As much as the components of this have been around for a long time, there are some people working very hard to avoid putting the pieces of the puzzle together.
Let us start with an edge piece:
i) A statement without context is meaningless.
ii) We define context with axioms.
iii) Axioms are statements.
There is nothing complicated here. There isn't any wriggle room.
This is a closed loop. There is no entry or exit point for definitions in this closed loop.
The loop neither creates or destroys. It is just a loop. The very essence of tautology.
You don't need someone else to do your thinking for you on this one. The only place a closed loop will take you is back to your starting point.
...
But what if we could have a definite definition without starting with some axioms? Maybe natural languages have access to some mechanism that breaks this loop?
Okay... let us assume that you have a definite fixed point; something that is unambiguously definable. A starting point from which we can build.
The first thing you need to do is to communicate that fixed point to me so that I have the same starting point as you.
So... you communicate an unambiguous set of axioms... oh... I don't have the fixed point from which we can define a definite meaning for axioms. Until you have communicated that fixed point... I don't have a fixed point which allows you to communicate that fixed point.
Hmm... Perhaps you can show me the fixed point...
This fixed point... this starting point for axioms... it needs to not just be a fixed point  it needs to be a fixed point that anyone (or anything) looking at it will instantly recognise it as a fixed point and know all the properties of that fixed point.
Even if axiomatic mathematics had an initial fixed point  it would be useless unless everyone fully understood that fixed point without explanation or justification.
It isn't just that we don't have a starting point for axiomatic mathematics. Even if there was a starting point, it would require that everyone intuitively understood that starting point in every detail for it to be of any use.
Axiomatic mathematics requires omniscience as a starting point... and if you have omniscience you don't need axiomatic mathematics.
There are no axioms. Without axioms there is no axiomatic logic and there is no axiomatic proof.
Whatever existing mathematics is doing it has nothing to do with axioms, it has nothing to do with axiomatic logic and it has nothing to do with axiomatic proof.
...
The structure of axioms is fundamentally flawed. It isn't a system that would have been neat if only we had a way to build that initial set of axioms... it simply isn't a workable approach to knowledge.
Having said that... we had to learn that it wasn't a workable approach to knowledge. Mistakes are an essential part of learning, and attempting to create axiomatic systems has educated us about what is possible.
So, where does that leave Set Theories (as represented by ZFC)?
The astute will have noticed that having made axiomatic mathematics disappear in a flash of logic, we still have computers, bridges that don't fall down in the first gust of wind, economics, global positioning satellites and a bunch of other neat stuff.
As much as axiomatic mathematics is not the answer  there are still processes to be understood.
We can still reason, albeit we cannot formalise reasoning into a specific, defined, logic.
What we can't do is build anything in an axiomatic fashion. Anytime you are under the impression that you are building something axiomatically... you are simply wrong.
The claim has been made that the axioms of ZFC are immune to the critique of axioms in general because the axioms are abstract  sufficiently indefinite as to not get caught up in the issue of axioms not being defined.
As much as it may feel like no assumptions are being made... if you are managing to build anything at all in an axiomatic fashion it is because you are assuming the existence of a fixed point. Without an initial definition, axiomatic mathematics cannot do anything. So, obviously, if you feel like you are doing something, you either aren't using axiomatic logic, or you are pretending that you have a fixed point.
gmalivuk wrote:Yeah, if you accept the notion that there can be things and that there can be relationships between them, then you can pretty much build set theory.
"pretty much"...
Yeah... "pretty much", as in "not at all, in any way".
Sets themselves are not a problem. As a generally abstract object they are just a way of expressing relationships between objects.
But sets are not the same as set theory.
Remove 'the empty set' and the axiom of regularity (and any other axiom that precludes self reference (identity) for sets) and you are left with a really weak system. You cannot distinguish between sets. You cannot construct V. You cannot construct cardinality.
You have to define things a certain way, but defining things isn't the same as making baseless propositions.
???
Yes it is. That is exactly what it is.
As much as you may think a given definition is justified or consistent or unambiguous... we know that, in fact, we cannot define anything in the axiomatic sense.
Agreement is not the same as definition. Your perception that a thing is obvious doesn't imply that you actually know what that thing is.
The empty set contains no other set because that's simply what I mean when I say "empty set".
What is what you mean?
I'm not being perverse or arbitrary  although I am being pedantic.
In an absolute sense... you
CANNOT know what a particular phrase means. As much as you feel otherwise... you do not know what "empty" means. Even if every other person on the planet appears to agree with you over the meaning of "empty"... you cannot distinguish between a single set with self reference (identity) and a ring of sets without presupposing that distinction.
And then the axioms about the empty set and the things you can do with the "contains" relationship are further definitions, not declarations that some real thing exists in a concrete way that you can expect to touch and manipulate with physics or whatever.
This distinction you are trying to make here is irrelevant. The degree of abstractness has no impact on whether the reasoning is justified.
z4lis wrote:If you're OK with talking about the "O from which the arrows leave" and distinguishing objects based on their relationships to other objects in a network, then you should be OK with some primitive models of set theory.
The "O from which the arrows leave" may well also be the O to which the arrow is pointing.
Identity relationships really mess with set theory  which is why set theories tend to exclude them.
It means that we cannot, a priori, distinguish one end of the relationship from the other.
There's actually some "model" of a very weak set theory where sets are represented as trees, but you disregard any symmetries.
I'd hardly call ZFC a very weak set theory. V  the von Neumann hierarchy is exactly what you describe  trees in which symmetry is disregarded.
But in order to create the trees, ZFC needs both to exclude self reference in sets and to specify an arbitrarily identified starting point (the empty set).
Without those  in a genuinely weak set theory  it isn't possible to distinguish any given set from any other set. This is why set theory goes to so much trouble to axiomatically specify distinctions between sets (e.g. empty set is distinguishable from all other sets as an axiom rather than being derived). Without an imposed distinction... there is no way to distinguish sets in ZF or ZFC.
So the tree
O > O

O
would represent the same tree as just O > O, since you aren't allowed to talk about the "left/right" or "top/bottom" parts of the tree. Intuitively, each node represents the set containing its children. So in this model, the tree O represents the empty set. O > O represents {empty set} and O > O > O represents {{empty set}}, while
O > O > O

O
represents {empty set, {empty set}}. Disregarding symmetries comes about from the fact that {empty set, empty set} should really be the same set as {empty set}.
This is the basic reasoning for disregarding symmetries that I was using... but allowing a set to have an identity relationship creates many more symmetries. Each of the trees you describe becomes indistinguishable. Specifically, a tree requires a definite start and end point. With identity relationships it becomes impossible to distinguish a start point from any other point.
PS I also need to point out lorb's post, above. Everything any of us have said in this thread has been debated, thought about, and written about for centuries by philosophers, scientists, mathematicians, writers,..., so please go find out what's out there on the topic!
You may have missed that lorb addressed the substantive portion of his post to "everyone".
It is those who are arguing for the status quo that do not understand how weak their position is.
arbiteroftruth wrote:We are allowed to talk about networks of relationships, and refer abstractly to objects that have these relationships. Since the relationships are directional, we can describe them as arrows and can distinguish between the property of being on the starting end of the arrow vs. the property of being on the destination end of the arrow. Let me know if you object to any of that so far.
Yes  objection.
ZFC specifically disallows identity relationships and arbitrarily distinguishes the empty set specifically so that a distinction can be made between the beginning and end of a relationship.
Without that assumed distinction  it is actually impossible to distinguish between two sets.
The axioms of ZFC aren't there for fun... they serve a purpose. A significant part of that purpose is to create a distinction between sets that doesn't arise by other means.
Category Theory recognises that there isn't a natural way to distinguish sets. Which is why it gives up on the effort entirely.
Given all that, is there any reason we can't talk about the property of an object "not having any arrows pointing toward it"?
Very much so.
This is why we have formal mathematics that pedantically tries to chase down all the details. As intuitive as the idea of distinguishing between objects based on the number of arrows pointing at them... it turns out that we can't create that distinction. If we assume that the distinction exists  then it exists... but if we don't start with that assumption then we can't build it from component parts. Things that we cannot build from component parts are assumptions of a fixed point  impossible assumptions (see also: axiom of choice).
Separately, is there any reason we can't describe restrictions on the properties of networks that we're interested in studying?
Umm...
We cannot describe anything in an absolute sense. We can and do describe all sorts of things subjectively. Whether a given restriction is well formed is not always trivial to determine.
If neither of those things are objectionable, then we can combine them. We can say that we're only interested in studying networks that have the property that it makes sense to talk about *the* object in the network with no arrows pointing toward it.
Ah. No  we can't.
While arrows are directional, they can also be circular. The from and to object might be the same object. Or they might not be. Unless we preclude the possibility of self reference, we cannot count the number of arrows in any degree.
We can further restrict our studies to networks in which all objects may be distinguished one way or another. In particular, we can focus on networks in which all objects may be distinguished purely in terms of the available paths to reach the object in question starting from the object with no incoming arrows. We can focus even more narrowly by requiring the network to always provide certain patterns of outgoing arrows.
This is true for something like ZFC... but ZFC achieves it by making impossible assumptions. Without those impossible assumptions all networks are similar in that we cannot definitely distinguish any set from any other set.
In short, if you accept the notion of describing properties of networks, and you accept the notion of restricting our studies to networks that have certain properties, then the act of doing so is what the phrase "axiomatic system" is intended to convey, in terms of the language you've established.
I agree that this is what people are claiming something like ZFC is. But the assumptions needed to build such systems are themselves violations of axiomatic mathematics.
So that leaves you four options that I can see.
1) Object to the act of describing properties of networks, in which case your entire attempt to use networks as a new form of mathematics breaks down.
The assumptions of axiomatic mathematics are really deeply implanted, aren't they?
You feel that if we can't describe a network in a certain way that there is nothing we can do. By excluding this option you are missing out on reality. Everything we know, understand and experience is contained within this option.
Axiomatic mathematics cannot understand this position... but frankly, axiomatic mathematics cannot understand anything.
Even within the fold of axiomatic mathematics... Category Theory completely disregards the distinction between sets. Objects are entirely irrelevant except to the degree that they provide notional anchors for relationships.
lorb wrote:@Treatid: Have you yet familiarized yourself with the work of your precursors? I pointed out some earlier that adress a few of the arguments you make. Another one that already has been mentioned on this thread is Quine. I think you will like this quote of him when writing about set theory and it's foundations:
Willard van Orman Quine wrote:We find ourselves making deliberate choices and setting them forth unaccompanied by any attempt at justification other than in terms of their elegance and convenience,
(
Source)
If you read further into his work you will find that he renders most of you objections to axiomatic maths moot, but arguing very similar to your own arguments.
I wouldn't say I have exhaustively worked my way through all philosophy inside and outside of mathematics... I'm always eager to discover a new perspective  please do keep the references coming.
In response to another's comments... just because earlier philosophers/mathematicians haven't been able to definitively solve the puzzle is no reason to believe that we can't. We have the advantage of standing on their shoulders  and the view from up here is fantastic.
Sidenote:
Treatid wrote: Clearly there are sufficient mathematicians who think that ZFC is sufficiently well defined that the question of whether P=NP is meaningful.
Are you saying PNP is independent from ZFC?
I was specifically saying that there are no axiomatic systems. ZFC isn't (cannot) be axiomatically defined. In the absence of a base, anything supposedly constructed on that base is moot.
However, even within the conventionally accepted constraints of ZFC... P=NP is still independent. This has already been proven with respect to Relativising proofs. It has been shown that Relativising proofs cannot resolve the question. Generally this is taken as showing that some other sort of proof is needed. This is due to a mistaken interpretation of an aspect of Godel's incompleteness. The inability of Relativising proofs to resolve the question is taken as evidence that alternate proofs are required. However, all alternate proofs are necessarily external to the system being considered. We can choose any external system and prove whatever we like.
Specifically... Godel proposes statements that are "true but unprovable" within the system. However, there is no possible way to distinguish between "true but unprovable" statements and statements that are independent of the axioms.
Without any way to distinguish between the two classes of statement there is no justification for viewing them as two distinct objects. "True but unprovable" can be considered equal to "independent of the axioms" until some mechanism is proposed to distinguish the two cases.
Dopefish wrote:I am somewhat uncomfortable with the use of "we" in statements like "We have now established/shown/proven..." when I haven't really seen anything that resembles you showing/proving anything despite the volume of words being used.
It has been established that the lack of foundation for axioms is well covered. I don't need to prove anything because it has already been done. I'm leaning on existing, known mathematics. lorb provides a fresh link to the foundational crises which covers a wide variety of approaches to the same basic issue.
I additionally don't think you should be making references to physics, and especially not quantum mechanics, as those topics are at best tangential to your point if not completely offtopic, and QM references frequently come across as "I don't understand this thing, and so no one understands this thing. QM is still real despite not being understood though, and so whatever I'm talking about is also real despite the lack of understanding.". (This is not to say you sound like that necessarily, but you should surely acknowledge there are many many cranks who make appeals to QM, so if you don't wish to be viewed as such you'd best avoid the topic if at all possible.)
Ultimately my aim is to discuss physics.
In order to communicate I needed to understand the language that other people use... but when I tried to find a firm foundation in mathematics, I discovered instead that there is no foundation of any kind. The only thing that exists is a set of agreed assumptions. These assumptions are not suitable for discussing fundamental physics.
While I agree that there are many crackpots who diss existing physics... I'm already here dissing the entirety of axiomatic mathematics... And I'm only doing that so I can construct a common basis for discussing fundamental physics.
I am fully aware that Quantum Mechanics is an extremely successful theory in terms of making confirmed predictions right up to the limit of our ability to measure. However, I do feel that it is a triumph of mathematics over rationality. It makes predictions  but it doesn't provide an intuitive insight into the structure of things. By contrast, I find General Relativity to be a masterful piece of physics.
I am still not convinced that there's not some self defeating logic being used here, since logic is part of mathematics, and so by saying something isn't logically valid (e.g. axiomatic mathematics), you're using that invalid system to reach that conclusion, and so your conclusion isn't valid.
Mathematics already uses paradox as an indicator that a set of axioms aren't consistent with each other.
Axiomatic mathematic's own logic shows itself to contain a paradox. This is a pretty conventional way to demonstrate that a system isn't a valid system.
Granted, having disposed of axioms, the way forward without axiomatic logic may be a bit murky.
The loss of axiomatic logic is unfortunate... but in practice, nothing has been lost. We never had axioms. We never had axiomatic logic. Yet we have managed to achieved our current technology despite this handicap.
We have reasoning and progress even without axiomatic logic. We will never have the kind of definite knowledge that axiomatic mathematics tried to create. Love it or hate it... this is the situation we have.
The Relativistic Mathematics we can have is different in nature to the axiomatic approach to knowledge. However, as a system that actually works it has the potential to be far more effective than axiomatic mathematics.
If the argument is that the logic still 'works' and it's just not for the reasons we might think, then that's a rather strong claim that logic just happens to actually work, but the rest of math doesn't necessarily.
No  logic doesn't work. It never did work in the axiomatic sense. But reason does work. There are consistencies that we can work with.
Nothing has changed beyond us, potentially, becoming a bit more selfaware. Axiomatic mathematics hasn't stopped working... it never worked. The things we were doing when we thought we were doing axiomatic mathematics still exist and are still as practically useful as we found them previously. By seeing past axiomatic mathematics we have a chance to better understand those systems and processes. Better understanding will give us more conscious control.
As attractive as the idea of definite knowledge is... as useful as it seems to be able to be absolute about... anything... there isn't, and never was, objective truth. We never could unambiguously define something as true or provable. Yet we can still construct computers that (mostly) reliably reproduce a given behaviour millions of times over.
If the rest of math does work, than we're fine, and things like the clay prizes are merely for people finding something that 'works' for the claims involved rather than any deeper claims.
For varying degrees of 'works'.
Calculating the pay cheque at the end of the month is going to carry on much the same for the forseeable future.
One of the reasons we want to know whether P=NP is because it directly impacts cryptography. We need to know whether there is a shortcut for a given algorithms... whether a given encryption is secure. That question is intimately tied to the nature of mathematics. Anyone seriously involved in encryption cannot afford to continue to pretend that axiomatic mathematics exists.
Forest Goose wrote:My hang up is on how if there is no meaning, no objective truth, and this makes axioms impossible: why is anything else possible  including: why can you state and specify your "results" as if they were meaningful, well defined, absolute, and objective; if their very conclusion is nothing is any of those things?
This is a good question. It goes right to the heart of our experience as humans.
This is why it isn't sufficient to shrug off the foundational crises as some obscure technicality of little relevance.
We absolutely do have subjective meaning. We do perceive significance. Yet the axiomatic approach to knowledge cannot work.
There is something to be described and understood... and the assumed tool of axiomatic mathematics cannot gain the slightest traction.
Approaching understanding without the use of axiomatic mathematics is going to take some learning. The first step of the process is to understand that the existing assumptions of axiomatic systems really are entirely the wrong approach. Fortunately for your sanity  you don't have to take my word on this point. The foundational crises contains a wealth of approaches to showing this. All you have to do is understand that the foundational crises is actually something important that is worthy of attention...
DR6 wrote:... but if category theory can be used to describe set theory, and category theory is a step better than sets, then set theory must be at least as well founded as category theory, since you can translate all statements about sets to category theory.
I was expecting someone to have picked up on this sooner.
First... basic Category Theory needs additional axioms in order to describe conventional set theories.
But mainly... a Universal Turing Machine can emulate any other Turing Machine. Both Category Theory and ZFC are attempts at Universal Turing Machine Equivalents. They are designed to be able to describe every other describable system.
In general, all systems can be described by a network of relationships. In this regard  Category Theory can describe that pattern of relationships (set theory too).
However, a pattern of relationships is not the same thing as an axiomatic system. To illustrate... there are infinitely many axioms for Peano arithmetic. Peano Arithmetic cannot be expressed as a definite network of relationships... even an infinite network. There are many systems that are axiomatically defined that cannot be expressed by a definite network of relationships. A prime reason is that the relevant systems are constructed using an impossible axiom (or several). As well formed as these system might appear they cannot be expressed as a network of relationships. To fix this  the same impossible axioms can be inserted into Category Theory.
So... Category Theory, and to a lesser extent, ZFC are Universal Turing Machine Equivalents that can emulate all describable systems. But not all axiomatic systems are describable. The feature that makes those systems not describable can be added to Category Theory (and tend to be built into ZFC already).