'absolute' is not as well formed as you are assuming.

Remember: Since we don't, actually, currently have a single instance of an absolute definition, everything we have achieved thus far has been done without a single absolute definition. Whatever the actual mechanism of mathematics - it doesn't require a single absolute definition. Because it has never had a single absolute definition.

Relative vs absolute

On the plus side - Since we have never had a single absolute definition, the loss of the idea of absolute definitions isn't that big a deal. Everything is already being done with an absence of absolute definitions.

I don't have to define what an entirely relative system looks like - because you have never experienced anything else. In the absence of absolutes, everything must be relative (unless you can come up with a third option that is neither absolute nor relative).

On the other hand, I'm as certain as it is possible to be in the absence of absolutes, that mathematics moving away from faith based reasoning to evidence based reasoning will reveal much that is currently obscured by the blinkers of religion belief in the existence of absolutes. I am not proposing a dose of nihilism. Merely a simple serving of Occam's Razor - don't invent unnecessary complications that (in this case) cannot be contributing to the solution.

Relative vs absolute

I can't prove that something you can't taste, touch, see, hear, smell or describe doesn't exist (not least because mathematical proof relies on the existence of absolute definitions).

“You Cannot Reason People Out of Something They Were Not Reasoned Into.”

The general belief in absolutes is not a rational one (in that nobody started from first principles and constructed the concept of absolutes).

[aside]Indeed, I suspect the enduring appeal of 'absolute' is that it hasn't been defined in its own terms. As such it can be all things to all people. And if some perception of 'absolute' is shown to be unreasonable... well... what better way to refute arguments than to point out that any proffered definition of your terms isn't/cannot be what you were thinking of ("but trust us - we know what we mean even if we can't quite work out how to communicate it in a definitive manner - honest guv'n'r.") (Witness: this thread in which we have spent an inordinate amount of time arguing over what is and isn't the definition of axiomatic mathematics before finally admitting that there isn't an absolute definition of anything (where 'anything' just happens to include axiomatic mathematics)).

An interesting irony is that a prime aim of axiomatic mathematics was to avoid the sloppy thinking due to improperly defined terms. But then mathematicians were so intent on defining terms they failed to take heed of all the results that told them it is impossible to properly define terms in the way that axiomatic mathematics attempts.[/aside]

Gödel's Incompleteness Theory, by itself, demonstrates that it is impossible to define (in an absolute sense) anything within an axiomatic system. We've argued this point. But you don't have to accept my view. The only thing you need is:

IF mathematics has never defined or described anything in an absolute fashion THEN nothing that has been achieved in mathematics relies on defining or describing things in an absolute fashion.

I honestly think you will be astonished at how quickly a purely relativistic world view starts to make sense and pays back dividends in genuine understanding (including why 'absolute' must be a null concept). All you have to do is give up on the idea that it is possible to define anything (to any degree) in an axiomatic sense.

The difficult bit is not 'trying to work out how to describe everything in purely relative terms'. You are already doing that. In the absence of an absolute definition, everything we communicate must be communicated in relative terms.

The difficult bit is to stop pretending: that absolutes exist, and that anything we have done is an approximation to an absolute.

You don't even have to assume that absolutes are impossible. You can start by just recognising that everything we have done thus far has been done without defining a single absolute.

Relative vs absolute

Once more with feeling:

Has mathematics ever created a single absolute statement, description or definition?

No?

Then the only option left is a purely relativistic system.

Cauchy
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### Re: Misunderstanding basic math concepts, help please?

Okay, axiomatically defined things are defined relative to the language they're written in, or something. So what? You're the only one who seems to think that the absolute vs. relative divide matters, or that axiomatic mathematics not being "absolute" damns it. The goal of axiomatic mathematics is to push the relativity into a small corner and box it up so that we can say "if we can agree on this, look how much we get out of it!". If that means that axiomatic mathematics is "a purely relativistic system", then I'm just fine with that, but your choice of words makes "purely relativistic system" sound like something else.
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### Re: Misunderstanding basic math concepts, help please?

Treatid, nobody here is pretending that mathematics holds "absolute truth" in the sense that you claim. I'm not sure why you're trying to fight this battle. If it makes you feel better to say that mathematics, as it exists now, is "purely relativistic" then sure, it is. You don't seem to be offering another route to "absolute truth", and so far your suggestions about a "better way" to know relative truths hasn't amounted to much.

As I (and others) have said before, the beauty and power of math is the ability to start with simple building blocks and create something that is robust and useful. If you don't like some of those building blocks, that's your choice - there are multiple ways to start with different initial assumptions and end up with different results. If those results are useful, we call that a branch of mathematics; if they are not useful or result in contradictions, we end up discarding them and focusing our attention elsewhere.

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### Re: Misunderstanding basic math concepts, help please?

I do believe you all are giving up on absolute truths way too fast. Though it is rather an epistemological question and doesn't really concern mathematics much, so it should also be discussed within an epistemological framework. Mathematics itself does not provide any clues about whether it's theories are absolute or relative. Instead of Gödels incompleteness theorem one should rather start looking at Descartes "je pense donc je suis" and from that starting point one can have a nice discussion about the existence of absolute truth/knowledge. Also, it's not even easy to define what 'truth' really should be. Just take the right definition and you will see that mathematics without a doubt generates absolute truths.
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### Re: Misunderstanding basic math concepts, help please?

No, Descartes is most certainly nonsense.
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### Re: Misunderstanding basic math concepts, help please?

I tend to agree, but it's a good starting point if one wants to get serious about epistemology. To understand Descartes attempt at Foundationalism yields valuable insights how to methodologically attempt to get hold of an absolute truth. Like in mathematics where you first learn to do euclidian division of integers ahead of learning about rationals, you should try to understand Descartes ahead of wrapping your head around Quine, Russel, Popper, Wittgenstein and the like.
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### Re: Misunderstanding basic math concepts, help please?

When people are saying that nonwellfounded sets don't exist, they mean that they don't exist in ZFC, right?

I don't think Aczel's anti-foundation axiom has been shown to cause a contradiction or anything like that.

I'm somewhat fond of mathematical platonism, and I personally find non-well-founded sets to be somewhat appealing to be able to occasionally use/talk-about (though I would prefer that when considering sets, the default would be assuming that all the sets being talked about are well founded.).

Maybe Treatid was saying (a bit ago now) , something like "lets consider ZFC except without the axiom of foundation" ?

(and, two distinct non-well-founded sets being "indistinguishable" in a certain sense seems reasonable. It seems like it might be fun to formally define. Though, I think it would have to be "indistinguishable using only <something>", because, where A and B are two such sets. {A,{A,B}} seems like it would (at least usually?) be "enough to distinguish between them". ok I'm getting more detailed than is relevant here.)

not to say that the stuff that followed after that about sets in general being indistinguishable (???) made sense, but, the idea of nonwellfounded sets seems sensible to me, so long as one is working in, uh, a system that has nonwellfounded sets.

But I probably just missed where it was being said that this is working in zfc, instead of somewhere where nonwellfounded sets would be sensible.
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### Re: Misunderstanding basic math concepts, help please?

You have a point: I said in my own words that since mathematics hasn't had an absolute then everything that mathematics has done is already relative.

If mathematics is already relative then what is all the fuss about not being absolute?

Theory vs practice

The practice of mathematics is necessarily relative.

The theory of axiomatic mathematics is not.

It is this disparity between the intention and the practice that gets my girdle all a twitter.

Generalising Special Relativity (no - not General Relativity - a different generalisation)

I submit that the results of Special Relativity are interpreted in too narrow a fashion.

In the following, Newtonian Mechanics is a direct corollary to axiomatic mathematics while the Relativistic mathematics of which I speak is a direct corollary of Special Relativity.

Let us assume a standard Newtonian Mechanics within Euclidean Geometry.

Newtonian Mechanics is known to be misleading in comparison to Special Relativity. However, for the most part, Newtonian Mechanics is a reasonable approximation to a single reference frame within Special Relativity. So long as you never accelerate or otherwise switch reference frames, Newtonian Mechanics is a useful approximation. The greater the delta-v of reference frame switching, the less applicable Newtonian Mechanics becomes.

In an environment where the delta-v of reference frame switching is generally constrained, it may take a while to notice that Newtonian Mechanics is a poor representation of the larger picture that incorporates all possible reference frames.

{Specifically: Axiomatic mathematics appears to be satisfactory because we generally apply it to a local, human, reference frame where the fixed observer is us and therefore does not need to be formally defined.}

Special Relativity as a purely mathematical object

Forget that Special Relativity is a Theory of Physics. We are currently only concerned with the mathematical aspect of Special Relativity.

Abstraction:

IF

1. We have a set of qualities that are related in some way. Distance/position, Velocity and Time in the case of Special Relativity - but the mathematics does not depend in any way on what we label the qualities.

2. We have relationships between those qualities. Again, the exact symbols and the rules they represent are arbitrary provided we are consistent in handling them.

3. There are distinct states for each of the qualities (i.e. distinguishable reference frames - in this regard a reference frame is with respect to the qualities specified and may well not be a space-time reference frame).

4. There is any value that is asserted fixed with respect to all reference frames (e.g. infinity, speed of light in vacuum, absolute zero kelvin, any specific integer).

THEN

Special Relativity applies.

...

An actual applicable bit of mathematics - whoo!

For every (x = y) there is a reference frame where x cannot possibly equal y.

In exactly the same way as has been done with Newtonian Mechanics vs Special Relativity; we can now apply Special Relativity to any instance of (x = y) to determine the range of reference frames over which (x = y) is a practical approximation.

Specific Example

Near the reference frame of integer zero: (1 + 1 = 2). However, if we apply Special Relativity to switch to a 'reference frame at infinity': (1 + 1 = 2) transforms to (0 + 0 = 0).

This isn't that surprising. We already know that (infinity + infinity = infinity). From the perspective of infinity, there is no pair of finite integers added together that equal anything other than a finite integer.

All that has changed is that by recognising reference frames we can see that the switch from 'integer behaviour' to 'infinite behaviour' happens gradually in accordance with the delta-v of movement from small integer reference frames up to large integer reference frames. Fortunately, for the typical relationships between integers we use, this change in behaviour from our local reference frame can be ignored most of the time.

Notes

Reference Frames: It probably appears odd to talk about integers as reference frames. The mathematics of Special Relativity does not directly define reference frames except by example. Specifically: a quality that has more than one possible state.

In the absence of an absolute definition of anything, there is no reason to treat one quality with multiple states differently to another quality with multiple states. The mathematics works the same irrespective of any label we apply to any given quality (except where there is no ordering of states).

Limits: Special Relativity remains unchanged whether we label the 'speed of light in vacuum' as being some 3.00×108 m/s or infinite m/s. The only thing that changes are the units. With respect to this; 'infinite' is a synonym for 'furthest reference frame from us (limit)' on whatever scale(s) are relevant.

As such it is never necessary to specify what the limit is. Provided there is ordering with respect to the quality, Special Relativity applies. I.e. Special Relativity applies to all ordered systems. (Order implies a relationship between the ordered elements - as such - it would be redundant to also state that some relationship between elements is required - So: If a system contains order - then Generalised Special Relativity applies).

All that is left is to decide on some arbitrary/convenient units at the end.

Ordering:
The specific mathematics of Special Relativity are constructed around counting (i.e. distance, velocity and time are assumed to be ordered in line with real numbers). For all ordering systems that can be reduced to (are equivalent to) the real numbers, it is just a matter of substituting labels within Special Relativity - the mathematics of Special Relativity is identical for all such systems.

Where an ordering is not directly equivalent to the ordering of real numbers (and/or integers), then the actual mathematics of Special Relativity would change - but only in accordance with the properties of the specific ordering.

{Whoops. Had to check. Special Relativity only describes the relationship of reference frames within a given ordering. The specific relationship of time, distance and velocity is irrelevant - only the presumed ordering of time, distance and velocity is relevant.}

To be clear

I'm not using a metaphor or simile.

It is necessarily the case that the principles of Special Relativity apply to anything that can be described. Only utterly chaotic systems with no pattern (order) are exempt - and they have their own drawbacks when it comes to description.

In order to speak, we need a reference frame (a language, if nothing else). A statement in one reference frame cannot apply equally to all reference frames (you already know this bit).

However, by taking Relativity seriously, it suddenly becomes possible to measure exactly how distorted a given statement is in other reference frames and even accurately translate statements between reference frames (to a limit short of... the ordering limit (the local infinity)).

doogly
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### Re: Misunderstanding basic math concepts, help please?

No this is all complete nonsense.
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Gwydion
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### Re: Misunderstanding basic math concepts, help please?

I agree with doogly. Where in the gosh-darned fuck did this stuff about relativity come from?
The practice of mathematics is necessarily relative.

The theory of axiomatic mathematics is not.
It starts right here. What does that mean? The "theory of axiomatic mathematics" is, in a manner of speaking, always relative - it is the idea that you can take a few statements as true, and see what else they imply. The "theory of axiomatic mathematics" doesn't purport that anything about ZFC is universally true in the sense that you keep implying/saying, but rather that if we accept a limited set of simple postulates/axioms/statements/rules as true, then a lot of other fantastic things follow from that and (so far) nobody has been able to prove that collection inconsistent.
*lots of other stuff*
Not a clue why you think special relativity applies to so many things. Is this like the time you thought the principle of explosion applied to English?

Maybe we should back up, because you were doing a little better for a while there. Let's go back to basic principles - why do you think relativistic mathematics is so critical? Explain in certain terms (used the way everyone else uses them, or explicitly defined) a situation in which axiomatic mathematics is insufficient to address a math problem. What specific aspects of it are troubling to you? It is not enough to simply say that mathematics is absolute and this is wrong - we're saying that it's not absolute and that's ok, so maybe try to explain why it's not ok?

doogly
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### Re: Misunderstanding basic math concepts, help please?

And there is no such thing as non-axiomatic mathematics. It doesn't exist, so no comparisons can be made to it.
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Xanthir
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### Re: Misunderstanding basic math concepts, help please?

Well, you've got the intuitive mathematics that people and many animals can do on their own. It's basically just small-number addition plus some statistics that only works for relatively large probabilities and short timescales.
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doogly
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### Re: Misunderstanding basic math concepts, help please?

You're still working off axioms though, you're just not talking about them as much.
It's like how every marriage has a prenup.
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Xanthir
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### Re: Misunderstanding basic math concepts, help please?

I mean, in the sense that everything *can* be modeled as a logical system, sure. You're not, like, actually *doing* axiomatic math in those cases, tho. Birds certainly aren't.
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Cauchy
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### Re: Misunderstanding basic math concepts, help please?

What is a reference frame in mathematics? You need to be more specific. You say "the reference frame at integer zero" and "the reference frame at infinity", but what do those mean? Showing one example of an equivalent equation in those two frames does very little to enlighten. Does "1+1=2" no longer hold true in the reference frame at infinity? That's what you seem to be implying when you say that "[f]or every (x = y) there is a reference frame where x cannot possibly equal y."

You're using this analogy of special relativity as if special relativity says something profound about all of human understanding instead of it (rightly) being about how physics actually works. I worry about your argument when it starts making analogies like that one.
(∫|p|2)(∫|q|2) ≥ (∫|pq|)2
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Treatid
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### Re: Misunderstanding basic math concepts, help please?

[Story]

A man is punching himself in the face {punch}.

Otherman: Hey! You are punching yourself in the face.

Man: {punch} no I'm not {punch} that would be silly {punch} therefore I'm not punching myself {punch} in the face {punch}.

[/Story]

We are at the point where you agree that it is futile to claim that anything has been defined in an absolute sense.

But {punch} you are still {punch} pretending that it is possible {punch} to define things in the sense {punch} that axiomatic mathematics {punch} describes.

{That was an analogy. The Special Relativity bit isn't an analogy.}

Axiomatic mathematics is fundamentally structured around the concept of absolutes. The First Law of Thought is a statement of the assumption of absolutes: (x = x). This straight up says that x is assumed to have an absolute value. That the value of x does not depend on its relationship with other things.

I am honestly staggered that any of you can claim that axiomatic mathematics is the slightest bit relative.

Right there, in the First Law, it says "We assume that x is absolute".

"Oh - but axiomatic mathematics isn't really absolute in any way - oh no - just ignore that first law of thought - it isn't as if it has any impact on how we do mathematics".

Jesus Fucking Christ on a pointy stick. At least religious fundamentalists don't claim that their beliefs are backed up by reason.

NO. Axiomatic mathematics is not even a tiny bit relativistic. (some of the mathematics done in the name of axiomatic mathematics is relativistic - but that is more a criticism of mathematicians being sloppy about what axiomatic mathematics is).

The reason I spent an entire post saying there are no absolutes is because axiomatic mathematics is a pure absolute system. It really, really, really isn't relative in the slightest. The lack of a single absolute definition is devastating to axiomatic mathematics.

Your "yeah... no... why are you getting all up in our faces about axiomatic mathematics... we've already conceded that there aren't absolute definitions... but that doesn't really have anything to do with axiomatic mathematics..." is the very height of internalised hypocrisy - the absolute(sic) pinnacle of two faced determined blind faith in the face of overwhelming evidence.

Axiomatic mathematics without the first law of thought is not axiomatic mathematics.

A relativistic system with the first law of thought is not relative.

Without the ability to define anything in an absolute sense, axiomatic mathematics doesn't do anything. Whatever you may have thought you were doing under the banner of axiomatic mathematics it was not axiomatic mathematics.

Once again: Intention vs practice

The intention of axiomatic mathematics is to be an absolute system. The practice is that we have never come close to having an absolute system.

"Oh yeah... axiomatic mathematics is totally relative man..."

What is the game plan here? Just pull complete bollocks out or your arses because you don't want to question the way things have always been done? Or are we back to pretending that the laws of thought aren't the basic underlying principles of mathematics? Or perhaps I don't understand that (x = x) really means that everything is relative and nothing is fixed?

It wouldn't be so bad if I thought you were just spoofing me for funsies. But I think that most of you are posting in earnest. You actually believe yourself when you say that axiomatic mathematics is relative for all intents and purposes.

Garghh!

Most of the time ranting isn't productive. Probably isn't now.

But if the bulk of your argument is: "Well... Axiomatic Mathematics is pretty much relative anyway... what's the big deal?" then we are clearly at a point where you are bypassing rationality and there is little option left but to shout loudly.

Axiomatic mathematics is not even slightly a relativistic system. It was designed from the ground up to be an absolute system.

That you could actually claim the contrary strikes me as bizarre.

Are you really so afraid of what I'm saying that you will spout demonstrable nonsense (in union no less)?

"Yeah... that (x = x) stuff... You took that seriously? Nah man... that was sarcasm... yeah... sarcasm."

A Sweetner

Actually it probably makes it even more galling...

I think most of you are sane enough to realise you've spouted something that seemed justified at the time but when called on it turns out to be somewhat less self evident than you initially assumed.

The first law of thought is a declaration of absolutes.

The fundamental basis of axiomatic mathematics is an assumption of absolutes.

You cannot wave that off.

Other bits

I'd like to make some constructive replies, particularly to Gwydion and Cauchy -

But if you genuinely believe that axiomatic mathematics is even slightly similar to a relativistic system then we are so drastically at cross purposes that we aren't going to make any progress until this point is cleared up. As noted: Any systems that start with the First Law of Thought as a fundamental assumption aren't, in any way, shape or form, relativistic.

arbiteroftruth
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### Re: Misunderstanding basic math concepts, help please?

Treatid wrote:Axiomatic mathematics is fundamentally structured around the concept of absolutes. The First Law of Thought is a statement of the assumption of absolutes: (x = x). This straight up says that x is assumed to have an absolute value. That the value of x does not depend on its relationship with other things.

It does no such thing. It says x is whatever x is. What x is might very well depend on other things. What x is might change. What x is might be inherently fuzzy and 'relative' and forever unknowable. But x is whatever it is.

x=x isn't really saying anything about x. It's saying something about what the symbol '=' means.

The rest of your rant is based on this odd misunderstanding of yours, so I'll stop there.